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+# ANSWERING COMPLEX OPEN-DOMAIN QUESTIONS WITH MULTI-HOP DENSE RETRIEVAL
+
+Wenhan Xiong1∗ Xiang Lorraine $\mathbf { L i } ^ { 2 * }$ Srinivasan Iyer‡ Jingfei Du‡
+
+Patrick Lewis‡† William Wang1 Yashar Mehdad‡ Wen-tau Yih‡
+
+Sebastian Riedel‡† Douwe Kiela‡ Barlas Oguz ˘ ‡
+
+1University of California, Santa Barbara   
+2University of Massachusetts Amherst   
+‡Facebook AI   
+†University College London   
+{xwhan, william}@cs.ucsb.edu, xiangl@cs.umass.edu,   
+{sviyer, jingfeidu, plewis, mehdad, scottyih, sriedel, dkiela, barlaso}@fb.com
+
+# ABSTRACT
+
+We propose a simple and efficient multi-hop dense retrieval approach for answering complex open-domain questions, which achieves state-of-the-art performance on two multi-hop datasets, HotpotQA and multi-evidence FEVER. Contrary to previous work, our method does not require access to any corpus-specific information, such as inter-document hyperlinks or human-annotated entity markers, and can be applied to any unstructured text corpus. Our system also yields a much better efficiency-accuracy trade-off, matching the best published accuracy on HotpotQA while being 10 times faster at inference time.1
+
+# 1 INTRODUCTION
+
+Open domain question answering is a challenging task where the answer to a given question needs to be extracted from a large pool of documents. The prevailing approach (Chen et al., 2017) tackles the problem in two stages. Given a question, a retriever first produces a list of $k$ candidate documents, and a reader then extracts the answer from this set. Until recently, retrieval models were dependent on traditional term-based information retrieval (IR) methods, which fail to capture the semantics of the question beyond lexical matching and remain a major performance bottleneck for the task. Recent work on dense retrieval methods instead uses pretrained encoders to cast the question and documents into dense representations in a vector space and relies on fast maximum inner-product search (MIPS) to complete the retrieval. These approaches (Lee et al., 2019; Guu et al., 2020; Karpukhin et al., 2020) have demonstrated significant retrieval improvements over traditional IR baselines.
+
+However, such methods remain limited to simple questions, where the answer to the question is explicit in a single piece of text evidence. In contrast, complex questions typically involve aggregating information from multiple documents, requiring logical reasoning or sequential (multihop) processing in order to infer the answer (see Figure 1 for an example). Since the process for answering such questions might be sequential in nature, single-shot approaches to retrieval are insufficient. Instead, iterative methods are needed to recursively retrieve new information at each step, conditioned on the information already at hand. Beyond further expanding the scope of existing textual open-domain QA systems, answering more complex questions usually involves multi-hop reasoning, which poses unique challenges for existing neural-based AI systems. With its practical and research values, multi-hop QA has been extensively studied recently (Talmor & Berant, 2018; Yang et al., 2018; Welbl et al., 2018) and remains an active research area in NLP (Qi et al., 2019; Nie et al., 2019; Min et al., 2019; Zhao et al., 2020; Asai et al., 2020; Perez et al., 2020).
+
+![](images/73a454659944835a7feee1f943a32694c385501a6c775db95eb427eff5cb8d0f.jpg)  
+Figure 1: An overview of the multi-hop dense retrieval approach.
+
+The main problem in answering multi-hop open-domain questions is that the search space grows exponentially with each retrieval hop. Most recent work tackles this issue by constructing a document graph utilizing either entity linking or existing hyperlink structure in the underlying Wikipedia corpus (Nie et al., 2019; Asai et al., 2020). The problem then becomes finding the best path in this graph, where the search space is bounded by the number of hyperlinks in each passage. However, such methods may not generalize to new domains, where entity linking might perform poorly, or where hyperlinks might not be as abundant as in Wikipedia. Moreover, efficiency remains a challenge despite using these data-dependent pruning heuristics, with the best model (Asai et al., 2020) needing hundreds of calls to large pretrained models to produce a single answer.
+
+In contrast, we propose to employ dense retrieval to the multi-hop setting with a simple recursive framework. Our method iteratively encodes the question and previously retrieved documents as a query vector and retrieves the next relevant documents using efficient MIPS methods. With highquality, dense representations derived from strong pretrained encoders, our work first demonstrates that the sequence of documents that provide sufficient information to answer the multi-hop question can be accurately discovered from unstructured text, without the help of corpus-specific hyperlinks. When evaluated on two multi-hop benchmarks, HotpotQA (Yang et al., 2018) and a multi-evidence subset of FEVER (Thorne et al., 2018), our approach improves greatly over the traditional linkingbased retrieval methods. More importantly, the better retrieval results also lead to state-of-the-art downstream results on both datasets. On HotpotQA, we demonstrate a vastly improved efficiencyaccuracy trade-off achieved by our system: by limiting the amount of retrieved contexts fed into downstream models, our system can match the best published result while being $1 0 \mathrm { x }$ faster.
+
+# 2 METHOD
+
+# 2.1 PROBLEM DEFINITION
+
+The retrieval task considered in this work can be described as follows (see also Figure 1). Given a multi-hop question $q$ and a large text corpus $\mathcal { C }$ , the retrieval module needs to retrieve a sequence of passages $\mathcal { P } _ { s e q } : \{ p _ { 1 } , p _ { 2 } , . . . , p _ { n } \}$ that provide sufficient information for answering $q$ . Practically, the retriever returns the $k$ best-scoring sequence candidates, ${ \{ \mathcal { P } _ { s e q } ^ { 1 } , \mathcal { P } _ { s e q } ^ { 2 } , . . . , \mathcal { P } _ { s e q } ^ { k } \} }$ $k \ll | \mathcal { C } | )$ , with the hope that at least one of them has the desired qualities. $k$ should be small enough for downstream modules to process in a reasonable time while maintaining adequate recall. In general, retrieval also needs to be efficient enough to handle real-world corpora containing millions of documents.
+
+# 2.2 MULTI-HOP DENSE RETRIEVAL
+
+Model Based on the sequential nature of the multi-hop retrieval problem, our system solves it in an iterative fashion. We model the probability of selecting a certain passage sequence as follows:
+
+$$
+P ( \mathcal P _ { s e q } | q ) = \prod _ { t = 1 } ^ { n } P ( p _ { t } | q , p _ { 1 } , . . . , p _ { t - 1 } ) ,
+$$
+
+where for $t = 1$ , we only condition on the original question for retrieval. At each retrieval step, we construct a new query representation based on previous results and the retrieval is implemented as maximum inner product search over the dense representations of the whole corpus:
+
+Here $\langle \cdot , \cdot \rangle$ is the inner product between the query and passage vectors. $h ( \cdot )$ and and $g ( \cdot )$ are passage and query encoders that produce the dense representations. In order to reformulate the query representation to account for previous retrieval results at time step $t$ , we simply concatenate the question and the retrieved passages as the inputs to $g ( \cdot )$ . Note that our formulation for each retrieval step is similar to existing single-hop dense retrieval methods (Lee et al., 2019; Guu et al., 2020; Karpukhin et al., 2020) except that we add the query reformulation process conditioned on previous retrieval results. Additionally, instead of using a bi-encoder architecture with separately parameterized encoders for queries and passages, we use a shared RoBERTa-base (Liu et al., 2019) encoder for both $h ( \cdot )$ and $g ( \cdot )$ . In $\ S 3 . 1 . 3$ , we show this simple modification yields considerable improvements. Specifically, we apply layer normalization over the start token’s representations from RoBERTa to get the final dense query/passage vectors.
+
+Training and Inference The retriever model is trained as in Karpukhin et al. (2020), where each input query (which at each step consists of a question and previously retrieved passages) is paired with a positive passage and $m$ negative passages to approximate the softmax over all passages. The positive passage is the gold annotated evidence at step $t$ . Negative passages are a combination of passages in the current batch which correspond to other questions (in-batch), and hard negatives which are false adversarial passages. In our experiments, we obtain hard negatives from TF-IDF retrieved passages and their linked pages in Wikipedia. We note that using hyperlinked pages as additional negatives is neither necessary nor critical for our approach. In fact we observe only a very small degradation in performance if we remove them from training (§3.1.3). In addition to in-batch negatives, we use a memory bank $( \mathcal { M } )$ mechanism (Wu et al., 2018) to further increase the number of negative examples for each question. The memory bank stores a large number of dense passage vectors. As we block the gradient back-propagation in the memory bank, its size $( | \mathcal { M } | \gg$ batch size) is less restricted by the GPU memory size. Specifically, after training to convergence with the shared encoder, we freeze a copy of the encoder as the new passage encoder and collect a bank of passage representations across multiple batches to serve as the set of negative passages. This simple extension results in further improvement in retrieval. $( \ S 3 . 1 . 3 )$ .
+
+For inference, we first encode the whole corpus into an index of passage vectors. Given a question, we use beam search to obtain top- $k$ passage sequence candidates, where the candidates to beam search at each step are generated by MIPS using the query encoder at step $t$ , and the beams are scored by the sum of inner products as suggested by the probabilistic formulation discussed above. Such inference relies only on the dense passage index and the query representations, and does not need explicit graph construction using hyperlinks or entity linking. The top- $k$ sequences will then be fed into task-specific downstream modules to produce the desired outputs.
+
+# 3 EXPERIMENTS
+
+Datasets Our experiments focus on two datasets: HotpotQA and Multi-evidence FEVER. HotpotQA (Yang et al., 2018) includes 113k multi-hop questions. Unlike other multi-hop QA datasets (Zhang et al., 2018; Talmor & Berant, 2018; Welbl et al., 2018), where the information sources of the answers are knowledge bases, HotpotQA uses documents in Wikipedia. Thus, its questions are not restricted by the fixed KB schema and can cover more diverse topics. Each question in HotpotQA is also provided with ground truth support passages, which enables us to evaluate the intermediate retrieval performance. Multi-evidence FEVER includes 20k claims from the FEVER (Thorne et al., 2018) fact verification dataset, where the claims can only be verified using multiple documents. We use this dataset to validate the general applicability of our method.
+
+Implementation Details All the experiments are conducted on a machine with 8 32GB V100 GPUs. Our code is based on Huggingface Transformers (Wolf et al., 2019). Our best retrieval results are predicted using the exact inner product search index (IndexFlatIP) in FAISS (Johnson et al., 2017).
+
+Table 1: Retrieval performance in recall at $k$ retrieved passages and precision/recall/F1.   
+
+<table><tr><td rowspan="2">Method</td><td colspan="3">HotpotQA</td><td colspan="3">FEVER</td></tr><tr><td>R@2</td><td>R@10</td><td>R@20</td><td>Precision</td><td>Recall</td><td>F1</td></tr><tr><td>TF-IDF</td><td>10.3</td><td>29.1</td><td>36.8</td><td>14.9</td><td>28.2</td><td>19.5</td></tr><tr><td>TF-IDF+Linked</td><td>17.3</td><td>50.0</td><td>62.7</td><td>18.6</td><td>35.8</td><td>24.5</td></tr><tr><td>DrKIT</td><td>38.3</td><td>67.2</td><td>71.0</td><td>-</td><td>1</td><td>-</td></tr><tr><td>Entity Linking</td><td>1</td><td>1</td><td>1</td><td>30.6</td><td>53.8</td><td>39.0</td></tr><tr><td>MDR</td><td>65.9</td><td>77.5</td><td>80.2</td><td>45.7</td><td>69.1</td><td>55.0</td></tr></table>
+
+Both datasets assume 2 hops, so we fix $n = 2$ for all experiments. Since HotpotQA does not provide the order of the passage sequences, as a heuristic, we consider the passage that includes the answer span as the final passage. 2 In $\ S 3 . 1 . 3$ , we show that the order of the passages is important for effective retriever training. The hyperparameters can be found in Appendix B.1.
+
+# 3.1 EXPERIMENTS: RETRIEVAL
+
+We evaluate our multi-hop dense retriever (MDR) in two different use cases: direct and reranking, where the former outputs the top- $k$ results directly using the retriever scores and the latter applies a task-specific reranking model to the initial results from MDR.
+
+# 3.1.1 DIRECT
+
+We first compare MDR with several efficient retrieval methods that can directly find the top- $k$ passage sequences from a large corpus, including TF-IDF, TF- $\mathrm { . I D F + }$ Linked, DrKIT and Entity Linking. TFIDF is the standard term-matching baseline, while TF-IDF $^ +$ Linked is a straightforward extension that also extracts the hyperlinked passages from TF-IDF passages, and then reranks both TF-IDF and hyperlinked passages with $\mathbf { B } \mathbf { M } 2 5 \mathbf { \Omega } ^ { 3 }$ scores. DrKIT (Dhingra et al., 2020) is a recently proposed dense retrieval approach, which builds a entity-level (mentions of entities) dense index for retrieval. It relies on hyperlinks to extract entity mentions and prunes the search space with a binary mask that restricts the next hop to using hyperlinked entities. On FEVER, we additionally consider an entity linking baseline (Hanselowski et al., 2018) that is commonly used in existing fact verification pipelines. This baseline first uses a constituency parser to extract potential entity mentions in the fact claim and then uses the MediaWiki API to search documents with titles that match the mentions.
+
+Table 1 shows the performance of different retrieval methods. On HotpotQA the metric is recall at the top $k$ paragraphs4, while on FEVER the metrics are precision, recall and $\mathrm { F _ { 1 } }$ in order to be consistent with previous results. On both datasets, MDR substantially outperforms all baselines.
+
+# 3.1.2 RERANKING
+
+Reranking documents returned by efficient retrieval methods with a more sophisticated model is a common strategy for improving retrieval quality. For instance, state-of-the-art multi-hop QA systems usually augment traditional IR techniques with large pretrained language models to select a more compact but precise passage set. On HotpotQA, we test the effectiveness of MDR after a simple BERT-based reranking: each of the top $k$ passage sequences from MDR is first prepended with the original question and then fed into a BERT-like encoder that predicts relevant scores. We train this reranking model with a binary cross-entropy loss, with the target being whether the passage sequence cover both groundtruth passages. We empirically compare our approach with two other existing reranking-based retrieval methods: Semantic Retrieval (Nie et al., 2019) uses BERT at both passage-level and sentence-level to select context from the initial TF-IDF and hyperlinked passages; Graph Recurrent Retriever (Asai et al., 2020) learns to recursively select the best passage sequence on top of a hyperlinked passage graph, where each passage node is encoded with BERT.
+
+Table 2 shows the reranking results. Following Asai et al. (2020), we use Answer Recall and Support Passage Exact Match $( S P E M ) ^ { 5 }$ as the evaluation metrics. Even without reranking, MDR is already better than Semantic Retrieval, which requires around 50 BERT encoding (where each encoding involves cross-attention over a concatenated question-passage pair). After we rerank the top-100 sequences from the dense retriever, our passage recall is better than the state-of-the-art Graph Recurrent Retriever, which uses BERT to process more than 500 passages. We do not compare the reranked results on FEVER, as most FEVER systems directly use BERT encoder to select the top evidence sentences from the retrieved documents, instead of the reranking the documents.
+
+Table 2: HotpotQA reranked retrieval results (input passages for final answer prediction).   
+
+<table><tr><td>Method</td><td>SP EM</td><td>Ans Recall</td></tr><tr><td>Semantic Retrieval</td><td>63.9</td><td>77.9</td></tr><tr><td>Graph Rec Retriever</td><td>75.7</td><td>87.5</td></tr><tr><td>MDR (direct)</td><td>65.9</td><td>75.4</td></tr><tr><td>MDR (reranking)</td><td>81.2</td><td>88.2</td></tr></table>
+
+Table 3: Retriever Model Ablation on HotpotQA retrieval. Single-hop here is equivalent to the DPR method (Karpukhin et al., 2020).   
+
+<table><tr><td>Retriever variants</td><td>R@2</td><td>R@10</td><td>R@20</td></tr><tr><td>Full Retrieval Model</td><td>65.9</td><td>77.5</td><td>80.2</td></tr><tr><td>- w/o linked negatives</td><td>64.6</td><td>76.8</td><td>79.6</td></tr><tr><td>- w/o memory bank</td><td>63.7</td><td>74.2</td><td>77.2</td></tr><tr><td>- w/o shared encoder</td><td>59.9</td><td>70.6</td><td>73.1</td></tr><tr><td>- w/o order</td><td>17.6</td><td>55.6</td><td>62.3</td></tr><tr><td>Single-hop</td><td>25.2</td><td>45.4</td><td>52.1</td></tr></table>
+
+# 3.1.3 ANALYSIS
+
+To understand the strengths and weaknesses of MDR, we conduct further analysis on HotpotQA dev.
+
+Retrieval Error Analysis HotpotQA contains two question categories: bridge questions in which an intermediate entity is missing and needs to be retrieved before inferring the answer; and comparison questions where two entities are mentioned simultaneously and compared in some way. In Figure 2, we show the retrieval performance of both question types. The case of comparison questions proves easier, since both entities needed for retrieval are present in the question.
+
+This case appears almost solved, confirming recent work demonstrating that dense retrieval is very effective at entity linking (Wu et al., 2019).
+
+For the case of bridge questions, we manually inspect 50 randomly sampled erroneous examples after reranking. We find that in half of these cases, our retrieval model predicts an alternative passage sequence that is also valid (see Appendix A.1 for examples).
+
+![](images/54c347cef2bfcd3a83e607929fe87c1ee3d212945badc8d68f52f331dad1176d.jpg)  
+Figure 2: The retrieval performance gap between comparison and bridge questions. Left: recall of groundtruth passage sequences without reranking. Right: Top-1 chain exact match after reranking.
+
+This gives an estimated top-1 passage sequence accuracy of about $90 \%$ . Other remaining errors are due to the dense method’s inability to capture the exact n-gram match between the question and passages. This is a known issue (Lee et al., 2019; Karpukhin et al., 2020) of dense retrieval methods when dealing with questions that have high lexical overlap with the passages. To this end, a hybrid multi-hop retrieval method with both term and dense index might be used to further improve the performance on bridge questions.
+
+Retriever Ablation Study In Table 3, we examine our model with different variations on HotpotQA to show the effectiveness of each proposed component. We see that further training with a memory bank results in modest gains, while using a shared encoder is crucial for the best performance. Respecting the ordering of passages in two hops is essential - training in an order-agnostic manner hardly works at all, and underperforms even the single-hop baseline. Finally, not using hyperlinked paragraphs from TF-IDF passages as additional negatives has only a minor impact on performance.
+
+Question Decomposition for Retrieval As multi-hop questions have more complex structures than simple questions, recent studies (Min et al., 2019; Perez et al., 2020) propose to use explicit question decomposition to simplify the problem. Wolfson et al. (2020) shows that with TF-IDF, using decomposed questions improves the retrieval results. We investigate whether the conclusion still holds with stronger dense retrieval methods. We use the human-annotated question decomposition from the QDMR dataset (Wolfson et al., 2020) for analysis. For a question like Q:Mick Carter is the landlord of a public house located at what address?, QDMR provides two subquestions, SubQ1: What is the public house that Mick Carter is the landlord of? and SubQ2: What is the address that #1 is located at?. We sample 100 bridge questions and replace $\# 1$ in SubQ2 with the correct answer (The Queen Victoria) to SubQ1. Note that this gives advantages to the decomposed method as we ignore any intermediate errors. We estimate the performance of potential decomposed methods with the state-of-the-art single-hop dense retrieval model (Karpukhin et al., 2020).
+
+As shown in Table 4, we did not observe any strong improvements from explicit question decomposition, which is contrary to the findings by Wolfson et al. (2020) when using term-based IR methods. Moreover, as shown in the third row of the table, when the 1st hop of the decomposed retrieval (i.e., SubQ1) is replaced with the original question, no performance degradation is observed. This suggests that strong pretrained encoders can effectively learn to select necessary information from the multi-hop question at each
+
+Table 4: Comparison with decomposed dense retrieval which uses oracle question decomposition (test on 100 bridge questions). See text for details about the decomposed settings.   
+
+<table><tr><td>Method</td><td>R@2</td><td>R@10</td><td>R@20</td></tr><tr><td>MDR</td><td>54.9</td><td>63.7</td><td>70.6</td></tr><tr><td>Decomp (SubQ1;SubQ2)</td><td>50.0</td><td>64.7</td><td>67.6</td></tr><tr><td>Decomp (Q;SubQ2)</td><td>51.0</td><td>64.7</td><td>68.6</td></tr></table>
+
+retrieval step. Regarding the performance drop when using explicit compositions, we hypothesize that it is because some information in one decomposed subquestion could be useful for the other retrieval hop. Examples supporting this hypothesis can be found in Appendix A.2. While this could potentially be addressed by a different style of decomposition, our analysis suggests that decomposition approaches might be sub-optimal in the context of dense retrieval with strong pretrained encoders.
+
+# 3.2 EXPERIMENTS: HOTPOTQA
+
+We evaluate how the better retrieval results of MDR improve multi-hop question answering in this section. As our retriever system is agnostic to downstream models, we test two categories of answer prediction architectures: the extractive span prediction models based on pretrained masked language models, such as BERT (Devlin et al., 2019) and ELECTRA (Clark et al., 2020), and the retrieval-augmented generative reader models (Lewis et al., 2020b; Izacard & Grave, 2020), which are based on pretrained sequence-to-sequence (seq2seq) models such as BART (Lewis et al., 2020a) and T5 (Raffel et al., 2019). Note that compared to more complicated graph reasoning models (Fang et al., 2019; Zhao et al., 2020), these two classes of models do not rely on hyperlinks and can be applied to any text.
+
+Extractive reader models learn to predict an answer span from the concatenation of the question and passage sequence $( [ q , p _ { 1 } , . . . , p _ { n } ] )$ . On top of the token representations produced by pretrained models, we add two prediction heads to predict the start and end position of the answer span.6 To predict the supporting sentences, we add another prediction head and predict a binary label at each sentence start. For simplicity, the same encoder is also responsible for reranking the top $k$ passage sequences. The reranking detail has been discussed in $\ S 3 . 1 . 2$ . Our best reader model is based on ELECTRA (Clark et al., 2020), which has achieved the best single-model performance on the standard SQuAD (Rajpurkar et al., 2018) benchmark. Additionally, we also report the performance of BERT-large with whole word masking (BERT-wwm) to fairly compare with Asai et al. (2020).
+
+Generative models, such as RAG (Lewis et al., 2020b) and FiD (Izacard & Grave, 2020), are based on pretrained seq2seq models. These methods finetune pretrained models with the concatenated questions and retrieved documents as inputs, and answer tokens as outputs. This generative paradigm has shown state-of-the-art performance on single-hop open-domain QA tasks. Specifically, FiD first uses the T5 encoder to process each retrieved passage sequence independently and then uses the decoder to perform attention over the representations of all input tokens while generating answers. RAG is built on the smaller BART model. Instead of only tuning the seq2seq model, it also jointly train the question encoder of the dense retriever. We modified it to allow multi-hop retrieval.
+
+More details about these two classes of reader models are described in Appendix B.2.
+
+3.2.1 RESULTS   
+Table 5: HotpotQA-fullwiki test results.   
+
+<table><tr><td rowspan="2">Methods</td><td colspan="2">Answer</td><td colspan="2">Support</td><td colspan="2">Joint</td></tr><tr><td>EM</td><td>F1</td><td>EM</td><td>F1</td><td>EM</td><td>F1</td></tr><tr><td>GoldEn Retriever (Qi et al.,2019)</td><td>37.9</td><td>48.6</td><td>30.7</td><td>64,2</td><td>18.9</td><td>39.1</td></tr><tr><td>Semantic Retrieval (Nie et al.,2019)</td><td>46.5</td><td>58.8</td><td>39.9</td><td>71.5</td><td>26.6</td><td>49.2</td></tr><tr><td>Transformer-XH (Zhao et al., 2020)</td><td>51.6</td><td>64.1</td><td>40.9</td><td>71.4</td><td>26.1</td><td>51.3</td></tr><tr><td>HGN (Fang et al., 2019)</td><td>56.7</td><td>69.2</td><td>50.0</td><td>76.4</td><td>35.6</td><td>59.9</td></tr><tr><td>DrKIT (Dhingra et al., 2020)</td><td>42.1</td><td>51.7</td><td>37.1</td><td>59.8</td><td>24.7</td><td>42.9</td></tr><tr><td>Graph Recurrent Retriever (Asai et al., 2020)</td><td>60.0</td><td>73.0</td><td>49.1</td><td>76.4</td><td>35.4</td><td>61.2</td></tr><tr><td>MDR (ELECTRA Reader)</td><td>62.3</td><td>75.3</td><td>57.5</td><td>80.9</td><td>41.8</td><td>66.6</td></tr></table>
+
+Comparison with Existing Systems Table 5 compares the HotpotQA test performance of our best ELECTRA reader with recently published systems, using the numbers from the official leaderboard, which measure answer and supporting sentence exact match (EM)/F1 and joint EM/F1. Among these methods, only GoldEn Retriever (Qi et al., 2019) does not exploit hyperlinks. In particular, Graph Recurrent Retriever trains a graph traversal model for chain retrieval; TransformerXH (Zhao et al., 2020) and HGN (Fang et al., 2019) explicitly encode the hyperlink graph structure within their answer prediction models. In fact, this particular inductive bias provides a perhaps unreasonably strong advantage in the specific context of HotpotQA, which by construction guarantees groundtruth passage sequences to follow hyperlinks. Despite not using such prior knowledge, our model outperforms all previous systems by large margins, especially on supporting fact prediction, which benefits more directly from better retrieval.
+
+Reader Model Variants Results for reader model variants are shown in Table 6.7 First, we see that the BERT-wwm reader is $1 \%$ worse than the ELECTRA reader when using enough passages. However, it still outperforms the results in (Asai et al., 2020) which also uses BERT-wwm for answer prediction. While RAG and FiD have shown strong improvements over extractive models on single-hop datasets such as NaturalQuestions (Kwiatkowski
+
+Table 6: Reader comparison on HotpotQA dev set.   
+
+<table><tr><td></td><td>Model</td><td>Topk</td><td>EM</td><td>F1</td></tr><tr><td rowspan="2">Extractive</td><td>ELECTRA ELECTRA</td><td>Top50</td><td>61.7</td><td>74.3</td></tr><tr><td>BERT-wwm</td><td>Top 250 Top250</td><td>63.4 61.5</td><td>76.2 74.7</td></tr><tr><td rowspan="2">Generative</td><td rowspan="2">Multi-hop RAG FiD</td><td>Top 4*4</td><td>51.2</td><td>63.9</td></tr><tr><td>Top 50</td><td>61.7</td><td>73.1</td></tr></table>
+
+et al., 2019), they do not show an advantage in the multi-hop case. Despite having twice as many parameters as ELECTRA, FiD fails to outperform it using the same amount of context (top 50). In contrast, on NaturalQuestions, FiD is 4 points better than a similar extractive reader when using the top 100 passages in both.8 We hypothesize that the improved performance on single-hop questions is due to the ability of larger pretrained models to more effectively memorize single-hop knowledge about real-world entities.9 Compared to multi-hop questions that involve multiple relations and missing entities, simple questions usually only ask about a certain property of an entity. It is likely that such simple entity-centric information is explicitly mentioned by a single text piece in the pretraining corpus, while the evidence for multihop questions is typically dispersed, making the complete reasoning chain nontrivial to memorize. More analysis on RAG can be found in Appendix A.3.
+
+Table 7: Multi-Evidence FEVER Fact Verification Results. Loose-Multi represents the subset that requires multiple evidence sentences. Strict-Multi is a subset of Loose-Multi that require multiple evidence sentences from different documents.   
+
+<table><tr><td>Method</td><td>Loose-Multi (1,960) LA FEVER</td><td>Strict-Multi (1,059) LA</td><td>FEVER</td></tr><tr><td>GEAR</td><td>66.4</td><td></td><td></td></tr><tr><td>GAT</td><td>66.1</td><td>1 -</td><td>1 =</td></tr><tr><td>KGAT with ESIM rerank</td><td>65.9</td><td>51.5</td><td>7.7</td></tr><tr><td>KGAT with BERT rerank</td><td>65.9</td><td>51.0</td><td>6.2</td></tr><tr><td>Ours + KGAT with BERT rerank</td><td>77.9</td><td>72.1</td><td>16.2</td></tr></table>
+
+Inference Efficiency To compare with existing multi-hop QA systems in terms of efficiency, we follow Dhingra et al. (2020) and measure the inference time with 16 CPU cores and batch size 1. We implement our system with a fast approximate nearest neighbor search method, i.e., HNSW (Malkov & Yashunin, 2018), which achieves nearly the same performance as exact search. With an in-memory index, we observe that the retrieval time is negligible compared to the forward pass of large pretrained models. Similarly, for systems that use term-based indices, the BERT calls for passage reranking cause the main efficiency bottleneck. Thus, for systems that do not release the end-to-end code, we estimate the running time based on the number of BERT cross-attention forward passes (the same estimation strategy used by Dhingra et al. (2020)), and ignore the overhead caused by ad
+
+![](images/71c9c5c48ff3e7e240450bcaa22e5bd13cfdaed09d05f4b9869bb2c235bfdc32.jpg)  
+Figure 3: Efficiency-performance trade-off comparison with published HotpotQA systems. The curve is plotted with different number of top $k$ $( k { = } 1 , 5 , 1 0 , 2 0 , 5 0 , 1 0 0 , 2 0 0 )$ passage sequences we feed into the reader model. seq/Q denotes the time required for each query.
+
+ditional processing such as TF-IDF or linking graph construction. As shown in Figure 3, our method is about 10 times faster than current state-of-the-art systems while achieving a similar level of performance. Compared to two efficient systems (DrKIT and GoldEn), we achieve over 10 points improvement while only using the top-1 retrieval result for answer and supporting sentence prediction.
+
+# 3.3 EXPERIMENTS: MULTI-EVIDENCE FEVER
+
+For FEVER claim verification, we reuse the best open-sourced verification system, i.e., KGAT (Liu et al., 2020), to show the benefit of our retrieval approach over existing retrieval methods. We report the results in verification label accuracy (LA) and the FEVER score10 in Table 7, where the numbers of competitive baselines, GEAR (Zhou et al., 2019), graph attention network (GAT) (Velickovi ˇ c´ et al., 2017) and variants of KGAT are from the KGAT (Liu et al., 2020) paper. All these baselines use entity linking for document retrieval, then rerank the sentences of the retrieved documents, and finally use different graph attention mechanisms over the fully-connected sentence graph to predict verification labels. Since some instances in the multi-evidence subset used by previous studies only needs multiple evidence sentences from the same document, we additionally test on a strict multi-hop subset with instances that need multiple documents. As shown by the results, even without finetuning the downstream modules, simply replacing the retrieval component with MDR leads to significant improvements, especially on the strict multi-evidence subset.
+
+# 4 RELATED WORK
+
+Open-domain QA with Dense Retrieval In contrast to sparse term-index IR methods that are widely used by existing open-domain QA systems (Chen et al., 2017; Wang et al., 2018; Yang et al., 2019), recent systems (Lee et al., 2019; Guu et al., 2020; Karpukhin et al., 2020) typically uses dense passage retrieval techniques that better capture the semantic matching beyond simple n-gram overlaps. To generate powerful dense question and passage representations, these methods either conduct large-scale pretraining with self-supervised tasks that are close to the underlying question-passage matching in retrieval, or directly use the human-labeled question-passage pairs to finetune pretrained masked language models. On single-hop information-seeking QA datasets such as NaturalQuestions (Kwiatkowski et al., 2019) or WebQuestions (Berant et al., 2013), these dense methods have achieved significant improvements over traditional IR methods. Prior to these methods based on pretrained models, Das et al. (2019) use RNN encoder to get dense representations of questions and passages. They also consider an iterative retrieval process and reformulate the query representation based on reader model’s hidden states. However, their method requires an initial round of TF-IDF/BM25 retrieval and a sophisticated RL-based training paradigm to work well. Finally, like the aforementioned methods, only single-hop datasets are considered in their experiments. More akin to our approach, Feldman & El-Yaniv (2019) use a similar recursive dense retrieval formulation for multi-hop QA. In contrast to their biattenional reformulation component applied on top of token query and passage representations, we adopt a more straightforward query reformulation strategy, by simply concatenating the original query and previous retrieval as the inputs to the query encoder. Together with stronger pretrained encoders and more effective training methods (in-batch $^ +$ memory bank negative sampling vs their binary ranking loss), MDR is able to double the accuracy of their system.
+
+Query Expansion Techniques in IR As our dense encoder augments the original question with the initial retrieved results to form the updated query representation, our work is also relevant to query expansion techniques (Rocchio, 1971; Voorhees, 1994; Ruthven & Lalmas, 2003) that are widely used in traditional IR systems. In particular, our system is similar in spirit to pseudo-relevance feedback techniques (Croft & Harper, 1979; Cao et al., 2008; Lv & Zhai, 2010), where no additional user interaction is required at the query reformulation stage. Existing studies mainly focus on alleviating the uncertainty of the user query (Collins-Thompson & Callan, 2007) by adding relevant terms from the first round of retrieval, where the retrieval target remains the same throughout the iterative process. In contrast, the query reformulation in our approach aims to follow the multi-hop reasoning chain and effectively retrieves different targets at each step. Furthermore, instead of explicitly selecting terms to expand the query, we simply concatenate the whole passage and rely on the pretrained encoder to choose useful information from the last retrieved passage.
+
+Other Multi-hop QA Work Apart from HotpotQA, other multi-hop QA datasets (Welbl et al., 2018; Talmor & Berant, 2018; Zhang et al., 2018) are mostly built from knowledge bases (KBs). Compared to questions in HotpotQA, questions in these datasets are rather synthetic and less diverse. As multi-hop relations in KBs could be mentioned together in a single text piece, these datasets are not designed for an open-domain setting which necessitates multi-hop retrieval. Existing methods on these datasets either retrieve passages from a small passage pool pruned based on the the specific dataset (Sun et al., 2019; Dhingra et al., 2020), or focus on a non-retrieval setting where a compact documents set is already given (De Cao et al., 2018; Zhong et al., 2019; Tu et al., 2019; Beltagy et al., 2020). Compared to these research, our work aims at building an efficient multi-hop retrieval model that easily scales to large real-world corpora that include millions of open-domain documents.
+
+# 5 CONCLUSION
+
+In this work, we generalized the recently proposed successful dense retrieval methods by extending them to the multi-hop setting. This allowed us to handle complex multi-hop queries with much better accuracy and efficiency than the previous best methods. We demonstrated the versatility of our approach by applying it to two different tasks, using a variety of downstream modules. In addition, the simplicity of the framework and the fact that it does not depend on a corpus-dependent graph structure opens the possibility of applying such multi-hop retrieval methods more easily and broadly cross different domains and settings.
+
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+
+# A QUALITATIVE ANALYSIS
+
+# A.1 FALSE BRIDGE QUESTION ERROR CASES
+
+As mentioned in $\ S 3 . 1 . 3$ , half of the errors of bridge questions are not real errors. In Table 8, we can see that the model predicts alternative passage sequences that could also be used to answer the questions.
+
+# Predicted:
+
+1. Museum of Human Beings: Museum of Human Beings, included in the National American Indian Heritage Month Booklist, November 2012 and 2013 is a novel written by Colin Sargent, which delves into the heart-rending life of Jean-Baptiste Charbonneau, the son of Sacagawea. Sacagawea was the Native American guide, who at 16 led the Lewis and Clark expedition.
+
+2. Jean Baptiste Charbonneau: Jean Baptiste Charbonneau (February 11, 1805 – May 16, 1866) was an American Indian explorer, guide, fur trapper trader, military scout during the MexicanAmerican War, ”alcalde” (mayor) of Mission San Luis Rey de Francia and a gold prospector and hotel operator in Northern California. He spoke French and English, and learned German and Spanish during his six years in Europe from 1823 to 1829. He spoke Shoshone, his mother tongue, and other western American Indian languages...
+
+Q: Altnahinch is located in a county that has a population density of how many per square mile? Ground-truth SP Passage Titles: Altnahinch Dam; County Antrim
+
+# Predicted:
+
+1. Altnahinch: Altnahinch is a townland in County Antrim, Northern Ireland.
+
+2. County Antrim: County Antrim (named after the town of Antrim, from Irish: ”Aontroim” , meaning ”lone ridge” , )) is one of six counties that form Northern Ireland. Adjoined to the north-east shore of Lough Neagh, the county covers an area of $3 0 4 6 \mathrm { k m } 2$ and has a population of about 618,000. County Antrim has a population density of 203 people per square kilometer / 526 people per square mile...
+
+Q: What foundation do scholars give for the likelihood of collaboration on a William Shakespeare Play written between 1588 and 1593?
+
+# Ground-truth SP Passage Titles:
+
+Authorship of Titus Andronicus, William Shakespeare’s collaborations
+
+# Predicted:
+
+1. Titus Andronicus: Titus Andronicus is a tragedy by William Shakespeare, believed to have been written between 1588 and 1593, probably in collaboration with George Peele. It is thought to be Shakespeare’s first tragedy, and is often seen as his attempt to emulate the violent and bloody revenge plays of his contemporaries, which were extremely popular with audiences throughout the 16th century.
+
+2. William Shakespeare’s collaborations: Like most playwrights of his period, William Shakespeare did not always write alone... Some of the following attributions, such as ”The Two Noble Kinsmen”, have well-attested contemporary documentation; others, such as ”Titus Andronicus”, are dependent on linguistic analysis by modern scholars...
+
+Q: Zach Parise’s father played in which league?
+
+Ground-truth SP Passage Titles: Jordan Parise; Zach Parise
+
+# Predicted:
+
+1. Zach Parise: Zachary Justin Parise (born July 28, 1984) is an American professional ice hockey left winger who is currently serving as an alternate captain for the Minnesota Wild in the National Hockey League (NHL). He has also played for the New Jersey Devils, where he served as team captain and led the team to the 2012 Stanley Cup Finals. Parise’s father, J. P. Parise... ´ 2. J. P. Parise´: Jean-Paul Joseph-Louis Parise (December 11, 1941 – January 7, 2015) was a ´ Canadian professional ice hockey coach and player. Parise played in the National Hockey League (NHL), most notably for the Minnesota North Stars and the New York Islanders.
+
+<table><tr><td>Table 9: Sampled retrieval errors (marked in red) only made by the decomposed system. These errors could be potentially avoided if the model has access to the full information in the original question or previous hop results. The important clue for correctly retrieving the documents or avoiding errors is marked in blue. Once decomposed,the marked information are not longer available in one of the decomposed retrieval hop. Multi-hop Question: What is the birthday of the author of &quot;She Walks These Hills&quot;?</td></tr><tr><td>Decomposed Questions: 1.Who is the author of She Walks These Hills? 2. What is the birthday of Sharyn McCrumb? Ground-truth SP Passages: She Walks These Hills: She Walks These Hills is a book written by Sharyn McCrumb and published by Charles Scribner&#x27;s Sons in 1994, which later went on to win the Anthony Award for Best Novel in 1995. Sharyn McCrumb: Sharyn McCrumb (born February 26,1948) is an American writer whose books celebrate the history and folklore of Appalachia.McCrumb is the winner of numerous</td></tr><tr><td>literary awards... Decomposed Error Case: 1. She Walks These Hills (√) 2. Tané McClure: Tané M. McClure (born June 8, 1958) is an American singer and actress.</td></tr><tr><td>Multi-hop Question: When was the album with the song Unbelievable by American rapper The Notorious B.I.G released? Decomposed Questions: 1. What is the album with the song Unbelievable by American rapper The Notorious B.I.G?</td></tr><tr><td>2.When was the album Ready to Die released? Ground-truth SP Passages: Unbelievable (The Notorious B.I.G. song): Unbelievable is a song by American rapper The Notorious B.I.G., recorded for his debut studio album Ready to Die... Ready to Die: Ready to Die is the debut studio album by American rapper The Notorious B.I.G.; it was released on September 13,1994,by Bad Boy Records and Arista Records.. Decomposed Error Case: 1.Unbelievable (The Notorious B.I.G. song) (√)</td></tr><tr><td>2. Ready to Die (The Stooges album): Ready to Die is the fifth and final studio album by Amer- ican rock band Iggy and the Stooges. The album was released on April 30, 2013... Multi-hop Question: Whose death dramatized in a stage play helped end the death penalty in Australia? Decomposed Questions: 1. What is the stage play that helped end the death penalty in Australia?</td></tr><tr><td>2. Whose death was dramatized in Remember Ronald Ryan? Ground-truth SP Passages: Barry Dickins: Barry Dickins (born 1949) is a prolific Australian playwright, author, artist, actor, educator and journalist.. His most well-known work is the award winning stage play &quot;Remember Ronald Ryan”,a dramatization of the life and subsequent death of Ronald Ryan, the last man executed in Australia... Ronald Ryan: Ronald Joseph Ryan (21 February 1925-3 February 1967) was the last person to</td></tr></table>
+
+# A.3 EXTRACTIVE & GENERATIVE READER MODEL
+
+Table 6 demonstrates the answer prediction performance for four different reader models. The extractive models predict answers given the top 250 retrieved passage sequences (pairs of passage from hop1 and hop2). Since generative models are generally heavier on the computation side, we can only use fewer passages. Besides the observations alredy discussed in $\ S 3 . 2 . 1$ , we hypothesize the worse performance of multi-hop RAG compared to FiD is partially due to the smaller pretrained model used in RAG, i.e., BART is only half the size of T5-large. Also, as RAG back-propagate the gradients to the query encoder, it needs more memory footprint and can only take in fewer retrieved contexts. Our RAG implementation largely follows the implementation of the original paper and we did not use the PyTorch checkpoint (as used by FiD) to trade computation for memory. We conjecture the multi-hop RAG performance will also improve if we augment the current implementation with memory-saving tricks. However, given the same amount of context and read model size, the multi-hop RAG is still worse than the extractive ELECTRA reader, i.e., with only the top 1 retrieved passage sequence, our ELECTRA reader gets $5 3 . 8 \mathrm { E M }$ compared to the 51.2 answer EM achieved by multi-hop RAG when using more context.
+
+Table 10: Answer EM using top 50 retrieved passage chains   
+
+<table><tr><td>Model</td><td>Overall</td><td>Comp (20%)</td><td>Bridge (80%)</td></tr><tr><td>ELECTRA</td><td>61.7</td><td>79.0</td><td>57.4</td></tr><tr><td>FiD</td><td>61.7</td><td>75.3</td><td>58.3</td></tr></table>
+
+Given the same number of retrieved passage sequences (top 50) as shown in table 10, FiD obtains similar performance to ELECTRA, despite that the generative model can generate arbitrary answers for the given input. (We tried constrained decoding for the generative model. However, no significant performance improvements were observed, indicating that the errors from the generative model are not due to the free-form generation task.) Further question type analysis in HotpotQA showed that the main difference comes from the comparison type of question, while for bridge question, FiD performs slightly better than ELECTRA. This finding might indicate that for generation models, numerical comparison is still a bigger issue compared to extractive models.
+
+# B MODEL DETAILS
+
+# B.1 BEST MODEL HYPERPARAMETERS
+
+Table 11: Hyperparameters of Retriever   
+
+<table><tr><td>learning rate batch size maximum passage length maximum query length at initial hop maximum query length at 2nd hop</td><td>2e-5 150 300 70 350</td></tr><tr><td>warmup ratio gradient clipping norm</td><td>0.1 2.0</td></tr><tr><td>traininig epoch weight decay</td><td>50 0</td></tr></table>
+
+Table 12: Hyperparameters of Extractive Reader (ELECTRA)   
+
+<table><tr><td>learning rate batch size</td><td>5e-5 128</td></tr><tr><td>maximum sequence length maximum answer length</td><td>512 30</td></tr><tr><td>Warmup ratio</td><td>0.1</td></tr><tr><td>gradient clipping norm</td><td>2.0</td></tr><tr><td>traininig epoch weight decay</td><td>7</td></tr><tr><td></td><td>0</td></tr><tr><td># of negative context per question weight of SP sentence prediction loss</td><td>5 0.025</td></tr></table>
+
+# B.2 FURTHER DETAILS ABOUT READER MODELS
+
+# B.2.1 EXTRACTIVE READER
+
+The extractive reader is trained with four loss functions. With the [CLS] token, we predict a reranking score based on whether the passage sequence match the groundtruth supporting passages. On top of the representation of each token, we predict a answer start score and answer end score. Finally, we prepend each sentence with the [unused0] special token and predict whether the sentence is one of the supporting sentences using the representations of the special token. At training time, we pair each question with 1 groundtruth passage sequence and 5 negative passage sequence which do not contain the answer. At inference time, we feed in the top 250 passage sequences from MDR. We rank the predicted answer for each sequence with a linear combination of the reranking score and the answer span score. The combination weight is selected based on the dev results.
+
+# B.2.2 FUSION-IN-DECODER
+
+The FiD model uses T5-large as the underlying seq2seq model. It is twice as large as the extractive models and has 770M parameters. We reuse the hyperparameters as described in Izacard & Grave (2020). The original FiD uses the top 100 passages for NaturalQuestions. In our case, we use the top 50 retrieved passage sequences and concatenate the passages in each sequence before feeding into T5. In order to fit this model into GPU, we make use of PyTorch checkpoint 11 for training.
+
+# B.2.3 MULTI-HOP RAG
+
+The RAG model aims to generate answer $y$ given question $x$ and the retrieved documents $z$ . Similarly, the goal of multi-hop RAG can be expressed as: generate answer $y$ given question $x$ and retrieved documents in hop one $z _ { 1 }$ and hop two $z _ { 2 }$ (Limiting to two hops for HotpotQA). The model has three components:
+
+• Hop-one retriever $p _ { \eta _ { 1 } } ( z _ { 1 } | x )$ with parameter $\eta _ { 1 }$ to represent the retrieved top- $\mathbf { \nabla } \cdot \mathbf { k }$ passage distribution (top-k truncated distribution) given the input question $x$ .   
+• Hop-two retriever $p _ { \eta _ { 2 } } ( z _ { 2 } | x , z _ { 1 } )$ with parameter $\eta _ { 2 }$ to represent the hop-two retrieved top- $\mathbf { \nabla } \cdot \mathbf { k }$ passage distribution given not only the question $x$ but also the retrieved document $z _ { 1 }$ from hop-one.   
+• A generator $p _ { \theta } ( y _ { i } | x , z _ { 1 } , z _ { 2 } , , y _ { 1 : i - 1 } )$ to represent the next token distribution given input question $x$ , hop-one retrieved document $z _ { 1 }$ , hop-two retrieved document $z _ { 2 }$ and previous predicted token $y _ { 1 : i - 1 }$ parametrized by $\theta$
+
+Multi-Hop RAG Sequence Model As the RAG Sequence model, this model generates the answer sequence given the fixed set of documents from hop-one retriever and hop-two retriever. In order to the get the probability of the generated sequence, we marginalize through the two latent variables corresponding to the two retrieval hops:
+
+$$
+\begin{array} { l } { { \displaystyle p _ { s e q u e n c e } ( y | x ) = } } \\ { { \displaystyle \sum _ { z _ { 1 } } p _ { \eta _ { 1 } } ( z _ { 1 } | x ) \sum _ { z _ { 2 } } p _ { \eta _ { 2 } } ( z _ { 2 } | x , z _ { 1 } ) \prod _ { i } ^ { N } p _ { \theta } ( y _ { i } | x , z _ { 1 } , z _ { 2 } , y _ { 1 : i - 1 } ) } } \\ { { \displaystyle \sum _ { z _ { 1 } } \sum _ { z _ { 2 } } p _ { \eta _ { 1 } } ( z _ { 1 } | x ) p _ { \eta _ { 2 } } ( z _ { 2 } | x , z _ { 1 } ) \prod _ { i } ^ { N } p _ { \theta } ( y _ { i } | x , z _ { 1 } , z _ { 2 } , y _ { 1 : i - 1 } ) } } \end{array}
+$$
+
+where $z _ { 1 }$ and $z _ { 2 }$ are top $\mathrm { k }$ document from the respective retrieval modules.
+
+Multi-Hop RAG Token Model Moreover, the model can make predictions based on different passage extracted at each token.
+
+$$
+\begin{array} { l } { { \displaystyle p _ { t o k e n } ( y | x ) = } } \\ { { \displaystyle \prod _ { i } ^ { N } \sum _ { z _ { 1 } } \sum _ { z _ { 2 } } p _ { \eta _ { 1 } } ( z _ { 1 } | x ) p _ { \eta _ { 2 } } ( z _ { 2 } | x , z _ { 1 } ) p _ { \theta } ( y _ { i } | x , z _ { 1 } , z _ { 2 } , y _ { 1 : i - 1 } ) } } \end{array}
+$$
+
+The predicted probability for each token is the following
+
+$$
+\begin{array} { l } { \displaystyle p _ { t o k e n } ( y _ { i } | ( x , y _ { j } ) ) = } \\ { \displaystyle \sum _ { z _ { 1 } } \sum _ { z _ { 2 } } p _ { \eta _ { 1 } } ( z _ { 1 } | x ) p _ { \eta _ { 2 } } ( z _ { 2 } | x , z _ { 1 } ) p _ { \theta } ( y _ { i } | x , z _ { 1 } , z _ { 2 } , y _ { 1 : i - 1 } ) } \end{array}
+$$
+
+# C RETRIEVAL-FREE APPROACHES
+
+Inspired by a recent work (Roberts et al., 2020) that trains the T5 seq2seq model to directly decode answers from questions (retrieval-free), we conduct similar experiments on HotpotQA using BART (Lewis et al., 2020a). As shown in Figure 4, the performance gap between retrieval-based methods and retrieval-free methods on multi-hop QA is much larger than the gap in the case of simple single-hop questions.
+
+![](images/0b4e83817ae46076ce6fb53037be9db31135934d4b4731e083ba622fb8d1e059.jpg)  
+Figure 4: Performance gap between retrieval-free and retrieval-based methods on different QA datasets.
+
+# D A UNIFIED QA RETRIEVAL SYSTEM
+
+In practice, when a fixed text corpus is given for open-domain systems, we do not know beforehand whether the incoming questions require single or multiple text evidence. Thus, it is essential to build a unified system that adaptively retrieves for multiple hops. Due to the simplicity of the approach, our method can easily be extended in the unified setup. To the best of our knowledge, only (Asai et al., 2020) test the same retrieval method on both single and multi-hop questions but with separate trained models. Here we take a further step and explore the possibility of using a single retrieval model for both types of questions.
+
+To enable adaptive retrieval, we add a binary prediction head on top of the question encoder. Once the retriever finishes the 1-hop retrieval, it encodes concatenation of $q$ and $p _ { 1 }$ and predicts whether to stop retrieval using the final hidden state of the first token. We construct this unified setting with NaturalQuestions-Open (Lee et al., 2019) (NQ) as single-hop and HotpotQA as multi-hop. As the two datasets use different corpora, we merge the two12 for easy comparison. As baselines, we use the retrieval models trained only on the respective dataset. For HotpotQA, the baseline is the best multi-hop retrieval model discussed in the main text. For NQ, we follow the training method in DPR (Karpukhin et al., 2020), but with a shared question and passage encoder, which achieves stronger results. As the NQ corpus includes multiple passages of the same document and the HotpotQA corpus only uses the introduction passage, we are not able to compute the strict title-based support passage recall for HotpotQA as in $\ S 3 . 2$ . Thus, we only evaluate answer recall. Results are in Table 13. In contrast to existing studies that train different models for each dataset, we show that a unified dense retrieval model can maintain competitive performance on both, despite the vastly different nature of both datasets. Note that the information-seeking questions in NQ is usually noisier and more ambiguous, while HotpotQA questions are more complicated and contains more lexical overlaps with the evidence passages. Specifically, for NQ, the unified retrieval model achieves very similar performance as the single-dataset DPR model, while the performance on HotpotQA decreases more. We conjecture that this is because the information-seeking questions in NQ cover more diverse patterns, and the added HotpotQA training questions do not cause a dramatic distribution shift from the NQ test data. We leave the development of a more general retrieval system that handles different styles of questions to future work.
+
+Table 13: Comparing the unified retrieval model with models specifically trained for each task. We test the retrieval performance with a single merged corpus. For easy comparison, all three models are based on BERT-base encoder which we find achieves stronger performance than RoBERTa-base on NQ. AR $@ \mathrm { K }$ denotes answer recall at top-K retrieved passage sequences.   
+
+<table><tr><td rowspan="2">Model</td><td colspan="2">NQ</td><td colspan="2">HotpotQA</td></tr><tr><td>AR@20</td><td>AR@100</td><td>AR@20</td><td>AR@100</td></tr><tr><td rowspan="2">single-hop only</td><td>80.7</td><td>87.3</td><td>-</td><td>1</td></tr><tr><td>=</td><td>-</td><td>83.4</td><td>89.4</td></tr><tr><td>multi-hop only unified</td><td>79.5</td><td>86.1</td><td>78.1</td><td>83.0</td></tr></table>

parse/train/SyjjD1WRb/SyjjD1WRb.md

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+# EVOLUTIONARY EXPECTATION MAXIMIZATION FOR GENERATIVE MODELS WITH BINARY LATENTS
+
+Anonymous authors Paper under double-blind review
+
+# ABSTRACT
+
+We establish a theoretical link between evolutionary algorithms and variational parameter optimization of probabilistic generative models with binary hidden variables. While the novel approach is independent of the actual generative model, here we use two such models to investigate its applicability and scalability: a noisy-OR Bayes Net (as a standard example of binary data) and Binary Sparse Coding (as a model for continuous data). Learning of probabilistic generative models is first formulated as approximate maximum likelihood optimization using variational expectation maximization (EM). We choose truncated posteriors as variational distributions in which discrete latent states serve as variational parameters. In the variational E-step, the latent states are then optimized according to a tractable free-energy objective. Given a data point, we can show that evolutionary algorithms can be used for the variational optimization loop by (A) considering the bit-vectors of the latent states as genomes of individuals, and by (B) defining the fitness of the individuals as the (log) joint probabilities given by the used generative model. As a proof of concept, we apply the novel evolutionary EM approach to the optimization of the parameters of noisy-OR Bayes nets and binary sparse coding on artificial and real data (natural image patches). Using point mutations and single-point cross-over for the evolutionary algorithm, we find that scalable variational EM algorithms are obtained which efficiently improve the data likelihood. In general we believe that, with the link established here, standard as well as recent results in the field of evolutionary optimization can be leveraged to address the difficult problem of parameter optimization in generative models.
+
+# 1 INTRODUCTION
+
+Evolutionary algorithms (EA) have been introduced (e.g. Fogel et al., 1966; Rechenberg, 1965) as a technique for function optimization using methods inspired by biological evolutionary processes such as mutation, recombination, and selection. As such EAs are of interest as tools to solve Machine Learning problems, and they have been frequently applied to a number of tasks such as clustering (Pernkopf & Bouchaffra, 2005; Hruschka et al., 2009), reinforcement learning (Salimans et al., 2017), and hierarchical unsupervised (Myers et al., 1999) or deep supervised learning (e.g., Stanley & Miikkulainen 2002 and Suganuma et al. 2017; Real et al. 2017 for recent examples). In some of these tasks EAs have been investigated as alternatives to standard procedures (Hruschka et al., 2009), but most frequently EAs are used to solve specific sub-problems. For example, for classification with Deep Neural Networks (DNNs LeCun et al., 2015; Schmidhuber, 2015), EAs are frequently applied to solve the sub-problem of selecting the best DNN architectures for a given task (e.g. Stanley & Miikkulainen, 2002; Suganuma et al., 2017) or more generally to find the best hyper-parameters of a DNN (e.g. Loshchilov & Hutter, 2016; Real et al., 2017).
+
+Inspired by these previous contributions, we here ask if EAs and learning algorithms can be linked more tightly. To address this question we make use of the theoretical framework of probabilistic generative models and expectation maximization (EM Dempster et al., 1977) approaches for parameter optimization. The probabilistic approach in combination with EM is appealing as it establishes a very general unifying framework able to encompass diverse algorithms from clustering and dimensionality reduction (Roweis, 1998; Tipping & Bishop, 1999) over feature learning and sparse coding (Olshausen & Field, 1997) to deep learning approaches (Patel et al., 2016). However, for most generative data models, EM is computationally intractable and requires approximations. Variational
+
+EM is a very prominent such approximation and is continuously further developed to become more efficient, more accurate and more autonomously applicable. Variational EM seeks to approximately solve optimization problems of functions with potentially many local optima in potentially very high dimensional spaces. The key observation exploited in this study is that a variational EM algorithm can be formulated such that latent states serve as variational parameters. If the latent states are then considered as genomes of individuals, EAs emerge as a very natural choice for optimization in the variational loop of EM.
+
+# 2 TRUNCATED VARIATIONAL EM
+
+A probabilistic generative model stochastically generates data points $\vec { y }$ using a set of hidden (or latent) variables $\vec { s } ,$ . The generative process can be formally expressed in the form of joint probability $p ( \vec { s } , \vec { y } | \Theta )$ , where $\Theta$ are the model parameters. Given a set of $N$ data points, $\vec { y } ^ { ( 1 ) } , \dotsc , \vec { y } ^ { ( N ) } =$ $\vec { y } ^ { ( 1 : N ) }$ , learning seeks to change the parameters $\Theta$ so that the data generated by the generative model becomes as similar as possible to the $N$ real data points. One of the most popular approaches to achieve this goal is to seek maximum likelihood (ML) parameters $\Theta ^ { * }$ , i.e., parameters that maximize the data log-likelihood for a given generative model:
+
+$$
+L ( \Theta ) : = \log ( \mathcal { L } ( \Theta ) ) = \sum _ { n } \log \big ( \sum _ { \{ \vec { s } \} } p \left( \vec { y } ^ { n } , \vec { s } \mid \Theta \right) \big )
+$$
+
+To efficiently find (approximate) ML parameters we follow Saul & Jordan (1996); Neal & Hinton (1998); Jordan et al. (1999) who reformulated the problem in terms of a maximization of a lower bound of the log-likelihood, the free energy $\mathcal { F } ( \vec { q } , \Theta )$ . Free energies are given by
+
+$$
+\mathcal { F } ( q ^ { ( 1 : N ) } , \Theta ) = \sum _ { n = 1 } ^ { N } \Big ( \sum _ { \{ \vec { s } \} } q ^ { ( n ) } ( \vec { s } ) \log \big ( p ( \vec { s } , \vec { y } ^ { ( n ) } | \Theta ) \big ) \Big ) + \sum _ { n = 1 } ^ { N } H ( q ^ { ( n ) } ( \vec { s } ) ) ,
+$$
+
+where $q ^ { ( n ) } ( \vec { s } )$ are variational distributions, and where $H ( q )$ denotes the entropy of a distribution $q$ . For the purposes of this study, we consider elementary generative models which are difficult to train because of exponentially large state spaces. These models serve well for illustrating the approach but we stress that any generative model which gives rise to a joint distribution $p ( \bar { \vec { s } } , \bar { y } | \Theta ) \bar { \vec { s } }$ can be trained with the approach discussed here as long as the latents $\vec { s }$ are binary.
+
+In order to find approximate maximum likelihood solutions, distributions $q ^ { ( n ) } ( \vec { s } )$ are sought that approximate the intractable posterior distributions $p ( \vec { s } | \vec { y } ^ { ( n ) } , \Theta )$ as well as possible, which results in the free-energy being as similar (or tight) as possible to the exact log-likelihood. At the same time variational distributions have to result in tractable parameter updates. Standard approaches include Gaussian variational distributions (e.g. Opper & Winther, 2005) or mean-field variational distributions (Jordan et al., 1999). If we denote the parameters of the variational distributions by $\Lambda$ , then a variational EM algorithm consists of iteratively maximizing $\mathcal { F } ( \Lambda , \Theta )$ w.r.t. $\Lambda$ in the variational E-step and w.r.t. $\Theta$ in the M-step. The M-step can hereby maintain the same functional form as for exact EM but the expectation values now have to be computed w.r.t. the variational distributions.
+
+Instead of using parametric functions such as Gaussians or factored (mean-field) distributions, for our purposes we choose truncated variational distributions defined as a function of a finite set of states (Lucke & Eggert, 2010; Sheikh et al., 2014; Shelton et al., 2017). These states will later serve ¨ as populations of evolutionary algorithms. If we denote $\kappa ^ { n }$ a population of hidden states for a given data point $\vec { y } ^ { ( n ) }$ , then variational distributions and their corresponding expectation values are given by (e.g. Lucke & Eggert, 2010; Sheikh et al., 2014): ¨
+
+$$
+q ^ { n } ( \vec { s } \mid K ^ { n } , \Theta ) : = \frac { p \left( \vec { s } \mid \vec { y ^ { n } } , \Theta \right) } { \sum _ { \vec { s ^ { \prime } } \in K ^ { n } } p \left( \vec { s } ^ { \prime } \mid \vec { y ^ { n } } , \Theta \right) } \delta ( \vec { s } \in K ^ { n } ) , \langle g ( \vec { s } ) \rangle _ { q ^ { n } } = \frac { \sum _ { \vec { s } \in K ^ { n } } p ( \vec { s } , \vec { y ^ { n } } \mid \Theta ) g ( \vec { s } ) } { \sum _ { \vec { s ^ { \prime } } \in K ^ { n } } p ( \vec { s } ^ { \prime } , \vec { y ^ { n } } \mid \Theta ) } .
+$$
+
+where $\delta ( \vec { s } \in \mathcal { K } ^ { n } )$ is 1 if $\textstyle { \mathcal { K } } ^ { n }$ contains the hidden state $\vec { s }$ , zero otherwise. If the set $K ^ { n }$ contains all states with significant posterior mass, then (3) approximates expectations w.r.t. full posteriors very well. By inserting truncated distributions as variational distribution of the free-energy (2), it can be
+
+shown (Lucke, 2016) that the free-energy takes a very compact simplified form given by: ¨
+
+$$
+\mathcal { F } ( \mathcal { K } , \Theta ) = \sum _ { n } \log \big ( \sum _ { \vec { s } \in \mathcal { K } ^ { n } } p \left( \vec { y } ^ { n } , \vec { s } \mid \Theta \right) \big ) , \mathrm { ~ w h e r e ~ } \mathcal { K } = ( \mathcal { K } ^ { 1 } , \ldots , \mathcal { K } ^ { N } ) .
+$$
+
+As the variational parameters of the variational distribution (3) are now given by populations of hidden states, a variational $\mathrm { E }$ -step now consists of finding for each data point $n$ the population $\textstyle { \mathcal { K } } ^ { n }$ that maximizes $\begin{array} { r } { \sum _ { \vec { s } \in \mathcal { K } ^ { n } } p ( \vec { y } ^ { n } , \vec { s }  \Theta ) } \end{array}$ .
+
+# 3 EVOLUTIONARY OPTIMIZATION
+
+For the generative models considered here, each latent state $\vec { s }$ takes the form of a bit vector. Hence, each population $\kappa ^ { n }$ is a collection of bit vectors. Because of the specific form (4), the free-energy is increased in the variational E-step if and only if we replace and individual $\vec { s }$ in population ${ \boldsymbol { \kappa } } ^ { ( n ) }$ by a new individual $\vec { s } ^ { \mathrm { n e w } }$ so far not in ${ \boldsymbol { \kappa } } ^ { ( n ) }$ such that:
+
+$$
+p ( \vec { s } ^ { \mathrm { n e w } } , \vec { y } ^ { n } | \Theta ) > p ( \vec { s } , \vec { y } ^ { n } | \Theta ) .
+$$
+
+More generally, this means that the free energy is maximized in the variational E-step if we find for each $n$ those $S$ individuals with the largest joints $p ( \vec { s } , \vec { y } ^ { n } | \Theta )$ , where $p ( \vec { s } , \vec { y } ^ { n } | \Theta )$ is given by the respective generative model (compare Lucke, 2016; Forster & L ¨ ucke, 2017, for formal derivations). ¨
+
+Full maximization of the free-energy is often a computationally much harder problem than increasing the free-energy; and in practice an increase is usually sufficient to finally approximately maximize the likelihood. As we increase the free-energy by applying (5) we can choose any fitness function $F ( \vec { s } ; \vec { y } ^ { n } , \Theta )$ for an evolutionary optimization which fulfils the property:
+
+$$
+\begin{array} { r l r } { F ( \bar { s } ^ { \mathrm { n e w } } ; \vec { y } ^ { n } , \Theta ) } & { > } & { F ( \vec { s } ; \vec { y } ^ { n } , \Theta ) \qquad \Leftrightarrow \qquad p ( \vec { s } ^ { \mathrm { n e w } } , \vec { y } ^ { n } | \Theta ) \quad > \quad p ( \vec { s } , \vec { y } ^ { n } | \Theta ) . } \end{array}
+$$
+
+Any mutations selected such that the fitness $F ( \vec { s } ; \vec { y } ^ { n } , \Theta )$ increases will result in provably increased free-energies. Together with M-step optimizations of model parameters, the resulting variational EM algorithm will monotonously increase the free-energy. The freedom in choosing a fitness function satisfying (6) leaves us free to pick a form that enables an efficient parent selection procedure. More concretely (while acknowledging that other choices are possible) we define the fitness $F ( \vec { s } ^ { \mathrm { n e w } } ; \vec { y } ^ { n } , \Theta )$ to be:
+
+$$
+F ( \vec { s } ) = F ( \vec { s } ; \vec { y } ^ { n } , \Theta ) = \widetilde { l o g P } ( \vec { s } ; \vec { y } ^ { n } , \Theta ) - 2 \operatorname* { m i n } _ { s } \widetilde { ( l o g P ( \vec { s } ; \vec { y } ^ { n } , \Theta ) ) }
+$$
+
+where $\widetilde { l o g P }$ is defined as the logarithm of the joint probability where summands that do not depend on the state $\vec { s }$ have been elided. $\widetilde { l o g P }$ is usually more efficiently computable than the joint probabilities and has better numerical stability, while being a monotonously increasing function of the joints when the data-point ${ \vec { y } } ^ { n }$ is considered fixed. As we will want to sample states proportionally to their fitness, an offset is applied to $\widetilde { l o g P }$ to make sure $F$ always takes positive values. As previously mentioned, other choices of $F$ are possible as long as (6) holds. From now on we will drop the argument ${ \vec { y } } ^ { n }$ or index $n$ (while keeping in mind that an optimization is performed for each data point ${ \vec { y } } ^ { n }$ ).
+
+Our applied EAs then seek to optimize $F ( \vec { s } )$ for a population of individual $\kappa$ (we also drop the index $n$ here). More concretely, given the current population $\kappa$ of unique individuals $\vec { s } ,$ , the EA iteratively seeks a new set $\kappa \prime$ with higher overall fitness. For our models, $\vec { s }$ are bit-vectors of length $H$ , and we usually require that populations $\kappa \prime$ and $\kappa$ to have the same size as is customary for truncated approximations (e.g. Lucke & Eggert, 2010; Shelton et al., 2017). Our example algorithm includes ¨ three common genetic operators, discussed in more detail below: parent selection, generation of children by single-point crossover and stochastic mutation of the children. We repeat this process over $N _ { g }$ generations in which subsequent iterations use the output of previous iterations as input population.
+
+Parent Selection. This step selects $N _ { p }$ parents from the population $\kappa$ . Ideally, the selection procedure should be balanced between exploitation of parents with high fitness (which will more likely produce children with high fitness) and exploration of mutations of poor performing parents (which might eventually produce children with high fitness while increasing population diversity). Diversity is crucial, as $\kappa$ is a set of unique individuals and therefore the improvement of the overall fitness of the population depends on generating different children with high fitness. In our numerical experiments we explored both fitness-proportional selection of parents (a classic strategy in which the probability of an individual being selected as a parent is proportional to its fitness) and random uniform selection of parents.
+
+![](images/53029bd277755c7b685af471115730cd4cdf9edfaee64e0c8b4864dfd5ae59c4.jpg)  
+Figure 1: Components of the genetic algorithm.   
+until $\mathcal { F }$ has increased sufficiently
+
+Crossover. During the crossover step, random pairs of parents are selected; then each pair is assigned a number $c$ from 1 to $H - 1$ with uniform probability (this is the single crossover point); finally the parents swap the last $H - c$ bits to produce the offspring. We denote $N _ { c }$ the number of children generated in this way. The crossover step can be skipped, making the EA more lightweight but decreasing variety in the offspring.
+
+Mutation. Finally, each of the $N _ { c }$ children undergoes one or more random bitflips to further increase offspring diversity. In our experiments we compare results of random uniform selection of the bits to flip with a more refined sparsity-driven bitflip algorithm. This latter bitflip schemes assignes to 0’s and 1’s different probabilities of being flipped in order to produce children with a sparsity compatible with the one learned by the model. In case the crossover step is skipped, a different bitflip mutation is performed on $N _ { c }$ identical copies of each parent.
+
+mization   
+choose initial model parameters $\Theta$ and initial sets ${ \boldsymbol { \kappa } } ^ { ( n ) }$   
+repeat for each data-point $n$ do candidates $= \{ \}$ for $g = 0$ to $N _ { g }$ do parents $=$ select parents children $=$ mutation(crossover(parents)) candidates $=$ candidates ∪ children K(n) select best(K(n) ∪ candidates) update $\Theta$ using M-steps with (3) and ${ \boldsymbol { \kappa } } ^ { ( n ) }$
+
+A full run of the evolutionary algorithm therefore produces $N _ { g } N _ { c } N _ { p }$ children (or new states $\vec { s } ^ { * }$ ). Finally we compute the union set of the original population $\kappa$ with all children and select the $S$ fittest individuals of the union as the new population $\kappa \prime$ .
+
+The EEM Algorithm. We now have all elements required to formulate a learning algorithm with EAs as its integral part. Alg. 1 summarizes the essential computational steps. Note that this E-step can be trivially parallelized over data-points. Finally, it is worth pointing out that algorithm 1, by construction, never decreases the free-energy.
+
+# 4 THE GENERATIVE MODELS
+
+We will use the EA formulated above as integral part of an unsupervised learning algorithm. The objective of the learning algorithm is the optimization of the log-likelihood 1. $D$ denotes the number of observed variables, $H$ the number of hidden units, and $N$ the number of data points.
+
+Noisy-OR. The noisy-OR model is a highly non-linear bipartite data model with all-to-all connectivity among hidden and observable variables. All variables take binary values. The model assumes a Bernoulli prior for the latents, and active latents are then combined via the actual noisy-OR rule.
+
+$$
+\begin{array} { c } { { p \displaystyle ( \vec { s } \mid \Theta ) = \prod _ { h } \pi _ { h } ^ { s _ { h } } ( 1 - \pi _ { h } ) ^ { 1 - s _ { h } } } } \\ { { p \displaystyle ( \vec { y } \mid \vec { s } , \Theta ) = \prod _ { d } N _ { d } ( \vec { s } ) ^ { y _ { d } } ( 1 - N _ { d } ( \vec { s } ) ) ^ { 1 - y _ { d } } \quad \mathrm { w h e r e } \quad N _ { d } ( \vec { s } ) : = 1 - \prod _ { h } ( 1 - { \cal W } _ { d h } s _ { h } ) } } \end{array}
+$$
+
+![](images/0c1ae83e3cdfa59494aa051b593db2826d3c3e15a1ec24ddd71320af2a1beb11.jpg)  
+Figure 2: A small Noisy-OR model. Each observable $y _ { d }$ is conditionally dependent on all $s _ { h }$ . The generative process first samples each $s _ { h }$ from a Bernoulli distribution; then each $y _ { d }$ is sampled from a Bernoulli distribution of parameter $N _ { d } ( \vec { s } )$ , generating a data-point.
+
+In the context of the Noisy-OR model, $\Theta = \{ \vec { \pi } , \vec { W } \}$ , where $\vec { \pi }$ is the set of values $\pi _ { h } ~ \in ~ [ 0 , 1 ]$ representing the prior activation probabilities for the hidden variables $s _ { h }$ and $\vec { W }$ is a $D \times H$ matrix of values $W _ { d h } \in [ 0 , 1 ]$ representing the probability that the latent $s _ { h }$ activates the observable $y _ { d }$ .
+
+Section A of the appendix contains the explicit forms of the free energies and the M-step update rules for noisy-OR.
+
+Binary Sparse Coding. As a second model and one for continuous data, we consider Binary Sparse Coding (BSC; Henniges et al., 2010). BSC differs from standard Sparse Coding in its use of binary latent variables. The latents are assumed to follow a univariate Bernoulli distribution which uses the same activation probability for each hidden unit. The combination of the latents is described by a linear superposition rule. Given the latents, the observables are independently and identically drawn from a Gaussian distribution:
+
+$$
+p \left( \vec { s } \mid \Theta \right) = \prod _ { h = 1 } ^ { H } \pi ^ { s _ { h } } \left( 1 - \pi \right) ^ { 1 - s _ { h } } , \qquad p \left( \vec { y } \mid \vec { s } , \Theta \right) = \prod _ { d = 1 } ^ { D } \mathcal { N } ( y _ { d } ; \sum _ { h = 1 } ^ { H } W _ { d h } s _ { h } , \sigma ^ { 2 } ) .
+$$
+
+The parameters of the model are $\Theta = ( \pi , W , \sigma ^ { 2 } )$ , where $W$ is a $D \times H$ matrix whose columns contain the weights associated with each hidden unit $s _ { h }$ and where $\sigma ^ { 2 }$ determines the variance of the Gaussian. M-step update rules for BSC can be derived in close-form by optimizing the free energy (2) wrt. all model parameters (compare, e.g., Henniges et al., 2010). We report the final expressions in appendix B.
+
+# 5 NUMERICAL EXPERIMENTS
+
+We describe numerical experiments performed to test the applicability and scalability of EEM. Throughout the section, the different evolutionary algorithms are named by indicating which parent selection procedure was used (“fitparents” for fitness-proportional selection, “randparents” for random uniform selection) and which bitflip algorithm (“sparseflips” or “randflips”). We add “cross” to the name of the EA when crossover was employed.
+
+# 5.1 ARTIFICIAL DATA
+
+First we investigate EMM using artificial data where the ground-truth components are known. We use the bars test as a standard setup for such purposes (Foldiak, 1990; Hoyer, 2003; L ¨ ucke & Sa- ¨ hani, 2008). In the standard setup, $H ^ { \mathrm { g e n } } / 2$ non-overlapping vertical and $H ^ { \mathrm { g e n } } / 2$ non-overlapping horizontal bars act as components on $\begin{array} { r } { D = H ^ { \mathrm { g e n } } \times H ^ { \mathrm { g e n } } } \end{array}$ pixel images. $N$ images are then generated by first selecting each bar with probability $\pi ^ { \mathrm { g e n } }$ . The bars are then superimposed according to the noisy-OR model (non-linear superposition) or according to the BSC model. In the case of BSC Gaussian noise is then added.
+
+Noisy-OR. Let us start with the standard bars test which uses a non-linear superposition (Foldiak, ¨ 1990) of 16 different bars (Spratling, 1999; Lucke & Sahani, 2008), and a standard average crowd- ¨ edness of two bars per images $\textstyle ( \pi ^ { \mathrm { g e n } } = { \frac { 2 } { H ^ { \mathrm { g e n } } } } ,$ ). We apply EEM for noisy-OR using different configurations of the EA. We use $H = 1 6$ generative fields. As a performance metric we here employ reliability (compare, e.g., Spratling, 1999; Lucke & Sahani, 2008), i.e., the fraction of runs whose ¨ learned free energies are above a certain minimum threshold and which learn the full dictionary of bars as well as the correct values for the prior probabilities $\pi$ .
+
+![](images/2ee372dab6240ca730aeac4968d233c96d21ac7c4d06f57ec876fc005479c271.jpg)  
+Figure 3: Reliability for the listed EAs over 10 runs of EEM for noisy-OR on 8x8 bars images. In this figure, black bars indicate both priors and bars were recovered correctly, grey bars indicate bars were recovered but not priors. For all runs $H = 1 6$ , $\mathbf { \dot { N } } = 1 0 ^ { 4 }$ , $N _ { g } = 2$ , $N _ { p } = 8$ , $N _ { c } = 7$ , $S = 1 2 0$ . Each run performed 100 iterations.   
+Figure 3 shows reliabilities over 10 different runs for each of the EAs. On $8 \mathrm { x } 8$ images the more exploitative nature of “fitparents-sparseflips” is advantageous over the simpler and more explorative “randparents-randflips”. Note that this is not necessarily true for lower dimensionalities or otherwise easier-to-explore state spaces, in which also a naive random search might quickly find high-fitness individuals. In this test the addition of crossover reduces the probability of finding all bars and leads to an overestimation of the crowdedness $\pi H$ .
+
+After the initial verification on a standard bars test, we now make the component extraction problem more difficult by increasing overlap among the bars. A highly non-linear generative model such as noisy-OR is a good candidate to model occlusion effects in images. Figure 4 shows the results of training noisy-OR with EEM on a bars data-set in which the latent causes have sensible overlaps. The test parameters were chosen to be equal to those in (Lucke & Sahani, 2008, Fig. 9). After ¨ applying EEM with noisy-OR $H = 3 2$ ) to $N = 4 0 0$ images with 16 strongly overlapping bars, we observed that all $H ^ { \mathrm { g e n } } = 1 6$ bars were recovered in 13 of 25 runs, which is competitive especially when keeping in mind that no additional assumptions (e.g., compared to other models applied to this test) are used by EEM for noisy-OR.
+
+![](images/d765aec9e4065fdfc7ad6bd7ff90e8a85f2f62b814d3bde904ec71706e4027a7.jpg)  
+Figure 4: Sample input (left) and learned generative fields (right) for a run on overlapping bars. Out of 25 runs, 13 recovered all 16 ground-truth generative components (14.92 recovered bars in average, median 16). As $H = 3 2$ , the extra generative fields are used to explain common overlaps and noise.
+
+BSC. Like for the non-linear generative model, we first evaluate EEM for the linear BSC model on a bars test. For BSC, the bars are superimposed linearly (Henniges et al., 2010), which makes the problem easier. As a consequence, standard bars test were solved with very high reliability using EEM for BSC even if merely random bitflips were used for the EA. In order to make the task more challenging, we therefore (A) increased the dimensionality of the data to $D = 1 0 \times 1 0$ bars images, (B) increased the number of components to $H ^ { \mathrm { g e n } } = 2 0$ , and (C) increased the average number of bars per data point from two (the standard setting) to five. We employed $N = 5 , 0 0 0$ training data points and tested the same five different configurations of the EA as were evaluated for noisy-OR. We set the number of hidden units to $H = H ^ { \mathrm { g e n } } = 2 0$ and used $S = 1 2 0$ variational states. Per data point and per iteration, in total 112 new states $N _ { p } = 8$ , $N _ { c } = 7$ , $N _ { g } = 2 $ ) were sampled to vary $\textstyle { \mathcal { K } } ^ { n }$ . Per configuration of the EA, we performed 20 independent runs, each with 300 iterations. The results of the experiment are depicted in Fig. 5. We observe that a basic approach such as random uniform selection of parents and random uniform bitflips for the EA works well. However, more sophisticated EAs improve performance. For instance, combining bitflips with crossover and selecting parents proportionally to their fitness shows to be very benefical. The results also show that sparseness-driven bitflips lead generally to very poor performance, even if crossover or fitnessproportional selection of the parents is included. This effect may be explained with the initialization of $\textstyle { \mathcal { K } } ^ { n }$ . The initial states are drawn from a Bernoulli distribution with parameter $\textstyle { \frac { 1 } { H } }$ which makes it more difficult for sparseness-driven EAs to explore and find solutions with higher crowdedness. Fig. 8 in appendix C depicts the averaged free energy values for this experiment.
+
+![](images/1eb7169f41b7206679046a2c5f7e241f43c693cc080c4c2dcb1cdb7045711fd6.jpg)  
+Figure 5: Reliability for the listed EAs over 20 runs of EEM for BSC on $1 0 \mathrm { x } 1 0$ bars images.
+
+# 5.2 NATURAL IMAGE PATCHES
+
+Next, we verify the approach on natural data. We use patches of natural images, which are known to have a multi-component structure, which are well investigated, and for which typically models with high-dimensional latent spaces are applied. The image patches used are extracted from the van Hateren image database (van Hateren & van der Schaaf, 1998).
+
+Noisy-OR. First we consider raw images patches, i.e., images without substantial pre-processing which directly reflect light intensities. Such image patches were generated by extracting random square subsections of a single $2 5 5 \mathrm { x } 2 5 5$ image of overlapping grass wires (part of image 2338 of the database). We removed the brightest $1 \%$ pixels from the data-set, scaled each data-point to have gray-scale values in the range $[ 0 , 1 ]$ and then created data points with binary entries by repeatedly choosing a random gray-scale image and sampling binary pixels from a Bernoulli distribution with parameter equal to the gray-scale value of the original pixel (cfr. figure 6). Note that components in such light-intensity images can be expected to superimpose non-linearly because of occlusion, which motivates the application of a non-linear generative model such as noisy-OR. We employ the “fitparents-sparseflips” evolutionary algorithm that was shown to perform best on artificial data (3). Parameters were $H = 1 0 0$ , $S = 1 2 0$ , $N _ { g } = 2$ , $N _ { p } = 8$ , $N _ { c } = 7$ . Figure 6 shows the generative fields learned over 200 iterations. EEM allows learning of generative fields resembling curved edges, in line with expectations and with the results obtained in (Lucke & Sahani, 2008). ¨
+
+![](images/acbc9688afb4d64469908082a30747851909a2d94c193380b2088e3c36816111.jpg)  
+Figure 6: 50 generative fields learned by applying EEM (“fitparents-sparseflips”) for noisy-OR to natural image patches. See Appendix F for a run at $H = 2 0 0$ .
+
+BSC. Finally, we consider pre-processed image patches using common whitening approaches as they are customary for sparse coding approaches (Olshausen & Field, 1997). We use $N = 1 0 0 , 0 0 0$ patches of size $D = 1 6 \times 1 6$ , randomly picked from the whole data set. The highest $2 \%$ of the amplitudes were clamped to compensate for light reflections and patches without significant structure were excluded for learning. ZCA whitening (Bell & Sejnowski, 1997) was applied retaining $9 5 \%$ of the variance (we used the procedure of a recent paper Exarchakis & Lucke, 2017). We trained the ¨ BSC model for 4,000 iterations using the “fitparents-cross-sparseflips” EA and employing $H = 3 0 0$ hidden units and $S = 2 0 0$ variational states. Per data point and per iteration, in total 360 new states $( N _ { p } = 1 0$ , $N _ { c } = 9$ , $N _ { g } = 4 \AA$ ) were sampled to vary $K ^ { n }$ . The results of the experiment are depicted in Fig. 7. The obtained generative fields primarily take the form of Gabor functions with different locations, orientations, phase, and spatial frequencies. This is a typical outcome of sparse coding being applied to images. On average more than five units were activated per data point showing that the learned code makes use of the generative model’s multiple causes structure. The generative fields converged faster than prior and noise parameters (similar effects are known from probabilistic PCA for the variance parameter). The finit slope of the free-energy after 4000 iterations is presumably due to these parameters still changing slowly.
+
+![](images/cf37fac785ad130d3f0c4c7df0f70d5308a78dbc60fe8689911eda0deaac5667.jpg)  
+Figure 7: Results on training the BSC model on natural images using the “fitparents-crosssparseflips” EA. $\mathbf { A } 6 0$ of the 300 generative fields obtained through training (see Appendix for all fields). B Evolution of the free energy per data point over iterations. C Evolution of the expected number of active hidden units per data point over iterations. D Evolution of the standard deviation over iterations.
+
+# 6 DISCUSSION
+
+The training of generative models is a very intensively studied branch of Machine Learning. If EM is applied for training, most non-elementary models require approximations. For this reason, sophisticated and mathematically grounded approaches such as sampling or variational EM have been developed in order to derive sufficiently precise and efficient learning algorithms.
+
+Evolutionary algorithms (EAs) have also been applied in conjunction with EM. Pernkopf & Bouchaffra (2005), for instance, have used EAs for clustering with Gaussian mixture models (GMMs). However, the GMM parameters are updated by their approach relatively conventionally using EM, while EAs are used to select the best GMM models for the clustering problem (using a min. description length criterion). Such a use of EAs is similar to DNN optimization where EAs optimize DNN hyperparameters in an outer optimization loop (Stanley & Miikkulainen, 2002; Loshchilov & Hutter, 2016; Real et al., 2017; Suganuma et al., 2017, etc), while the DNNs themselves are optimized using standard error-minimization algorithms. Still other approaches have used EAs to directly optimize, e.g., a clustering objective. But in these cases EAs replace EM approaches for optimization (compare Hruschka et al., 2009). In contrast to all such previous applications, we have here shown that EAs and EM can be combined directly and intimately: Alg. 1 defines EAs as an integral part of EM, and as such EAs address the key optimization problem arising in the training of generative models.
+
+We see the main contribution of our study in the establishment of this close theoretical link between EAs and EM. This novel link will make it possible to leverage an extensive body of knowledge and experience from the community of evolutionary approaches for learning algorithms. Our numerical experiments are a proof of concept which shows that EAs are indeed able to train generative models with large hidden spaces and local optima. For this purpose we used very basic EAs with elementary selection, mutation, cross-over operators.
+
+EAs more specialized to the specific optimization problems arising in the training of generative models have great potentials in future improvements of accuracy and scalability, we believe. In our experiments, we have only just started to exploit the abilities of EAs for learning algorithms. Still, our results represent, to the knowledge of the authors, the first examples of noisy-OR or sparse coding models trained with EAs (although both models have been studied very extensively before). Most importantly, we have pointed out a novel mathematically grounded way how EAs can be used for generative models with binary latents in general. The approach here established is, moreover, not only very generically formulated using the models’ joint probabilities but it is also very straightforward to apply.
+
+# REFERENCES
+
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+Peter Foldiak. Forming sparse representations by local anti-hebbian learning. ¨ Biological cybernetics, 64(2):165–170, 1990.   
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+I. Rechenberg. Cybernetic solution path of an experimental problem. 1965.   
+S. Roweis. EM algorithms for PCA and SPCA. NIPS, pp. 626–32, 1998.   
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+
+# APPENDIX
+
+A: NOISY-OR
+
+The truncated free energy takes on the following form for Noisy-OR:
+
+$$
+\begin{array} { l } { \mathcal { F } _ { N O R } ( K , \Theta ) = N \displaystyle \sum _ { h } \log \left( 1 - \pi _ { h } \right) + \displaystyle \sum _ { n } \log \displaystyle \sum _ { \tilde { s } \in \mathcal { K } ^ { ( n ) } } \exp \tilde { \mathcal { F } } } \\ { \displaystyle \qquad \tilde { \mathcal { F } } ( \vec { s } , \Theta ) : = \sum _ { h } s _ { h } \log \left( \frac { \pi _ { h } } { 1 - \pi _ { h } } \right) } \\ { \displaystyle \qquad + \sum _ { d } y _ { d } ^ { n } \log \left( \frac { 1 } { \prod _ { h } \left( 1 - W _ { d h } s _ { h } \right) } - 1 \right) } \\ { \displaystyle \qquad + \sum _ { h } \log \left( 1 - W _ { d h } s _ { h } \right) } \end{array}
+$$
+
+The M-step equations for noisy-OR are obtained by taking derivatives of the free energy, equating them to zero and solving the resulting set of equations. We report the results here for completeness:
+
+$$
+\pi _ { h } ^ { n e w } = \frac { 1 } { N } \sum _ { n } \left. s _ { h } \right. _ { q ^ { n } }
+$$
+
+$$
+W _ { d h } ^ { n e w } = 1 + \frac { \sum _ { n } ( y _ { d } ^ { n } - 1 ) \left. D _ { d h } ( \vec { s } ) \right. _ { q ^ { n } } } { \sum _ { n } \left. C _ { d h } ( \vec { s } ) \right. _ { q ^ { n } } }
+$$
+
+where
+
+$$
+D _ { d h } ( \vec { s } ) : = \frac { \widetilde { W } _ { d h } ( \vec { s } ) s _ { h } } { N _ { d } ( \vec { s } ) ( 1 - N _ { d } ( \vec { s } ) ) }
+$$
+
+$$
+C _ { d h } ( \vec { s } ) : = \widetilde { W } _ { d h } ( \vec { s } ) D _ { d h } ( \vec { s } )
+$$
+
+$$
+\widetilde { W } _ { d h } ( \vec { s } ) : = \prod _ { h ^ { \prime } \neq h } ( 1 - W _ { d h ^ { \prime } } s _ { h ^ { \prime } } )
+$$
+
+The update rule for $\vec { \pi }$ is quite straightforward. The update equations for the weights $W _ { d h }$ , on the other hand, do not allow a closed form solution (i.e. no exact M-step equation can be derived). The rule presented here, instead, expresses each $W _ { d h } ^ { n e w }$ as a function of all current $\vec { W }$ ; this is a fixedpoint equation whose fixed point would be the exact solution of the maximization step. Rather than solving the equation numerically at each step of the learning algorithm, we exploit the fact that in practice one single evaluation of 13 is enough to (noisily, not optimally) move towards convergence. Since TV-EM is guaranteed to never decrease $\mathcal { F }$ , drops of the free-energy during training can only be ascribed to this fixed-point equation; this provides a simple mechanism to check and possibly correct for misbehaviors of 13 if needed.
+
+# B: M-STEP UPDATE RULES FOR BSC
+
+The free energy for BSC follows from inserting (10) into (2). Update rules can be obtained by optimizing the resulting expression separately for the model parameters $\pi , \sigma ^ { 2 }$ and $W$ (compare, e.g., Henniges et al., 2010). For the sake of completeness, we show the result here:
+
+$$
+\pi = \frac { 1 } { N } \sum _ { n = 1 } ^ { N } \sum _ { h = 1 } ^ { H } \left. s _ { h } \right. _ { q ^ { n } }
+$$
+
+$$
+\sigma ^ { 2 } = \frac { 1 } { N D } \sum _ { n = 1 } ^ { N } \left. | | \vec { y } ^ { ( n ) } - W \vec { s } | | ^ { 2 } \right. _ { q ^ { n } }
+$$
+
+$$
+W = \left( \sum _ { n = 1 } ^ { N } \vec { y } ^ { ( n ) } \langle \vec { s } \rangle _ { q ^ { n } } ^ { T } \right) \left( \sum _ { n ^ { \prime } = 1 } ^ { N } \langle \vec { s } \vec { s } ^ { T } \rangle _ { q ^ { n ^ { \prime } } } \right) ^ { - 1 }
+$$
+
+Exact EM can be obtained by setting $q ^ { n }$ to the exact posterior $p ( \vec { s } | \vec { y } ^ { ( n ) } , \Theta )$ . As this quickly becomes computational intractable with higher latent dimensionality, we approximate exact posteriors by truncated variational distributions (3). For BSC, the truncated free energy (4) takes the form
+
+$$
+\mathcal { F } ( \mathcal { K } , \Theta ) = - \frac { N D } { 2 } \log \left( 2 \pi \sigma ^ { 2 } \right) + N H \log \left( 1 - \pi \right) + \sum _ { n } \log \left( \sum _ { \bar { s } \in \mathcal { K } _ { n } } \exp \left( \widetilde { \log p } \left( \bar { y } ^ { ( n ) } , \bar { s } | \Theta \right) \right) \right)
+$$
+
+where
+
+$$
+\widetilde { \log p } ( \vec { y } , \vec { s } | \Theta ) = - \frac { 1 } { 2 \sigma ^ { 2 } } ( \vec { y } - W \vec { s } ) ^ { T } ( \vec { y } - W \vec { s } ) + | \vec { s } | \log \left( \frac { \pi } { 1 - \pi } \right)
+$$
+
+C: FURTHER EXPERIMENTAL RESULTS FOR BSC
+
+![](images/e5cbc7870a7653deb74fa450f9a6f296bf198247008f8fc45e8ac1439eac82e0.jpg)  
+Figure 8: Results of the experiment with artificial data ( $1 0 \mathrm { x } 1 0$ bars) for the BSC model. Depicted is the evolution of the free energy for different EAs averaged over 20 independent runs. Dots and vertical errorbars show the mean and the standard deviation, respectively.
+
+![](images/350763eb871d2cd60f159d4a33bc9ffa81ecc79dc1f4d64bd435946813832bee.jpg)  
+Figure 9: Full dictionary learned from natural images by the BSC model trained with the “fitparentscross-sparseflips” EA. Depicted is the dictionary at iteration 4,000. The generative fields are ordered according to their activation, starting with most active fields.
+
+# D: SPARSITY-DRIVEN BITFLIPS
+
+When performing sparsity-driven bitflips, we flip each bit of a particular child ${ \vec { s } } ^ { * }$ with probability $p _ { 0 }$ if it is 0, with probability $p _ { 1 }$ otherwise. We call $p _ { b f }$ the average probability of flipping any bit in $\vec { s } ^ { * }$ . We impose the following constraints on $p _ { 0 }$ and $p _ { 1 }$ :
+
+• $p _ { 1 } = \alpha p _ { 0 }$ for some constant $\alpha$ • the average number of on bits after mutation is set at $\widetilde { s }$
+
+which yield the following expressions for $p _ { 0 }$ and $p _ { 1 }$
+
+$$
+\alpha = \frac { ( H - | \vec { s } | ) \cdot ( ( H p _ { b f } ) - ( \widetilde { s } - | \vec { s } | ) ) } { ( \widetilde { s } - | \vec { s } | + H p _ { b f } ) | \vec { s } | } 
+$$
+
+Trivially, random uniform bitflips correspond to the case $p _ { 0 } = p _ { 1 } = p _ { b f }$
+
+# E: RELIABILITY OF EEM FOR NOISY-OR ON OVERLAPPING BARS
+
+With respect to the tests shown in figure 4 and discussed in section 5.1, it is worth to spend a few more words on comparisons with the other algorithms shown (Lucke & Sahani, 2008, Fig. 9). Quan- ¨ titative comparison to NMF approaches, neural nets (DI Spratling et al., 2009), and MCA (Lucke ¨ & Sahani, 2008) shows that EMM for noisy-OR performs well but there are also approaches with higher reliability. Of all the approaches which recover more than 15 bars on average, most require additional assumptions. E.g., all NMF approaches, non-negative sparse coding (Hoyer, 2004) and $\mathbf { R { - } M C A _ { 2 } }$ require constraints on weights and/or latent activations. Only $\mathbf { M C A } _ { 3 }$ does not require constraints and presumably neither DI. DI is a neural network approach, which makes the used assumptions difficult to infer. $\mathbf { M C A } _ { 3 }$ is a generative model with a max-non-linearity as superposition model. For learning it explores all sparse combinations with up to 3 components. Applied with $H = 3 2$ latents, it hence evaluates more than 60000 states per data point per iteration for learning. For comparison, EEM for noisy-OR evaluates on the order of $S = 1 0 0$ states per data point per iteration.
+
+F: HIGHER-SCALE NATURAL IMAGE PATCHES FOR NOISY-OR
+
+Figure 10: Generative fields learned running EEM for noisy-OR (“fitparents-sparseflips”) for 175 iterations with $H = 2 0 0$ latent variables. Learned crowdedness $\pi H$ was 1.6.

parse/train/SyjjD1WRb/SyjjD1WRb_content_list.json

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+[
+    {
+        "type": "text",
+        "text": "EVOLUTIONARY EXPECTATION MAXIMIZATION FOR GENERATIVE MODELS WITH BINARY LATENTS ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "Anonymous authors Paper under double-blind review ",
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+        "type": "text",
+        "text": "ABSTRACT ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "We establish a theoretical link between evolutionary algorithms and variational parameter optimization of probabilistic generative models with binary hidden variables. While the novel approach is independent of the actual generative model, here we use two such models to investigate its applicability and scalability: a noisy-OR Bayes Net (as a standard example of binary data) and Binary Sparse Coding (as a model for continuous data). Learning of probabilistic generative models is first formulated as approximate maximum likelihood optimization using variational expectation maximization (EM). We choose truncated posteriors as variational distributions in which discrete latent states serve as variational parameters. In the variational E-step, the latent states are then optimized according to a tractable free-energy objective. Given a data point, we can show that evolutionary algorithms can be used for the variational optimization loop by (A) considering the bit-vectors of the latent states as genomes of individuals, and by (B) defining the fitness of the individuals as the (log) joint probabilities given by the used generative model. As a proof of concept, we apply the novel evolutionary EM approach to the optimization of the parameters of noisy-OR Bayes nets and binary sparse coding on artificial and real data (natural image patches). Using point mutations and single-point cross-over for the evolutionary algorithm, we find that scalable variational EM algorithms are obtained which efficiently improve the data likelihood. In general we believe that, with the link established here, standard as well as recent results in the field of evolutionary optimization can be leveraged to address the difficult problem of parameter optimization in generative models. ",
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+    {
+        "type": "text",
+        "text": "1 INTRODUCTION ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "Evolutionary algorithms (EA) have been introduced (e.g. Fogel et al., 1966; Rechenberg, 1965) as a technique for function optimization using methods inspired by biological evolutionary processes such as mutation, recombination, and selection. As such EAs are of interest as tools to solve Machine Learning problems, and they have been frequently applied to a number of tasks such as clustering (Pernkopf & Bouchaffra, 2005; Hruschka et al., 2009), reinforcement learning (Salimans et al., 2017), and hierarchical unsupervised (Myers et al., 1999) or deep supervised learning (e.g., Stanley & Miikkulainen 2002 and Suganuma et al. 2017; Real et al. 2017 for recent examples). In some of these tasks EAs have been investigated as alternatives to standard procedures (Hruschka et al., 2009), but most frequently EAs are used to solve specific sub-problems. For example, for classification with Deep Neural Networks (DNNs LeCun et al., 2015; Schmidhuber, 2015), EAs are frequently applied to solve the sub-problem of selecting the best DNN architectures for a given task (e.g. Stanley & Miikkulainen, 2002; Suganuma et al., 2017) or more generally to find the best hyper-parameters of a DNN (e.g. Loshchilov & Hutter, 2016; Real et al., 2017). ",
+        "bbox": [
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Inspired by these previous contributions, we here ask if EAs and learning algorithms can be linked more tightly. To address this question we make use of the theoretical framework of probabilistic generative models and expectation maximization (EM Dempster et al., 1977) approaches for parameter optimization. The probabilistic approach in combination with EM is appealing as it establishes a very general unifying framework able to encompass diverse algorithms from clustering and dimensionality reduction (Roweis, 1998; Tipping & Bishop, 1999) over feature learning and sparse coding (Olshausen & Field, 1997) to deep learning approaches (Patel et al., 2016). However, for most generative data models, EM is computationally intractable and requires approximations. Variational ",
+        "bbox": [
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+        "type": "text",
+        "text": "EM is a very prominent such approximation and is continuously further developed to become more efficient, more accurate and more autonomously applicable. Variational EM seeks to approximately solve optimization problems of functions with potentially many local optima in potentially very high dimensional spaces. The key observation exploited in this study is that a variational EM algorithm can be formulated such that latent states serve as variational parameters. If the latent states are then considered as genomes of individuals, EAs emerge as a very natural choice for optimization in the variational loop of EM. ",
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+        "page_idx": 1
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+    {
+        "type": "text",
+        "text": "2 TRUNCATED VARIATIONAL EM ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 1
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+    {
+        "type": "text",
+        "text": "A probabilistic generative model stochastically generates data points $\\vec { y }$ using a set of hidden (or latent) variables $\\vec { s } ,$ . The generative process can be formally expressed in the form of joint probability $p ( \\vec { s } , \\vec { y } | \\Theta )$ , where $\\Theta$ are the model parameters. Given a set of $N$ data points, $\\vec { y } ^ { ( 1 ) } , \\dotsc , \\vec { y } ^ { ( N ) } =$ $\\vec { y } ^ { ( 1 : N ) }$ , learning seeks to change the parameters $\\Theta$ so that the data generated by the generative model becomes as similar as possible to the $N$ real data points. One of the most popular approaches to achieve this goal is to seek maximum likelihood (ML) parameters $\\Theta ^ { * }$ , i.e., parameters that maximize the data log-likelihood for a given generative model: ",
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+        "page_idx": 1
+    },
+    {
+        "type": "equation",
+        "img_path": "images/00aa0db539a7b9004bd2de4210d5e26d48674d3524a4ab95c4eaefdd3112207c.jpg",
+        "text": "$$\nL ( \\Theta ) : = \\log ( \\mathcal { L } ( \\Theta ) ) = \\sum _ { n } \\log \\big ( \\sum _ { \\{ \\vec { s } \\} } p \\left( \\vec { y } ^ { n } , \\vec { s } \\mid \\Theta \\right) \\big )\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "To efficiently find (approximate) ML parameters we follow Saul & Jordan (1996); Neal & Hinton (1998); Jordan et al. (1999) who reformulated the problem in terms of a maximization of a lower bound of the log-likelihood, the free energy $\\mathcal { F } ( \\vec { q } , \\Theta )$ . Free energies are given by ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "equation",
+        "img_path": "images/cc176ae24ccc0993a2a8944acdae9ce9b107cd68e43b93dd9328bda7f3dedb46.jpg",
+        "text": "$$\n\\mathcal { F } ( q ^ { ( 1 : N ) } , \\Theta ) = \\sum _ { n = 1 } ^ { N } \\Big ( \\sum _ { \\{ \\vec { s } \\} } q ^ { ( n ) } ( \\vec { s } ) \\log \\big ( p ( \\vec { s } , \\vec { y } ^ { ( n ) } | \\Theta ) \\big ) \\Big ) + \\sum _ { n = 1 } ^ { N } H ( q ^ { ( n ) } ( \\vec { s } ) ) ,\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "where $q ^ { ( n ) } ( \\vec { s } )$ are variational distributions, and where $H ( q )$ denotes the entropy of a distribution $q$ . For the purposes of this study, we consider elementary generative models which are difficult to train because of exponentially large state spaces. These models serve well for illustrating the approach but we stress that any generative model which gives rise to a joint distribution $p ( \\bar { \\vec { s } } , \\bar { y } | \\Theta ) \\bar { \\vec { s } }$ can be trained with the approach discussed here as long as the latents $\\vec { s }$ are binary. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "In order to find approximate maximum likelihood solutions, distributions $q ^ { ( n ) } ( \\vec { s } )$ are sought that approximate the intractable posterior distributions $p ( \\vec { s } | \\vec { y } ^ { ( n ) } , \\Theta )$ as well as possible, which results in the free-energy being as similar (or tight) as possible to the exact log-likelihood. At the same time variational distributions have to result in tractable parameter updates. Standard approaches include Gaussian variational distributions (e.g. Opper & Winther, 2005) or mean-field variational distributions (Jordan et al., 1999). If we denote the parameters of the variational distributions by $\\Lambda$ , then a variational EM algorithm consists of iteratively maximizing $\\mathcal { F } ( \\Lambda , \\Theta )$ w.r.t. $\\Lambda$ in the variational E-step and w.r.t. $\\Theta$ in the M-step. The M-step can hereby maintain the same functional form as for exact EM but the expectation values now have to be computed w.r.t. the variational distributions. ",
+        "bbox": [
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+        "type": "text",
+        "text": "Instead of using parametric functions such as Gaussians or factored (mean-field) distributions, for our purposes we choose truncated variational distributions defined as a function of a finite set of states (Lucke & Eggert, 2010; Sheikh et al., 2014; Shelton et al., 2017). These states will later serve ¨ as populations of evolutionary algorithms. If we denote $\\kappa ^ { n }$ a population of hidden states for a given data point $\\vec { y } ^ { ( n ) }$ , then variational distributions and their corresponding expectation values are given by (e.g. Lucke & Eggert, 2010; Sheikh et al., 2014): ¨ ",
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+        "type": "equation",
+        "img_path": "images/7c22fc7f0d6b7e994a40c2349bd6db71a5a3a55945d12a26a89fb89827165b86.jpg",
+        "text": "$$\nq ^ { n } ( \\vec { s } \\mid K ^ { n } , \\Theta ) : = \\frac { p \\left( \\vec { s } \\mid \\vec { y ^ { n } } , \\Theta \\right) } { \\sum _ { \\vec { s ^ { \\prime } } \\in K ^ { n } } p \\left( \\vec { s } ^ { \\prime } \\mid \\vec { y ^ { n } } , \\Theta \\right) } \\delta ( \\vec { s } \\in K ^ { n } ) , \\langle g ( \\vec { s } ) \\rangle _ { q ^ { n } } = \\frac { \\sum _ { \\vec { s } \\in K ^ { n } } p ( \\vec { s } , \\vec { y ^ { n } } \\mid \\Theta ) g ( \\vec { s } ) } { \\sum _ { \\vec { s ^ { \\prime } } \\in K ^ { n } } p ( \\vec { s } ^ { \\prime } , \\vec { y ^ { n } } \\mid \\Theta ) } .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "where $\\delta ( \\vec { s } \\in \\mathcal { K } ^ { n } )$ is 1 if $\\textstyle { \\mathcal { K } } ^ { n }$ contains the hidden state $\\vec { s }$ , zero otherwise. If the set $K ^ { n }$ contains all states with significant posterior mass, then (3) approximates expectations w.r.t. full posteriors very well. By inserting truncated distributions as variational distribution of the free-energy (2), it can be ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "shown (Lucke, 2016) that the free-energy takes a very compact simplified form given by: ¨ ",
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+        "page_idx": 2
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+    {
+        "type": "equation",
+        "img_path": "images/b5fa15b82b6390b2c18626599e1ec2096809678ca2e963ccca8c6eebbaf67848.jpg",
+        "text": "$$\n\\mathcal { F } ( \\mathcal { K } , \\Theta ) = \\sum _ { n } \\log \\big ( \\sum _ { \\vec { s } \\in \\mathcal { K } ^ { n } } p \\left( \\vec { y } ^ { n } , \\vec { s } \\mid \\Theta \\right) \\big ) , \\mathrm { ~ w h e r e ~ } \\mathcal { K } = ( \\mathcal { K } ^ { 1 } , \\ldots , \\mathcal { K } ^ { N } ) .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "type": "text",
+        "text": "As the variational parameters of the variational distribution (3) are now given by populations of hidden states, a variational $\\mathrm { E }$ -step now consists of finding for each data point $n$ the population $\\textstyle { \\mathcal { K } } ^ { n }$ that maximizes $\\begin{array} { r } { \\sum _ { \\vec { s } \\in \\mathcal { K } ^ { n } } p ( \\vec { y } ^ { n } , \\vec { s }  \\Theta ) } \\end{array}$ . ",
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+    {
+        "type": "text",
+        "text": "3 EVOLUTIONARY OPTIMIZATION ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "For the generative models considered here, each latent state $\\vec { s }$ takes the form of a bit vector. Hence, each population $\\kappa ^ { n }$ is a collection of bit vectors. Because of the specific form (4), the free-energy is increased in the variational E-step if and only if we replace and individual $\\vec { s }$ in population ${ \\boldsymbol { \\kappa } } ^ { ( n ) }$ by a new individual $\\vec { s } ^ { \\mathrm { n e w } }$ so far not in ${ \\boldsymbol { \\kappa } } ^ { ( n ) }$ such that: ",
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+    },
+    {
+        "type": "equation",
+        "img_path": "images/c95e32cac2d9b75e3b20bf45ebf90183b784b260b0ec3a8e2748d6e248259fda.jpg",
+        "text": "$$\np ( \\vec { s } ^ { \\mathrm { n e w } } , \\vec { y } ^ { n } | \\Theta ) > p ( \\vec { s } , \\vec { y } ^ { n } | \\Theta ) .\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "More generally, this means that the free energy is maximized in the variational E-step if we find for each $n$ those $S$ individuals with the largest joints $p ( \\vec { s } , \\vec { y } ^ { n } | \\Theta )$ , where $p ( \\vec { s } , \\vec { y } ^ { n } | \\Theta )$ is given by the respective generative model (compare Lucke, 2016; Forster & L ¨ ucke, 2017, for formal derivations). ¨ ",
+        "bbox": [
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+        "text": "Full maximization of the free-energy is often a computationally much harder problem than increasing the free-energy; and in practice an increase is usually sufficient to finally approximately maximize the likelihood. As we increase the free-energy by applying (5) we can choose any fitness function $F ( \\vec { s } ; \\vec { y } ^ { n } , \\Theta )$ for an evolutionary optimization which fulfils the property: ",
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+    {
+        "type": "equation",
+        "img_path": "images/4dbbca4a786b7f5acf902c3b5b3405873a2bfe149640e5556f352197101d272f.jpg",
+        "text": "$$\n\\begin{array} { r l r } { F ( \\bar { s } ^ { \\mathrm { n e w } } ; \\vec { y } ^ { n } , \\Theta ) } & { > } & { F ( \\vec { s } ; \\vec { y } ^ { n } , \\Theta ) \\qquad \\Leftrightarrow \\qquad p ( \\vec { s } ^ { \\mathrm { n e w } } , \\vec { y } ^ { n } | \\Theta ) \\quad > \\quad p ( \\vec { s } , \\vec { y } ^ { n } | \\Theta ) . } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Any mutations selected such that the fitness $F ( \\vec { s } ; \\vec { y } ^ { n } , \\Theta )$ increases will result in provably increased free-energies. Together with M-step optimizations of model parameters, the resulting variational EM algorithm will monotonously increase the free-energy. The freedom in choosing a fitness function satisfying (6) leaves us free to pick a form that enables an efficient parent selection procedure. More concretely (while acknowledging that other choices are possible) we define the fitness $F ( \\vec { s } ^ { \\mathrm { n e w } } ; \\vec { y } ^ { n } , \\Theta )$ to be: ",
+        "bbox": [
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+    {
+        "type": "equation",
+        "img_path": "images/7c83613462a16b8c7fd7573237d73f2980ce6309cfe530cc6da050934e70bf74.jpg",
+        "text": "$$\nF ( \\vec { s } ) = F ( \\vec { s } ; \\vec { y } ^ { n } , \\Theta ) = \\widetilde { l o g P } ( \\vec { s } ; \\vec { y } ^ { n } , \\Theta ) - 2 \\operatorname* { m i n } _ { s } \\widetilde { ( l o g P ( \\vec { s } ; \\vec { y } ^ { n } , \\Theta ) ) }\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "where $\\widetilde { l o g P }$ is defined as the logarithm of the joint probability where summands that do not depend on the state $\\vec { s }$ have been elided. $\\widetilde { l o g P }$ is usually more efficiently computable than the joint probabilities and has better numerical stability, while being a monotonously increasing function of the joints when the data-point ${ \\vec { y } } ^ { n }$ is considered fixed. As we will want to sample states proportionally to their fitness, an offset is applied to $\\widetilde { l o g P }$ to make sure $F$ always takes positive values. As previously mentioned, other choices of $F$ are possible as long as (6) holds. From now on we will drop the argument ${ \\vec { y } } ^ { n }$ or index $n$ (while keeping in mind that an optimization is performed for each data point ${ \\vec { y } } ^ { n }$ ). ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Our applied EAs then seek to optimize $F ( \\vec { s } )$ for a population of individual $\\kappa$ (we also drop the index $n$ here). More concretely, given the current population $\\kappa$ of unique individuals $\\vec { s } ,$ , the EA iteratively seeks a new set $\\kappa \\prime$ with higher overall fitness. For our models, $\\vec { s }$ are bit-vectors of length $H$ , and we usually require that populations $\\kappa \\prime$ and $\\kappa$ to have the same size as is customary for truncated approximations (e.g. Lucke & Eggert, 2010; Shelton et al., 2017). Our example algorithm includes ¨ three common genetic operators, discussed in more detail below: parent selection, generation of children by single-point crossover and stochastic mutation of the children. We repeat this process over $N _ { g }$ generations in which subsequent iterations use the output of previous iterations as input population. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Parent Selection. This step selects $N _ { p }$ parents from the population $\\kappa$ . Ideally, the selection procedure should be balanced between exploitation of parents with high fitness (which will more likely produce children with high fitness) and exploration of mutations of poor performing parents (which might eventually produce children with high fitness while increasing population diversity). Diversity is crucial, as $\\kappa$ is a set of unique individuals and therefore the improvement of the overall fitness of the population depends on generating different children with high fitness. In our numerical experiments we explored both fitness-proportional selection of parents (a classic strategy in which the probability of an individual being selected as a parent is proportional to its fitness) and random uniform selection of parents. ",
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+    {
+        "type": "image",
+        "img_path": "images/53029bd277755c7b685af471115730cd4cdf9edfaee64e0c8b4864dfd5ae59c4.jpg",
+        "image_caption": [
+            "Figure 1: Components of the genetic algorithm. ",
+            "until $\\mathcal { F }$ has increased sufficiently "
+        ],
+        "image_footnote": [],
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+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Crossover. During the crossover step, random pairs of parents are selected; then each pair is assigned a number $c$ from 1 to $H - 1$ with uniform probability (this is the single crossover point); finally the parents swap the last $H - c$ bits to produce the offspring. We denote $N _ { c }$ the number of children generated in this way. The crossover step can be skipped, making the EA more lightweight but decreasing variety in the offspring. ",
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+    {
+        "type": "text",
+        "text": "Mutation. Finally, each of the $N _ { c }$ children undergoes one or more random bitflips to further increase offspring diversity. In our experiments we compare results of random uniform selection of the bits to flip with a more refined sparsity-driven bitflip algorithm. This latter bitflip schemes assignes to 0’s and 1’s different probabilities of being flipped in order to produce children with a sparsity compatible with the one learned by the model. In case the crossover step is skipped, a different bitflip mutation is performed on $N _ { c }$ identical copies of each parent. ",
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+    {
+        "type": "text",
+        "text": "mization   \nchoose initial model parameters $\\Theta$ and initial sets ${ \\boldsymbol { \\kappa } } ^ { ( n ) }$   \nrepeat for each data-point $n$ do candidates $= \\{ \\}$ for $g = 0$ to $N _ { g }$ do parents $=$ select parents children $=$ mutation(crossover(parents)) candidates $=$ candidates ∪ children K(n) select best(K(n) ∪ candidates) update $\\Theta$ using M-steps with (3) and ${ \\boldsymbol { \\kappa } } ^ { ( n ) }$ ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "A full run of the evolutionary algorithm therefore produces $N _ { g } N _ { c } N _ { p }$ children (or new states $\\vec { s } ^ { * }$ ). Finally we compute the union set of the original population $\\kappa$ with all children and select the $S$ fittest individuals of the union as the new population $\\kappa \\prime$ . ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "The EEM Algorithm. We now have all elements required to formulate a learning algorithm with EAs as its integral part. Alg. 1 summarizes the essential computational steps. Note that this E-step can be trivially parallelized over data-points. Finally, it is worth pointing out that algorithm 1, by construction, never decreases the free-energy. ",
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+    {
+        "type": "text",
+        "text": "4 THE GENERATIVE MODELS ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "We will use the EA formulated above as integral part of an unsupervised learning algorithm. The objective of the learning algorithm is the optimization of the log-likelihood 1. $D$ denotes the number of observed variables, $H$ the number of hidden units, and $N$ the number of data points. ",
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+        "type": "text",
+        "text": "Noisy-OR. The noisy-OR model is a highly non-linear bipartite data model with all-to-all connectivity among hidden and observable variables. All variables take binary values. The model assumes a Bernoulli prior for the latents, and active latents are then combined via the actual noisy-OR rule. ",
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+        "img_path": "images/3bbd6171abc9ad9f47487fb8825c2e0591cfc6f3c70f56eaec34e5d6d3ac7027.jpg",
+        "text": "$$\n\\begin{array} { c } { { p \\displaystyle ( \\vec { s } \\mid \\Theta ) = \\prod _ { h } \\pi _ { h } ^ { s _ { h } } ( 1 - \\pi _ { h } ) ^ { 1 - s _ { h } } } } \\\\ { { p \\displaystyle ( \\vec { y } \\mid \\vec { s } , \\Theta ) = \\prod _ { d } N _ { d } ( \\vec { s } ) ^ { y _ { d } } ( 1 - N _ { d } ( \\vec { s } ) ) ^ { 1 - y _ { d } } \\quad \\mathrm { w h e r e } \\quad N _ { d } ( \\vec { s } ) : = 1 - \\prod _ { h } ( 1 - { \\cal W } _ { d h } s _ { h } ) } } \\end{array}\n$$",
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+            "Figure 2: A small Noisy-OR model. Each observable $y _ { d }$ is conditionally dependent on all $s _ { h }$ . The generative process first samples each $s _ { h }$ from a Bernoulli distribution; then each $y _ { d }$ is sampled from a Bernoulli distribution of parameter $N _ { d } ( \\vec { s } )$ , generating a data-point. "
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+        "type": "text",
+        "text": "In the context of the Noisy-OR model, $\\Theta = \\{ \\vec { \\pi } , \\vec { W } \\}$ , where $\\vec { \\pi }$ is the set of values $\\pi _ { h } ~ \\in ~ [ 0 , 1 ]$ representing the prior activation probabilities for the hidden variables $s _ { h }$ and $\\vec { W }$ is a $D \\times H$ matrix of values $W _ { d h } \\in [ 0 , 1 ]$ representing the probability that the latent $s _ { h }$ activates the observable $y _ { d }$ . ",
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+        "type": "text",
+        "text": "Section A of the appendix contains the explicit forms of the free energies and the M-step update rules for noisy-OR. ",
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+        "type": "text",
+        "text": "Binary Sparse Coding. As a second model and one for continuous data, we consider Binary Sparse Coding (BSC; Henniges et al., 2010). BSC differs from standard Sparse Coding in its use of binary latent variables. The latents are assumed to follow a univariate Bernoulli distribution which uses the same activation probability for each hidden unit. The combination of the latents is described by a linear superposition rule. Given the latents, the observables are independently and identically drawn from a Gaussian distribution: ",
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+        "text": "$$\np \\left( \\vec { s } \\mid \\Theta \\right) = \\prod _ { h = 1 } ^ { H } \\pi ^ { s _ { h } } \\left( 1 - \\pi \\right) ^ { 1 - s _ { h } } , \\qquad p \\left( \\vec { y } \\mid \\vec { s } , \\Theta \\right) = \\prod _ { d = 1 } ^ { D } \\mathcal { N } ( y _ { d } ; \\sum _ { h = 1 } ^ { H } W _ { d h } s _ { h } , \\sigma ^ { 2 } ) .\n$$",
+        "text_format": "latex",
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+        "type": "text",
+        "text": "The parameters of the model are $\\Theta = ( \\pi , W , \\sigma ^ { 2 } )$ , where $W$ is a $D \\times H$ matrix whose columns contain the weights associated with each hidden unit $s _ { h }$ and where $\\sigma ^ { 2 }$ determines the variance of the Gaussian. M-step update rules for BSC can be derived in close-form by optimizing the free energy (2) wrt. all model parameters (compare, e.g., Henniges et al., 2010). We report the final expressions in appendix B. ",
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+        "type": "text",
+        "text": "5 NUMERICAL EXPERIMENTS ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "We describe numerical experiments performed to test the applicability and scalability of EEM. Throughout the section, the different evolutionary algorithms are named by indicating which parent selection procedure was used (“fitparents” for fitness-proportional selection, “randparents” for random uniform selection) and which bitflip algorithm (“sparseflips” or “randflips”). We add “cross” to the name of the EA when crossover was employed. ",
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+        "type": "text",
+        "text": "5.1 ARTIFICIAL DATA ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "First we investigate EMM using artificial data where the ground-truth components are known. We use the bars test as a standard setup for such purposes (Foldiak, 1990; Hoyer, 2003; L ¨ ucke & Sa- ¨ hani, 2008). In the standard setup, $H ^ { \\mathrm { g e n } } / 2$ non-overlapping vertical and $H ^ { \\mathrm { g e n } } / 2$ non-overlapping horizontal bars act as components on $\\begin{array} { r } { D = H ^ { \\mathrm { g e n } } \\times H ^ { \\mathrm { g e n } } } \\end{array}$ pixel images. $N$ images are then generated by first selecting each bar with probability $\\pi ^ { \\mathrm { g e n } }$ . The bars are then superimposed according to the noisy-OR model (non-linear superposition) or according to the BSC model. In the case of BSC Gaussian noise is then added. ",
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+        "text": "Noisy-OR. Let us start with the standard bars test which uses a non-linear superposition (Foldiak, ¨ 1990) of 16 different bars (Spratling, 1999; Lucke & Sahani, 2008), and a standard average crowd- ¨ edness of two bars per images $\\textstyle ( \\pi ^ { \\mathrm { g e n } } = { \\frac { 2 } { H ^ { \\mathrm { g e n } } } } ,$ ). We apply EEM for noisy-OR using different configurations of the EA. We use $H = 1 6$ generative fields. As a performance metric we here employ reliability (compare, e.g., Spratling, 1999; Lucke & Sahani, 2008), i.e., the fraction of runs whose ¨ learned free energies are above a certain minimum threshold and which learn the full dictionary of bars as well as the correct values for the prior probabilities $\\pi$ . ",
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+        "img_path": "images/2ee372dab6240ca730aeac4968d233c96d21ac7c4d06f57ec876fc005479c271.jpg",
+        "image_caption": [
+            "Figure 3: Reliability for the listed EAs over 10 runs of EEM for noisy-OR on 8x8 bars images. In this figure, black bars indicate both priors and bars were recovered correctly, grey bars indicate bars were recovered but not priors. For all runs $H = 1 6$ , $\\mathbf { \\dot { N } } = 1 0 ^ { 4 }$ , $N _ { g } = 2$ , $N _ { p } = 8$ , $N _ { c } = 7$ , $S = 1 2 0$ . Each run performed 100 iterations. ",
+            "Figure 3 shows reliabilities over 10 different runs for each of the EAs. On $8 \\mathrm { x } 8$ images the more exploitative nature of “fitparents-sparseflips” is advantageous over the simpler and more explorative “randparents-randflips”. Note that this is not necessarily true for lower dimensionalities or otherwise easier-to-explore state spaces, in which also a naive random search might quickly find high-fitness individuals. In this test the addition of crossover reduces the probability of finding all bars and leads to an overestimation of the crowdedness $\\pi H$ . "
+        ],
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+    {
+        "type": "text",
+        "text": "After the initial verification on a standard bars test, we now make the component extraction problem more difficult by increasing overlap among the bars. A highly non-linear generative model such as noisy-OR is a good candidate to model occlusion effects in images. Figure 4 shows the results of training noisy-OR with EEM on a bars data-set in which the latent causes have sensible overlaps. The test parameters were chosen to be equal to those in (Lucke & Sahani, 2008, Fig. 9). After ¨ applying EEM with noisy-OR $H = 3 2$ ) to $N = 4 0 0$ images with 16 strongly overlapping bars, we observed that all $H ^ { \\mathrm { g e n } } = 1 6$ bars were recovered in 13 of 25 runs, which is competitive especially when keeping in mind that no additional assumptions (e.g., compared to other models applied to this test) are used by EEM for noisy-OR. ",
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+    {
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+        "img_path": "images/d765aec9e4065fdfc7ad6bd7ff90e8a85f2f62b814d3bde904ec71706e4027a7.jpg",
+        "image_caption": [
+            "Figure 4: Sample input (left) and learned generative fields (right) for a run on overlapping bars. Out of 25 runs, 13 recovered all 16 ground-truth generative components (14.92 recovered bars in average, median 16). As $H = 3 2$ , the extra generative fields are used to explain common overlaps and noise. "
+        ],
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+        "text": "BSC. Like for the non-linear generative model, we first evaluate EEM for the linear BSC model on a bars test. For BSC, the bars are superimposed linearly (Henniges et al., 2010), which makes the problem easier. As a consequence, standard bars test were solved with very high reliability using EEM for BSC even if merely random bitflips were used for the EA. In order to make the task more challenging, we therefore (A) increased the dimensionality of the data to $D = 1 0 \\times 1 0$ bars images, (B) increased the number of components to $H ^ { \\mathrm { g e n } } = 2 0$ , and (C) increased the average number of bars per data point from two (the standard setting) to five. We employed $N = 5 , 0 0 0$ training data points and tested the same five different configurations of the EA as were evaluated for noisy-OR. We set the number of hidden units to $H = H ^ { \\mathrm { g e n } } = 2 0$ and used $S = 1 2 0$ variational states. Per data point and per iteration, in total 112 new states $N _ { p } = 8$ , $N _ { c } = 7$ , $N _ { g } = 2 $ ) were sampled to vary $\\textstyle { \\mathcal { K } } ^ { n }$ . Per configuration of the EA, we performed 20 independent runs, each with 300 iterations. The results of the experiment are depicted in Fig. 5. We observe that a basic approach such as random uniform selection of parents and random uniform bitflips for the EA works well. However, more sophisticated EAs improve performance. For instance, combining bitflips with crossover and selecting parents proportionally to their fitness shows to be very benefical. The results also show that sparseness-driven bitflips lead generally to very poor performance, even if crossover or fitnessproportional selection of the parents is included. This effect may be explained with the initialization of $\\textstyle { \\mathcal { K } } ^ { n }$ . The initial states are drawn from a Bernoulli distribution with parameter $\\textstyle { \\frac { 1 } { H } }$ which makes it more difficult for sparseness-driven EAs to explore and find solutions with higher crowdedness. Fig. 8 in appendix C depicts the averaged free energy values for this experiment. ",
+        "bbox": [
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+    {
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+        "img_path": "images/1eb7169f41b7206679046a2c5f7e241f43c693cc080c4c2dcb1cdb7045711fd6.jpg",
+        "image_caption": [
+            "Figure 5: Reliability for the listed EAs over 20 runs of EEM for BSC on $1 0 \\mathrm { x } 1 0$ bars images. "
+        ],
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+    {
+        "type": "text",
+        "text": "5.2 NATURAL IMAGE PATCHES ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "Next, we verify the approach on natural data. We use patches of natural images, which are known to have a multi-component structure, which are well investigated, and for which typically models with high-dimensional latent spaces are applied. The image patches used are extracted from the van Hateren image database (van Hateren & van der Schaaf, 1998). ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Noisy-OR. First we consider raw images patches, i.e., images without substantial pre-processing which directly reflect light intensities. Such image patches were generated by extracting random square subsections of a single $2 5 5 \\mathrm { x } 2 5 5$ image of overlapping grass wires (part of image 2338 of the database). We removed the brightest $1 \\%$ pixels from the data-set, scaled each data-point to have gray-scale values in the range $[ 0 , 1 ]$ and then created data points with binary entries by repeatedly choosing a random gray-scale image and sampling binary pixels from a Bernoulli distribution with parameter equal to the gray-scale value of the original pixel (cfr. figure 6). Note that components in such light-intensity images can be expected to superimpose non-linearly because of occlusion, which motivates the application of a non-linear generative model such as noisy-OR. We employ the “fitparents-sparseflips” evolutionary algorithm that was shown to perform best on artificial data (3). Parameters were $H = 1 0 0$ , $S = 1 2 0$ , $N _ { g } = 2$ , $N _ { p } = 8$ , $N _ { c } = 7$ . Figure 6 shows the generative fields learned over 200 iterations. EEM allows learning of generative fields resembling curved edges, in line with expectations and with the results obtained in (Lucke & Sahani, 2008). ¨ ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "image",
+        "img_path": "images/acbc9688afb4d64469908082a30747851909a2d94c193380b2088e3c36816111.jpg",
+        "image_caption": [
+            "Figure 6: 50 generative fields learned by applying EEM (“fitparents-sparseflips”) for noisy-OR to natural image patches. See Appendix F for a run at $H = 2 0 0$ . "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "BSC. Finally, we consider pre-processed image patches using common whitening approaches as they are customary for sparse coding approaches (Olshausen & Field, 1997). We use $N = 1 0 0 , 0 0 0$ patches of size $D = 1 6 \\times 1 6$ , randomly picked from the whole data set. The highest $2 \\%$ of the amplitudes were clamped to compensate for light reflections and patches without significant structure were excluded for learning. ZCA whitening (Bell & Sejnowski, 1997) was applied retaining $9 5 \\%$ of the variance (we used the procedure of a recent paper Exarchakis & Lucke, 2017). We trained the ¨ BSC model for 4,000 iterations using the “fitparents-cross-sparseflips” EA and employing $H = 3 0 0$ hidden units and $S = 2 0 0$ variational states. Per data point and per iteration, in total 360 new states $( N _ { p } = 1 0$ , $N _ { c } = 9$ , $N _ { g } = 4 \\AA$ ) were sampled to vary $K ^ { n }$ . The results of the experiment are depicted in Fig. 7. The obtained generative fields primarily take the form of Gabor functions with different locations, orientations, phase, and spatial frequencies. This is a typical outcome of sparse coding being applied to images. On average more than five units were activated per data point showing that the learned code makes use of the generative model’s multiple causes structure. The generative fields converged faster than prior and noise parameters (similar effects are known from probabilistic PCA for the variance parameter). The finit slope of the free-energy after 4000 iterations is presumably due to these parameters still changing slowly. ",
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+        "type": "text",
+        "text": "",
+        "bbox": [
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+    },
+    {
+        "type": "image",
+        "img_path": "images/cf37fac785ad130d3f0c4c7df0f70d5308a78dbc60fe8689911eda0deaac5667.jpg",
+        "image_caption": [
+            "Figure 7: Results on training the BSC model on natural images using the “fitparents-crosssparseflips” EA. $\\mathbf { A } 6 0$ of the 300 generative fields obtained through training (see Appendix for all fields). B Evolution of the free energy per data point over iterations. C Evolution of the expected number of active hidden units per data point over iterations. D Evolution of the standard deviation over iterations. "
+        ],
+        "image_footnote": [],
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+    {
+        "type": "text",
+        "text": "6 DISCUSSION ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "The training of generative models is a very intensively studied branch of Machine Learning. If EM is applied for training, most non-elementary models require approximations. For this reason, sophisticated and mathematically grounded approaches such as sampling or variational EM have been developed in order to derive sufficiently precise and efficient learning algorithms. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Evolutionary algorithms (EAs) have also been applied in conjunction with EM. Pernkopf & Bouchaffra (2005), for instance, have used EAs for clustering with Gaussian mixture models (GMMs). However, the GMM parameters are updated by their approach relatively conventionally using EM, while EAs are used to select the best GMM models for the clustering problem (using a min. description length criterion). Such a use of EAs is similar to DNN optimization where EAs optimize DNN hyperparameters in an outer optimization loop (Stanley & Miikkulainen, 2002; Loshchilov & Hutter, 2016; Real et al., 2017; Suganuma et al., 2017, etc), while the DNNs themselves are optimized using standard error-minimization algorithms. Still other approaches have used EAs to directly optimize, e.g., a clustering objective. But in these cases EAs replace EM approaches for optimization (compare Hruschka et al., 2009). In contrast to all such previous applications, we have here shown that EAs and EM can be combined directly and intimately: Alg. 1 defines EAs as an integral part of EM, and as such EAs address the key optimization problem arising in the training of generative models. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "We see the main contribution of our study in the establishment of this close theoretical link between EAs and EM. This novel link will make it possible to leverage an extensive body of knowledge and experience from the community of evolutionary approaches for learning algorithms. Our numerical experiments are a proof of concept which shows that EAs are indeed able to train generative models with large hidden spaces and local optima. For this purpose we used very basic EAs with elementary selection, mutation, cross-over operators. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "EAs more specialized to the specific optimization problems arising in the training of generative models have great potentials in future improvements of accuracy and scalability, we believe. In our experiments, we have only just started to exploit the abilities of EAs for learning algorithms. Still, our results represent, to the knowledge of the authors, the first examples of noisy-OR or sparse coding models trained with EAs (although both models have been studied very extensively before). Most importantly, we have pointed out a novel mathematically grounded way how EAs can be used for generative models with binary latents in general. The approach here established is, moreover, not only very generically formulated using the models’ joint probabilities but it is also very straightforward to apply. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "REFERENCES ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            150,
+            287,
+            165
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "A. J. Bell and T. J. Sejnowski. The “independent components” of natural scenes are edge filters. Vision Research, 37(23):3327–38, 1997.   \nA. P. Dempster, N. M. Laird, and D. B. Rubin. Maximum likelihood from incomplete data via the EM algorithm (with discussion). Journal of the Royal Statistical Society B, 39:1–38, 1977.   \nG. Exarchakis and J. Lucke. Discrete sparse coding. ¨ Neural Computation, 29:2979–3013, 2017.   \nL. J. Fogel, A. J. Owens, and M. J. Walsh. Artificial intelligence through simulated evolution. 1966.   \nPeter Foldiak. Forming sparse representations by local anti-hebbian learning. ¨ Biological cybernetics, 64(2):165–170, 1990.   \nD. Forster and J. Lucke. Truncated variational EM for semi-supervised Neural Simpletrons. In ¨ IJCNN, pp. 3769–3776, 2017.   \nM. Henniges, G. Puertas, J. Bornschein, J. Eggert, and J. Lucke. Binary sparse coding. In ¨ Proceedings LVA/ICA, LNCS 6365, pp. 450–57. Springer, 2010.   \nP. O. Hoyer. Modeling receptive fields with non-negative sparse coding. Neurocomputing, 52-54: 547–52, June 2003. ISSN 09252312. doi: 10.1016/S0925-2312(02)00782-8.   \nP. O. Hoyer. Non-negative matrix factorization with sparseness constraints. Journal of Machine Learning Research, 5:1457–69, 2004.   \nE. R. Hruschka, R. JGB Campello, A. A. Freitas, et al. A survey of evolutionary algorithms for clustering. IEEE Trans. on Systems, Man, and Cybernetics, 39(2):133–155, 2009.   \nM. Jordan, Z. Ghahramani, T. Jaakkola, and L. Saul. An introduction to variational methods for graphical models. Machine Learning, 37:183–233, 1999.   \nY. LeCun, Y. Bengio, and G. Hinton. Deep learning. Nature, 521(7553):436–444, 2015.   \nI. Loshchilov and F. Hutter. CMA-ES for hyperparameter optimization of deep neural networks. In ICLR Workshop, pp. 513–520, 2016.   \nJ. Lucke. Truncated variational expectation maximization. ¨ arXiv preprint, arXiv:1610.03113, 2016.   \nJ. Lucke and J. Eggert. Expectation truncation and the benefits of preselection in training generative ¨ models. JMLR, 11:2855–900, 2010.   \nJ. Lucke and M. Sahani. Maximal causes for non-linear component extraction. ¨ Journal of Machine Learning Research, 9:1227–67, 2008.   \nJ. W. Myers, K. B. Laskey, and K. A. DeJong. Learning Bayesian networks from incomplete data using evolutionary algorithms. In Proc. Annual Conference on Genetic and Evolutionary Computation, pp. 458–465, 1999.   \nR. Neal and G. Hinton. A view of the EM algorithm that justifies incremental, sparse, and other variants. In M. I. Jordan (ed.), Learning in Graphical Models. Kluwer, 1998.   \nB. A. Olshausen and D. J. Field. Sparse coding with an overcomplete basis set: A strategy employed by V1? Vision Research, 37(23):3311–3325, 1997.   \nM. Opper and O. Winther. Expectation consistent approximate inference. JMLR, 6:2177–04, 2005.   \nA. B. Patel, T. Nguyen, and R. G. Baraniuk. A probabilistic theory of deep learning. In NIPS, pp. 2558–2566, 2016.   \nF. Pernkopf and D. Bouchaffra. Genetic-based em algorithm for learning gaussian mixture models. IEEE Trans. on Pattern Analysis and Machine Intelligence, 27(8):1344–1348, 2005.   \nE. Real, S. Moore, A. Selle, S. Saxena, Y. L. Suematsu, J. Tan, Q. V. Le, and A. Kurakin. Large-scale evolution of image classifiers. In ICML, pp. 2902–2911, 2017.   \nI. Rechenberg. Cybernetic solution path of an experimental problem. 1965.   \nS. Roweis. EM algorithms for PCA and SPCA. NIPS, pp. 626–32, 1998.   \nT. Salimans, J. Ho, X. Chen, and I. Sutskever. Evolution strategies as a scalable alternative to reinforcement learning. arXiv preprint arXiv:1703.03864, 2017.   \nL. K. Saul and M. Jordan. Exploiting tractable substructures in intractable networks. NIPS, pp. 486–492, 1996.   \nJ. Schmidhuber. Deep learning in neural networks. Neural networks, 61:85–117, 2015.   \nA.-S. Sheikh, J. A. Shelton, and J. Lucke. A truncated EM approach for spike-and-slab sparse ¨ coding. Journal of Machine Learning Research, 15:2653–2687, 2014.   \nJ. A. Shelton, J. Gasthaus, Z. Dai, J. Lucke, and A. Gretton. Gp-select: Accelerating em using ¨ adaptive subspace preselection. Neural Computation, 29(8):2177–2202, 2017.   \nM. W. Spratling. Pre-synaptic lateral inhibition provides a better architecture for self-organising neural networks. Network: Computation in Neural Systems, 10:285 – 301, 1999.   \nM. W. Spratling, K. De Meyer, and R. Kompass. Unsupervised learning of overlapping image components using divisive input modulation. Computational Intelligence and Neuroscience, pp. 1–19, 2009. ISSN 1687-5265.   \nK. O. Stanley and R. Miikkulainen. Evolving neural networks through augmenting topologies. Evolutionary Computing, 10(2):99–127, 2002. ISSN 1063-6560.   \nM. Suganuma, S. Shirakawa, and T. Nagao. A genetic programming approach to designing convolutional neural network architectures. In GECCO, pp. 497–504, 2017.   \nM. Tipping and C. Bishop. Probabilistic principal component analysis. Journal of the Royal Statistical Society. Series B, 61, 1999.   \nJ. H. van Hateren and A. van der Schaaf. Independent component filters of natural images compared with simple cells in primary visual cortex. Proceedings of the Royal Society of London B, 265: 359–66, 1998. ",
+        "bbox": [
+            169,
+            156,
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+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            168,
+            82,
+            826,
+            647
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "APPENDIX ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            103,
+            264,
+            117
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "A: NOISY-OR ",
+        "bbox": [
+            174,
+            133,
+            272,
+            148
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "The truncated free energy takes on the following form for Noisy-OR: ",
+        "bbox": [
+            173,
+            159,
+            627,
+            174
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "equation",
+        "img_path": "images/cd89100d963bf703ffc4f7289d56e81ed799901e57229702db407e0d06aefce7.jpg",
+        "text": "$$\n\\begin{array} { l } { \\mathcal { F } _ { N O R } ( K , \\Theta ) = N \\displaystyle \\sum _ { h } \\log \\left( 1 - \\pi _ { h } \\right) + \\displaystyle \\sum _ { n } \\log \\displaystyle \\sum _ { \\tilde { s } \\in \\mathcal { K } ^ { ( n ) } } \\exp \\tilde { \\mathcal { F } } } \\\\ { \\displaystyle \\qquad \\tilde { \\mathcal { F } } ( \\vec { s } , \\Theta ) : = \\sum _ { h } s _ { h } \\log \\left( \\frac { \\pi _ { h } } { 1 - \\pi _ { h } } \\right) } \\\\ { \\displaystyle \\qquad + \\sum _ { d } y _ { d } ^ { n } \\log \\left( \\frac { 1 } { \\prod _ { h } \\left( 1 - W _ { d h } s _ { h } \\right) } - 1 \\right) } \\\\ { \\displaystyle \\qquad + \\sum _ { h } \\log \\left( 1 - W _ { d h } s _ { h } \\right) } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            295,
+            181,
+            697,
+            334
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "The M-step equations for noisy-OR are obtained by taking derivatives of the free energy, equating them to zero and solving the resulting set of equations. We report the results here for completeness: ",
+        "bbox": [
+            174,
+            347,
+            825,
+            376
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "equation",
+        "img_path": "images/1beb074dcd75e78a02a875df782e139766962de68e87c0d95cde3d5b947bc85e.jpg",
+        "text": "$$\n\\pi _ { h } ^ { n e w } = \\frac { 1 } { N } \\sum _ { n } \\left. s _ { h } \\right. _ { q ^ { n } }\n$$",
+        "text_format": "latex",
+        "bbox": [
+            421,
+            391,
+            573,
+            428
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "equation",
+        "img_path": "images/0e30624a2ea7cbd14a4ecb06531d2788c132bedaee6a93afacb23d2d6140a9ea.jpg",
+        "text": "$$\nW _ { d h } ^ { n e w } = 1 + \\frac { \\sum _ { n } ( y _ { d } ^ { n } - 1 ) \\left. D _ { d h } ( \\vec { s } ) \\right. _ { q ^ { n } } } { \\sum _ { n } \\left. C _ { d h } ( \\vec { s } ) \\right. _ { q ^ { n } } }\n$$",
+        "text_format": "latex",
+        "bbox": [
+            367,
+            431,
+            630,
+            470
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "where ",
+        "bbox": [
+            173,
+            473,
+            217,
+            488
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "equation",
+        "img_path": "images/14f2625c944ab62f34efcfe250141bdb1e3d4fe2183abdd77700658a6d95aa84.jpg",
+        "text": "$$\nD _ { d h } ( \\vec { s } ) : = \\frac { \\widetilde { W } _ { d h } ( \\vec { s } ) s _ { h } } { N _ { d } ( \\vec { s } ) ( 1 - N _ { d } ( \\vec { s } ) ) }\n$$",
+        "text_format": "latex",
+        "bbox": [
+            397,
+            483,
+            601,
+            520
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "equation",
+        "img_path": "images/fae2e3422bfef36693e23c0238074f92e498e22ff01d504a644c5205ad8842e6.jpg",
+        "text": "$$\nC _ { d h } ( \\vec { s } ) : = \\widetilde { W } _ { d h } ( \\vec { s } ) D _ { d h } ( \\vec { s } )\n$$",
+        "text_format": "latex",
+        "bbox": [
+            395,
+            535,
+            573,
+            556
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "equation",
+        "img_path": "images/10a18e09d36b1c6ff714507283672673ca64d3ca09b79a2daf97457380e6ca54.jpg",
+        "text": "$$\n\\widetilde { W } _ { d h } ( \\vec { s } ) : = \\prod _ { h ^ { \\prime } \\neq h } ( 1 - W _ { d h ^ { \\prime } } s _ { h ^ { \\prime } } )\n$$",
+        "text_format": "latex",
+        "bbox": [
+            395,
+            574,
+            602,
+            611
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "The update rule for $\\vec { \\pi }$ is quite straightforward. The update equations for the weights $W _ { d h }$ , on the other hand, do not allow a closed form solution (i.e. no exact M-step equation can be derived). The rule presented here, instead, expresses each $W _ { d h } ^ { n e w }$ as a function of all current $\\vec { W }$ ; this is a fixedpoint equation whose fixed point would be the exact solution of the maximization step. Rather than solving the equation numerically at each step of the learning algorithm, we exploit the fact that in practice one single evaluation of 13 is enough to (noisily, not optimally) move towards convergence. Since TV-EM is guaranteed to never decrease $\\mathcal { F }$ , drops of the free-energy during training can only be ascribed to this fixed-point equation; this provides a simple mechanism to check and possibly correct for misbehaviors of 13 if needed. ",
+        "bbox": [
+            173,
+            613,
+            825,
+            741
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "B: M-STEP UPDATE RULES FOR BSC ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            758,
+            428,
+            772
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "The free energy for BSC follows from inserting (10) into (2). Update rules can be obtained by optimizing the resulting expression separately for the model parameters $\\pi , \\sigma ^ { 2 }$ and $W$ (compare, e.g., Henniges et al., 2010). For the sake of completeness, we show the result here: ",
+        "bbox": [
+            174,
+            784,
+            823,
+            827
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "equation",
+        "img_path": "images/f9cf85e800729f0bd02ae9ba70110420dd4f2b5d3dbc37d0841d416c14eeaf7d.jpg",
+        "text": "$$\n\\pi = \\frac { 1 } { N } \\sum _ { n = 1 } ^ { N } \\sum _ { h = 1 } ^ { H } \\left. s _ { h } \\right. _ { q ^ { n } }\n$$",
+        "text_format": "latex",
+        "bbox": [
+            421,
+            833,
+            575,
+            878
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "equation",
+        "img_path": "images/823f36913dec95294b5af95688300c43a2a0805dbe7f17c515a18a4db4b1548d.jpg",
+        "text": "$$\n\\sigma ^ { 2 } = \\frac { 1 } { N D } \\sum _ { n = 1 } ^ { N } \\left. | | \\vec { y } ^ { ( n ) } - W \\vec { s } | | ^ { 2 } \\right. _ { q ^ { n } }\n$$",
+        "text_format": "latex",
+        "bbox": [
+            375,
+            885,
+            620,
+            928
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "equation",
+        "img_path": "images/8751fee3005de59f2fd6bf866f7f5f68d1ea201e8a36dc30a773f28804d2dfd6.jpg",
+        "text": "$$\nW = \\left( \\sum _ { n = 1 } ^ { N } \\vec { y } ^ { ( n ) } \\langle \\vec { s } \\rangle _ { q ^ { n } } ^ { T } \\right) \\left( \\sum _ { n ^ { \\prime } = 1 } ^ { N } \\langle \\vec { s } \\vec { s } ^ { T } \\rangle _ { q ^ { n ^ { \\prime } } } \\right) ^ { - 1 }\n$$",
+        "text_format": "latex",
+        "bbox": [
+            346,
+            114,
+            650,
+            161
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Exact EM can be obtained by setting $q ^ { n }$ to the exact posterior $p ( \\vec { s } | \\vec { y } ^ { ( n ) } , \\Theta )$ . As this quickly becomes computational intractable with higher latent dimensionality, we approximate exact posteriors by truncated variational distributions (3). For BSC, the truncated free energy (4) takes the form ",
+        "bbox": [
+            173,
+            167,
+            825,
+            210
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "equation",
+        "img_path": "images/1c29571cb23e9309656894c7b3a21d656e67e8491cc3f0af8be893b0522983cd.jpg",
+        "text": "$$\n\\mathcal { F } ( \\mathcal { K } , \\Theta ) = - \\frac { N D } { 2 } \\log \\left( 2 \\pi \\sigma ^ { 2 } \\right) + N H \\log \\left( 1 - \\pi \\right) + \\sum _ { n } \\log \\left( \\sum _ { \\bar { s } \\in \\mathcal { K } _ { n } } \\exp \\left( \\widetilde { \\log p } \\left( \\bar { y } ^ { ( n ) } , \\bar { s } | \\Theta \\right) \\right) \\right)\n$$",
+        "text_format": "latex",
+        "bbox": [
+            179,
+            217,
+            789,
+            268
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "where ",
+        "bbox": [
+            173,
+            275,
+            217,
+            289
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "equation",
+        "img_path": "images/d5b370db07635a6fb3adcab8583ac2b19658a0e6afb6a3603ffcb7193ca98e1d.jpg",
+        "text": "$$\n\\widetilde { \\log p } ( \\vec { y } , \\vec { s } | \\Theta ) = - \\frac { 1 } { 2 \\sigma ^ { 2 } } ( \\vec { y } - W \\vec { s } ) ^ { T } ( \\vec { y } - W \\vec { s } ) + | \\vec { s } | \\log \\left( \\frac { \\pi } { 1 - \\pi } \\right)\n$$",
+        "text_format": "latex",
+        "bbox": [
+            285,
+            286,
+            710,
+            320
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "C: FURTHER EXPERIMENTAL RESULTS FOR BSC ",
+        "bbox": [
+            173,
+            337,
+            511,
+            352
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "image",
+        "img_path": "images/e5cbc7870a7653deb74fa450f9a6f296bf198247008f8fc45e8ac1439eac82e0.jpg",
+        "image_caption": [
+            "Figure 8: Results of the experiment with artificial data ( $1 0 \\mathrm { x } 1 0$ bars) for the BSC model. Depicted is the evolution of the free energy for different EAs averaged over 20 independent runs. Dots and vertical errorbars show the mean and the standard deviation, respectively. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            194,
+            373,
+            803,
+            598
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "image",
+        "img_path": "images/350763eb871d2cd60f159d4a33bc9ffa81ecc79dc1f4d64bd435946813832bee.jpg",
+        "image_caption": [
+            "Figure 9: Full dictionary learned from natural images by the BSC model trained with the “fitparentscross-sparseflips” EA. Depicted is the dictionary at iteration 4,000. The generative fields are ordered according to their activation, starting with most active fields. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            183,
+            683,
+            816,
+            859
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "D: SPARSITY-DRIVEN BITFLIPS ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            103,
+            390,
+            117
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "When performing sparsity-driven bitflips, we flip each bit of a particular child ${ \\vec { s } } ^ { * }$ with probability $p _ { 0 }$ if it is 0, with probability $p _ { 1 }$ otherwise. We call $p _ { b f }$ the average probability of flipping any bit in $\\vec { s } ^ { * }$ . We impose the following constraints on $p _ { 0 }$ and $p _ { 1 }$ : ",
+        "bbox": [
+            173,
+            128,
+            825,
+            172
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "• $p _ { 1 } = \\alpha p _ { 0 }$ for some constant $\\alpha$ • the average number of on bits after mutation is set at $\\widetilde { s }$ ",
+        "bbox": [
+            215,
+            184,
+            591,
+            217
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "which yield the following expressions for $p _ { 0 }$ and $p _ { 1 }$ ",
+        "bbox": [
+            176,
+            228,
+            516,
+            243
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "equation",
+        "img_path": "images/0edbac77f1508e3922e09f3c0f9a36c4055e3342cf73ed036e4afa70afe043ea.jpg",
+        "text": "$$\n\\alpha = \\frac { ( H - | \\vec { s } | ) \\cdot ( ( H p _ { b f } ) - ( \\widetilde { s } - | \\vec { s } | ) ) } { ( \\widetilde { s } - | \\vec { s } | + H p _ { b f } ) | \\vec { s } | } \n$$",
+        "text_format": "latex",
+        "bbox": [
+            370,
+            248,
+            627,
+            337
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "Trivially, random uniform bitflips correspond to the case $p _ { 0 } = p _ { 1 } = p _ { b f }$ ",
+        "bbox": [
+            173,
+            349,
+            648,
+            366
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "E: RELIABILITY OF EEM FOR NOISY-OR ON OVERLAPPING BARS ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            382,
+            629,
+            396
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "With respect to the tests shown in figure 4 and discussed in section 5.1, it is worth to spend a few more words on comparisons with the other algorithms shown (Lucke & Sahani, 2008, Fig. 9). Quan- ¨ titative comparison to NMF approaches, neural nets (DI Spratling et al., 2009), and MCA (Lucke ¨ & Sahani, 2008) shows that EMM for noisy-OR performs well but there are also approaches with higher reliability. Of all the approaches which recover more than 15 bars on average, most require additional assumptions. E.g., all NMF approaches, non-negative sparse coding (Hoyer, 2004) and $\\mathbf { R { - } M C A _ { 2 } }$ require constraints on weights and/or latent activations. Only $\\mathbf { M C A } _ { 3 }$ does not require constraints and presumably neither DI. DI is a neural network approach, which makes the used assumptions difficult to infer. $\\mathbf { M C A } _ { 3 }$ is a generative model with a max-non-linearity as superposition model. For learning it explores all sparse combinations with up to 3 components. Applied with $H = 3 2$ latents, it hence evaluates more than 60000 states per data point per iteration for learning. For comparison, EEM for noisy-OR evaluates on the order of $S = 1 0 0$ states per data point per iteration. ",
+        "bbox": [
+            173,
+            407,
+            825,
+            588
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "F: HIGHER-SCALE NATURAL IMAGE PATCHES FOR NOISY-OR ",
+        "bbox": [
+            173,
+            606,
+            599,
+            619
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            176,
+            638,
+            823,
+            743
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "Figure 10: Generative fields learned running EEM for noisy-OR (“fitparents-sparseflips”) for 175 iterations with $H = 2 0 0$ latent variables. Learned crowdedness $\\pi H$ was 1.6. ",
+        "bbox": [
+            173,
+            753,
+            823,
+            782
+        ],
+        "page_idx": 12
+    }
+]

parse/train/SyxCqGbRZ/SyxCqGbRZ.md

@@ -0,0 +1,178 @@
+# LEARNING TO TREAT SEPSIS WITH MULTI-OUTPUTGAUSSIAN PROCESS DEEP RECURRENT Q-NETWORKS
+
+Anonymous authors Paper under double-blind review
+
+# ABSTRACT
+
+Sepsis is a life-threatening complication from infection and a leading cause of mortality in hospitals. While early detection of sepsis improves patient outcomes, there is little consensus on exact treatment guidelines, and treating septic patients remains an open problem. In this work we present a new deep reinforcement learning method that we use to learn optimal personalized treatment policies for septic patients. We model patient continuous-valued physiological time series using multi-output Gaussian processes, a probabilistic model that easily handles missing values and irregularly spaced observation times while maintaining estimates of uncertainty. The Gaussian process is directly tied to a deep recurrent Q-network that learns clinically interpretable treatment policies, and both models are learned together end-to-end. We evaluate our approach on a heterogeneous dataset of septic spanning 15 months from our university health system, and find that our learned policy could reduce patient mortality by as much as $8 . 2 \%$ from an overall baseline mortality rate of $1 3 . 3 \%$ . Our algorithm could be used to make treatment recommendations to physicians as part of a decision support tool, and the framework readily applies to other reinforcement learning problems that rely on sparsely sampled and frequently missing multivariate time series data.
+
+# 1 INTRODUCTION
+
+Sepsis is a poorly understood complication arising from infection, and is both a leading cause in patient mortality (Epstein et al. (2016)) and in associated healthcare costs (Torio & Moore (2016)). Early detection is imperative, as earlier treatment is associated with better outcomes (Seymour et al. (2017), Kumar et al. (2006)). However, even among patients with recognized sepsis, there is no standard consensus on the best treatment. There is a pressing need for personalized treatment strategies tailored to the unique physiology of individual patients. Guidelines on sepsis treatment previously centered on early goal directed therapy (EGDT) and more recently have focused on sepsis care bundles, but none of these approaches are individualized.
+
+Before the landmark publication on the use of early goal directed therapy (Rivers et al. (2001)), there was no standard management for severe sepsis and septic shock. EGDT consists of early identification of high-risk patients, appropriate cultures, infection source control, antibiotics administration, and hemodynamic optimization. The study compared a 6-hour protocol of EGDT promoting use of central venous catheterization to guide administration of fluids, vasopressors, inotropes, and packed red-blood cell transfusions, and was found to significantly lower mortality. Following the initial trial, EGDT became the cornerstone of the sepsis resuscitation bundle for the Surviving Sepsis Campaign (SCC) and the Centers for Medicare and Medicaid Services (CMS) (Dellinger et al. (2013)).
+
+Despite the promising results of EGDT, concerns arose. External validity outside the single center study was unclear, it required significant resources for implementation, and the elements needed to achieve pre-specified hemodynamic targets held potential risks. Between 2014–2017, a trio of trials reported an all-time low sepsis mortality, and questioned the continued need for all elements of EGDT for patients with severe and septic shock (ProCESS et al. (2014), ARISE & Group (2014), PRISM (2017)). The trial authors concluded EGDT did not improve patient survival compared to usual care but was associated with increased ICU admissions (Angus et al. (2015)). As a result, they did not recommend it be included in the updated SCC guidelines (Rhodes et al. (2017)).
+
+Although the SSC guidelines provide an overarching framework for sepsis treatment, there is renewed interest in targeting treatment and disassembling the bundle (Lewis (2010)). A recent metaanalysis evaluated 12 randomized trials and 31 observational studies and found that time to first antibiotics explained $9 6 - 9 9 \%$ of the survival benefit (Kalil et al. (2017)). Likewise, a study of 50,000 patients across the state of New York found mortality benefit for early antibiotic administration, but not intravenous fluids (Seymour et al. (2017)). Beyond narrowing the bundle, there is emerging evidence that a patient’s baseline risk plays an important role in response to treatment, as survival benefit was significantly reduced for patients with more severe disease (Kalil et al. (2017)).
+
+Taken together, the poor performance of EGDT compared to standard-of-care and improved understanding of individual treatment effects calls for re-envisioning sepsis treatment recommendations. Though general consensus in critical care is that the individual elements of the sepsis bundle are typically useful, it is unclear exactly when each element should be administered and in what quantity.
+
+In this paper, we aim to directly address this problem using deep reinforcement learning. We develop a novel framework for applying deep reinforcement learning to clinical data, and use it to learn optimal treatments for sepsis. With the widespread adoption of Electronic Health Records, hospitals are already automatically collecting the relevant data required to learn such models. However, real-world operational healthcare data present many unique challenges and motivate the need for methodologies designed with their structure in mind. In particular, clinical time series are typically irregularly sampled and exhibit large degrees of missing values that are often informatively missing, necessitating careful modeling. The high degree of heterogeneity presents an additional difficulty, as patients with similar symptoms may respond very differently to treatments due to unmeasured sources of variation. Alignment of patient time series can also be a potential issue, as patients admitted to the hospital may have very different unknown clinical states and can develop sepsis at any time throughout their stay (with many already septic upon admission).
+
+Part of the novelty in our approach hinges on the use of a Multi-output Gaussian process (MGP) as a preprocessing step that is jointly learned with the reinforcement learning model. We use an MGP to interpolate and to impute missing physiological time series values used by the downstream reinforcement learning algorithm, while importantly maintaining uncertainty about the clinical state. The MGP hyperparameters are learned end-to-end during training of the reinforcement learning model by optimizing an expectation of the standard Q-learning loss. Additionally, the MGP allows for estimation of uncertainty in the learned Q-values. For the model architecture we use a deep recurrent Q-network, in order to account for the potential for non-Markovian dynamics and allow the model to have memory of past states and actions. In our experiments utilizing EHR data from septic patients spanning 15 months from our university health system, we found that both the use of the MGP and the deep recurrent Q-network offered improved performance over simpler approaches.
+
+# 2 BACKGROUND
+
+In this section we outline important background that motivates our improvements on prior work.
+
+# 2.1 DEEP Q-LEARNING
+
+Reinforcement learning (RL) considers learning policies for agents interacting with unknown environments, and are typically formulated as a Markov decision process (MDP) (Sutton $\&$ Barto (1998)). At each time $t$ , an agent observes the state of the environment, $s _ { t } \in S$ , takes an action $a _ { t } \in \mathcal A$ , and receives a reward $r _ { t } \in \mathbb { R }$ , at which time the environment transitions to a new state $s _ { t + 1 }$ . The state space $s$ and action space $\mathcal { A }$ may be continuous or discrete. The goal of an RL agent is to select actions in order to maximize its return, or expected discounted future reward, defined as $\begin{array} { r } { R _ { t } = \sum _ { t ^ { \prime } = t } ^ { T } \gamma ^ { t ^ { \prime } - t } r _ { t ^ { \prime } } } \end{array}$ , where $\gamma$ captures tradeoff between immediate and future rewards.
+
+Q-Learning (Watkins & Dayan (1992)) is a model-free off-policy algorithm for estimating the expected return from executing an action in a given state. The optimal action value function is the maximum discounted expected reward obtained by executing action $a$ in state $s$ and acting optimally afterwards, defined as $\begin{array} { r } { Q ^ { * } ( s , a ) = \operatorname* { m a x } _ { \pi } \mathbb { E } [ R _ { t } | s _ { t } = s , a _ { t } = a , \pi ] } \end{array}$ , where $\pi$ is a policy that maps states to actions. Given $Q ^ { * }$ , an optimal policy is to act by selecting argmax ${ } _ { a } Q ^ { * } ( s , a )$ . In Q-learning, the Bellman equation is used to iteratively update the current estimate of the optimal action value function according to $Q ( s , a ) \doteq Q ( s , a ) + \alpha \bar { ( } r + \gamma \mathrm { m a x } _ { a } Q ( s ^ { \prime } , a ^ { \prime } ) - Q ( s , a ) )$ , adjusting towards the observed reward plus the maximal Q-value at the next state $s ^ { \prime }$ . In Deep Q-learning a deep neural network is used to approximate $\mathrm { Q }$ -values (Mnih et al. (2015)), overcoming the issue that there may be infinitely many states if the state space is continuous. Denoting the parameters of the neural network by $\theta$ , Q-values $Q ( s , a | \theta )$ are now estimated by performing a forward pass through the network. Updates to the parameters are obtained by minimizing a differentiable loss function, $\begin{array} { r } { L ( s , a | \theta _ { i } ) = ( \dot { r } + \gamma \mathrm { m a x } _ { a ^ { \prime } } \dot { Q ( s ^ { \prime } , a ^ { \prime } | \theta _ { i } ) } - Q ( s , a | \theta _ { i } ) ) ^ { 2 } } \end{array}$ , and training is usually accomplished with stochastic gradient descent.
+
+# 2.2 PARTIAL OBSERVABILITY AND DEEP RECURRENT Q-NETWORKS
+
+A fundamental limiting assumption of Markov decision processes is the Markov property, which is rarely satisfied in real-world problems. In medical applications such as our problem of learning optimal sepsis treatments, it is unlikely that a patient’s full clinical state will be measured. A Partially Observable Markov Decision Process (POMDP) better captures the dynamics of these types of realworld environments. An extension of an MDP, a POMDP assumes that an agent does not receive the true state of the system, instead receiving only observations $o \in \Omega$ generated from the underlying system state according to some unknown observation model $o \sim \mathcal { O } ( s )$ . Deep Q-learning has no reliable way to learn the underlying state of the POMDP, as in general $Q ( o , a | \bar { \theta } ) \neq Q ( s , \bar { a } | \theta )$ , and will only perform well if the observations well reflect the underlying state. Returning to our medical application, the system state might be the patient’s unknown clinical status or disease severity, and our observations in the form of vitals or laboratory measurements offer some insight into the state.
+
+The Deep Recurrent Q-Network (DRQN) (Hausknecht & Stone (2015)) extends vanilla Deep Qnetworks (DQN) by using recurrent LSTM layers (Hochreiter & Schmidhuber (1997)), which are well known to capture long-term dependencies. LSTM recurrent neural network (RNN) models have frequently been used in past applications to medical time series, such as Lipton et al. (2016). In our experiments we investigate the effect of replacing fully connected neural network layers with LSTM layers in our Q-network architecture in order to test how realistic the Markov assumption is in our application.
+
+# 2.3 MGPS: MULTI-OUTPUT GAUSSIAN PROCESSES
+
+Multi-output Gaussian processes (MGPs) are commonly used probabilistic models for irregularly sampled multivariate time series, as they seamlessly handle variable spacing, differing numbers of observations per series, and missing values. In addition, they maintain estimates of uncertainty about the state of the series. MGPs have been frequently applied to model patient physiological time series, e.g. Ghassemi et al. (2015), Durichen et al. (2015), Cheng et al. (2017).
+
+Given $M$ time series (physiological labs/vitals), an MGP is specified by a mean function for each series $\{ \mu _ { m } ( t ) \} _ { m = 1 } ^ { M }$ , commonly assumed to be zero, and a covariance function or kernel $K$ . Letting $f _ { m } ( t )$ denote the latent function for series $m$ at time , then $K ( t , t ^ { \prime } , m , m ^ { \prime } ) = \operatorname { c o v } ( f _ { m } ( t ) , f _ { m ^ { \prime } } ( t ^ { \prime } ) ) \big ]$ . Typicallytion, e.g. $y _ { m } ( t ) \sim \mathcal { N } ( f _ { m } ( t ) , \sigma _ { m } ^ { 2 } )$ re cewith $\lbrace \sigma _ { m } ^ { 2 } \rbrace _ { m = 1 } ^ { M }$ he latent functions according to some distribu-noise parameters. We use the linear model of coregionalization covariance function with an Ornstein-Uhlenbeck base kernels $k ( t , t ^ { \prime } ) = e ^ { - | t - t ^ { \prime } | / l }$ to flexibly model temporal correlations in time as well as covariance structure between different physiological variables. For each patient, letting t denote the complete set of measurement times across all observations, the full joint kernel is $\begin{array} { r } { \bar { K } ( \mathbf { t } , \mathbf { t } ^ { \prime } ) = \sum _ { p = 1 } ^ { P } \mathbf { B } _ { p } \otimes k _ { p } ( \mathbf { t } _ { * } , \mathbf { t } _ { * } ^ { \prime } ) } \end{array}$ , where $P$ denotes the number of mixture kernel. $\mathbf { t } _ { * }$ denotes the time vector for each physiological sign, assumed here to be the same for notational convenience, but in practice the full kernel need only be computed at the observed variables. Each $\mathbf { B } _ { p } \in \mathbb { R } ^ { M \times M }$ encodes the scale covariance between different time series. We found that $P = 2$ works well in practice and allows learning of correlations on both short and long time scales. Given the MGP kernel hyperparameters shared across all patients, collectively referred to as $\eta$ , imputation and interpolation at arbitrary times can be computed either using the posterior mean or the full posterior distribution over unknown function values.
+
+# 2.4 MGP-RNNS: MULTI-OUTPUT GAUSSIAN PROCESS RECURRENT NEURAL NETWORKS
+
+Multi-output Gaussian processes and recurrent neural networks can be combined and trained endto-end (MGP-RNNs), in order to solve supervised learning problems for sequential data (Futoma et al. (2017a), Futoma et al. (2017b)). This methodology was shown to exhibit superior predictive performance at early detection of sepsis from clinical time series data, when compared with vanilla RNNs with last-one-carried-forward imputation. In fitting the two models end-to-end, the MGP hyperparameters are learned discriminatively, in essence learning an imputation and interpolation mechanism tuned for the supervised task at hand.
+
+Learning an MGP-RNN consists of minimizing an expectation of some loss function, with respect to the posterior distribution of the MGP. Letting z denote a set of latent time series values distributed according to an MGP posterior, and $g ( \mathbf { z } )$ denote the prediction(s) made by an RNN from this time series, then the goal is to minimize $\mathbb { E } _ { \mathbf { z } \sim \mathcal { M G P } } [ l ( \mathbf { 0 } , g ( \mathbf { z } ) ) ]$ , where $l$ is some loss function (e.g. crossentropy for a classification task) and o is the true label(s). We can express the MGP distributed latent variable $\mathbf { z }$ as ${ \bf z } = \mu _ { z } + R _ { z } \boldsymbol { \xi }$ , where $\mu _ { z }$ is the posterior mean and $R _ { z } { \bf \bar { \cal R } _ { z } ^ { \top } } = \Sigma _ { z }$ with $\Sigma _ { z }$ the posterior covariance, and $\xi \sim ( 0 , I )$ . This allows us to apply the reparameterization trick (Kingma & Welling (2014)) and use Monte Carlo sampling to compute approximate gradients of this expectation with respect to both MGP hyperparameters $\eta$ and RNN parameters $\theta$ , so that the loss can be minimized via stochastic gradient descent. The stochasticity in this learning procedure introduced from the Monte Carlo sampling additionally acts as a form of regularization, and helps prevent the RNN from overfitting. In Section 3 we show how this can be applied to a reinforcement learning task.
+
+# 2.5 RELATED WORK FROM REINFORCEMENT LEARNING IN HEALTHCARE
+
+There has been substantial recent interest in development of machine learning methodologies motivated by healthcare data. However, most prior work in clinical machine learning focuses on supervised tasks, such as diagnosis (Esteva et al. (2017)) or risk stratification (Futoma et al. (2017a)). Many recent papers have developed models for early detection of sepsis, a related problem to our task of learning treatments for sepsis, e.g. Soleimani et al. (2017), Henry et al. (2015), Futoma et al. (2017b). However, as supervised problems rely on ground truth they cannot be applied to treatment recommendation, unless the assumption is made that past training examples of treatments represent optimal behavior. Instead, it is preferable to frame the problem using reinforcement learning in order to learn optimal treatment actions from data collected from potentially suboptimal actions.
+
+While deep reinforcement learning has seen huge success over the past few years, only very recently have reinforcement learning methods been designed with healthcare applications in mind. Applying reinforcement learning methods to healthcare data is difficult, as it requires careful consideration to set up the problem, especially the rewards. Furthermore, it is typically not possible to collect additional data and so evaluating learned policies on retrospective data presents a challenge.
+
+Most related to this paper are Raghu et al. (2017) and Komorowski et al. (2016), who also look at the problem of learning optimal sepsis treatments. We build off of their work by using a more sophisticated network architecture that takes into account both memory through the use of DRQNs and uncertainty in time series imputation and interpolation using MGPs. Other relevant work includes Prasad et al. (2017), who use a simpler learning algorithms to learn optimal strategies for ventilator weaning, and Nemati et al. (2016), who also use a deep RL approach for modeling ICU heparin dosing as a POMDP with discriminative hidden Markov models and Q-networks. There also exists a rich set of work from the statistics and causal inference literature on learning dynamic treatment regimes, e.g. Chakraborty & Moodie (2013), Shortreed et al. (2010), although the models are typically fairly simple for ease of interpretability.
+
+# 3 MGP-DRQN: MULTI-OUTPUT GAUSSIAN PROCESS DEEP RECURRENTQ-NETWORKS
+
+We now introduce Multi-Output Gaussian Process Deep Recurrent Q-Networks, or MGP-DRQNs, a novel reinforcement learning algorithm for learning optimal treatment policies from noisy, sparsely sampled, and frequently missing clinical time series data. We assume a discrete action space, $a \in$ $\mathcal { A } = \{ 1 , \ldots , A \}$ . Let $\mathbf { X }$ denote $T$ regularly spaced grid times at which we would like to learn optimal treatment decisions. Given a set of clinical physiological time series $\mathbf { y }$ that we assume to be distributed according to an MGP, we can compute a posterior distribution for $z _ { t } | \mathbf { y }$ , the latent unobserved time series values at each grid time.
+
+The loss function we optimize is similar to in normal deep Q-learning, with the addition of the expectation due to the MGP and the fact that we compute the loss over full patient trajectories. In particular, we learn optimal DRQN parameters $\theta ^ { * }$ and MGP hyperparameters $\eta ^ { * }$ via:
+
+$$
+\theta ^ { * } , \eta ^ { * } = \operatorname * { a r g m i n } _ { \theta , \eta } \mathbb { E } \left[ \mathbb { E } _ { p ( \mathbf { z } | \mathbf { y } ; \eta ) } \left\{ \frac { 1 } { T } \sum _ { t = 1 } ^ { T } ( Q _ { t a r g e t } ^ { ( t ) } - Q ( [ z _ { t } , s _ { t } ] ^ { \top } , a ; \theta ) ) ^ { 2 } \right\} \right] ,
+$$
+
+where the $t ^ { \prime }$ ’th target value is $\begin{array} { r } { Q _ { t a r g e t } ^ { ( t ) } = r _ { t } + \gamma \mathrm { m a x } _ { a ^ { \prime } } Q ( [ z _ { t + 1 } , s _ { t + 1 } ] , a ^ { \prime } ) } \end{array}$ , the outer expectation is concatenate the two separate types of model inputs at time $t$ , with $z _ { t }$ denoting latent variables distributed according to an MGP posterior from other relevant inputs to the model denoted $s _ { t }$ , such as static baseline covariates. In Section 4.1 we go into detail on the particular variables included in $s _ { t }$ .
+
+We use a Dueling Double-Deep Q-network architecture, similar to Raghu et al. (2017). The Dueling Q-network architecture Wang et al. (2016) has separate value and advantage streams to separate the effect of a patient being in a good underlying state from a good action being taken. The Double-Deep Q-network architecture (van Hasselt et al. (2016)) helps correct overestimation of Q-values by using a second target network to compute the Q-values in the target $\boldsymbol { Q } _ { t a r g e t }$ . Finally, we use Prioritized Experience Replay in order to speed learning, so that patient encounters with higher training error will be resampled more frequently. We use 2 LSTM layers with 64 hidden units each that feed to a final fully connected layer with 64 hidden units, before splitting into equally sized value and advantage streams that are finally then projected onto the action space to obtain $\mathrm { Q }$ -value estimates.
+
+We implemented our methods in Tensorflow using the Adam optimizers (Kingma & Ba (2015)) with minibatches of 50 encounters sampled at a time, a learning rate of 0.001, and $L _ { 2 }$ regularization on weights. We use 25 Monte Carlo samples from the MGP for each sampled encounter in order to approximate the expected loss and compute approximate gradients, and these samples and other inputs are fed in a forward pass through the DRQN to get predictions $Q ( s , a )$ . We will release source code via Github after the review period.
+
+# 4 EXPERIMENTS, EVALUATION, AND RESULTS
+
+In this section we first describe the details of our dataset of septic patients before highlighting how the experiments were set up and how the algorithms were evaluated.
+
+# 4.1 DATASET AND PREPROCESSING
+
+Our dataset consists of information collected during 9,255 patient encounters resulting in sepsis at our university hospital, spanning a period of 15 months. We define sepsis to be the first time at which a patient simultaneously had persistently abnormal vitals (as measured by a $^ { 2 + }$ SIRS score, Bone et al. (1992)), a suspicion of infection (as measured by an order for a blood culture), and an abnormal laboratory value indicative of organ damage. This differs from the new Sepsis-3 definition (Seymour et al. (2016)), which has since been largely criticized for its detection of sepsis late in the clinical course (Cortes-Puch & Hartog (2016)). We break the full dataset into 7867 training patient encounters and reserve the remaining 1388 for testing.
+
+We discretize the data to learn actions in 4 hour windows. We emphasize that the raw data itself is not down-sampled; rather, we use the MGP to learn a posterior for the time series values every 4 hours. Actions for the RL setup consist of 3 treatments commonly given to septic patients: antibiotics, vasopressors, and IV fluids. Antibiotics and vasopressors are broken down into 3 categories, based on whether 0, 1, or $^ { 2 + }$ were administered in each 4 hour window. For IV Fluids, we consider 5 discrete categories: either 0, or one of 4 aggregate doses based on empirical quartiles of total fluid volumes. This yields a discrete action space with $3 \times 3 \times 5 = 4 5$ distinct actions.
+
+Our data consists of 36 longitudinal physiological variables (e.g. blood pressure, pulse, white blood cell count), 2 longitudinal categorical variables, and 38 variables available at baseline (e.g. age, previous medical conditions). 8 medications tangential to sepsis treatment are included as inputs to MGP-DRQN, as well as an indicator for which of the 45 actions was administered at the last time. Additionally, 36 indicator variables for whether or not each lab/vital was recently sampled allows the model to learn from informative sampling due to non-random missingness. In total, there are 165 input observation variables to each of the Q-network models at each time.
+
+Our outcome of interest is mortality within 30 days of onset of sepsis. We use a sparse reward function in this initial work, so that the reward at every non-terminal time point is 0, with a reward of $\pm 1 0$ at the end of a trajectory based on patient survival/death. Although this presents a challenging credit assignment problem, this allows for data to inform what actions should be taken to reduce chance of death without being overly prescriptive.
+
+# 4.2 BASELINE METHODS
+
+We use SARSA, an on-policy algorithm, to estimate state-action values for the physician policy.
+
+We compare a number of different architectures for learning optimal sepsis treatments. In addition to our proposed MGP-DRQN, we compare against MGP-mean-DRQN, a variant where we move the posterior expectation inside the DRQN loss function, meaning we use the posterior mean of the MGP rather than use Monte Carlo samples from the MGP. We also compare against a DRQN with identical architecture, but replace the MGP with last-one-carried-forward imputation to fill in any missing values, and use the mean if there are multiple measurements. We also compare against a vanilla DQN, a MGP-DQN, and a MGP-mean-DQN, with an equivalent number of layers and parameters, to test the effect of the recurrence in the DRQN models.
+
+# 4.3 OFF-POLICY VALUE EVALUATION
+
+We use Doubly Robust Off-policy Value Evaluation (Jiang & Li (2016)) to compute unbiased estimates of each learned optimal policy using our observed off-policy data. For each patient trajectory in the test set we estimate its value using this method, and the average results. In order to apply this method we train an MGP-RNN to estimate the action probabilities of the physician policy.
+
+# 4.4 QUANTITATIVE RESULTS
+
+In Figure 1 we show the results of using SARSA to estimate expected returns for the physician policy on the test data. The Q-values appear to be well calibrated with mortality, as patients who were estimated to have higher expected returns tended to have lower mortality. Due to small sample sizes for very low expected returns, the mortality rate does not always monotonically decrease.
+
+![](images/db3a0f5163cac607345f2838b1203f651d12f015dfb2c92b1abe2cb9e4a5f19d.jpg)  
+Figure 1: For the 1388 patients in the test set we show the expected returns as computed by SARSA, against 30-day mortality among patients with similar Q-values. Our model appears to be well calibrated, as higher returns are associated with lower mortality.
+
+We can estimate the potential reduction in mortality a learned policy might have by computing an unbiased estimate of the policy value, as described in Section 4.3, and then use the results in Figure 1. Table 1 contains the policy value estimates for each algorithm considered, along with estimated mortality rates. The physician policy has an estimated value of 5.52 and corresponding mortality of $1 3 . 3 \%$ , matching the observed mortality in the test set of $1 3 . 3 \%$ . Overall the MGPDRQN performs and might reduce mortality by as much as $8 \%$ . The DRQN architectures tended to yield higher expected returns, probably because they are able to retain some memory of past clinical states and actions taken. The MGP consistently improved results as well, and the additional uncertainty information contained in the full MGP posterior appeared to do better than the policies that only used the posterior mean.
+
+<table><tr><td rowspan=1 colspan=2>Policy</td><td rowspan=1 colspan=1>Expected Return</td><td rowspan=1 colspan=1>EstimatedMortality</td></tr><tr><td rowspan=7 colspan=2>PhysicianMGP-DRQNMGP-mean-DRQNDRQNMGP-DQNMGP-mean-DQNDQN</td><td rowspan=1 colspan=1>5.52</td><td rowspan=6 colspan=1>13.3 ± 0.7%5.1 ± 0.5%6.6 ± 0.4%8.4 ± 0.4%6.6 ± 0.4%7.5 ± 0.4%</td></tr><tr><td rowspan=1 colspan=1>7.51</td></tr><tr><td rowspan=1 colspan=1>6.97</td></tr><tr><td rowspan=1 colspan=1>6.63</td></tr><tr><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>7.05</td></tr><tr><td rowspan=1 colspan=1>6.73</td></tr><tr><td rowspan=1 colspan=1>6.09</td><td rowspan=1 colspan=1>10.6 ± 0.5%</td></tr></table>
+
+Table 1: Expected returns for the various policies considered. For the 6 reinforcement learning algorithms considered, we estimate their expected returns using an off-policy value evaluation algorithm. Using the results from Figure 1, we estimate the potential expected mortality reduction associated with each policy.
+
+# 4.5 QUALITATIVE RESULTS
+
+We also qualitatively evaluate the results of the policy from our best performing learning algorithm, the MGP-DRQN. In Figure 2 we compare the number of times each type of action was actually taken by physicians, and how many times the learned policy selected that action. The MGP-DRQN policy tended to recommend more use of antibiotics and more vasopressors than were actually used by physician, while strangely recommending somewhat less use of IV fluids.
+
+![](images/3fdc39dc3509052c29a0d73b9f7364d8cbe8516eba0b95136bcd0c49edacea0f.jpg)  
+Figure 2: Comparison of physician actions with the actions that would have been taken by the MGP-DRQN policy, with actions separated according to the 3 types of treatments considered.
+
+In Figure 3, we show how mortality rates differ on the test set as a function of how different the observed physician action was from what the MGP-DRQN would have recommended. For all 3 types of treatments, there appears to be a local minimum at 0 and we observe a $\mathrm { v }$ shape, indicating that empirically, mortality tended to be lowest when the clinicians took the same actions that the MGP-DRQN would have. Uncertainty tends to be higher due to smaller sample sizes for situations where there is larger disparity.
+
+![](images/91d661b22376fb336538862913236ef6c2aa8babdbd0353457bb34fedec0b587.jpg)  
+Figure 3: Empirical mortality rates as a function of how much the MGP-DRQN policy’s actions differed from the observed physician actions. Minimal mortality is observed for all 3 treatment types at 0, where the physicians and MGP-DRQN agreed.
+
+Finally, in Figure 4 we show clinical data from a sample patient case. In the top pane of the figure we show five representative vital signs and lab measurements to illustrate the patient’s clinical status, while the bottom shows both what actions physicians actually took and what actions the model recommended. The patient was admitted to the Emergency Department for altered mental status, and the MGP-DRQN quickly recognizes the need for antibiotics and IV fluids. The patient is admitted to the hospital and around hour 6 the clinical team becomes aware of sepsis. However, antibiotics are not first administered until hour 18, about 16 hours after the model recommended treating with them. After the patient is transferred to the Intensive Care Unit, their white blood cell count continues to rise (a sign of worsening infection) and their blood pressure continues to fall (a sign of worsening shock). By hour 14, the RL model starts and continues to recommend use of vasopressors to attempt to increase blood pressure, but they are not actually administered for about another 16 hours at hour 30. Ultimately, by hour 45 care was withdrawn and the patient passed away at hour 50. Cases such as this one illustrate the potential benefits of using our learned treatment policy in a decision support tool to recommend treatments to providers. If such a tool were used in this situation, it is possible that earlier treatments and more aggressive interventions might have resulted in a different outcome.
+
+![](images/2a99595a10a77663252693bd28c4531b89c26bd8a65175621dfff5e5a5f49322.jpg)  
+Figure 4: Top: clinical data from a patient who acquired sepsis, decompensated in the Intensive Care Unit while progressing to septic shock, and ultimately did not survive. Bottom: shaded symbols denote treatments that the learned MGP-DRQN policy would have recommended, while open symbols denote the treatment actions actually taken by physicians caring for this patient.
+
+# 5 CONCLUSION
+
+In this paper we presented a new framework combining multi-output Gaussian processes and deep reinforcement learning for clinical problems, and found that our approach performed well in estimating optimal treatment strategies for septic patients. The use of recurrent structure in the Q-network architecture yielded higher expected returns than a standard Q-network, accounting for the nonMarkovian nature of real-world medical data. The multi-output Gaussian process also improved performance by offering a more principled method for interpolation and imputation, and use of the full MGP posterior improved upon the results from just using the posterior mean.
+
+In the future, we could include treatment recommendations from our learned policies into our dashboard application we have developed for early detection of sepsis. The treatment recommendations might help providers better care for septic patients after sepsis has been properly identified, and start treatments faster. There are many potential avenues for future work. One promising direction is to investigate the use of more complex reward functions, rather than the sparse rewards used in this work. More sophisticated rewards might take into account clinical targets for maintaining hemodynamic stability, and penalize an overzealous model that recommends too many unnecessary actions. Our modeling framework is fairly generalizable, and can easily be applied to other medical applications where there is a need for data-driven decision support tools. In future work we plan to use similar methods to learn optimal treatment strategies for treating patients with cardiogenic shock, and to learn effective insulin dosing regimes for patients on high-dose steroids.
+
+REFERENCES   
+D. C. Angus, A. E. Barnato, D. Bell, and et al. A systematic review and meta-analysis of early goal-directed therapy for septic shock: the arise, process and promise investigators. Intensive Care Medicine, 41(9): 1549–1560, 2015.   
+Investigators ARISE and Anzics Clinical Trials Group. Goal-directed resuscitation for patients with early septic shock. N Engl J Med, 371(16):1496–1506, 2014.   
+R. C. Bone, R. A. Balk, and F. B. et al. Cerra. Definitions for sepsis and organ failure and guidelines for the use of innovative therapies in sepsis. Chest, 101(6):1644–55, 1992.   
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+M. Ghassemi, M. A. F. Pimentel, and T. et al. Naumann. A multivariate timeseries modeling approach to severity of illness assessment and forecasting in icu with sparse, heterogeneous clinical data. AAAI, 2015.   
+M. Hausknecht and P. Stone. Deep recurrent q-learning for partially observable mdps. AAAI, 2015.   
+K. E. Henry, D. N. Hager, P. J. Pronovost, and S. Saria. A targeted real-time early warning score (trewscore) for septic shock. Science Translational Medicine, 7(299), 2015.   
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+Investigators PRISM. Early, goal-directed therapy for septic shock a patient-level meta-analysis. N Engl J Med, 376:2223–2234, 2017.   
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+C. J. C. H. Watkins and P. Dayan. Q-learning. Machine Learning, 8(3-4):279–292, 1992.

parse/train/SyxCqGbRZ/SyxCqGbRZ_content_list.json

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+[
+    {
+        "type": "text",
+        "text": "LEARNING TO TREAT SEPSIS WITH MULTI-OUTPUTGAUSSIAN PROCESS DEEP RECURRENT Q-NETWORKS",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Anonymous authors Paper under double-blind review ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "ABSTRACT ",
+        "text_level": 1,
+        "bbox": [
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+        "type": "text",
+        "text": "Sepsis is a life-threatening complication from infection and a leading cause of mortality in hospitals. While early detection of sepsis improves patient outcomes, there is little consensus on exact treatment guidelines, and treating septic patients remains an open problem. In this work we present a new deep reinforcement learning method that we use to learn optimal personalized treatment policies for septic patients. We model patient continuous-valued physiological time series using multi-output Gaussian processes, a probabilistic model that easily handles missing values and irregularly spaced observation times while maintaining estimates of uncertainty. The Gaussian process is directly tied to a deep recurrent Q-network that learns clinically interpretable treatment policies, and both models are learned together end-to-end. We evaluate our approach on a heterogeneous dataset of septic spanning 15 months from our university health system, and find that our learned policy could reduce patient mortality by as much as $8 . 2 \\%$ from an overall baseline mortality rate of $1 3 . 3 \\%$ . Our algorithm could be used to make treatment recommendations to physicians as part of a decision support tool, and the framework readily applies to other reinforcement learning problems that rely on sparsely sampled and frequently missing multivariate time series data. ",
+        "bbox": [
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "1 INTRODUCTION ",
+        "text_level": 1,
+        "bbox": [
+            176,
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Sepsis is a poorly understood complication arising from infection, and is both a leading cause in patient mortality (Epstein et al. (2016)) and in associated healthcare costs (Torio & Moore (2016)). Early detection is imperative, as earlier treatment is associated with better outcomes (Seymour et al. (2017), Kumar et al. (2006)). However, even among patients with recognized sepsis, there is no standard consensus on the best treatment. There is a pressing need for personalized treatment strategies tailored to the unique physiology of individual patients. Guidelines on sepsis treatment previously centered on early goal directed therapy (EGDT) and more recently have focused on sepsis care bundles, but none of these approaches are individualized. ",
+        "bbox": [
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Before the landmark publication on the use of early goal directed therapy (Rivers et al. (2001)), there was no standard management for severe sepsis and septic shock. EGDT consists of early identification of high-risk patients, appropriate cultures, infection source control, antibiotics administration, and hemodynamic optimization. The study compared a 6-hour protocol of EGDT promoting use of central venous catheterization to guide administration of fluids, vasopressors, inotropes, and packed red-blood cell transfusions, and was found to significantly lower mortality. Following the initial trial, EGDT became the cornerstone of the sepsis resuscitation bundle for the Surviving Sepsis Campaign (SCC) and the Centers for Medicare and Medicaid Services (CMS) (Dellinger et al. (2013)). ",
+        "bbox": [
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Despite the promising results of EGDT, concerns arose. External validity outside the single center study was unclear, it required significant resources for implementation, and the elements needed to achieve pre-specified hemodynamic targets held potential risks. Between 2014–2017, a trio of trials reported an all-time low sepsis mortality, and questioned the continued need for all elements of EGDT for patients with severe and septic shock (ProCESS et al. (2014), ARISE & Group (2014), PRISM (2017)). The trial authors concluded EGDT did not improve patient survival compared to usual care but was associated with increased ICU admissions (Angus et al. (2015)). As a result, they did not recommend it be included in the updated SCC guidelines (Rhodes et al. (2017)). ",
+        "bbox": [
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Although the SSC guidelines provide an overarching framework for sepsis treatment, there is renewed interest in targeting treatment and disassembling the bundle (Lewis (2010)). A recent metaanalysis evaluated 12 randomized trials and 31 observational studies and found that time to first antibiotics explained $9 6 - 9 9 \\%$ of the survival benefit (Kalil et al. (2017)). Likewise, a study of 50,000 patients across the state of New York found mortality benefit for early antibiotic administration, but not intravenous fluids (Seymour et al. (2017)). Beyond narrowing the bundle, there is emerging evidence that a patient’s baseline risk plays an important role in response to treatment, as survival benefit was significantly reduced for patients with more severe disease (Kalil et al. (2017)). ",
+        "bbox": [
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Taken together, the poor performance of EGDT compared to standard-of-care and improved understanding of individual treatment effects calls for re-envisioning sepsis treatment recommendations. Though general consensus in critical care is that the individual elements of the sepsis bundle are typically useful, it is unclear exactly when each element should be administered and in what quantity. ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "In this paper, we aim to directly address this problem using deep reinforcement learning. We develop a novel framework for applying deep reinforcement learning to clinical data, and use it to learn optimal treatments for sepsis. With the widespread adoption of Electronic Health Records, hospitals are already automatically collecting the relevant data required to learn such models. However, real-world operational healthcare data present many unique challenges and motivate the need for methodologies designed with their structure in mind. In particular, clinical time series are typically irregularly sampled and exhibit large degrees of missing values that are often informatively missing, necessitating careful modeling. The high degree of heterogeneity presents an additional difficulty, as patients with similar symptoms may respond very differently to treatments due to unmeasured sources of variation. Alignment of patient time series can also be a potential issue, as patients admitted to the hospital may have very different unknown clinical states and can develop sepsis at any time throughout their stay (with many already septic upon admission). ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Part of the novelty in our approach hinges on the use of a Multi-output Gaussian process (MGP) as a preprocessing step that is jointly learned with the reinforcement learning model. We use an MGP to interpolate and to impute missing physiological time series values used by the downstream reinforcement learning algorithm, while importantly maintaining uncertainty about the clinical state. The MGP hyperparameters are learned end-to-end during training of the reinforcement learning model by optimizing an expectation of the standard Q-learning loss. Additionally, the MGP allows for estimation of uncertainty in the learned Q-values. For the model architecture we use a deep recurrent Q-network, in order to account for the potential for non-Markovian dynamics and allow the model to have memory of past states and actions. In our experiments utilizing EHR data from septic patients spanning 15 months from our university health system, we found that both the use of the MGP and the deep recurrent Q-network offered improved performance over simpler approaches. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "2 BACKGROUND ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            589,
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+            606
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "In this section we outline important background that motivates our improvements on prior work. ",
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+    {
+        "type": "text",
+        "text": "2.1 DEEP Q-LEARNING ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Reinforcement learning (RL) considers learning policies for agents interacting with unknown environments, and are typically formulated as a Markov decision process (MDP) (Sutton $\\&$ Barto (1998)). At each time $t$ , an agent observes the state of the environment, $s _ { t } \\in S$ , takes an action $a _ { t } \\in \\mathcal A$ , and receives a reward $r _ { t } \\in \\mathbb { R }$ , at which time the environment transitions to a new state $s _ { t + 1 }$ . The state space $s$ and action space $\\mathcal { A }$ may be continuous or discrete. The goal of an RL agent is to select actions in order to maximize its return, or expected discounted future reward, defined as $\\begin{array} { r } { R _ { t } = \\sum _ { t ^ { \\prime } = t } ^ { T } \\gamma ^ { t ^ { \\prime } - t } r _ { t ^ { \\prime } } } \\end{array}$ , where $\\gamma$ captures tradeoff between immediate and future rewards. ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Q-Learning (Watkins & Dayan (1992)) is a model-free off-policy algorithm for estimating the expected return from executing an action in a given state. The optimal action value function is the maximum discounted expected reward obtained by executing action $a$ in state $s$ and acting optimally afterwards, defined as $\\begin{array} { r } { Q ^ { * } ( s , a ) = \\operatorname* { m a x } _ { \\pi } \\mathbb { E } [ R _ { t } | s _ { t } = s , a _ { t } = a , \\pi ] } \\end{array}$ , where $\\pi$ is a policy that maps states to actions. Given $Q ^ { * }$ , an optimal policy is to act by selecting argmax ${ } _ { a } Q ^ { * } ( s , a )$ . In Q-learning, the Bellman equation is used to iteratively update the current estimate of the optimal action value function according to $Q ( s , a ) \\doteq Q ( s , a ) + \\alpha \\bar { ( } r + \\gamma \\mathrm { m a x } _ { a } Q ( s ^ { \\prime } , a ^ { \\prime } ) - Q ( s , a ) )$ , adjusting towards the observed reward plus the maximal Q-value at the next state $s ^ { \\prime }$ . In Deep Q-learning a deep neural network is used to approximate $\\mathrm { Q }$ -values (Mnih et al. (2015)), overcoming the issue that there may be infinitely many states if the state space is continuous. Denoting the parameters of the neural network by $\\theta$ , Q-values $Q ( s , a | \\theta )$ are now estimated by performing a forward pass through the network. Updates to the parameters are obtained by minimizing a differentiable loss function, $\\begin{array} { r } { L ( s , a | \\theta _ { i } ) = ( \\dot { r } + \\gamma \\mathrm { m a x } _ { a ^ { \\prime } } \\dot { Q ( s ^ { \\prime } , a ^ { \\prime } | \\theta _ { i } ) } - Q ( s , a | \\theta _ { i } ) ) ^ { 2 } } \\end{array}$ , and training is usually accomplished with stochastic gradient descent. ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "2.2 PARTIAL OBSERVABILITY AND DEEP RECURRENT Q-NETWORKS",
+        "text_level": 1,
+        "bbox": [
+            176,
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "A fundamental limiting assumption of Markov decision processes is the Markov property, which is rarely satisfied in real-world problems. In medical applications such as our problem of learning optimal sepsis treatments, it is unlikely that a patient’s full clinical state will be measured. A Partially Observable Markov Decision Process (POMDP) better captures the dynamics of these types of realworld environments. An extension of an MDP, a POMDP assumes that an agent does not receive the true state of the system, instead receiving only observations $o \\in \\Omega$ generated from the underlying system state according to some unknown observation model $o \\sim \\mathcal { O } ( s )$ . Deep Q-learning has no reliable way to learn the underlying state of the POMDP, as in general $Q ( o , a | \\bar { \\theta } ) \\neq Q ( s , \\bar { a } | \\theta )$ , and will only perform well if the observations well reflect the underlying state. Returning to our medical application, the system state might be the patient’s unknown clinical status or disease severity, and our observations in the form of vitals or laboratory measurements offer some insight into the state. ",
+        "bbox": [
+            174,
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "The Deep Recurrent Q-Network (DRQN) (Hausknecht & Stone (2015)) extends vanilla Deep Qnetworks (DQN) by using recurrent LSTM layers (Hochreiter & Schmidhuber (1997)), which are well known to capture long-term dependencies. LSTM recurrent neural network (RNN) models have frequently been used in past applications to medical time series, such as Lipton et al. (2016). In our experiments we investigate the effect of replacing fully connected neural network layers with LSTM layers in our Q-network architecture in order to test how realistic the Markov assumption is in our application. ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "2.3 MGPS: MULTI-OUTPUT GAUSSIAN PROCESSES ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            478,
+            545,
+            492
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Multi-output Gaussian processes (MGPs) are commonly used probabilistic models for irregularly sampled multivariate time series, as they seamlessly handle variable spacing, differing numbers of observations per series, and missing values. In addition, they maintain estimates of uncertainty about the state of the series. MGPs have been frequently applied to model patient physiological time series, e.g. Ghassemi et al. (2015), Durichen et al. (2015), Cheng et al. (2017). ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Given $M$ time series (physiological labs/vitals), an MGP is specified by a mean function for each series $\\{ \\mu _ { m } ( t ) \\} _ { m = 1 } ^ { M }$ , commonly assumed to be zero, and a covariance function or kernel $K$ . Letting $f _ { m } ( t )$ denote the latent function for series $m$ at time , then $K ( t , t ^ { \\prime } , m , m ^ { \\prime } ) = \\operatorname { c o v } ( f _ { m } ( t ) , f _ { m ^ { \\prime } } ( t ^ { \\prime } ) ) \\big ]$ . Typicallytion, e.g. $y _ { m } ( t ) \\sim \\mathcal { N } ( f _ { m } ( t ) , \\sigma _ { m } ^ { 2 } )$ re cewith $\\lbrace \\sigma _ { m } ^ { 2 } \\rbrace _ { m = 1 } ^ { M }$ he latent functions according to some distribu-noise parameters. We use the linear model of coregionalization covariance function with an Ornstein-Uhlenbeck base kernels $k ( t , t ^ { \\prime } ) = e ^ { - | t - t ^ { \\prime } | / l }$ to flexibly model temporal correlations in time as well as covariance structure between different physiological variables. For each patient, letting t denote the complete set of measurement times across all observations, the full joint kernel is $\\begin{array} { r } { \\bar { K } ( \\mathbf { t } , \\mathbf { t } ^ { \\prime } ) = \\sum _ { p = 1 } ^ { P } \\mathbf { B } _ { p } \\otimes k _ { p } ( \\mathbf { t } _ { * } , \\mathbf { t } _ { * } ^ { \\prime } ) } \\end{array}$ , where $P$ denotes the number of mixture kernel. $\\mathbf { t } _ { * }$ denotes the time vector for each physiological sign, assumed here to be the same for notational convenience, but in practice the full kernel need only be computed at the observed variables. Each $\\mathbf { B } _ { p } \\in \\mathbb { R } ^ { M \\times M }$ encodes the scale covariance between different time series. We found that $P = 2$ works well in practice and allows learning of correlations on both short and long time scales. Given the MGP kernel hyperparameters shared across all patients, collectively referred to as $\\eta$ , imputation and interpolation at arbitrary times can be computed either using the posterior mean or the full posterior distribution over unknown function values. ",
+        "bbox": [
+            174,
+            582,
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "2.4 MGP-RNNS: MULTI-OUTPUT GAUSSIAN PROCESS RECURRENT NEURAL NETWORKS ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            827,
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Multi-output Gaussian processes and recurrent neural networks can be combined and trained endto-end (MGP-RNNs), in order to solve supervised learning problems for sequential data (Futoma et al. (2017a), Futoma et al. (2017b)). This methodology was shown to exhibit superior predictive performance at early detection of sepsis from clinical time series data, when compared with vanilla RNNs with last-one-carried-forward imputation. In fitting the two models end-to-end, the MGP hyperparameters are learned discriminatively, in essence learning an imputation and interpolation mechanism tuned for the supervised task at hand. ",
+        "bbox": [
+            174,
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            173,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Learning an MGP-RNN consists of minimizing an expectation of some loss function, with respect to the posterior distribution of the MGP. Letting z denote a set of latent time series values distributed according to an MGP posterior, and $g ( \\mathbf { z } )$ denote the prediction(s) made by an RNN from this time series, then the goal is to minimize $\\mathbb { E } _ { \\mathbf { z } \\sim \\mathcal { M G P } } [ l ( \\mathbf { 0 } , g ( \\mathbf { z } ) ) ]$ , where $l$ is some loss function (e.g. crossentropy for a classification task) and o is the true label(s). We can express the MGP distributed latent variable $\\mathbf { z }$ as ${ \\bf z } = \\mu _ { z } + R _ { z } \\boldsymbol { \\xi }$ , where $\\mu _ { z }$ is the posterior mean and $R _ { z } { \\bf \\bar { \\cal R } _ { z } ^ { \\top } } = \\Sigma _ { z }$ with $\\Sigma _ { z }$ the posterior covariance, and $\\xi \\sim ( 0 , I )$ . This allows us to apply the reparameterization trick (Kingma & Welling (2014)) and use Monte Carlo sampling to compute approximate gradients of this expectation with respect to both MGP hyperparameters $\\eta$ and RNN parameters $\\theta$ , so that the loss can be minimized via stochastic gradient descent. The stochasticity in this learning procedure introduced from the Monte Carlo sampling additionally acts as a form of regularization, and helps prevent the RNN from overfitting. In Section 3 we show how this can be applied to a reinforcement learning task. ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "2.5 RELATED WORK FROM REINFORCEMENT LEARNING IN HEALTHCARE ",
+        "text_level": 1,
+        "bbox": [
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+        "type": "text",
+        "text": "There has been substantial recent interest in development of machine learning methodologies motivated by healthcare data. However, most prior work in clinical machine learning focuses on supervised tasks, such as diagnosis (Esteva et al. (2017)) or risk stratification (Futoma et al. (2017a)). Many recent papers have developed models for early detection of sepsis, a related problem to our task of learning treatments for sepsis, e.g. Soleimani et al. (2017), Henry et al. (2015), Futoma et al. (2017b). However, as supervised problems rely on ground truth they cannot be applied to treatment recommendation, unless the assumption is made that past training examples of treatments represent optimal behavior. Instead, it is preferable to frame the problem using reinforcement learning in order to learn optimal treatment actions from data collected from potentially suboptimal actions. ",
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+        "text": "While deep reinforcement learning has seen huge success over the past few years, only very recently have reinforcement learning methods been designed with healthcare applications in mind. Applying reinforcement learning methods to healthcare data is difficult, as it requires careful consideration to set up the problem, especially the rewards. Furthermore, it is typically not possible to collect additional data and so evaluating learned policies on retrospective data presents a challenge. ",
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+        "text": "Most related to this paper are Raghu et al. (2017) and Komorowski et al. (2016), who also look at the problem of learning optimal sepsis treatments. We build off of their work by using a more sophisticated network architecture that takes into account both memory through the use of DRQNs and uncertainty in time series imputation and interpolation using MGPs. Other relevant work includes Prasad et al. (2017), who use a simpler learning algorithms to learn optimal strategies for ventilator weaning, and Nemati et al. (2016), who also use a deep RL approach for modeling ICU heparin dosing as a POMDP with discriminative hidden Markov models and Q-networks. There also exists a rich set of work from the statistics and causal inference literature on learning dynamic treatment regimes, e.g. Chakraborty & Moodie (2013), Shortreed et al. (2010), although the models are typically fairly simple for ease of interpretability. ",
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+        "type": "text",
+        "text": "3 MGP-DRQN: MULTI-OUTPUT GAUSSIAN PROCESS DEEP RECURRENTQ-NETWORKS",
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+        "text": "We now introduce Multi-Output Gaussian Process Deep Recurrent Q-Networks, or MGP-DRQNs, a novel reinforcement learning algorithm for learning optimal treatment policies from noisy, sparsely sampled, and frequently missing clinical time series data. We assume a discrete action space, $a \\in$ $\\mathcal { A } = \\{ 1 , \\ldots , A \\}$ . Let $\\mathbf { X }$ denote $T$ regularly spaced grid times at which we would like to learn optimal treatment decisions. Given a set of clinical physiological time series $\\mathbf { y }$ that we assume to be distributed according to an MGP, we can compute a posterior distribution for $z _ { t } | \\mathbf { y }$ , the latent unobserved time series values at each grid time. ",
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+        "text": "The loss function we optimize is similar to in normal deep Q-learning, with the addition of the expectation due to the MGP and the fact that we compute the loss over full patient trajectories. In particular, we learn optimal DRQN parameters $\\theta ^ { * }$ and MGP hyperparameters $\\eta ^ { * }$ via: ",
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+        "text": "$$\n\\theta ^ { * } , \\eta ^ { * } = \\operatorname * { a r g m i n } _ { \\theta , \\eta } \\mathbb { E } \\left[ \\mathbb { E } _ { p ( \\mathbf { z } | \\mathbf { y } ; \\eta ) } \\left\\{ \\frac { 1 } { T } \\sum _ { t = 1 } ^ { T } ( Q _ { t a r g e t } ^ { ( t ) } - Q ( [ z _ { t } , s _ { t } ] ^ { \\top } , a ; \\theta ) ) ^ { 2 } \\right\\} \\right] ,\n$$",
+        "text_format": "latex",
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+        "type": "text",
+        "text": "where the $t ^ { \\prime }$ ’th target value is $\\begin{array} { r } { Q _ { t a r g e t } ^ { ( t ) } = r _ { t } + \\gamma \\mathrm { m a x } _ { a ^ { \\prime } } Q ( [ z _ { t + 1 } , s _ { t + 1 } ] , a ^ { \\prime } ) } \\end{array}$ , the outer expectation is concatenate the two separate types of model inputs at time $t$ , with $z _ { t }$ denoting latent variables distributed according to an MGP posterior from other relevant inputs to the model denoted $s _ { t }$ , such as static baseline covariates. In Section 4.1 we go into detail on the particular variables included in $s _ { t }$ . ",
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+        "text": "We use a Dueling Double-Deep Q-network architecture, similar to Raghu et al. (2017). The Dueling Q-network architecture Wang et al. (2016) has separate value and advantage streams to separate the effect of a patient being in a good underlying state from a good action being taken. The Double-Deep Q-network architecture (van Hasselt et al. (2016)) helps correct overestimation of Q-values by using a second target network to compute the Q-values in the target $\\boldsymbol { Q } _ { t a r g e t }$ . Finally, we use Prioritized Experience Replay in order to speed learning, so that patient encounters with higher training error will be resampled more frequently. We use 2 LSTM layers with 64 hidden units each that feed to a final fully connected layer with 64 hidden units, before splitting into equally sized value and advantage streams that are finally then projected onto the action space to obtain $\\mathrm { Q }$ -value estimates. ",
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+        "text": "We implemented our methods in Tensorflow using the Adam optimizers (Kingma & Ba (2015)) with minibatches of 50 encounters sampled at a time, a learning rate of 0.001, and $L _ { 2 }$ regularization on weights. We use 25 Monte Carlo samples from the MGP for each sampled encounter in order to approximate the expected loss and compute approximate gradients, and these samples and other inputs are fed in a forward pass through the DRQN to get predictions $Q ( s , a )$ . We will release source code via Github after the review period. ",
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+        "text": "4 EXPERIMENTS, EVALUATION, AND RESULTS ",
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+        "text": "In this section we first describe the details of our dataset of septic patients before highlighting how the experiments were set up and how the algorithms were evaluated. ",
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+        "text": "4.1 DATASET AND PREPROCESSING ",
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+        "text": "Our dataset consists of information collected during 9,255 patient encounters resulting in sepsis at our university hospital, spanning a period of 15 months. We define sepsis to be the first time at which a patient simultaneously had persistently abnormal vitals (as measured by a $^ { 2 + }$ SIRS score, Bone et al. (1992)), a suspicion of infection (as measured by an order for a blood culture), and an abnormal laboratory value indicative of organ damage. This differs from the new Sepsis-3 definition (Seymour et al. (2016)), which has since been largely criticized for its detection of sepsis late in the clinical course (Cortes-Puch & Hartog (2016)). We break the full dataset into 7867 training patient encounters and reserve the remaining 1388 for testing. ",
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+        "text": "We discretize the data to learn actions in 4 hour windows. We emphasize that the raw data itself is not down-sampled; rather, we use the MGP to learn a posterior for the time series values every 4 hours. Actions for the RL setup consist of 3 treatments commonly given to septic patients: antibiotics, vasopressors, and IV fluids. Antibiotics and vasopressors are broken down into 3 categories, based on whether 0, 1, or $^ { 2 + }$ were administered in each 4 hour window. For IV Fluids, we consider 5 discrete categories: either 0, or one of 4 aggregate doses based on empirical quartiles of total fluid volumes. This yields a discrete action space with $3 \\times 3 \\times 5 = 4 5$ distinct actions. ",
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+        "type": "text",
+        "text": "Our data consists of 36 longitudinal physiological variables (e.g. blood pressure, pulse, white blood cell count), 2 longitudinal categorical variables, and 38 variables available at baseline (e.g. age, previous medical conditions). 8 medications tangential to sepsis treatment are included as inputs to MGP-DRQN, as well as an indicator for which of the 45 actions was administered at the last time. Additionally, 36 indicator variables for whether or not each lab/vital was recently sampled allows the model to learn from informative sampling due to non-random missingness. In total, there are 165 input observation variables to each of the Q-network models at each time. ",
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+        "type": "text",
+        "text": "Our outcome of interest is mortality within 30 days of onset of sepsis. We use a sparse reward function in this initial work, so that the reward at every non-terminal time point is 0, with a reward of $\\pm 1 0$ at the end of a trajectory based on patient survival/death. Although this presents a challenging credit assignment problem, this allows for data to inform what actions should be taken to reduce chance of death without being overly prescriptive. ",
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+        "text": "4.2 BASELINE METHODS",
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+        "type": "text",
+        "text": "We use SARSA, an on-policy algorithm, to estimate state-action values for the physician policy. ",
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+        "text": "We compare a number of different architectures for learning optimal sepsis treatments. In addition to our proposed MGP-DRQN, we compare against MGP-mean-DRQN, a variant where we move the posterior expectation inside the DRQN loss function, meaning we use the posterior mean of the MGP rather than use Monte Carlo samples from the MGP. We also compare against a DRQN with identical architecture, but replace the MGP with last-one-carried-forward imputation to fill in any missing values, and use the mean if there are multiple measurements. We also compare against a vanilla DQN, a MGP-DQN, and a MGP-mean-DQN, with an equivalent number of layers and parameters, to test the effect of the recurrence in the DRQN models. ",
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+        "text": "4.3 OFF-POLICY VALUE EVALUATION ",
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+        "text": "We use Doubly Robust Off-policy Value Evaluation (Jiang & Li (2016)) to compute unbiased estimates of each learned optimal policy using our observed off-policy data. For each patient trajectory in the test set we estimate its value using this method, and the average results. In order to apply this method we train an MGP-RNN to estimate the action probabilities of the physician policy. ",
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+        "type": "text",
+        "text": "4.4 QUANTITATIVE RESULTS ",
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+        "type": "text",
+        "text": "In Figure 1 we show the results of using SARSA to estimate expected returns for the physician policy on the test data. The Q-values appear to be well calibrated with mortality, as patients who were estimated to have higher expected returns tended to have lower mortality. Due to small sample sizes for very low expected returns, the mortality rate does not always monotonically decrease. ",
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+        "img_path": "images/db3a0f5163cac607345f2838b1203f651d12f015dfb2c92b1abe2cb9e4a5f19d.jpg",
+        "image_caption": [
+            "Figure 1: For the 1388 patients in the test set we show the expected returns as computed by SARSA, against 30-day mortality among patients with similar Q-values. Our model appears to be well calibrated, as higher returns are associated with lower mortality. "
+        ],
+        "image_footnote": [],
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+        "type": "text",
+        "text": "We can estimate the potential reduction in mortality a learned policy might have by computing an unbiased estimate of the policy value, as described in Section 4.3, and then use the results in Figure 1. Table 1 contains the policy value estimates for each algorithm considered, along with estimated mortality rates. The physician policy has an estimated value of 5.52 and corresponding mortality of $1 3 . 3 \\%$ , matching the observed mortality in the test set of $1 3 . 3 \\%$ . Overall the MGPDRQN performs and might reduce mortality by as much as $8 \\%$ . The DRQN architectures tended to yield higher expected returns, probably because they are able to retain some memory of past clinical states and actions taken. The MGP consistently improved results as well, and the additional uncertainty information contained in the full MGP posterior appeared to do better than the policies that only used the posterior mean. ",
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+        "table_body": "<table><tr><td rowspan=1 colspan=2>Policy</td><td rowspan=1 colspan=1>Expected Return</td><td rowspan=1 colspan=1>EstimatedMortality</td></tr><tr><td rowspan=7 colspan=2>PhysicianMGP-DRQNMGP-mean-DRQNDRQNMGP-DQNMGP-mean-DQNDQN</td><td rowspan=1 colspan=1>5.52</td><td rowspan=6 colspan=1>13.3 ± 0.7%5.1 ± 0.5%6.6 ± 0.4%8.4 ± 0.4%6.6 ± 0.4%7.5 ± 0.4%</td></tr><tr><td rowspan=1 colspan=1>7.51</td></tr><tr><td rowspan=1 colspan=1>6.97</td></tr><tr><td rowspan=1 colspan=1>6.63</td></tr><tr><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>7.05</td></tr><tr><td rowspan=1 colspan=1>6.73</td></tr><tr><td rowspan=1 colspan=1>6.09</td><td rowspan=1 colspan=1>10.6 ± 0.5%</td></tr></table>",
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+        "text": "Table 1: Expected returns for the various policies considered. For the 6 reinforcement learning algorithms considered, we estimate their expected returns using an off-policy value evaluation algorithm. Using the results from Figure 1, we estimate the potential expected mortality reduction associated with each policy. ",
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+        "text": "4.5 QUALITATIVE RESULTS ",
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+        "text": "We also qualitatively evaluate the results of the policy from our best performing learning algorithm, the MGP-DRQN. In Figure 2 we compare the number of times each type of action was actually taken by physicians, and how many times the learned policy selected that action. The MGP-DRQN policy tended to recommend more use of antibiotics and more vasopressors than were actually used by physician, while strangely recommending somewhat less use of IV fluids. ",
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+        "image_caption": [
+            "Figure 2: Comparison of physician actions with the actions that would have been taken by the MGP-DRQN policy, with actions separated according to the 3 types of treatments considered. "
+        ],
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+        "type": "text",
+        "text": "In Figure 3, we show how mortality rates differ on the test set as a function of how different the observed physician action was from what the MGP-DRQN would have recommended. For all 3 types of treatments, there appears to be a local minimum at 0 and we observe a $\\mathrm { v }$ shape, indicating that empirically, mortality tended to be lowest when the clinicians took the same actions that the MGP-DRQN would have. Uncertainty tends to be higher due to smaller sample sizes for situations where there is larger disparity. ",
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+        "img_path": "images/91d661b22376fb336538862913236ef6c2aa8babdbd0353457bb34fedec0b587.jpg",
+        "image_caption": [
+            "Figure 3: Empirical mortality rates as a function of how much the MGP-DRQN policy’s actions differed from the observed physician actions. Minimal mortality is observed for all 3 treatment types at 0, where the physicians and MGP-DRQN agreed. "
+        ],
+        "image_footnote": [],
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+    {
+        "type": "text",
+        "text": "Finally, in Figure 4 we show clinical data from a sample patient case. In the top pane of the figure we show five representative vital signs and lab measurements to illustrate the patient’s clinical status, while the bottom shows both what actions physicians actually took and what actions the model recommended. The patient was admitted to the Emergency Department for altered mental status, and the MGP-DRQN quickly recognizes the need for antibiotics and IV fluids. The patient is admitted to the hospital and around hour 6 the clinical team becomes aware of sepsis. However, antibiotics are not first administered until hour 18, about 16 hours after the model recommended treating with them. After the patient is transferred to the Intensive Care Unit, their white blood cell count continues to rise (a sign of worsening infection) and their blood pressure continues to fall (a sign of worsening shock). By hour 14, the RL model starts and continues to recommend use of vasopressors to attempt to increase blood pressure, but they are not actually administered for about another 16 hours at hour 30. Ultimately, by hour 45 care was withdrawn and the patient passed away at hour 50. Cases such as this one illustrate the potential benefits of using our learned treatment policy in a decision support tool to recommend treatments to providers. If such a tool were used in this situation, it is possible that earlier treatments and more aggressive interventions might have resulted in a different outcome. ",
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+        "image_caption": [
+            "Figure 4: Top: clinical data from a patient who acquired sepsis, decompensated in the Intensive Care Unit while progressing to septic shock, and ultimately did not survive. Bottom: shaded symbols denote treatments that the learned MGP-DRQN policy would have recommended, while open symbols denote the treatment actions actually taken by physicians caring for this patient. "
+        ],
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+    {
+        "type": "text",
+        "text": "5 CONCLUSION ",
+        "text_level": 1,
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+        "text": "In this paper we presented a new framework combining multi-output Gaussian processes and deep reinforcement learning for clinical problems, and found that our approach performed well in estimating optimal treatment strategies for septic patients. The use of recurrent structure in the Q-network architecture yielded higher expected returns than a standard Q-network, accounting for the nonMarkovian nature of real-world medical data. The multi-output Gaussian process also improved performance by offering a more principled method for interpolation and imputation, and use of the full MGP posterior improved upon the results from just using the posterior mean. ",
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+        "text": "In the future, we could include treatment recommendations from our learned policies into our dashboard application we have developed for early detection of sepsis. The treatment recommendations might help providers better care for septic patients after sepsis has been properly identified, and start treatments faster. There are many potential avenues for future work. One promising direction is to investigate the use of more complex reward functions, rather than the sparse rewards used in this work. More sophisticated rewards might take into account clinical targets for maintaining hemodynamic stability, and penalize an overzealous model that recommends too many unnecessary actions. Our modeling framework is fairly generalizable, and can easily be applied to other medical applications where there is a need for data-driven decision support tools. In future work we plan to use similar methods to learn optimal treatment strategies for treating patients with cardiogenic shock, and to learn effective insulin dosing regimes for patients on high-dose steroids. ",
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+        "type": "text",
+        "text": "REFERENCES   \nD. C. Angus, A. E. Barnato, D. Bell, and et al. A systematic review and meta-analysis of early goal-directed therapy for septic shock: the arise, process and promise investigators. Intensive Care Medicine, 41(9): 1549–1560, 2015.   \nInvestigators ARISE and Anzics Clinical Trials Group. Goal-directed resuscitation for patients with early septic shock. N Engl J Med, 371(16):1496–1506, 2014.   \nR. C. Bone, R. A. Balk, and F. B. et al. Cerra. Definitions for sepsis and organ failure and guidelines for the use of innovative therapies in sepsis. Chest, 101(6):1644–55, 1992.   \nB. Chakraborty and E. E. M. Moodie. Statistical Methods for Dynamic Treatment Regimes: Reinforcement Learning, Causal Inference and Personalized Medicine. Springer, 2013.   \nL. Cheng, G. Darnell, C. Chivers, M. E. Draugelis, K. Li, and B. E. Engelhardt. Sparse multi-output Gaussian processes for medical time series prediction. arXiv preprint arXiv:1703.09112, pp. 1–36, March 2017. URL https://arxiv.org/abs/1703.09112.   \nI. Cortes-Puch and C. $\\mathrm { \\Delta } \\mathrm { S } _ { \\mathrm { \\Delta } }$ Hartog. Change is not necessarily progress: Revision of the sepsis definition should be based on new scientific insights. Am J Respir Crit Care Med, 194(1):16–18, 2016.   \nR. P. Dellinger, M. M. Levy, Surviving Sepsis Campaign Guidelines Committee, and et al. Surviving sepsis campaign: International guidelines for management of severe sepsis and septic shock. Intensive Care Medicine, 39(2):165–228, 2013.   \nR. Durichen, M. A. F. Pimentel, and L. et al. Clifton. Multitask gaussian processes for multivariate physiological time-series analysis. IEEE Transactions on Biomedical Engineering, 61(1), 2015.   \nL. Epstein, R. Dantes, S. Magill, and A. Fiore. Varying estimates of sepsis mortality using death certificates and administrative codes united states, 19992014. MMWR Morb Mortal Wkly Rep., 65(13):342–345, 2016.   \nA. Esteva, B. Kuprel, and R. A. Novoa. Dermatologist-level classification of skin cancer with deep neural networks. Nature, 542:115–118, 2017.   \nJ. Futoma, S. Hariharan, and K. Heller. Learning to detect sepsis with a multitask gaussian process rnn classifier. ICML, 2017a.   \nJ. Futoma, S. Hariharan, M. Sendak, and et al. An improved multi-output gaussian process rnn with real-time validation for early sepsis detection. MLHC, 2017b.   \nM. Ghassemi, M. A. F. Pimentel, and T. et al. Naumann. A multivariate timeseries modeling approach to severity of illness assessment and forecasting in icu with sparse, heterogeneous clinical data. AAAI, 2015.   \nM. Hausknecht and P. Stone. Deep recurrent q-learning for partially observable mdps. AAAI, 2015.   \nK. E. Henry, D. N. Hager, P. J. Pronovost, and S. Saria. A targeted real-time early warning score (trewscore) for septic shock. Science Translational Medicine, 7(299), 2015.   \nS. Hochreiter and J. Schmidhuber. Long short-term memory. Neural Computation, 9(8):1735–80, 1997.   \nN. Jiang and L. Li. Doubly robust off-policy value evaluation for reinforcement learning. ICML, 2016.   \nA. C. Kalil, D. W. Johnson, S. J. Lisco, and J. Sun. Early goal-directed therapy for sepsis: A novel solution for discordant survival outcomes in clinical trials. Critical Care Medicine, 45(4):607–614, 2017.   \nD. P. Kingma and J. Ba. Adam: A method for stochastic optimization. ICLR, 2015.   \nD. P. Kingma and M. Welling. Auto-encoding variational bayes. ICLR, 2014.   \nM. Komorowski, A. Gordon, L. A. Celi, and A. Faisal. A markov decision process to suggest optimal treatment of severe infections in intensive care. NIPS Workshop on Machine Learning for Health, 2016.   \nA. Kumar, D. Roberts, K. E. Wood, and et al. Duration of hypotension before initiation of effective antimicrobial therapy is the critical determinant of survival in human septic shock. Crit Care Med., 34(6):1589–96, 2006.   \nR. J. Lewis. Disassembling goal-directed therapy for sepsis: a first step. JAMA, 303(8):777–779, 2010.   \nZ. C. Lipton, D. C. Kale, C. Elkan, and R. Wetzel. Learning to diagnose with lstm recurrent neural networks. ICLR, 2016.   \nV. Mnih, K. Kavukcuoglu, D. Silver, and et al. Human-level control through deep reinforcement learning. Nature, 518(7540):529–533, 2015.   \nS. Nemati, M. M. Ghassemi, and G. D. Clifford. Optimal medication dosing from suboptimal clinical examples: A deep reinforcement learning approach. International Conference of IEEE-EMBC, 2016.   \nN. Prasad, L. F. Cheng, C. Chivers, and et al. A reinforcement learning approach to weaning of mechanical ventilation in intensive care units. UAI, 2017.   \nInvestigators PRISM. Early, goal-directed therapy for septic shock a patient-level meta-analysis. N Engl J Med, 376:2223–2234, 2017.   \nInvestigators ProCESS, D. M. Yealy, J. A. Kellum, and et al. A randomized trial of protocol-based care for early septic shock. N Engl J Med, 370(18):1683–1693, 2014.   \nA. Raghu, M. Komorowski, and L. A. et al. Celi. Continuous state-space models for optimal sepsis treatment - a deep reinforcement learning approach. MLHC, 2017.   \nA. Rhodes, L. E. Evans, W. Alhazzani, and et al. Surviving sepsis campaign: International guidelines for management of sepsis and septic shock: 2016. Intensive Care Medicine, 43(3):304–377, 2017.   \nE. Rivers, B. Nguyen, S. Havstad, and et al. Early goal-directed therapy in the treatment of severe sepsis and septic shock. N Engl J Med., 345(19):1368–1377, 2001.   \nC. W. Seymour, V. X. Liu, T. J. Iwashyna, and et al. Assessment of clinical criteria for sepsis: For the third international consensus definitions for sepsis and septic shock (sepsis-3). JAMA, 315(8):762, 2016.   \nC. W. Seymour, F. Gesten, and H. C. et al. Prescott. Time to treatment and mortality during mandated emergency care for sepsis. New England Journal of Medicine, 2017.   \nS. M. Shortreed, E. Laber, and D. J. et al. Lizotte. Informing sequential decision-making through reinforcement learning: an empirical study. Machine Learning, 2010.   \nH. Soleimani, J. Hensman, and S. Saria. Scalable joint models for reliable uncertainty-aware event prediction. IEEE Transactions on Pattern Analysis and Machine Intelligence, 2017.   \nRichard S. Sutton and Andrew G. Barto. Introduction to Reinforcement Learning. MIT Press, Cambridge, MA, USA, 1st edition, 1998.   \nC. M. Torio and B. J. Moore. National inpatient hospital costs: The most expensive conditions by pay, 2013. Healthcare Cost and Utilization Project (HCUP) Statistical Briefs, 204, 2016.   \nH. van Hasselt, A. Guez, and D. Silver. Deep reinforcement learning with double q-learning. AAAI, 2016.   \nZ. Wang, T. Schaul, M. Hessel, and et al. Dueling network architectures for deep reinforcement learning. ICML, 2016.   \nC. J. C. H. Watkins and P. Dayan. Q-learning. Machine Learning, 8(3-4):279–292, 1992. ",
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parse/train/ak06J5jNR4/ak06J5jNR4.md

@@ -0,0 +1,236 @@
+# Revisiting Model Stitching to Compare Neural Representations
+
+Yamini Bansal Harvard University ybansal@g.harvard.edu
+
+Preetum Nakkiran Harvard University preetum@cs.harvard.edu
+
+Boaz Barak Harvard University b@boazbarak.org
+
+# Abstract
+
+We revisit and extend model stitching (Lenc & Vedaldi 2015) as a methodology to study the internal representations of neural networks. Given two trained and frozen models $A$ and $B$ , we consider a “stitched model” formed by connecting the bottom-layers of $A$ to the top-layers of $B$ , with a simple trainable layer between them. We argue that model stitching is a powerful and perhaps under-appreciated tool, which reveals aspects of representations that measures such as centered kernel alignment (CKA) cannot. Through extensive experiments, we use model stitching to obtain quantitative verifications for intuitive statements such as “good networks learn similar representations”, by demonstrating that good networks of the same architecture, but trained in very different ways (e.g.: supervised vs. self-supervised learning), can be stitched to each other without drop in performance. We also give evidence for the intuition that “more is better” by showing that representations learnt with (1) more data, (2) bigger width, or (3) more training time can be “plugged in” to weaker models to improve performance. Finally, our experiments reveal a new structural property of SGD which we call “stitching connectivity”, akin to mode-connectivity: typical minima reached by SGD can all be stitched to each other with minimal change in accuracy.
+
+# 1 Introduction
+
+The success of deep neural networks can, arguably, be attributed to the intermediate features or representations learnt by them [Rumelhart et al., 1985]. While neural networks are trained in an end-to-end fashion with no explicit constraints on their intermediate representations, there is a body of evidence that suggests that they learn rich a representation of the data along the way [Goh et al., 2021, Olah et al., 2017]. However, theoretically we understand very little about how to formally characterize these representations, let alone why representation learning occurs. For instance, there are various ad-hoc pretraining methods [Chen et al., 2020a,b], that are purported to perform well by learning good representations, but it is unclear which aspects of these methods (objective, training algorithm, architecture) are crucial for representation learning and if these methods learn qualitatively different representations at all.
+
+Moreover, we have an incomplete understanding of the relations between different representations. Are all “good representations” essentially the same, or is each representation “good” in its own unique way? That is, even when we train good end-to-end models (i.e., small test loss), the internals of these models could potentially be very different from one another. A priori, the training process could evolve in either one of the following extreme scenarios (see Figure 1):
+
+(1) In the “snowflakes” scenario, training with different initialization, architectures, and objectives (e.g., supervised vs self-supervised) will result in networks with very different internals, which are completely incompatible with one another. For example, even if we train two models with identical data, architecture, and task, but starting from two different initializations, we may end up at local minima with very different properties (e.g., Liu et al. [2020]). If models are trained with different data (e.g., different samples), different architecture (e.g., different width), or different task (e.g., self-supervised vs supervised) then they could end up being even more different from one another.
+
+(2) In the “Anna Karenina” scenario1, all successful models end up learning roughly the same internal representations. For example, all models for vision tasks will have internal representation corresponding to curve detectors, and models that are better (for example, trained on more data, are bigger, or trained for more time) will have better curve detectors.
+
+![](images/aa8480a5baabe7d33703c381c6c758cd48d3364589842a1511fd6fa6dc46fa27.jpg)  
+Figure 1: Two extreme “cartoons” for training dynamics of neural networks. In the “snowflakes” scenario, there are exponentially many well-performing neural networks with highly diverging internals. In the “Anna Karenina” scenario all well-performing networks end up learning similar representations, even if their initialization, architecture, data, and objectives differ. Image credits: Li et al. [2018], Olah et al. [2017, 2020], Komarechka [2021].
+
+The “Anna Karenina” scenario implies the following predictions:
+
+“All roads lead to Rome:” Successful models learned with different initializations, architectures, and tasks, should learn similar internal representations, and so if $A$ and $B$ are two such models, it should be possible to “plug in” the internals from $A$ into $B$ without a significant loss in performance. See Figure 2A-B.
+
+“More is better:” Better models trained using more data, bigger size, or more compute, should learn better versions of the same internal representations. Hence if $A$ is a more successful model than $B$ , it should be possible to “plug in” $A$ ’s internals to $B$ and obtain improved performance. See Figure 2C.
+
+In this work, we revisit the empirical methodology of “model stitching” to test the above predictions. Initially proposed by Lenc and Vedaldi [2015], model stitching is natural way of “plugging in” the bottom layers of one network into the top layers of another network, thus forming a stitched network (however care must be taken in the way it is performed, see Section 2). We show that model stitching has some unique advantages that make it more suitable for studying representations than representational similarity measures such as CKA [Kornblith et al., 2019] and SVCCA [Raghu et al., 2017]. Our work provides quantitative evidence for the intuition, shared by many practitioners, that the internals of neural networks often end up being very similar in a certain sense, even when they are trained under different settings.
+
+# 1.1 Summary of Results
+
+Model stitching as an experimental tool. We establish model stitching as a way of studying the representations of neural networks. A version of model stitching was proposed in Lenc and Vedaldi [2015] to study the equivalence between representations. In this work, we argue that the idea behind model stitching is more powerful than has been appreciated: we analyze the benefits of modelstitching over other methods to study representations, and we then use model-stitching to establish an number of intuitive properties, including new results on the properties of SGD.
+
+![](images/6eda4b77ee50a5fd6733742618f6b6447635351e8858d8cc8c6f329fd8a8717f.jpg)  
+Figure 2: Summary of main results (A) Various models trained on CIFAR-10 identically except with different random initializations are “stitching connected”: can be stitched at all layers with minimal performance drop (see Section 4). Stitching with a random bottom network shown for reference. (B) Models of the same architecture and similar test error, but trained on ImageNet with end-to-end supervised learning versus self-supervised learning can be stitched with good performance (see Section 5). (C) Better representation obtained by training the network with more samples can be "plugged-in" with stitching to improve performance (see Section 6). In all figures, stitching penalty is the difference in error between the stitched model and the base top model.
+
+In this paper, we use model stitching in the following way. Suppose we have a neural network $A$ (which we’ll think of as the “top model”) for some task with loss function $\mathcal { L }$ (e.g. the CIFAR-10 or ImageNet test error). Let $r : \mathcal { X } \overset { } { \to } \mathbb { R } ^ { d }$ be a candidate “representation” function, which can come from the first (bottom-most) layers of some “bottom model” $B$ . Our intuition is that $r$ has better quality than the first \` layers of network $A$ if “swapping out” these layers with $r$ will improve performance.
+
+“Swapping out” is performed by introducing an additional trainable stitching layer to $r$ with $A$ (defined more formally in Section 2). The stitching layers have very low capacity, and are only meant to “align” representations, rather than improving the model.
+
+Comparison to Representational Similarity Metrics. Much of the current work studying representations focuses on similarity metrics such as CKA. However, we argue that model stitching can be a better suited tool to study representations in various scenarios, and can give qualitatively different conclusions about the behavior of neural representations compared to these metrics. We analyze the differences between model stitching and prior similarity metrics in Section 3.
+
+Quantitative evidence for intuitions. Using model stitching, we are able to provide formal and quantitative evidence to the intuitions mentioned above. In particular we give evidence for the “all roads lead to rome” intuition by showing compatibility of networks with representation that are trained using (1) different initializations, (2) different subsets of the dataset, (3) different tasks (e.g., self-supervised or coarse labels). See Figure 2A-B for results. We also show that network with different random initialization enjoy a property which we call stitching connectivity, wherein almost all minima reachable via SGD can be “stitched” to each other with minimal loss of accuracy. We also give evidence for the “more is better” intuition by showing that we can improve the performance of a network $A$ by plugging in a representation $r$ that was trained with (1) more data, (2) larger width, or (3) more training epochs. See example with more samples in Figure 2C.
+
+The results above are not surprising, in the sense that they confirm intuitions that practitioners might already have. However model stitching allows us to obtain quantitative and formal measures of these in a way that is not achievable by prior representation measures.
+
+Organization. We first define model stitching formally in Section 2. We compare it with prior work on representational similarity measures in Section 3. Then, we formally define stitching connectivity and provide experimental evidence for it in Section 4. Finally, in Section 5 and 6, we provide quantitative evidence for the "all roads lead to Rome" and "more is better" intuitions respectively.
+
+Related works. As mentioned above, stitching was introduced by Lenc and Vedaldi [2015] who studied equivalence of representations. They showed that certain early layers in a network trained on the Places dataset [Zhou et al., 2014] are compatible with AlexNet (see Table 4 in Lenc and Vedaldi [2015]). After completing this work, we were made aware of concurrent work Csiszárik et al. [2021] that also proposes model stitching to compare neural representations. The results from this work complement ours by studying the effect of changing the stitching layer in various ways (for instance, imposing a sparsity penalty on the stitching layer), and further clarifying the relationship of stitching with other representational similarity measures. In contrast, our work demonstrates stitching compatibility under different scenarios, like the comparison between supervised and self-supervised methods and experiments that show "more is better".
+
+Our work on stitching connectivity is related to the work on mode connectivity [Garipov et al., 2018, Freeman and Bruna, 2017, Draxler et al., 2018]. These works show that the local minima found by SGD are often connected through low-loss paths. These paths are generally non-linear, though it was shown that there are linear paths between these minima if they are identically initialized but then use different SGD noise (order of samples) after a certain point in training [Frankle et al., 2020]. Stitching connectivity is complementary to mode connectivity. Stitching connectivity corresponds to a discrete path (with as many steps as layers) but one where the intermediate steps are interpretable. We also show stitching connectivity of networks that are trained on different tasks. Finally, most of the prior work on concrete metrics for relating representations was in the context of representation similarity measures. We describe this work and compare it to ours in Section 3.
+
+# 2 Model Stitching
+
+Let $A$ be a neural network of some architecture $\mathcal { A }$ , and let $r : \mathcal { X } \to \mathbb { R } ^ { d }$ be a candidate “representation” function. We consider a family $s$ of stitching layers which are simple (e.g. linear $1 \times 1$ convolutional layers for a convolutional network $A$ ) functions mapping $\mathbb { R } ^ { d }$ to $\mathbb { R } ^ { \hat { d } _ { \ell } }$ where $d _ { \ell }$ is the width of $A$ ’s $\ell$ -th layer. Given some loss function $\mathcal { L }$ (e.g., CIFAR-10 or ImageNet test accuracy) we define
+
+$$
+\mathcal { L } _ { \ell } ( r ; A ) = \operatorname* { i n f } _ { s \in \mathcal { S } } \mathcal { L } ( A _ { > \ell } \circ s \circ r )
+$$
+
+where $A _ { > \ell }$ denotes the function mapping the activations of $A$ ’s \`th layer to the final output, and $\circ$ denotes function composition. That is, $\mathcal { L } _ { \ell } ( r ; A )$ is the smallest loss obtained by stitching $r$ into all but the first $\ell$ layers of $A$ using a stitching layer from $s$ . We define the stitching penalty of the representation $r$ with respect to $A$ (as well as $\ell$ and the loss $\mathcal { L }$ ) as $\mathcal { L } _ { \ell } ( r ; A ) - \mathcal { L } ( A )$ . If the penalty is non-negative, then we say that the representation $r$ is at least as good as the first $\ell$ layers of $A$ .
+
+Consider the simple case of a fully-connected network and a stitching family of linear functions. In this case, model stitching tells us if there is a way to linearly transform the representation $r$ into that of the first $\ell$ layers of $A$ , but only in the subspace that is relevant to the achieving low loss. In practice, we approximate the infimum in Equation 1 by using gradient methods— concretely, by optimizing the stitching-layer using the train set of the task $\mathcal { L }$ . The stitching penality itself is then estimated on the test set.
+
+Choosing the stitching family $s$ : The stitching family can be chosen flexibly depending on the desired invariances between representations, as long as it is simple (see below). For instance, if we choose $S$ to be all permutations or orthogonal matrices, the stitching penalty will be invariant up to permutations or orthogonal transformations respectively. In this work, we mainly consider cases where $r = B _ { \leq l }$ for $B$ with architecture similar to $\mathcal { A }$ . We restrict the stitching family per architecture $\mathcal { A }$ such that the composed model $A _ { > \ell } \circ s \circ r$ lies in $\mathcal { A }$ . This way the stitched model consists of layers identical to the layers of either $A$ or $B$ , with the exception of just one layer. For example, for convolutional networks we consider a $1 \times 1$ convolution and for transformers we consider a token-wise linear function between transformer blocks. We perform an ablation with kernels of different sizes in Appendix B.1.
+
+Stitching is not learning: One concern that may arise is that if the stitching family $s$ is sufficiently powerful, it can learn to transform any representation into any other representation. This would defeat the aim of faithfully studying the original representations. To avoid this, the family of stitchers should be chosen to be simple (e.g., linear). Nevertheless, to verify that our experiments are not in such a regime, we stitch an untrained, randomly initialized network with a fully trained network. Figure 2A-B shows that the penalty of such a stitched network is high, specially for high $l$ . CKA has the same trend for the same networks (See Appendix B.2)
+
+Model stitching as a “happy middle”. Model stitching can be considered as a compromise between the following extremes: (1) Direct plugging in: the most naive interpretation of “plugging in” $r$ into $A$ would be to use no stitching at all. However, even in cases where networks are identical up to a permutation of neurons, plugging in $r$ into $A$ would not work. (2) Full fine tuning: the other extreme interpretation is to perform full fine tuning. That is, start with the initialized network $A { \mathord { > } } \ell \circ r$ and optimize over all choices of $A$ . The problem with this approach is that there are so many degrees of freedom in the choices for $A _ { > \ell }$ that the resulting network could achieve strong performance regardless of the quality of $r$ . For example, in B.3 we show that fine tuning can fail to distinguish between a trained network and a random network. (3) Linear probe: a popular way to define quality of representations is to use linear probes [Alain and Bengio, 2016]. However, linear probes are not as well suited for studying the representation of early layers, which have low linear separability. In particular, the linear probe accuracy of an early layer of network $A$ would generally always be much worse than a later layer of network $B$ , even if $A$ was of “higher quality” (e.g., trained with more data). Linear probes also don’t have the operational interpretation of compatibility.
+
+By choosing a trainable but low-capacity layer, model stitching “threads the needle” between simply plugging in, and full fine tuning, and unlike linear probes, enables the study of early layers using powerful nonlinear decoders.
+
+Experimental setup. Unless specified otherwise, the CIFAR-10 experiments are conducted on the ResNet-18 architecture (with first layer width 64) and the ImageNet experiments are conducted on the ResNet-50 architecture [He et al., 2015]. The ResNets are trained with the standard hyperparameters (See Appendix A.1 for training parameters of all base models). The stitching layer in the the convolutional networks consist of a $1 \times 1$ convolutional layer with input features equal to the number of channels in $r$ , and output features equal to the output channels of $A _ { \leq l } ( x )$ . We add a BatchNorm (BN) layer before and after this convolutional layer. Note that the BN layer does not change the representation capacity of the stitching layer and only aides with optimization. We perform stitching only between ResNet blocks (and not inside a block), but note that it is possible to stitch within the block as well. We use the Adam optimizer with the cosine learning rate decay and an initial learning rate of 0.001. Full experimental details for each experiment are described in Appendix A.
+
+# 3 Stitching vs. representational similarity
+
+Much prior work studying representations focused on representation similarity measures. These have been studied in both the neuroscience and machine learning communities [Kriegeskorte et al., 2008, Kornblith et al., 2019]. Examples of such measures include canonical correlation analysis $( C C A )$ [Hardoon et al., 2004] and its singular-vector and projection-weighted variants such as SVCCA and PWCCA [Raghu et al., 2017, Morcos et al., 2018]. Recently Kornblith, Norouzi, Lee, and Hinton [2019] proposed centered kernel alignment $( C K A )$ that addressed several issues with CCA. CKA was further explored by Nguyen et al. [2021].
+
+For two representations functions $\phi : \mathcal { X }  \mathbb { R } ^ { d _ { 1 } }$ and $\sigma : \mathcal { X }  \mathbb { R } ^ { d _ { 2 } }$ , the linear CKA is defined $\begin{array} { r } { \mathbf { C K A } ( \phi , \sigma ) : = \frac { | | \mathbf { C o v } ( \phi ( x ) , \sigma ( x ) ) | | _ { F } ^ { 2 } } { | | \mathbf { C o v } ( \phi ( x ) ) | | _ { F } \cdot | | \mathbf { C o v } ( \sigma ( x ) ) | | _ { F } } } \end{array}$ where all covariances are with respect to the test distribution on inputs $x \sim \mathcal { D }$ [Kornblith et al., 2019].
+
+Table 1: Qualitative results of our experiments, comparing CKA to stitching. $\mathbf { C K A } \approx 1$ means representations are close according to CKA. Error $\approx 0 \%$ means representations are close according to stitching. “Varies” means no consistent conclusion across different architectures and tasks.   
+
+<table><tr><td>Method</td><td>Stitching Connectivity Different Initialization</td><td>“All roads” Self-Supervision</td><td>“More is better” More Data /Time /Width</td></tr><tr><td>CKA Stitching</td><td>Varies (can be O) Close (up to 3% error)</td><td>Varies (0.35- 0.9) Close (up to 5% error)</td><td>Far (can be O for data, O.7 for width) Better</td></tr></table>
+
+Table 1 contains a qualitative summary of our results, comparing stitching with CKA. While in some cases the results of stitching and CKA agree, in several cases, stitching obtains results that align more closely with the intuitions that well-performing networks learn similar representations, and that more resources results in better versions of the same representations. In particular, in experiments where stitching indicates that a certain representation is better than another, CKA by its design can only indicate that the two representations are far from each other. Moreover, in some experiments CKA indicates that representations are far where we intuitively believe that they should be close, e.g. when networks only differ by two random initializations, or when one is trained with a supervised and another with a self-supervised task.2 In contrast, in these settings, model-stitching reveals that the two models have nearly equivalent representations, in the sense that they can be stitched to each other with low penalty. The precise experimental results appear in Sections 4-6 and Appendix B.2.
+
+Compared to representation-similarity measures, model stitching has several advantages:
+
+Ignoring spurious features: Measures such as CCA and CKA ultimately boil down to distances between the feature vectors, but they do not distinguish between features that are learned and relevant for downstream tasks, and spurious features, that may be completely useless or even random. For example, suppose we augment a representation $\phi : \dot { \mathcal { X } }  \dot { \mathbb { R } ^ { d } }$ by concatenating 1000 “useless” coordinates, with random gaussian features, to form a new representation $\phi ^ { \prime } : \mathcal { X } \overset { \mathbf { \bar { \Delta } } } { \to } \mathbb { R } ^ { d + 1 0 0 0 }$ . This would reduce the CKA, but representation $\phi ^ { \prime }$ is not different from $\phi$ in a meaningful way. Modelstitching resolves this, since we can stitch $\phi ^ { \prime }$ in place of $\phi$ by simply throwing away the useless coordinates. In general, model-stitching focuses only on aspects of the representation which are relevant for the downstream task, as opposed to aspects which are spurious or irrelevant. The price we pay for this is that, unlike measures such as CKA, model stitching depends on the downstream task. However, our results indicate that neural networks tend to learn similar representations for a variety of natural tasks.
+
+Asymmetry: Our intuition is that some representations are better than others. For example, we believe that with more data, neural networks learn better representations. However, by design, such comparisons cannot be demonstrated by representation similarity measures that only measure the distance between two representations. In contrast, we are able to demonstrate that “more is better” using stitching-based measures. Concretely, CKA and other similarity measures are symmetric, while stitching is not: it may be the case that a representation $\phi$ can be stitched in place of a representation $\sigma$ , but not vice-versa. Also, stitching a representation $\phi$ can (and sometimes does) improve performance.
+
+Interpretable units: Measures such as CCA/CKA give a number between 0 and 1 for the distance between representations, but it is hard to interpret what is the difference, for example, between a CKA value of 0.9 and value of 0.8. In contrast, if the loss function $\mathcal { L }$ has meaningful units, then the stitching penalty inherits those, and (for example) a representation $r$ having a penalty of $3 \%$ in CIFAR-10 accuracy has an operational meaning: if you replace the first layers of the network with $r$ the decrease in accuracy is at most $3 \%$ .
+
+Invariance: A representation similarity measure should be invariant to operations that do not modify the “quality” of the representation, but it is not always clear what these operations are. For example, CCA and CKA are invariant under orthogonal linear transformations, and some variants of CCA are also invariant under general invertible linear transformations (see Table 1 in Kornblith et al. [2019]). However, it is unclear if these are the natural families. For example, randomly permuting the order of pixels (either at the input or in the latents) is an orthogonal transform, and thus does not affect the CKA. However, this completely destroys the spatial structure of the input, and intuitively should affect the “representation quality.” On the other hand, certain non-orthogonal transforms may still preserve representation quality– for example, the non-invertible transformation of projecting out spurious coordinates. Thus, orthogonal transforms may not be the right invariance class to consider representations. Using stitching we can explicitly ensure invariance under any given family of transformations by adding it to the stitching layer. Concretely, our choice of using a $1 \times 1$ convolutional stitcher yields much weaker invariance than general orthogonal transforms– and thus, model stitching can predict that shuffling pixels leads to a “worse” representation.
+
+# 4 Stitching Connectivity
+
+We first focus on a special case of model-stitching, wherein we stitch two identically distributed networks to each other. That is, we train two networks of same architecture, on the same data distribution, but with independent random seeds and independent train samples. This question has been studied before with varied conclusions [Li et al., 2016, Wang et al., 2018]. We find that empirically, two such networks can be stitched to each other at all layers, with close to 0 penalty. This is a new empirical property of SGD trained networks, which we term “stitching connectivity.”
+
+Formally, let $A , B$ be two trained networks of identical architectures with $L$ layers, for a task with objective function $\mathcal { L }$ . For all $i \in \{ 0 , 1 , . . . , L \}$ , define $S _ { i }$ to be the stitched model where we replace the first $i$ layers of $A$ with those same layers in $B$ (and optimize the stitching layer as usual). That is, $\begin{array} { r } { S _ { i } : = \arg \operatorname* { m i n } _ { S = A _ { > i } \circ s \circ B _ { \leq i } } { \mathcal { L } } ( S ) } \end{array}$ . Observe that $S _ { 0 } = A$ and $S _ { L } = B$ , so the sequence of models $\{ S _ { 0 } , S _ { 1 } , \ldots , S _ { L } \}$ gives a kind of “path” between models $A$ and $B$ . Further, due to our family of stitching layer and network architecture $1 \times 1$ conv stitchers and conv-nets), the stitching layer $s$ can be folded into the adjacent model. Thus, all models $S _ { i }$ have identical architecture as $A$ and $B$ . We say $A$ and $B$ are “stitching-connected” if all the intermediate models $S _ { i }$ have low test loss.
+
+Definition 1 (Stitching Connectivity). Let $A , B$ be two networks of identical architectures. We say $A$ and $B$ are stitching-connected if they can be stitched to each other at all layers, with low penalty. That is, if all stitched models $S _ { i }$ , defined as above, have test loss comparable to $A$ .
+
+Stitching connectivity is not a trivial property: two networks with identical architectures, but very different internal representations, would fail to be stitching connected. Our main claim is that for a fixed data distribution, almost all minima reached by SGD are stitching-connected to each other.
+
+Conjecture 2 (Stitching Connectivity of SGD, informal). Let $A _ { 1 } , A _ { 2 }$ be two independent and identically-trained networks. That is, networks of the same architecture, trained by SGD with independent random seeds and independent train sets from the same distribution. Then, for natural architectures and data-distributions, the trained models $A _ { 1 }$ and $A _ { 2 }$ are stitching-connected.
+
+This conjecture states a structural property of models that are likely to be output by SGD. If we run the identical training procedure twice, we will almost certainly not produce models with identical parameters. However, these two different parameter settings yield essentially equivalent internal representations– this is what it means to be stitching-connected.
+
+Discussion. Stitching connectivity is similar in spirit to mode connectivity [Garipov et al., 2018, Freeman and Bruna, 2017, Draxler et al., 2018], in that they are both structural properties of the set of typical SGD minimas. Mode connectivity states that typical minima are connected by a low-test-loss path in parameter space. Stitching connectivity does not technically define a path in parameter space– rather, it defines a sequence $S _ { 0 } , S _ { 1 } , \ldots , S _ { L }$ of low-loss models connecting two endpoint models. For each model in this sequence, all but one layer is identical to one of the endpoint models. Thus, we can informally think of the stitching sequence as a different way of “interpolating” between two models.
+
+The stitching connectivity of SGD is especially interesting for overparameterized models, since it sheds light on the implicit bias of SGD. In this case, there are exponentially-many global minima of the train loss, even modulo permutation-symmetry. Apriori, it could be the case that each of these minima compute the classification decision in different ways (e.g. by memorizing a particular train set). However, empirically we find that SGD is “biased” towards minima with essentially the same internal representations– in that typical minima can be stitched to each other.
+
+Experiments. Figure 2A demonstrates the stitching-connectivity of SGD via the following experiment. We train a variety of model architectures on CIFAR-10, with two different randomly initialized models per architecture. We consider a ResNet-18, two variants of ResNet-18 with $0 . 5 \times$ and $2 \times$ width, a significantly deeper ResNet-164, a feed-forward convolutional network Myrtle-CNN [Page, 2018] and a Vision Transformer [Dosovitskiy et al., 2020] pretrained on CIFAR- $\cdot 5 \mathrm { m }$ [Nakkiran et al., 2021]. We stitch the two randomly initialized models at various intermediate layers, forming the stitching sequence $S _ { 0 } , \ldots S _ { L }$ . Figure 2A plots the test errors of these intermediate stitched models $S _ { i }$ with the first and last point showing the errors of the base models. For all layers $i$ , the test error of the stitched model $S _ { i }$ is close to the error of the base models. A similar result holds for networks trained on disjoint train sets. Full experimental details are provided in Appendix A.
+
+# 5 All Roads Lead to Rome
+
+In this section, we quantify the intuition that “all roads lead to Rome” in representation learning: many diverse choices of train method, label quality, and objective function all lead to similar representations in early layers. However, we find that such training details can affect the representation at later layers.
+
+Comparing self-supervised and supervised methods. If we train models with the same architecture and train set, but alter the training method significantly, what do we expect from the representations? To explore this, we compare the representations of two very different training methods — standard end-to-end supervised training (E2E) versus self-supervised $^ +$ simple classifiers (SSS), that first learn a representation from unlabeled data and then train a simple linear classifier on this representation.3
+
+SSS algorithms like SimCLR [Chen et al., 2020b], SwAV [Caron et al., 2021a] and DINO [Caron et al., 2021b] etc have recently emerged as prominent paradigm for training neural networks that achieve comparable accuracy to E2E networks. We stitch the representations of these SSS algorithms to an E2E supervised network (all with a ResNet-50 backbone) trained on ImageNet. While all of these methods achieve similar test accuracy on a ResNet-50 backbone of $7 5 \% \pm 1$ (except SimCLR at $6 8 . 8 \%$ ) [Goyal et al., 2021], they are trained very differently. Figure 2B shows that the SSS trained networks are stitching connected at all layers to the E2E network. This suggests that while the advances in SSL have been significant for learning features without labels, the features themselves are similar in both cases. This is in agreement with prior work which shows that SSS and E2E networks have similar texture-shape bias and make similar errors [Geirhos et al., 2020]. Since certain SSS algorithms have been proven to have small generalization gap [Bansal et al., 2021], the similarity of representations between SSS and E2E algorithms may yield some clues into the generalization mystery of E2E algorithms. A similar experiment for SimCLR with ResNet-18 trained on CIFAR-10 is shown in Appendix B.2, along with the CKA for the same networks. We find that the CKA varies between $0 . 3 5 - 0 . 9$ for different layers, while stitching gets maximum stitching penalty of $3 \%$ .
+
+Changing the label distribution $\mathbf { p } ( \mathbf { y } \vert \mathbf { x } )$ : How much does the representation quality depend on label quality? To explore this, we compare the representations of networks with the same input distribution $p ( x )$ , but different label distributions $p ( y | x )$ . We take the CIFAR-10 distribution on inputs $p ( x )$ , and consider several “less informative” label distributions: (1) Coarse labels: We super-class CIFAR-10 classes into a binary task, of Objects (Ship, Truck, etc.) vs. Animals (Cat, Dog, etc.) (2) Label noise: We set $p = \{ 0 . 1 , 0 . 5 , 1 . \}$ fraction of labels to random labels. We then stitch these “weak” networks to a standard CIFAR-10 network at varying layers, and measure the stitching penalty incurred in Figure 3A. We find that even with poor label quality, the first half of layers in the “weak” model are “as good as” layers in the standard model (with the exception of the weak network trained on $100 \%$ noisy labels). These experiments align with the results of Nakkiran and Bansal [2020], which show that neural networks can be sensitive to aspects of the input distribution that are not explicitly encoded in the labels. Full experimental details appear in Appendix A.
+
+These results are in agreement with prior work on vision [Olah et al., 2017], which suggests that the first few layers of a neural network learn general purpose features (such as curve detectors) that are likely to be useful a large variety of tasks. Formalizing the set of pre-training tasks for which such similar representations are learnt is an important direction for future work to understand pretraining.
+
+# 6 More is Better
+
+Now, we use stitching to quantify the intuition that “more is better” for representations. That is, larger sample size, model size, or train time lead to progressively better representations of the same type.
+
+Number of samples: When scaled appropriately, neural network performance improves predictably with the number of training samples (e.g. [Kaplan et al., 2020]). But how does this improvement manifest in their representations? We investigate this by stitching the lower parts of models trained $\{ 5 K , 1 0 K , 2 5 K \}$ samples of CIFAR-10 to the upper layers of a model trained with $1 0 K$ samples.
+
+First, as Figure 2C shows, we observe that the models are all stitching compatible— the better representation trained with $2 5 K$ samples can simply be "plugged in" to the model trained with fewer samples, and this improves the performance of the stitched network relative to the $1 0 K$ network. Note that there is no theoretical reason to expect this compatibility, better networks could have learnt fundamentally different features from their weaker counterparts. For example, a network trained on few samples could have learnt only “simple features” (presence of sky in the image / presence of horizontal lines), while one trained on many samples may learn only “complex features” (presence of an blue eye / presence of a furry ear), which are not decodable by the weaker network.
+
+Secondly, we find that some layers are more data hungry than others. When stitched at the first few layers, all the models have similar performance, but the performance degrades rapidly with fewer samples in the mid-layers of the network. This suggests that each layer of a neural network has its own sample complexity. As a corollary of this finding, we predict that we can train some of the layers with few samples, freeze them and train the rest of the model with larger number of samples. Indeed, we find that we can recover most of the accuracy of the network by training the first three, and the last three layers of this model (about half of the layers) with just $5 K$ samples, freezing them and training the rest of the model with all the samples to obtain a network within $2 \%$ accuracy of the original network. See Appendix B.4 for details of the experiment and training plots. This is similar to the "freeze-training" experiments suggested by Raghu et al. [2017].
+
+![](images/7c005a86efaa130cc232bb2c14cff6bbfbdc163b0a8a9b9b512e17ee41195811.jpg)  
+Figure 3: (A) Changing the label distribution: Representations trained on CIFAR-10 with the Object vs. Animals task or with $\{ 1 0 \% , 5 0 \% , 1 0 0 \% \}$ label noise and stitched to a network trained on original CIFAR-10 labels. Early layers learn similar representations even when the label distribution is "less informative" than training with all labels (B) Increasing training time: Representations at different epochs during training are stitching compatible and early layer converge faster (B) Increasing width: Better representations from a wider network can be stitched with a thinner network to improve performance. All experiments were performed with ResNet-18 on CIFAR-10.
+
+Training time: We observe similar results when we stitch networks trained for different number of epochs. We train a ResNet-18 trained on CIFAR-10 and stitch the representations from the model at $\{ 4 0 , 8 0 , 1 6 0 \}$ epochs to the model at the 80-th epoch. As training time increases, Figure 3B shows that the representations improve in a manner that is compatible with the earlier training times. Note that this is a nontrivial statement about neural network training dynamics: It could have been the case that, once a network is trained for very long, its representations move “far away” from its representations near initialization – and thus, stitching would fail. However, we find that the representations at the end of training remain compatible with those in the early stage of training. We also observe that earlier layers converge faster with time, as was shown by Raghu et al. [2017].
+
+Width: Similarly, we train models with the same architecture but varying width multipliers $\{ 0 . 2 5 \times , 1 \times , 2 \times \}$ (Figure 3C). We find that “better” models with higher width models can be stitched to those with lower width and improve performance, but not vice versa. The CKA between representations with $0 . 2 5 \times$ and $2 \times$ width multiplier is in the range $0 . 7 - 0 . 9$ (See Appendix B.2)
+
+Taken together, these results suggest that neural networks obey a certain kind of modularity — better layers can be plugged in without needing to re-train the whole network from scratch.
+
+# 7 Conclusion and Future Work
+
+As our work demonstrates, model stitching can be a very useful tool to compare representations in interpretable units. Model stitching does have its limitations: it requires training a network, making it more expensive in computation than other measures such as CKA. Additionally, stitching representations from two different architectures can be tricky and requires a careful choice of the stitching family. While we restricted ourselves to stitching the first $l$ layers of a network, stitching can be used to also plug in intermediate layers or parts of layers, and in general to “assemble” a new model from a collection of pre-trained components.
+
+There are various avenues for future research. All of our results are in the vision domain, but it would be interesting to compare representations in natural language processing, since language tasks tend to have more variety than vision tasks. It would also be interesting to study the representations of adversarially trained networks to diagnose why they lose performance in comparison to standard training. In general, we hope that model stitching will become a part of the standard diagnostic repertoire of the deep learning community. Societal impacts: This paper makes methodological and foundational contributions that do not have direct impact on society. Model stitching can potentially be used to understand and develop better representation learning mechanisms. While this could indirectly lead to future applications, it is premature to predict their positive or negative impacts.
+
+# 8 Acknowledgements
+
+YB is supported by IBM Global University awards program and NSF Awards IIS 1409097. PN is supported in part by a Google PhD Fellowship, the Simons Investigator Awards of Boaz Barak and Madhu Sudan, and NSF Awards under grants CCF 1565264, CCF 1715187. BB is supported by NSF award CCF 1565264, a Simons Investigator Fellowship and DARPA grant W911NF2010021. We thank MIT-IBM Watson AI Lab and John Cohn for providing access and support for the Satori compute cluster.
+
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+        "text": "We revisit and extend model stitching (Lenc & Vedaldi 2015) as a methodology to study the internal representations of neural networks. Given two trained and frozen models $A$ and $B$ , we consider a “stitched model” formed by connecting the bottom-layers of $A$ to the top-layers of $B$ , with a simple trainable layer between them. We argue that model stitching is a powerful and perhaps under-appreciated tool, which reveals aspects of representations that measures such as centered kernel alignment (CKA) cannot. Through extensive experiments, we use model stitching to obtain quantitative verifications for intuitive statements such as “good networks learn similar representations”, by demonstrating that good networks of the same architecture, but trained in very different ways (e.g.: supervised vs. self-supervised learning), can be stitched to each other without drop in performance. We also give evidence for the intuition that “more is better” by showing that representations learnt with (1) more data, (2) bigger width, or (3) more training time can be “plugged in” to weaker models to improve performance. Finally, our experiments reveal a new structural property of SGD which we call “stitching connectivity”, akin to mode-connectivity: typical minima reached by SGD can all be stitched to each other with minimal change in accuracy. ",
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+        "text": "The success of deep neural networks can, arguably, be attributed to the intermediate features or representations learnt by them [Rumelhart et al., 1985]. While neural networks are trained in an end-to-end fashion with no explicit constraints on their intermediate representations, there is a body of evidence that suggests that they learn rich a representation of the data along the way [Goh et al., 2021, Olah et al., 2017]. However, theoretically we understand very little about how to formally characterize these representations, let alone why representation learning occurs. For instance, there are various ad-hoc pretraining methods [Chen et al., 2020a,b], that are purported to perform well by learning good representations, but it is unclear which aspects of these methods (objective, training algorithm, architecture) are crucial for representation learning and if these methods learn qualitatively different representations at all. ",
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+        "text": "(2) In the “Anna Karenina” scenario1, all successful models end up learning roughly the same internal representations. For example, all models for vision tasks will have internal representation corresponding to curve detectors, and models that are better (for example, trained on more data, are bigger, or trained for more time) will have better curve detectors. ",
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+        "img_path": "images/aa8480a5baabe7d33703c381c6c758cd48d3364589842a1511fd6fa6dc46fa27.jpg",
+        "image_caption": [
+            "Figure 1: Two extreme “cartoons” for training dynamics of neural networks. In the “snowflakes” scenario, there are exponentially many well-performing neural networks with highly diverging internals. In the “Anna Karenina” scenario all well-performing networks end up learning similar representations, even if their initialization, architecture, data, and objectives differ. Image credits: Li et al. [2018], Olah et al. [2017, 2020], Komarechka [2021]. "
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+        "text": "The “Anna Karenina” scenario implies the following predictions: ",
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+        "text": "“All roads lead to Rome:” Successful models learned with different initializations, architectures, and tasks, should learn similar internal representations, and so if $A$ and $B$ are two such models, it should be possible to “plug in” the internals from $A$ into $B$ without a significant loss in performance. See Figure 2A-B. ",
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+        "text": "“More is better:” Better models trained using more data, bigger size, or more compute, should learn better versions of the same internal representations. Hence if $A$ is a more successful model than $B$ , it should be possible to “plug in” $A$ ’s internals to $B$ and obtain improved performance. See Figure 2C. ",
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+        "type": "text",
+        "text": "In this work, we revisit the empirical methodology of “model stitching” to test the above predictions. Initially proposed by Lenc and Vedaldi [2015], model stitching is natural way of “plugging in” the bottom layers of one network into the top layers of another network, thus forming a stitched network (however care must be taken in the way it is performed, see Section 2). We show that model stitching has some unique advantages that make it more suitable for studying representations than representational similarity measures such as CKA [Kornblith et al., 2019] and SVCCA [Raghu et al., 2017]. Our work provides quantitative evidence for the intuition, shared by many practitioners, that the internals of neural networks often end up being very similar in a certain sense, even when they are trained under different settings. ",
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+        "text": "1.1 Summary of Results ",
+        "text_level": 1,
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+        "text": "Model stitching as an experimental tool. We establish model stitching as a way of studying the representations of neural networks. A version of model stitching was proposed in Lenc and Vedaldi [2015] to study the equivalence between representations. In this work, we argue that the idea behind model stitching is more powerful than has been appreciated: we analyze the benefits of modelstitching over other methods to study representations, and we then use model-stitching to establish an number of intuitive properties, including new results on the properties of SGD. ",
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+            "Figure 2: Summary of main results (A) Various models trained on CIFAR-10 identically except with different random initializations are “stitching connected”: can be stitched at all layers with minimal performance drop (see Section 4). Stitching with a random bottom network shown for reference. (B) Models of the same architecture and similar test error, but trained on ImageNet with end-to-end supervised learning versus self-supervised learning can be stitched with good performance (see Section 5). (C) Better representation obtained by training the network with more samples can be \"plugged-in\" with stitching to improve performance (see Section 6). In all figures, stitching penalty is the difference in error between the stitched model and the base top model. "
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+        "text": "In this paper, we use model stitching in the following way. Suppose we have a neural network $A$ (which we’ll think of as the “top model”) for some task with loss function $\\mathcal { L }$ (e.g. the CIFAR-10 or ImageNet test error). Let $r : \\mathcal { X } \\overset { } { \\to } \\mathbb { R } ^ { d }$ be a candidate “representation” function, which can come from the first (bottom-most) layers of some “bottom model” $B$ . Our intuition is that $r$ has better quality than the first \\` layers of network $A$ if “swapping out” these layers with $r$ will improve performance. ",
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+        "text": "“Swapping out” is performed by introducing an additional trainable stitching layer to $r$ with $A$ (defined more formally in Section 2). The stitching layers have very low capacity, and are only meant to “align” representations, rather than improving the model. ",
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+        "type": "text",
+        "text": "Comparison to Representational Similarity Metrics. Much of the current work studying representations focuses on similarity metrics such as CKA. However, we argue that model stitching can be a better suited tool to study representations in various scenarios, and can give qualitatively different conclusions about the behavior of neural representations compared to these metrics. We analyze the differences between model stitching and prior similarity metrics in Section 3. ",
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+        "type": "text",
+        "text": "Quantitative evidence for intuitions. Using model stitching, we are able to provide formal and quantitative evidence to the intuitions mentioned above. In particular we give evidence for the “all roads lead to rome” intuition by showing compatibility of networks with representation that are trained using (1) different initializations, (2) different subsets of the dataset, (3) different tasks (e.g., self-supervised or coarse labels). See Figure 2A-B for results. We also show that network with different random initialization enjoy a property which we call stitching connectivity, wherein almost all minima reachable via SGD can be “stitched” to each other with minimal loss of accuracy. We also give evidence for the “more is better” intuition by showing that we can improve the performance of a network $A$ by plugging in a representation $r$ that was trained with (1) more data, (2) larger width, or (3) more training epochs. See example with more samples in Figure 2C. ",
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+        "type": "text",
+        "text": "The results above are not surprising, in the sense that they confirm intuitions that practitioners might already have. However model stitching allows us to obtain quantitative and formal measures of these in a way that is not achievable by prior representation measures. ",
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+        "type": "text",
+        "text": "Organization. We first define model stitching formally in Section 2. We compare it with prior work on representational similarity measures in Section 3. Then, we formally define stitching connectivity and provide experimental evidence for it in Section 4. Finally, in Section 5 and 6, we provide quantitative evidence for the \"all roads lead to Rome\" and \"more is better\" intuitions respectively. ",
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+        "type": "text",
+        "text": "Related works. As mentioned above, stitching was introduced by Lenc and Vedaldi [2015] who studied equivalence of representations. They showed that certain early layers in a network trained on the Places dataset [Zhou et al., 2014] are compatible with AlexNet (see Table 4 in Lenc and Vedaldi [2015]). After completing this work, we were made aware of concurrent work Csiszárik et al. [2021] that also proposes model stitching to compare neural representations. The results from this work complement ours by studying the effect of changing the stitching layer in various ways (for instance, imposing a sparsity penalty on the stitching layer), and further clarifying the relationship of stitching with other representational similarity measures. In contrast, our work demonstrates stitching compatibility under different scenarios, like the comparison between supervised and self-supervised methods and experiments that show \"more is better\". ",
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+        "text": "Our work on stitching connectivity is related to the work on mode connectivity [Garipov et al., 2018, Freeman and Bruna, 2017, Draxler et al., 2018]. These works show that the local minima found by SGD are often connected through low-loss paths. These paths are generally non-linear, though it was shown that there are linear paths between these minima if they are identically initialized but then use different SGD noise (order of samples) after a certain point in training [Frankle et al., 2020]. Stitching connectivity is complementary to mode connectivity. Stitching connectivity corresponds to a discrete path (with as many steps as layers) but one where the intermediate steps are interpretable. We also show stitching connectivity of networks that are trained on different tasks. Finally, most of the prior work on concrete metrics for relating representations was in the context of representation similarity measures. We describe this work and compare it to ours in Section 3. ",
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+        "type": "text",
+        "text": "2 Model Stitching ",
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+        "text": "Let $A$ be a neural network of some architecture $\\mathcal { A }$ , and let $r : \\mathcal { X } \\to \\mathbb { R } ^ { d }$ be a candidate “representation” function. We consider a family $s$ of stitching layers which are simple (e.g. linear $1 \\times 1$ convolutional layers for a convolutional network $A$ ) functions mapping $\\mathbb { R } ^ { d }$ to $\\mathbb { R } ^ { \\hat { d } _ { \\ell } }$ where $d _ { \\ell }$ is the width of $A$ ’s $\\ell$ -th layer. Given some loss function $\\mathcal { L }$ (e.g., CIFAR-10 or ImageNet test accuracy) we define ",
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+    {
+        "type": "equation",
+        "img_path": "images/93fcf5bda878f2ce4d7fdcf3974242f41a67bf1969494ada123dc03f1cb8f3e7.jpg",
+        "text": "$$\n\\mathcal { L } _ { \\ell } ( r ; A ) = \\operatorname* { i n f } _ { s \\in \\mathcal { S } } \\mathcal { L } ( A _ { > \\ell } \\circ s \\circ r )\n$$",
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+        "type": "text",
+        "text": "where $A _ { > \\ell }$ denotes the function mapping the activations of $A$ ’s \\`th layer to the final output, and $\\circ$ denotes function composition. That is, $\\mathcal { L } _ { \\ell } ( r ; A )$ is the smallest loss obtained by stitching $r$ into all but the first $\\ell$ layers of $A$ using a stitching layer from $s$ . We define the stitching penalty of the representation $r$ with respect to $A$ (as well as $\\ell$ and the loss $\\mathcal { L }$ ) as $\\mathcal { L } _ { \\ell } ( r ; A ) - \\mathcal { L } ( A )$ . If the penalty is non-negative, then we say that the representation $r$ is at least as good as the first $\\ell$ layers of $A$ . ",
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+        "type": "text",
+        "text": "Consider the simple case of a fully-connected network and a stitching family of linear functions. In this case, model stitching tells us if there is a way to linearly transform the representation $r$ into that of the first $\\ell$ layers of $A$ , but only in the subspace that is relevant to the achieving low loss. In practice, we approximate the infimum in Equation 1 by using gradient methods— concretely, by optimizing the stitching-layer using the train set of the task $\\mathcal { L }$ . The stitching penality itself is then estimated on the test set. ",
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+        "text": "Choosing the stitching family $s$ : The stitching family can be chosen flexibly depending on the desired invariances between representations, as long as it is simple (see below). For instance, if we choose $S$ to be all permutations or orthogonal matrices, the stitching penalty will be invariant up to permutations or orthogonal transformations respectively. In this work, we mainly consider cases where $r = B _ { \\leq l }$ for $B$ with architecture similar to $\\mathcal { A }$ . We restrict the stitching family per architecture $\\mathcal { A }$ such that the composed model $A _ { > \\ell } \\circ s \\circ r$ lies in $\\mathcal { A }$ . This way the stitched model consists of layers identical to the layers of either $A$ or $B$ , with the exception of just one layer. For example, for convolutional networks we consider a $1 \\times 1$ convolution and for transformers we consider a token-wise linear function between transformer blocks. We perform an ablation with kernels of different sizes in Appendix B.1. ",
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+        "type": "text",
+        "text": "Stitching is not learning: One concern that may arise is that if the stitching family $s$ is sufficiently powerful, it can learn to transform any representation into any other representation. This would defeat the aim of faithfully studying the original representations. To avoid this, the family of stitchers should be chosen to be simple (e.g., linear). Nevertheless, to verify that our experiments are not in such a regime, we stitch an untrained, randomly initialized network with a fully trained network. Figure 2A-B shows that the penalty of such a stitched network is high, specially for high $l$ . CKA has the same trend for the same networks (See Appendix B.2) ",
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+        "type": "text",
+        "text": "Model stitching as a “happy middle”. Model stitching can be considered as a compromise between the following extremes: (1) Direct plugging in: the most naive interpretation of “plugging in” $r$ into $A$ would be to use no stitching at all. However, even in cases where networks are identical up to a permutation of neurons, plugging in $r$ into $A$ would not work. (2) Full fine tuning: the other extreme interpretation is to perform full fine tuning. That is, start with the initialized network $A { \\mathord { > } } \\ell \\circ r$ and optimize over all choices of $A$ . The problem with this approach is that there are so many degrees of freedom in the choices for $A _ { > \\ell }$ that the resulting network could achieve strong performance regardless of the quality of $r$ . For example, in B.3 we show that fine tuning can fail to distinguish between a trained network and a random network. (3) Linear probe: a popular way to define quality of representations is to use linear probes [Alain and Bengio, 2016]. However, linear probes are not as well suited for studying the representation of early layers, which have low linear separability. In particular, the linear probe accuracy of an early layer of network $A$ would generally always be much worse than a later layer of network $B$ , even if $A$ was of “higher quality” (e.g., trained with more data). Linear probes also don’t have the operational interpretation of compatibility. ",
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+        "type": "text",
+        "text": "By choosing a trainable but low-capacity layer, model stitching “threads the needle” between simply plugging in, and full fine tuning, and unlike linear probes, enables the study of early layers using powerful nonlinear decoders. ",
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+        "type": "text",
+        "text": "Experimental setup. Unless specified otherwise, the CIFAR-10 experiments are conducted on the ResNet-18 architecture (with first layer width 64) and the ImageNet experiments are conducted on the ResNet-50 architecture [He et al., 2015]. The ResNets are trained with the standard hyperparameters (See Appendix A.1 for training parameters of all base models). The stitching layer in the the convolutional networks consist of a $1 \\times 1$ convolutional layer with input features equal to the number of channels in $r$ , and output features equal to the output channels of $A _ { \\leq l } ( x )$ . We add a BatchNorm (BN) layer before and after this convolutional layer. Note that the BN layer does not change the representation capacity of the stitching layer and only aides with optimization. We perform stitching only between ResNet blocks (and not inside a block), but note that it is possible to stitch within the block as well. We use the Adam optimizer with the cosine learning rate decay and an initial learning rate of 0.001. Full experimental details for each experiment are described in Appendix A. ",
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+        "type": "text",
+        "text": "3 Stitching vs. representational similarity ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "Much prior work studying representations focused on representation similarity measures. These have been studied in both the neuroscience and machine learning communities [Kriegeskorte et al., 2008, Kornblith et al., 2019]. Examples of such measures include canonical correlation analysis $( C C A )$ [Hardoon et al., 2004] and its singular-vector and projection-weighted variants such as SVCCA and PWCCA [Raghu et al., 2017, Morcos et al., 2018]. Recently Kornblith, Norouzi, Lee, and Hinton [2019] proposed centered kernel alignment $( C K A )$ that addressed several issues with CCA. CKA was further explored by Nguyen et al. [2021]. ",
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+        "text": "For two representations functions $\\phi : \\mathcal { X }  \\mathbb { R } ^ { d _ { 1 } }$ and $\\sigma : \\mathcal { X }  \\mathbb { R } ^ { d _ { 2 } }$ , the linear CKA is defined $\\begin{array} { r } { \\mathbf { C K A } ( \\phi , \\sigma ) : = \\frac { | | \\mathbf { C o v } ( \\phi ( x ) , \\sigma ( x ) ) | | _ { F } ^ { 2 } } { | | \\mathbf { C o v } ( \\phi ( x ) ) | | _ { F } \\cdot | | \\mathbf { C o v } ( \\sigma ( x ) ) | | _ { F } } } \\end{array}$ where all covariances are with respect to the test distribution on inputs $x \\sim \\mathcal { D }$ [Kornblith et al., 2019]. ",
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+        "table_caption": [
+            "Table 1: Qualitative results of our experiments, comparing CKA to stitching. $\\mathbf { C K A } \\approx 1$ means representations are close according to CKA. Error $\\approx 0 \\%$ means representations are close according to stitching. “Varies” means no consistent conclusion across different architectures and tasks. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Method</td><td>Stitching Connectivity Different Initialization</td><td>“All roads” Self-Supervision</td><td>“More is better” More Data /Time /Width</td></tr><tr><td>CKA Stitching</td><td>Varies (can be O) Close (up to 3% error)</td><td>Varies (0.35- 0.9) Close (up to 5% error)</td><td>Far (can be O for data, O.7 for width) Better</td></tr></table>",
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+        "type": "text",
+        "text": "Table 1 contains a qualitative summary of our results, comparing stitching with CKA. While in some cases the results of stitching and CKA agree, in several cases, stitching obtains results that align more closely with the intuitions that well-performing networks learn similar representations, and that more resources results in better versions of the same representations. In particular, in experiments where stitching indicates that a certain representation is better than another, CKA by its design can only indicate that the two representations are far from each other. Moreover, in some experiments CKA indicates that representations are far where we intuitively believe that they should be close, e.g. when networks only differ by two random initializations, or when one is trained with a supervised and another with a self-supervised task.2 In contrast, in these settings, model-stitching reveals that the two models have nearly equivalent representations, in the sense that they can be stitched to each other with low penalty. The precise experimental results appear in Sections 4-6 and Appendix B.2. ",
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+        "type": "text",
+        "text": "Compared to representation-similarity measures, model stitching has several advantages: ",
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+        "type": "text",
+        "text": "Ignoring spurious features: Measures such as CCA and CKA ultimately boil down to distances between the feature vectors, but they do not distinguish between features that are learned and relevant for downstream tasks, and spurious features, that may be completely useless or even random. For example, suppose we augment a representation $\\phi : \\dot { \\mathcal { X } }  \\dot { \\mathbb { R } ^ { d } }$ by concatenating 1000 “useless” coordinates, with random gaussian features, to form a new representation $\\phi ^ { \\prime } : \\mathcal { X } \\overset { \\mathbf { \\bar { \\Delta } } } { \\to } \\mathbb { R } ^ { d + 1 0 0 0 }$ . This would reduce the CKA, but representation $\\phi ^ { \\prime }$ is not different from $\\phi$ in a meaningful way. Modelstitching resolves this, since we can stitch $\\phi ^ { \\prime }$ in place of $\\phi$ by simply throwing away the useless coordinates. In general, model-stitching focuses only on aspects of the representation which are relevant for the downstream task, as opposed to aspects which are spurious or irrelevant. The price we pay for this is that, unlike measures such as CKA, model stitching depends on the downstream task. However, our results indicate that neural networks tend to learn similar representations for a variety of natural tasks. ",
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+        "type": "text",
+        "text": "Asymmetry: Our intuition is that some representations are better than others. For example, we believe that with more data, neural networks learn better representations. However, by design, such comparisons cannot be demonstrated by representation similarity measures that only measure the distance between two representations. In contrast, we are able to demonstrate that “more is better” using stitching-based measures. Concretely, CKA and other similarity measures are symmetric, while stitching is not: it may be the case that a representation $\\phi$ can be stitched in place of a representation $\\sigma$ , but not vice-versa. Also, stitching a representation $\\phi$ can (and sometimes does) improve performance. ",
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+        "type": "text",
+        "text": "Interpretable units: Measures such as CCA/CKA give a number between 0 and 1 for the distance between representations, but it is hard to interpret what is the difference, for example, between a CKA value of 0.9 and value of 0.8. In contrast, if the loss function $\\mathcal { L }$ has meaningful units, then the stitching penalty inherits those, and (for example) a representation $r$ having a penalty of $3 \\%$ in CIFAR-10 accuracy has an operational meaning: if you replace the first layers of the network with $r$ the decrease in accuracy is at most $3 \\%$ . ",
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+        "text": "Invariance: A representation similarity measure should be invariant to operations that do not modify the “quality” of the representation, but it is not always clear what these operations are. For example, CCA and CKA are invariant under orthogonal linear transformations, and some variants of CCA are also invariant under general invertible linear transformations (see Table 1 in Kornblith et al. [2019]). However, it is unclear if these are the natural families. For example, randomly permuting the order of pixels (either at the input or in the latents) is an orthogonal transform, and thus does not affect the CKA. However, this completely destroys the spatial structure of the input, and intuitively should affect the “representation quality.” On the other hand, certain non-orthogonal transforms may still preserve representation quality– for example, the non-invertible transformation of projecting out spurious coordinates. Thus, orthogonal transforms may not be the right invariance class to consider representations. Using stitching we can explicitly ensure invariance under any given family of transformations by adding it to the stitching layer. Concretely, our choice of using a $1 \\times 1$ convolutional stitcher yields much weaker invariance than general orthogonal transforms– and thus, model stitching can predict that shuffling pixels leads to a “worse” representation. ",
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+        "type": "text",
+        "text": "4 Stitching Connectivity ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "We first focus on a special case of model-stitching, wherein we stitch two identically distributed networks to each other. That is, we train two networks of same architecture, on the same data distribution, but with independent random seeds and independent train samples. This question has been studied before with varied conclusions [Li et al., 2016, Wang et al., 2018]. We find that empirically, two such networks can be stitched to each other at all layers, with close to 0 penalty. This is a new empirical property of SGD trained networks, which we term “stitching connectivity.” ",
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+        "type": "text",
+        "text": "Formally, let $A , B$ be two trained networks of identical architectures with $L$ layers, for a task with objective function $\\mathcal { L }$ . For all $i \\in \\{ 0 , 1 , . . . , L \\}$ , define $S _ { i }$ to be the stitched model where we replace the first $i$ layers of $A$ with those same layers in $B$ (and optimize the stitching layer as usual). That is, $\\begin{array} { r } { S _ { i } : = \\arg \\operatorname* { m i n } _ { S = A _ { > i } \\circ s \\circ B _ { \\leq i } } { \\mathcal { L } } ( S ) } \\end{array}$ . Observe that $S _ { 0 } = A$ and $S _ { L } = B$ , so the sequence of models $\\{ S _ { 0 } , S _ { 1 } , \\ldots , S _ { L } \\}$ gives a kind of “path” between models $A$ and $B$ . Further, due to our family of stitching layer and network architecture $1 \\times 1$ conv stitchers and conv-nets), the stitching layer $s$ can be folded into the adjacent model. Thus, all models $S _ { i }$ have identical architecture as $A$ and $B$ . We say $A$ and $B$ are “stitching-connected” if all the intermediate models $S _ { i }$ have low test loss. ",
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+        "type": "text",
+        "text": "Definition 1 (Stitching Connectivity). Let $A , B$ be two networks of identical architectures. We say $A$ and $B$ are stitching-connected if they can be stitched to each other at all layers, with low penalty. That is, if all stitched models $S _ { i }$ , defined as above, have test loss comparable to $A$ . ",
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+        "type": "text",
+        "text": "Stitching connectivity is not a trivial property: two networks with identical architectures, but very different internal representations, would fail to be stitching connected. Our main claim is that for a fixed data distribution, almost all minima reached by SGD are stitching-connected to each other. ",
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+        "type": "text",
+        "text": "Conjecture 2 (Stitching Connectivity of SGD, informal). Let $A _ { 1 } , A _ { 2 }$ be two independent and identically-trained networks. That is, networks of the same architecture, trained by SGD with independent random seeds and independent train sets from the same distribution. Then, for natural architectures and data-distributions, the trained models $A _ { 1 }$ and $A _ { 2 }$ are stitching-connected. ",
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+    {
+        "type": "text",
+        "text": "This conjecture states a structural property of models that are likely to be output by SGD. If we run the identical training procedure twice, we will almost certainly not produce models with identical parameters. However, these two different parameter settings yield essentially equivalent internal representations– this is what it means to be stitching-connected. ",
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+        "type": "text",
+        "text": "Discussion. Stitching connectivity is similar in spirit to mode connectivity [Garipov et al., 2018, Freeman and Bruna, 2017, Draxler et al., 2018], in that they are both structural properties of the set of typical SGD minimas. Mode connectivity states that typical minima are connected by a low-test-loss path in parameter space. Stitching connectivity does not technically define a path in parameter space– rather, it defines a sequence $S _ { 0 } , S _ { 1 } , \\ldots , S _ { L }$ of low-loss models connecting two endpoint models. For each model in this sequence, all but one layer is identical to one of the endpoint models. Thus, we can informally think of the stitching sequence as a different way of “interpolating” between two models. ",
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+    {
+        "type": "text",
+        "text": "The stitching connectivity of SGD is especially interesting for overparameterized models, since it sheds light on the implicit bias of SGD. In this case, there are exponentially-many global minima of the train loss, even modulo permutation-symmetry. Apriori, it could be the case that each of these minima compute the classification decision in different ways (e.g. by memorizing a particular train set). However, empirically we find that SGD is “biased” towards minima with essentially the same internal representations– in that typical minima can be stitched to each other. ",
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+    {
+        "type": "text",
+        "text": "Experiments. Figure 2A demonstrates the stitching-connectivity of SGD via the following experiment. We train a variety of model architectures on CIFAR-10, with two different randomly initialized models per architecture. We consider a ResNet-18, two variants of ResNet-18 with $0 . 5 \\times$ and $2 \\times$ width, a significantly deeper ResNet-164, a feed-forward convolutional network Myrtle-CNN [Page, 2018] and a Vision Transformer [Dosovitskiy et al., 2020] pretrained on CIFAR- $\\cdot 5 \\mathrm { m }$ [Nakkiran et al., 2021]. We stitch the two randomly initialized models at various intermediate layers, forming the stitching sequence $S _ { 0 } , \\ldots S _ { L }$ . Figure 2A plots the test errors of these intermediate stitched models $S _ { i }$ with the first and last point showing the errors of the base models. For all layers $i$ , the test error of the stitched model $S _ { i }$ is close to the error of the base models. A similar result holds for networks trained on disjoint train sets. Full experimental details are provided in Appendix A. ",
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+    {
+        "type": "text",
+        "text": "5 All Roads Lead to Rome ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "In this section, we quantify the intuition that “all roads lead to Rome” in representation learning: many diverse choices of train method, label quality, and objective function all lead to similar representations in early layers. However, we find that such training details can affect the representation at later layers. ",
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+    {
+        "type": "text",
+        "text": "Comparing self-supervised and supervised methods. If we train models with the same architecture and train set, but alter the training method significantly, what do we expect from the representations? To explore this, we compare the representations of two very different training methods — standard end-to-end supervised training (E2E) versus self-supervised $^ +$ simple classifiers (SSS), that first learn a representation from unlabeled data and then train a simple linear classifier on this representation.3 ",
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+    {
+        "type": "text",
+        "text": "",
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+    {
+        "type": "text",
+        "text": "SSS algorithms like SimCLR [Chen et al., 2020b], SwAV [Caron et al., 2021a] and DINO [Caron et al., 2021b] etc have recently emerged as prominent paradigm for training neural networks that achieve comparable accuracy to E2E networks. We stitch the representations of these SSS algorithms to an E2E supervised network (all with a ResNet-50 backbone) trained on ImageNet. While all of these methods achieve similar test accuracy on a ResNet-50 backbone of $7 5 \\% \\pm 1$ (except SimCLR at $6 8 . 8 \\%$ ) [Goyal et al., 2021], they are trained very differently. Figure 2B shows that the SSS trained networks are stitching connected at all layers to the E2E network. This suggests that while the advances in SSL have been significant for learning features without labels, the features themselves are similar in both cases. This is in agreement with prior work which shows that SSS and E2E networks have similar texture-shape bias and make similar errors [Geirhos et al., 2020]. Since certain SSS algorithms have been proven to have small generalization gap [Bansal et al., 2021], the similarity of representations between SSS and E2E algorithms may yield some clues into the generalization mystery of E2E algorithms. A similar experiment for SimCLR with ResNet-18 trained on CIFAR-10 is shown in Appendix B.2, along with the CKA for the same networks. We find that the CKA varies between $0 . 3 5 - 0 . 9$ for different layers, while stitching gets maximum stitching penalty of $3 \\%$ . ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Changing the label distribution $\\mathbf { p } ( \\mathbf { y } \\vert \\mathbf { x } )$ : How much does the representation quality depend on label quality? To explore this, we compare the representations of networks with the same input distribution $p ( x )$ , but different label distributions $p ( y | x )$ . We take the CIFAR-10 distribution on inputs $p ( x )$ , and consider several “less informative” label distributions: (1) Coarse labels: We super-class CIFAR-10 classes into a binary task, of Objects (Ship, Truck, etc.) vs. Animals (Cat, Dog, etc.) (2) Label noise: We set $p = \\{ 0 . 1 , 0 . 5 , 1 . \\}$ fraction of labels to random labels. We then stitch these “weak” networks to a standard CIFAR-10 network at varying layers, and measure the stitching penalty incurred in Figure 3A. We find that even with poor label quality, the first half of layers in the “weak” model are “as good as” layers in the standard model (with the exception of the weak network trained on $100 \\%$ noisy labels). These experiments align with the results of Nakkiran and Bansal [2020], which show that neural networks can be sensitive to aspects of the input distribution that are not explicitly encoded in the labels. Full experimental details appear in Appendix A. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "These results are in agreement with prior work on vision [Olah et al., 2017], which suggests that the first few layers of a neural network learn general purpose features (such as curve detectors) that are likely to be useful a large variety of tasks. Formalizing the set of pre-training tasks for which such similar representations are learnt is an important direction for future work to understand pretraining. ",
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+    {
+        "type": "text",
+        "text": "6 More is Better ",
+        "text_level": 1,
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+    },
+    {
+        "type": "text",
+        "text": "Now, we use stitching to quantify the intuition that “more is better” for representations. That is, larger sample size, model size, or train time lead to progressively better representations of the same type. ",
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+    {
+        "type": "text",
+        "text": "Number of samples: When scaled appropriately, neural network performance improves predictably with the number of training samples (e.g. [Kaplan et al., 2020]). But how does this improvement manifest in their representations? We investigate this by stitching the lower parts of models trained $\\{ 5 K , 1 0 K , 2 5 K \\}$ samples of CIFAR-10 to the upper layers of a model trained with $1 0 K$ samples. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "First, as Figure 2C shows, we observe that the models are all stitching compatible— the better representation trained with $2 5 K$ samples can simply be \"plugged in\" to the model trained with fewer samples, and this improves the performance of the stitched network relative to the $1 0 K$ network. Note that there is no theoretical reason to expect this compatibility, better networks could have learnt fundamentally different features from their weaker counterparts. For example, a network trained on few samples could have learnt only “simple features” (presence of sky in the image / presence of horizontal lines), while one trained on many samples may learn only “complex features” (presence of an blue eye / presence of a furry ear), which are not decodable by the weaker network. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Secondly, we find that some layers are more data hungry than others. When stitched at the first few layers, all the models have similar performance, but the performance degrades rapidly with fewer samples in the mid-layers of the network. This suggests that each layer of a neural network has its own sample complexity. As a corollary of this finding, we predict that we can train some of the layers with few samples, freeze them and train the rest of the model with larger number of samples. Indeed, we find that we can recover most of the accuracy of the network by training the first three, and the last three layers of this model (about half of the layers) with just $5 K$ samples, freezing them and training the rest of the model with all the samples to obtain a network within $2 \\%$ accuracy of the original network. See Appendix B.4 for details of the experiment and training plots. This is similar to the \"freeze-training\" experiments suggested by Raghu et al. [2017]. ",
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+    },
+    {
+        "type": "image",
+        "img_path": "images/7c005a86efaa130cc232bb2c14cff6bbfbdc163b0a8a9b9b512e17ee41195811.jpg",
+        "image_caption": [
+            "Figure 3: (A) Changing the label distribution: Representations trained on CIFAR-10 with the Object vs. Animals task or with $\\{ 1 0 \\% , 5 0 \\% , 1 0 0 \\% \\}$ label noise and stitched to a network trained on original CIFAR-10 labels. Early layers learn similar representations even when the label distribution is \"less informative\" than training with all labels (B) Increasing training time: Representations at different epochs during training are stitching compatible and early layer converge faster (B) Increasing width: Better representations from a wider network can be stitched with a thinner network to improve performance. All experiments were performed with ResNet-18 on CIFAR-10. "
+        ],
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+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Training time: We observe similar results when we stitch networks trained for different number of epochs. We train a ResNet-18 trained on CIFAR-10 and stitch the representations from the model at $\\{ 4 0 , 8 0 , 1 6 0 \\}$ epochs to the model at the 80-th epoch. As training time increases, Figure 3B shows that the representations improve in a manner that is compatible with the earlier training times. Note that this is a nontrivial statement about neural network training dynamics: It could have been the case that, once a network is trained for very long, its representations move “far away” from its representations near initialization – and thus, stitching would fail. However, we find that the representations at the end of training remain compatible with those in the early stage of training. We also observe that earlier layers converge faster with time, as was shown by Raghu et al. [2017]. ",
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Width: Similarly, we train models with the same architecture but varying width multipliers $\\{ 0 . 2 5 \\times , 1 \\times , 2 \\times \\}$ (Figure 3C). We find that “better” models with higher width models can be stitched to those with lower width and improve performance, but not vice versa. The CKA between representations with $0 . 2 5 \\times$ and $2 \\times$ width multiplier is in the range $0 . 7 - 0 . 9$ (See Appendix B.2) ",
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Taken together, these results suggest that neural networks obey a certain kind of modularity — better layers can be plugged in without needing to re-train the whole network from scratch. ",
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+    {
+        "type": "text",
+        "text": "7 Conclusion and Future Work ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "As our work demonstrates, model stitching can be a very useful tool to compare representations in interpretable units. Model stitching does have its limitations: it requires training a network, making it more expensive in computation than other measures such as CKA. Additionally, stitching representations from two different architectures can be tricky and requires a careful choice of the stitching family. While we restricted ourselves to stitching the first $l$ layers of a network, stitching can be used to also plug in intermediate layers or parts of layers, and in general to “assemble” a new model from a collection of pre-trained components. ",
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "There are various avenues for future research. All of our results are in the vision domain, but it would be interesting to compare representations in natural language processing, since language tasks tend to have more variety than vision tasks. It would also be interesting to study the representations of adversarially trained networks to diagnose why they lose performance in comparison to standard training. In general, we hope that model stitching will become a part of the standard diagnostic repertoire of the deep learning community. Societal impacts: This paper makes methodological and foundational contributions that do not have direct impact on society. Model stitching can potentially be used to understand and develop better representation learning mechanisms. While this could indirectly lead to future applications, it is premature to predict their positive or negative impacts. ",
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+    {
+        "type": "text",
+        "text": "",
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+        "page_idx": 9
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+    {
+        "type": "text",
+        "text": "8 Acknowledgements ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "YB is supported by IBM Global University awards program and NSF Awards IIS 1409097. PN is supported in part by a Google PhD Fellowship, the Simons Investigator Awards of Boaz Barak and Madhu Sudan, and NSF Awards under grants CCF 1565264, CCF 1715187. BB is supported by NSF award CCF 1565264, a Simons Investigator Fellowship and DARPA grant W911NF2010021. We thank MIT-IBM Watson AI Lab and John Cohn for providing access and support for the Satori compute cluster. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "References ",
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parse/train/rJUYGxbCW/rJUYGxbCW.md

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+# PIXELDEFEND: LEVERAGING GENERATIVE MODELS TO UNDERSTAND AND DEFEND AGAINST ADVERSARIAL EXAMPLES
+
+Yang Song   
+Stanford University   
+yangsong@cs.stanford.edu
+
+Taesup Kim Université de Montréal taesup.kim@umontreal.ca
+
+Sebastian Nowozin Microsoft Research nowozin@microsoft.com
+
+Stefano Ermon Stanford University ermon@cs.stanford.edu
+
+Nate Kushman   
+Microsoft Research   
+nkushman@microsoft.com
+
+# ABSTRACT
+
+Adversarial perturbations of normal images are usually imperceptible to humans, but they can seriously confuse state-of-the-art machine learning models. What makes them so special in the eyes of image classifiers? In this paper, we show empirically that adversarial examples mainly lie in the low probability regions of the training distribution, regardless of attack types and targeted models. Using statistical hypothesis testing, we find that modern neural density models are surprisingly good at detecting imperceptible image perturbations. Based on this discovery, we devised PixelDefend, a new approach that purifies a maliciously perturbed image by moving it back towards the distribution seen in the training data. The purified image is then run through an unmodified classifier, making our method agnostic to both the classifier and the attacking method. As a result, PixelDefend can be used to protect already deployed models and be combined with other model-specific defenses. Experiments show that our method greatly improves resilience across a wide variety of state-of-the-art attacking methods, increasing accuracy on the strongest attack from $63 \%$ to $84 \%$ for Fashion MNIST and from $32 \%$ to $70 \%$ for CIFAR-10.
+
+# 1 INTRODUCTION
+
+Recent work has shown that small, carefully chosen modifications to the inputs of a neural network classifier can cause the model to give incorrect labels (Szegedy et al., 2013; Goodfellow et al., 2014). This weakness of neural network models is particularly surprising because the modifications required are often imperceptible, or barely perceptible, to humans. As deep neural networks are being deployed in safety-critical applications such as self-driving cars (Amodei et al., 2016), it becomes increasingly important to develop techniques to handle these kinds of inputs.
+
+Rethinking adversarial examples The existence of such adversarial examples seems quite surprising. A neural network classifier can get super-human performance (He et al., 2015) on clean test images, but will give embarrassingly wrong predictions on the same set of images if some imperceptible noise is added. What makes this noise so special to deep neural networks?
+
+In this paper, we propose and empirically evaluate the following hypothesis: Even though they have very small deviations from clean images, adversarial examples largely lie in the low probability regions of the distribution that generated the data used to train the model. Therefore, they fool classifiers mainly due to covariate shift. This is analogous to training models on MNIST (LeCun et al., 1998) but testing them on Street View House Numbers (Netzer et al., 2011).
+
+To study this hypothesis, we first need to estimate the probability density of the underlying training distribution. To this end, we leverage recent developments in generative models. Specifically, we choose a PixelCNN (van den Oord et al., 2016b) model for its state-of-the-art performance in modeling image distributions (van den Oord et al., 2016a; Salimans et al., 2017) and tractability of evaluating the data likelihood. In the first part of the paper, we show that a well-trained PixelCNN generative model is very sensitive to adversarial inputs, typically giving them several orders of magnitude lower likelihoods compared to those of training and test images.
+
+Detecting adversarial examples An important step towards handling adversarial images is the ability to detect them. In order to catch any kind of threat, existing work has utilized confidence estimates from Bayesian neural networks (BNNs) or dropout (Li & Gal, 2017; Feinman et al., 2017). However, if their model is misspecified, the uncertainty estimates can be affected by covariate shift (Shimodaira, 2000). This is problematic in an adversarial setting, since the attacker might be able to make use of the inductive bias from the misspecified classifier to bypass the detection.
+
+Protection against the strongest adversary requires a pessimistic perspective—our assumption is that the classifier cannot give reliable predictions for any input outside of the training distribution. Therefore, instead of relying on label uncertainties given by the classifier, we leverage statistical hypothesis testing to detect any input not drawn from the same distribution as training images.
+
+Specifically, we first compute the probabilities of all training images under the generative model. Afterwards, for a novel input we compute the probability density at the input and evaluate its rank (in ascending order) among the density values of all training examples. Next, the rank can be used as a test statistic and gives us a $p$ -value for whether or not the image was drawn from the training distribution. This method is general and practical and we show that the $p$ -value enables us to detect adversarial images across a large number of different attacking methods with high probability, even when they differ from clean images by only a few pixel values.
+
+Purifying adversarial examples Since adversarial examples are generated from clean images by adding imperceptible perturbations, it is possible to decontaminate them by searching for more probable images within a small distance of the original ones. By limiting the $L ^ { \infty }$ distance1, this image purification procedure generates only imperceptible modifications to the original input, so that the true labels of the purified images remain the same. The resulting purified images have higher probability under the training distribution, so we can expect that a classifier trained on the clean images will have more reliable predictions on the purified images. Moreover, for inputs which are not corrupted by adversarial perturbations the purified results remain in a high density region.
+
+We use this intuition to build PixelDefend, an image purification procedure which requires no knowledge of the attack nor the targeted classifier. PixelDefend approximates the training distribution using a PixelCNN model. The constrained optimization problem of finding the highest probability image within an $\epsilon$ -ball of the original is computationally intractable, however, so we approximate it using a greedy decoding procedure. Since PixelDefend does not change the classification model, it can be combined with other adversarial defense techniques, including adversarial training (Goodfellow et al., 2014), to provide synergistic improvements. We show experimentally that PixelDefend performs exceptionally well in practice, leading to state-of-the art results against a large number of attacks, especially when combined with adversarial training.
+
+Contributions Our main contributions are as follows:
+
+• We show that generative models can be used for detecting adversarially perturbed images and observe that most adversarial examples lie in low probability regions.   
+• We introduce a novel family of methods for defending against adversarial attacks based on the idea of purification.   
+We show that a defensive technique from this family, PixelDefend, can achieve state-of-theart results on a large number of attacking techniques, improving the accuracy against the strongest adversary on the CIFAR-10 dataset from $32 \%$ to $70 \%$ .
+
+# 2 BACKGROUND
+
+# 2.1 ATTACKING METHODS
+
+Given a test image $\mathbf { X }$ , an attacking method tries to find a small perturbation $\pmb { \Delta }$ with $\| \pmb { \Delta } \| _ { \infty } \le \epsilon _ { \mathrm { a t t a c k } }$ such that a classifier $f$ gives different predictions on $\mathbf { X } ^ { a d v } \triangleq \mathbf { X } + \Delta$ and $\mathbf { X }$ . Here colors in the image are represented by integers from 0 to 255. Each attack method is controlled by a configurable $\epsilon _ { \mathrm { a t t a c k } }$ parameter which sets the maximum perturbation allowed for each pixel in integer increments on the color scale. We only consider white-box attacks in this paper, i.e., the attack methods can get access to weights of the classifier. In the following, we give an introduction to all the attacking methods used in our experiments.
+
+Random perturbation (RAND) Random perturbation is arguably the weakest attacking method, and we include it as the simplest baseline. Formally, the randomly perturbed image is given by
+
+$$
+{ \bf X } ^ { a d v } = { \bf X } + \mathcal { U } ( - \lfloor \epsilon _ { \mathrm { a t t a c k } } \rfloor , \lfloor \epsilon _ { \mathrm { a t t a c k } } \rfloor ) ,
+$$
+
+where $\textstyle { \mathcal { U } } ( a , b )$ denotes an element-wise uniform distribution of integers from $[ a , b ]$
+
+Fast gradient sign method (FGSM) Goodfellow et al. (2014) proposed the generation of malicious perturbations in the direction of the loss gradient $\nabla _ { \mathbf { X } } L ( \mathbf { X } , y )$ , where $L ( \mathbf { X } , y )$ is the loss function used to train the model. The adversarial examples are computed by
+
+$$
+{ \bf X } ^ { a d v } = { \bf X } + \epsilon _ { \mathrm { a t t a c k } } \mathrm { s i g n } ( \nabla _ { \bf X } L ( { \bf X } , y ) ) .
+$$
+
+Basic iterative method (BIM) Kurakin et al. (2016) tested a simple variant of the fast gradient sign method by applying it multiple times with a smaller step size. Formally, the adversarial examples are computed as
+
+$$
+{ \bf X } _ { 0 } ^ { a d v } = { \bf X } , { \bf X } _ { n + 1 } ^ { a d v } = \mathrm { C l i p } _ { \bf x } ^ { \epsilon _ { \mathrm { a t a c k } } } \left\{ { \bf X } _ { n } ^ { a d v } + \alpha \mathrm { s i g n } ( \nabla _ { \bf X } L ( { \bf X } _ { n } ^ { a d v } , y ) ) \right\} ,
+$$
+
+where $\mathrm { C l i p } _ { \mathbf { X } } ^ { \epsilon _ { \mathrm { a t t a c k } } }$ means we clip the resulting image to be within the $\epsilon _ { \mathrm { a t t a c k } }$ -ball of $\mathbf { X }$ . Following Kurakin et al. (2016), we set $\alpha = 1$ and the number of iterations to be $\left\lfloor \operatorname* { m i n } ( \epsilon _ { \mathrm { a t t a c k } } + 4 , 1 . 2 5 \epsilon _ { \mathrm { a t t a c k } } ) \right\rfloor$ This method is also called Projected Gradient Descent (PGD) in Madry et al. (2017).
+
+DeepFool DeepFool (Moosavi-Dezfooli et al., 2016) works by iteratively linearizing the decision boundary and finding the closest adversarial examples with geometric formulas. However, compared to FGSM and BIM, this method is much slower in practice. We clip the resulting image so that its perturbation is no larger than $\epsilon _ { \mathrm { a t t a c k } }$ .
+
+Carlini-Wagner (CW) Carlini & Wagner (2017b) proposed an efficient optimization objective for iteratively finding the adversarial examples with the smallest perturbations. As with DeepFool, we clip the output image to make sure the perturbations are limited by $\epsilon _ { \mathrm { a t t a c k } }$ .
+
+# 2.2 DEFENSE METHODS
+
+Current defense methods generally fall into two classes. They either (1) change the network architecture or training procedure to make it more robust, or (2) modify adversarial examples to reduce their harm. In this paper, we take the following defense methods into comparison.
+
+Adversarial training This defense works by generating adversarial examples on-the-fly during training and including them into the training set. FGSM adversarial examples are the most commonly used ones for adversarial training, since they are fast to generate and easy to train. Although training with higher-order adversarial examples (e.g., BIM) has witnessed some success in small datasets (Madry et al., 2017), other work has reported failure in larger ones (Kurakin et al., 2016). We consider both variants in our work.
+
+Label smoothing In contrast to adversarial training, label smoothing (Warde-Farley & Goodfellow, 2016) is agnostic to the attack method. It converts one-hot labels to soft targets, where the correct class has value $1 - \epsilon$ while the other (wrong) classes have value $\epsilon / ( N - \bar { 1 } )$ . Here $\epsilon$ is a small constant and $N$ is the number of classes. When the classifier is re-trained on these soft targets rather than the one-hot labels it is significantly more robust to adversarial examples. This method was originally devised to achieve a similar effect as defensive distillation (Papernot et al., 2016c), and their performance is comparable. We didn’t compare to defensive distillation since it is more computationally expensive.
+
+Feature squeezing Feature squeezing $\mathrm { { X u } }$ et al., 2017a) is both attack-agnostic and modelagnostic. Given any input image, it first reduces the color range from [0, 255] to a smaller value, and then smooths the image with a median filter. The resulting image is then passed to a classifier for predictions. Since this technique does not depend on attacking methods and classifiers, it can be combined with other defensive methods such as adversarial training, similar to PixelDefend.
+
+# 2.3 EXPERIMENT METHODOLOGIES
+
+Datasets Two datasets are used in our experiments: Fashion MNIST (Xiao et al., 2017) and CIFAR-10 (Krizhevsky et al.). Fashion MNIST was designed as a more difficult, but drop-in replacement for MNIST (LeCun et al., 1998). Thus it shares all of MNIST’s characteristics, i.e., 60, 000 training examples and 10, 000 test examples where each example is a $2 8 \times 2 8$ gray-scale image associated with a label from 1 of 10 classes. CIFAR-10 is another dataset that is also broadly used for image classification tasks. It consists of 60, 000 examples, where 50, 000 are used for training and 10, 000 for testing, and each sample is a $3 2 \times 3 2$ color image associated with 1 of 10 classes.
+
+Models We examine two state-of-the-art deep neural network image classifiers: ResNet (He et al., 2016) and VGG (Simonyan & Zisserman, 2014). The architectures are described in Appendix C.
+
+PixelCNN The PixelCNN (van den Oord et al., 2016b; Salimans et al., 2017) is a generative model with tractable likelihood especially designed for images. The model defines the joint distribution over all pixels by factorizing it into a product of conditional distributions.
+
+$$
+p _ { \mathrm { C N N } } ( \mathbf { X } ) = \prod _ { i } p _ { \mathrm { C N N } } ( x _ { i } | x _ { 1 : ( i - 1 ) } ) .
+$$
+
+The pixel dependencies are in raster scan order (row by row and column by column within each row). We train the PixelCNN model for each dataset using only clean (not perturbed) image samples. In Appendix D, we provide clean sample images from the datasets as well as generated image samples from PixelCNN (see Figure 8 and Figure 9).
+
+As a convenient representation of $p _ { \mathrm { C N N } } ( \mathbf { X } )$ for images, we also use the concept of bits per dimension, which is defined as $\mathrm { B P D } ( \mathbf { X } ) \triangleq - \log p _ { \mathrm { C N N } } ( \mathbf { X } ) / ( I \times J \times K \times \log 2 )$ for an image of resolution $I \times J$ and $K$ channels.
+
+# 3 DETECTING ADVERSARIAL EXAMPLES
+
+Adversarial images are defined with respect to a specific classifier. Intuitively, a maliciously perturbed image that causes one network to give a highly confident incorrect prediction might not fool another network. However, recent work (Papernot et al., 2016a; Liu et al., 2016; Tramèr et al., 2017) has shown that adversarial images can transfer across different classifiers. This indicates that there are some intrinsic properties of adversarial examples that are independent of classifiers.
+
+One possibility is that, compared to normal training and test images, adversarial examples have much lower probability densities under the image distribution. As a result, classifiers do not have enough training instances to get familiarized with this part of the input space. The resulting prediction task suffers from covariate shift, and since all of the classifiers are trained on the same dataset, this covariate shift will affect all of them similarly and will likely lead to misclassifications.
+
+To empirically verify this hypothesis, we train a PixelCNN model on the CIFAR-10 (Krizhevsky & Hinton, 2009) dataset and use its log-likelihood as an approximation to the true underlying probability density. The adversarial examples are generated with respect to a ResNet (He et al., 2016), which gets $92 \%$ accuracy on the test images. We generate adversarial examples from RAND, FGSM, BIM, DeepFool and CW methods with $\epsilon _ { \mathrm { a t t a c k } } = 8$ . Note that as shown in Figure 1, the resulting adversarial perterbations are barely perceptible to humans.
+
+![](images/f08adab06239984a46d4b77ee6fabaca7d40d3ac138ce08f629e31bcb656bf1d.jpg)  
+Figure 1: An image sampled from the CIFAR-10 test dataset and various adversarial examples generated from it. The text above shows the attacking method while the text below shows the predicted label of the ResNet.
+
+![](images/7e219f80073784145f4f5b8e698056f47d0e1bc7934c8580b00b3f76bfe5f436.jpg)  
+Figure 2: (a) Likelihoods of different perturbed images with $\epsilon _ { \mathrm { a t t a c k } } = 8$ . (b) Test errors of a ResNet on different adversarial examples.
+
+However, the distribution of log-likelihoods show considerable difference between perturbed images and clean images. As summarized in Figure 2, even a $3 \%$ perturbation can lead to systematic decrease of log-likelihoods. Note that the PixelCNN model has no information about the attacking methods for producing those adversarial examples, and no information about the ResNet model either.
+
+We can see from Figure 3(b) that random perturbations also push the images outside of the training distribution, even though they do not have the same adverse effect on accuracy. We believe this is due to an inductive bias that is shared by many neural network models but not inherent to all models, as discussed further in Appendix A.
+
+Besides qualitative analysis, the log-likelihoods from PixelCNN also provide a quantitative measure for detecting adversarial examples. Combined with permutation test (Efron & Tibshirani, 1994), we can provide a uncertainty value for each input about whether it comes from the training distribution or not. Specifically, let the input $\mathbf { X } ^ { \prime } \overset { \mathrm { i . i . d . } } { \sim } q ( \mathbf { X } )$ and training images $\mathbf { X } _ { 1 } , \cdots , \mathbf { X } _ { N } \overset { \mathrm { i . i . d . } } { \sim } p ( \mathbf { X } )$ . The null hypothesis is $H _ { 0 } : p ( \mathbf { X } ) = { \\overset { \cdot } { q } } ( \mathbf { X } )$ while the alternative is $H _ { 1 } : p ( \mathbf { X } ) \neq q ( \mathbf { X } )$ . We first compute the probabilities give by a PixelCNN for $\mathbf { X } ^ { \prime }$ and $\mathbf { X } _ { 1 } , \cdots , \mathbf { X } _ { N }$ , then use the rank of $p _ { \mathrm { C N N } } ( \mathbf { X } ^ { \prime } )$ in $\{ p _ { \mathrm { C N N } } ( \mathbf { X } _ { 1 } ) , \cdots , \bar { p } _ { \mathrm { C N N } } ( \mathbf { \bar { X } } _ { N } ) \}$ as our test statistic:
+
+$$
+T = T ( \mathbf { X } ^ { \prime } ; \mathbf { X } _ { 1 } , \cdots , \mathbf { X } _ { N } ) \triangleq \sum _ { i = 1 } ^ { N } \mathbb { I } [ p _ { \mathrm { C N N } } ( \mathbf { X } _ { i } ) \leq p _ { \mathrm { C N N } } ( \mathbf { X } ^ { \prime } ) ] .
+$$
+
+Here $\mathbb { I } [ \cdot ]$ is the indicator function, which equals 1 when the condition inside brackets is true and otherwise equals 0. Let $T _ { i } = T ( { \bf X } _ { i } ; { \bf X } _ { 1 } , \cdot \cdot \cdot , { \bf X } _ { i - 1 } , { \bf X } ^ { \prime } , { \bf X } _ { i + 1 } , \cdot \cdot \cdot , { \bf X } _ { N } )$ . According to the permutation principle, $T _ { i }$ has the same distribution as $T$ under the null hypothesis $H _ { 0 }$ . We can therefore compute the $p$ -value exactly by
+
+$$
+p = \frac { 1 } { N + 1 } \left( \sum _ { i = 1 } ^ { N } \mathbb { I } [ T _ { i } \leq T ] + 1 \right) = \frac { T + 1 } { N + 1 } = \frac { 1 } { N + 1 } \left( \sum _ { i = 1 } ^ { N } \mathbb { I } [ p _ { \mathrm { C N N } } ( \mathbf { X } _ { i } ) \leq p _ { \mathrm { C N N } } ( \mathbf { X } ^ { \prime } ) ] + 1 \right) .
+$$
+
+![](images/841fb2606539c7e4c79789b143754a98d8357f97e4cec03ca413c7385fcfd8ef.jpg)  
+Figure 3: The distribution of $p$ -values under the PixelCNN generative model. The inputs are more outside of the training distribution if their $p$ -value distribution has a larger deviation from uniform. Here “clean” means clean test images. From definition, the $p$ -values of clean training images have a uniform distribution.
+
+![](images/84c36078978cc3206f8239f356b8f2bdca0f5fa11b8e53dcc42b5f50501efca7.jpg)  
+Figure 4: An example of how purification works. The above row shows an image from CIFAR10 test set and various attacking images generated from it. The bottom row shows corresponding purified images. The text below each image is the predicted label given by our ResNet.
+
+For CIFAR-10, we provide histograms of $p$ -values for different adversarial examples in Figure 3 and ROC curves of using $p$ -values for detection in Figure 6(a). Note that in the ideal case, the $p$ -value distribution of clean test images should be uniform. The method works especially well for attacks producing larger perturbations, such as RAND, FGSM, and BIM. For DeepFool and CW adversarial examples, we can also observe significant deviations from uniform. As shown in Figure 3(a), the $p$ - value distribution of test images are almost uniform, indicating good generalization of the PixelCNN model.
+
+# 4 PURIFYING IMAGES WITH PIXELDEFEND
+
+In many circumstances, simply detecting adversarial images is not sufficient. It is often critical to be able to correctly classify images despite such adversarial modifications. In this section we introduce PixelDefend, a specific instance of a new family of defense methods that significantly improves the state-of-the-art performance on advanced attacks, while simultaneously performing well against all other attacks.
+
+# Algorithm 1 PixelDefend
+
+Input: Image X, Defense parameter $\epsilon _ { \mathrm { d e f e n d } }$ , Pre-trained PixelCNN model $p _ { \mathrm { C N N } }$   
+Output: Purified Image $\mathbf { X } ^ { * }$   
+1: $\bar { \mathbf { X } } ^ { * }  \mathbf { X }$   
+2: for each row $i$ do   
+3: for each column $j$ do   
+4: for each channel $k$ do   
+5: $x \gets \mathbf { X } [ i , j , k ]$   
+6: Set feasible range $R \gets [ \mathrm { m a x } ( x - \epsilon _ { \mathrm { d e f e n d } } , 0 ) , \mathrm { m i n } ( x + \epsilon _ { \mathrm { d e f e n d } } , 2 5 5 ) ]$   
+7: Compute the 256-way softmax $p _ { \mathrm { C N N } } ( \mathbf { X } ^ { * } )$ .   
+8: Update $\mathbf { X } ^ { * } [ i , j , k ] \gets \mathrm { a r g } \operatorname* { m a x } _ { z \in R } p _ { \mathrm { C N N } } [ i , j , k , z ]$   
+9: end for   
+10: end for   
+11: end for
+
+![](images/9cd8d3d299ff4ff7eed038f5178135903d715fc07f90ba51f32ac6907fe381b5.jpg)  
+Figure 5: The bits-per-dimension distributions of purified images from FGSM adversarial examples. We tested two purification methods, L-BFGS-B and greedy decoding, the latter of which is used in PixelDefend. A good purification method should give images that have lower bits per dimension compared to FGSM images and ideally similar bits per dimension compared to clean ones.
+
+# 4.1 RETURNING IMAGES TO THE TRAINING DISTRIBUTION
+
+The basic idea behind PixelDefend is to purify input images, by making small changes to them in order to move them back towards the training distribution, i.e., move the images towards a highprobability region. We then classify the purified image using any existing classifier. As the example in Figure 4 shows, the purified images can usually be classified correctly.
+
+Formally, we have training image distribution $p ( \mathbf { X } )$ , and input image $\mathbf { X }$ of resolution $I \times J$ with $\mathbf { X } [ i , j , k ]$ the pixel at location $( i , j )$ and channel $k \in \{ 1 , \cdots , C \}$ . We wish to find an image $\mathbf { X } ^ { * }$ that maximizes $p ( \mathbf { X } )$ subject to the constraint that $\mathbf { X } ^ { * }$ is within the $\epsilon _ { \mathrm { d e f e n d } }$ -ball of $\mathbf { X }$ :
+
+$$
+\begin{array} { c } { \displaystyle { \operatorname* { m a x } _ { \mathbf { X } ^ { * } } p ( \mathbf { X } ^ { * } ) } } \\ { \mathrm { s . t . } \quad \left\| \mathbf { X } ^ { * } - \mathbf { X } \right\| _ { \infty } \leq \epsilon _ { \mathrm { d e f e n d } } . } \end{array}
+$$
+
+Here $\epsilon _ { \mathrm { d e f e n d } }$ reflects a trade-off, since large $\epsilon _ { \mathrm { d e f e n d } }$ may change the meaning of $\mathbf { X }$ while small ϵdefend may not be sufficient for returning $\mathbf { X }$ to the correct distribution. In practice, we choose ϵdefend to be some value that overestimates $\epsilon _ { \mathrm { a t t a c k } }$ but still keeps high accuracies on clean images. As in Section 3, we approximate $p ( \mathbf { X } )$ with the PixelCNN distribution $p _ { \mathrm { C N N } } ( \mathbf { X } )$ , which is trained on the same training set as the classifier.
+
+However, exact constrained optimization of $p _ { \mathrm { C N N } } ( \mathbf { X } )$ is computationally intractable. Surprisingly, even gradient-based optimization faces great difficulty on that problem. We found that one advanced methods in gradient-based constrained optimization, L-BFGS-B (Byrd et al., 1995) (we use the scipy implementation based on Zhu et al. (1997)), actually decreases $p _ { \mathrm { C N N } } ( \mathbf { X } )$ for most random initializations within the $\epsilon _ { \mathrm { d e f e n d } }$ -ball.
+
+For efficient optimization, we instead use a greedy technique described in Algorithm 1, which is similar to the greedy decoding process typically used in sequence-to-sequence models (Sutskever et al., 2014). The method is similar to generating images from PixelCNN, with the additional constraint that the generated image should be within an $\epsilon _ { \mathrm { d e f e n d } }$ -ball of a perturbed image. As an autoregressive model, PixelCNN is slow in image generation. Nonetheless, by caching redundant calculation, Ramachandran et al. (2017) proposes a very fast generation algorithm for PixelCNN. In our experiments, adoption of Ramachandran et al. (2017)’s method greatly increases the speed of PixelDefend. For CIFAR-10 images, PixelDefend on average processes 3.6 images per second on one NVIDIA TITAN Xp GPU.
+
+![](images/9541592e4ea68f210ba02c6038cd122b98e392a4788d54a8ad822f5b222e631c.jpg)  
+Figure 6: ROC curves showing the efficacy of using $p$ -values as scores to detect adversarial examples. For computing the ROC, we assign negative labels to training images and positive labels to adversarial images (or clean test images). (a) Original adversarial examples. (b) Purified adversarial examples after PixelDefend.
+
+To show the effectiveness of this greedy method compared to L-BFGS-B, we take the first 10 images from CIFAR-10 test set, attack them by FGSM with $\epsilon _ { \mathrm { a t t a c k } } = 8$ , and purify them with L-BFGS-B and PixelDefend respectively. We used random start points for L-BFGS-B and repeated 100 times for each image. As depicted in Figure 5, most L-BFGS-B attempts failed at minimizing the bits per dimension of FGSM adversarial examples. Because of the rugged gradient landscape of PixelCNN, L-BFGS-B even results in images that have lower probabilities. In contrast, PixelDefend works much better in increasing the probabilities of purified images, although their probabilities are still lower compared to clean ones.
+
+In Figure 6 and Figure 7, we empirically show that after PixelDefend, purified images are more likely to be drawn from the training distribution. Specifically, Figure 6 shows that the detecting power of $p$ -values greatly decreases for purified images. For DeepFool and CW examples, purification makes them barely distinguishable from normal samples of the data distribution. This is also manifested by Figure 7, as the $p$ -value distributions of purified examples are closer to uniform. Visually, purified images indeed look much cleaner than adversarially perturbed ones. In Appendix E, we provide sampled purified images from Fashion MNIST and CIFAR-10.
+
+# 4.2 ADAPTIVE PIXELDEFEND
+
+One concern with the approach of purifying images is what happens when we purify a clean image. More generally, we will never know $\epsilon _ { \mathrm { a t t a c k } }$ and if we set $\epsilon _ { \mathrm { d e f e n d } }$ too large for a given attack, then we will modify all images to become the mode image, which would mostly result in misclassifications. One way to avoid this problem is to tune $\epsilon _ { \mathrm { d e f e n d } }$ adaptively based on the probability of the input image under the generative model. In this way, images that already have high probability under the training distribution would have a very low $\epsilon _ { \mathrm { d e f e n d } }$ preventing significant modification, while low probability images would have a high $\epsilon _ { \mathrm { d e f e n d } }$ thus allowing significant modifications. We implemented a very simple thresholding version of this, which sets $\epsilon _ { \mathrm { d e f e n d } }$ to zero if the input image probability is below a threshold value, and otherwise leaves it fixed at a manually chosen setting. In practice, we set this threshold based on knowledge of the set of possible attacks, so strictly speaking, the adaptive version of our technique is no longer attack-agnostic.
+
+![](images/d8ac12f892812f533f4b6b1ae8304e24aeb92180d4e2963b938837985ba1b75c.jpg)  
+Figure 7: The distributions of $p$ -values under the PixelCNN model after PixelDefend purification.
+
+Table 1: Fashion MNIST $( \epsilon _ { \mathrm { a t t a c k } } = 8 / 2 5 $ , $\epsilon _ { \mathrm { d e f e n d } } = 3 2$ )   
+
+<table><tr><td>NETWORK</td><td>TRAINING TECHNIQUE</td><td>CLEAN</td><td>RAND</td><td>FGSM</td><td>BIM</td><td>DEEP FOOL</td><td>Cw</td><td>STRONGEST ATTACK</td></tr><tr><td>ResNet</td><td>Normal</td><td>93/93</td><td>89/71</td><td>38/24</td><td>00/00</td><td>06/06</td><td>20/01</td><td>00/00</td></tr><tr><td>VGG</td><td>Normal</td><td>92/92</td><td>91/87</td><td>73/58</td><td>36/08</td><td>49/14</td><td>43/23</td><td>36/08</td></tr><tr><td rowspan="5">ResNet</td><td>Adversarial FGSM</td><td>93/93</td><td>92/89</td><td>85/85</td><td>51/00</td><td>63/07</td><td>67/21</td><td>51/00</td></tr><tr><td>Adversarial BIM</td><td>92/91</td><td>92/91</td><td>84/79</td><td>76/63</td><td>82/72</td><td>81/70</td><td>76/63</td></tr><tr><td>Label Smoothing</td><td>93/93</td><td>91/76</td><td>73/45</td><td>16/00</td><td>29/06</td><td>33/14</td><td>16/00</td></tr><tr><td>Feature Squeezing</td><td>84/84</td><td>84/70</td><td>70/28</td><td>56/25</td><td>83/83</td><td>83/83</td><td>56/25</td></tr><tr><td>Adversarial FGSM + Feature Squeezing</td><td>88/88</td><td>87/82</td><td>80/77</td><td>70/46</td><td>86/82</td><td>84/85</td><td>70/46</td></tr><tr><td>ResNet</td><td>Normal +PixelDefend</td><td>88/88</td><td>88/89</td><td>85/74</td><td>83/76</td><td>87/87</td><td>87/87</td><td>83/74</td></tr><tr><td>VGG</td><td>Normal+PixelDefend</td><td>89/89</td><td>89/89</td><td>87/82</td><td>85/83</td><td>88/88</td><td>88/88</td><td>85/82</td></tr><tr><td rowspan="2">ResNet</td><td>Adversarial FGSM +PixelDefend</td><td>90/89</td><td>91/90</td><td>88/82</td><td>85/76</td><td>90/88</td><td>89/88</td><td>85/76</td></tr><tr><td>Adversarial FGSM +Adaptive PixelDefend</td><td>91/91</td><td>91/91</td><td>88/88</td><td>85/84</td><td>89/90</td><td>89/84</td><td>85/84</td></tr></table>
+
+# 4.3 PIXELDEFEND RESULTS
+
+We carried out a comprehensive set of experiments to test various defenses versus attacks. Detailed information on experimental settings is provided in Appendix B. All experimental results are summarized in Tab. 1 and Tab. 2. In the upper part of the tables, we show how the various baseline defenses fare against each of the attacks, while in the lower part of the tables we show how our PixelDefend technique works. Each table cell contains accuracies on adversarial examples generated with different $\epsilon _ { \mathrm { a t t a c k } }$ . More specifically, for Fashion MNIST (Tab. 1), we tried $\epsilon _ { \mathrm { a t t a c k } } = 8$ and 25. The cells in Tab. 1 is formated as $x / y$ , where $x$ denotes the accuracy $( \% )$ on images attacked with $\epsilon _ { \mathrm { a t t a c k } } = 8$ , while $y$ denotes the accuracy when $\epsilon _ { \mathrm { a t t a c k } } = 2 5 $ . For CIFAR-10 (Tab. 2), we tried $\epsilon _ { \mathrm { a t t a c k } } = 2$ , 8, and 16, and the cells are formated in a similar way. We use the same $\epsilon _ { \mathrm { d e f e n d } }$ for different ϵattack’s to show that PixelDefend is insensitive to ϵattack.
+
+From the tables we observe that adversarial training successfully defends against the basic FGSM attack, but cannot defend against the more advanced ones. This is expected, as training on simple adversarial examples does not guarantee robustness to more complicated attacking techniques. Consistent with Madry et al. (2017), adversarial training with BIM examples is more successful at preventing a wider spectrum of attacks. For example, it improves the accuracy on strongest attack from $2 \%$ to $32 \%$ on CIFAR-10 when $\epsilon _ { \mathrm { a t t a c k } } = 8$ . But the numbers are still not ideal even with respect to BIM attack itself. As in Tab. 2, it only gets $6 \%$ on BIM and $8 \%$ on CW when $\epsilon _ { \mathrm { a t t a c k } } = 1 6$ . We also observe that label smoothing, which learns smoothed predictions so that the gradient $\nabla _ { \mathbf { X } } L ( \mathbf { X } , y )$ becomes very small, is only effective against simple FGSM attack. Model-agnostic methods, such as feature squeezing, can be combined with other defenses for strengthened performance. We observe that combining it with adversarial training indeed makes it more robust. Actually, Tab. 1 and Tab. 2 show that feature squeezing combined with adversarial training dominates using feature squeezing along in all settings. It also gets good performance on DeepFool and CW attacks. However, for iterative attacks with larger perturbations, i.e., BIM, feature squeezing performs poorly. On CIFAR-10, it only gets $2 \%$ and $0 \%$ accuracy on BIM with $\epsilon _ { \mathrm { a t t a c k } } = 8$ and 16 respectively.
+
+Table 2: CIFAR-10 $\mathcal { C } _ { \mathrm { a t t a c k } } = 2 / 8 / 1 6$ , $\epsilon _ { \mathrm { d e f e n d } } = 1 6$ )   
+
+<table><tr><td>NETWORK</td><td>TRAINING TECHNIQUE</td><td>CLEAN</td><td>RAND</td><td>FGSM</td><td>BIM</td><td>DEEP FOOL</td><td>CW</td><td>STRONGEST ATTACK</td></tr><tr><td>ResNet</td><td>Normal</td><td>92/92/92</td><td>92/87/76</td><td>33/15/11</td><td>10/00/00</td><td>12/06/06</td><td>07/00/00</td><td>07/00/00</td></tr><tr><td>VGG</td><td>Normal</td><td>89/89/89</td><td>89/88/80</td><td>60/46/30</td><td>44/02/00</td><td>57/25/11</td><td>37/00/00</td><td>37/00/00</td></tr><tr><td rowspan="5">ResNet</td><td>Adversarial FGSM</td><td>91/91/91</td><td>90/88/84</td><td>88/91/91</td><td>24/07/00</td><td>45/00/00</td><td>20/00/07</td><td>20/00/00</td></tr><tr><td>Adversarial BIM</td><td>87/87/87</td><td>87/87/86</td><td>80/52/34</td><td>74/32/06</td><td>79/48/25</td><td>76/42/08</td><td>74/32/06</td></tr><tr><td>Label Smoothing</td><td>92/92/92</td><td>91/88/77</td><td>73/54/28</td><td>59/08/01</td><td>56/20/10</td><td>30/02/02</td><td>30/02/01</td></tr><tr><td>Feature Squeezing</td><td>84/84/84</td><td>83/82/76</td><td>31/20/18</td><td>13/00/00</td><td>75/75/75</td><td>78/78/78</td><td>13/00/00</td></tr><tr><td>Adversarial FGSM + Feature Squeezing</td><td>86/86/86</td><td>85/84/81</td><td>73/67/55</td><td>55/02/00</td><td>85/85/85</td><td>83/83/83</td><td>55/02/00</td></tr><tr><td>ResNet</td><td>Normal+PixelDefend</td><td>85/85/88</td><td>82/83/84</td><td>73/46/24</td><td>71/46/25</td><td>80/80/80</td><td>78/78/78</td><td>71/46/24</td></tr><tr><td>VGG</td><td>Normal+PixelDefend</td><td>82/82/82</td><td>82/82/84</td><td>80/62/52</td><td>80/61/48</td><td>81/76/76</td><td>81/79/79</td><td>80/61/48</td></tr><tr><td rowspan="2">ResNet</td><td>Adversarial FGSM +PixelDefend</td><td>88/88/86</td><td>86/86/87</td><td>81/68/67</td><td>81/69/56</td><td>85/85/85</td><td>84/84/84</td><td>81/69/56</td></tr><tr><td>Adversarial FGSM +Adaptive PixelDefend</td><td>90/90/90</td><td>86/87/87</td><td>81/70/67</td><td>81/70/56</td><td>82/81/82</td><td>81/80/81</td><td>81/70/56</td></tr></table>
+
+PixelDefend, our model-agnostic and attack-agnostic method, performs well on different classifiers (ResNet and VGG) and different attacks without modification. In addition, we can see that augmenting basic adversarial training with PixelDefend can sometimes double the accuracies. We hypothesize that the purified images from PixelDefend are still not perfect, and adversarially trained networks have more toleration for perturbations. This also corroborates the plausibility and benefit of combining PixelDefend with other defenses.
+
+Furthermore, PixelDefend can simultaneously obtain accuracy above $70 \%$ for all other attacking techniques, while ensuring that performance on clean images only declines slightly. Models with PixelDefend consistently outperform other methods with respect to the strongest attack. On Fashion MNIST, PixelDefend methods improve the accuracy on strongest attack from $76 \%$ to $8 5 \%$ and $63 \%$ to $84 \%$ . On CIFAR-10, the improvements are even more significant, i.e., from $74 \%$ to $81 \%$ , $32 \%$ to $70 \%$ and $6 \%$ to $56 \%$ , for $\epsilon _ { \mathrm { a t t a c k } } = 2$ , 8, and 16 respectively. In a security-critical scenario, the weakest part of a system determines the overall reliability. Therefore, the outstanding performance of PixelDefend on the strongest attack makes it a valuable and useful addition for improving AI security.
+
+# 4.4 END-TO-END ATTACK OF PIXELDEFEND
+
+A natural question that arises is whether we can generate a new class of adversarial examples targeted specifically at the combined PixelDefend architecture of first purifying the image and then using an existing classifier to predict the label of the purified image. We have three pieces of empirical evidence to believe that such adversarial examples are hard to find in general. First, we attempted to apply the iterative BIM attack to an end-to-end differentiable version of PixelDefend generated by unrolling the PixelCNN purification process. However we found the resulting network was too deep and led to problems with vanishing gradients (Bengio et al., 1994), resulting in adversarial images that were identical to the original images. Moreover, attacking the whole system is very time consuming. Empirically, it took about 10 hours to generate 100 attacking images with one TITAN $\mathrm { X p }$ GPU which failed to fool PixelDefend. Secondly, we found the optimization problem in Eq. (4.1) was not amenable to gradient descent, as indicated in Figure 5. This makes gradient-based attacks especially difficult. Last but not least, the generative model and classifier are trained separately and have independent parameters. Therefore, the perturbation direction that leads to higher probability images has a smaller correlation with the perturbation direction that results in misclassification. Accordingly, it is harder to find adversarial examples that can fool both of them together. However, we will open source our codes and look forward to any possible attack from the community.
+
+# 5 RELATED WORK
+
+Most recent work on detecting adversarial examples focuses on adding an outlier class detection module to the classifier, such as Grosse et al. (2017), Gong et al. (2017) and Metzen et al. (2017). Those methods require the classification model to be changed, and are thus not model-agnostic. Feinman et al. (2017) also presents a detection method based on kernel density estimation and Bayesian neural network uncertainty. However, Carlini & Wagner (2017a) shows that all those methods can be bypassed.
+
+Grosse et al. (2017) also studied the distribution of adversarial examples from a statistical testing perspective. They reported the same discovery that adversarial examples are outside of the training distribution. However, our work is different from theirs in several important aspects. First, the kernel-based two-sample test used in their paper needs a large number of suspicious inputs, while our method only requires one data point. Second, they mainly tested on first-order methods such as FGSM and JSMA (Papernot et al., 2016b). We show the efficacy of PixelCNN on a wider range of attacking methods (see Figure 3), including both first-order and iterative methods. Third, we further demonstrate that random perturbed inputs are also outside of the training distribution.
+
+Some other work has focused on modifying the classifier architecture to increase its robustness, e.g., Gu & Rigazio (2014), Cisse et al. (2017) and Nayebi & Ganguli (2017). Although they have witnessed some success, such modifications of models might limit their representative power and are also not model-agnostic.
+
+Our basic idea of moving points to higher-density regions is also present in other machine learning methods not specifically designed for handling adversarial data; for example, the manifold denoising method of Hein & Maier (2007), the direct density gradient estimation of Sasaki et al. (2014), and the denoising autoencoders of Vincent et al. (2008) all move data points from low to high-density regions. In the future some of these methods could be adapted to amortize the purification process directly, that is, to learn a purification network.
+
+# 6 CONCLUSION
+
+In this work, we discovered that state-of-the-art neural density models, e.g., PixelCNN, can detect small perturbations with high sensitivity. This sensitivity broadly exists for a large number of perturbations generated with different methods. An interesting fact is that PixelCNN is only sensitive in one direction—it is relatively easy to detect perturbations that lead to lower probabilities rather than higher probabilities.
+
+Based on the sensitivity of PixelCNN, we utilized statistical hypothesis testing to verify that adversarial examples lie outside of the training distribution. With the permutation test, we give exact $p$ -values which can be used as a uncertainty measure for detecting outlier perturbations.
+
+Furthermore, we make use of the sensitivity of generative models to explore the idea of purifying adversarial examples. We propose the PixelDefend algorithm, and experimentally show that returning adversarial examples to high probability regions of the training distribution can significantly decrease their damage to classifiers. Different from many other defensive techniques, PixelDefend is model-agnostic and attack-agnostic, which means it can be combined with other defenses to improve robustness without modifying the classification model. As a result PixelDefend is a practical and effective defense against adversarial inputs.
+
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+Nicolas Papernot, Patrick McDaniel, Xi Wu, Somesh Jha, and Ananthram Swami. Distillation as a defense to adversarial perturbations against deep neural networks. In Security and Privacy (SP), 2016 IEEE Symposium on, pp. 582–597. IEEE, 2016c.
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+Prajit Ramachandran, Tom Le Paine, Pooya Khorrami, Mohammad Babaeizadeh, Shiyu Chang, Yang Zhang, Mark A Hasegawa-Johnson, Roy H Campbell, and Thomas S Huang. Fast generation for convolutional autoregressive models. arXiv preprint arXiv:1704.06001, 2017.
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+Tim Salimans, Andrej Karpathy, Xi Chen, and Diederik P Kingma. Pixelcnn $^ { + + }$ : Improving the pixelcnn with discretized logistic mixture likelihood and other modifications. arXiv preprint arXiv:1701.05517, 2017.
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+Han Xiao, Kashif Rasul, and Roland Vollgraf. Fashion-mnist: a novel image dataset for benchmarking machine learning algorithms, 2017.   
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+
+# APPENDIX A ON RANDOM PERTURBATIONS
+
+One may observe from Figure 3(b) that random perturbations have very low $p$ -values, and thus also live outside of the high density area. Although many classifiers are robust to random noise, it is not a property granted by the dataset. The fact is that robustness to random noise could be from model inductive bias, and there exist classifiers which have high generalization performance on clean images, but can be attacked by small random perturbations.
+
+It is easy to construct a concrete classifier that are susceptible to random perturbations. Our ResNet on CIFAR-10 gets $9 2 . 0 \%$ accuracy on the test set and $8 7 . 3 \%$ on randomly perturbed test images with $\epsilon _ { \mathrm { a t t a c k } } = 8$ . According to our PixelCNN, 175 of 10000 test images have a bits per dimension (BPD) larger than 4.5, while the number for random images is 9874. Therefore, we can define a new classifier
+
+$$
+\mathrm { R e s N e t ^ { \prime } ( X ) } \triangleq \left\{ \begin{array} { l l } { \mathrm { R e s N e t ( X ) } , } & { \mathrm { B P D } ( \mathbf { X } ) < 4 . 5 } \\ { \mathrm { r a n d o m \ l a b e l } , } & { \mathrm { B P D } ( \mathbf { X } ) \geq 4 . 5 } \end{array} \right. ,
+$$
+
+which will get roughly $9 2 \% \times 9 8 2 5 / 1 0 0 0 0 + 1 0 \% \times 1 7 5 / 1 0 0 0 0 \approx 9 0 . 6 \%$ accuracy on the test set, while only $8 7 . 3 \% \times 1 2 6 / 1 0 0 0 0 + 1 0 \% \times 9 8 7 4 / 1 0 0 0 0 \approx 1 1 . 0 \%$ accuracy on the randomly perturbed images. This classifier has comparable generalization performance to the original ResNet, but will give incorrect labels to most randomly perturbed images.
+
+# APPENDIX B EXPERIMENTAL SETTINGS
+
+Adversarial Training We have tested adversarial training with both FGSM and BIM examples. During training, we take special care of the label leaking problem as noted in Kurakin et al. (2016)— we use the predicted labels of the model to generate adversarial examples, instead of using the true labels. This prevents the adversarially trained network to perform better on adversarial examples than clean images by simply retrieving ground-truth labels. Following Kurakin et al. (2016), we also sample $\epsilon _ { \mathrm { a t t a c k } }$ from a truncated Gaussian distribution for generating FGSM or BIM adversarial examples, so that the adversarially trained network won’t overfit to any specific $\epsilon _ { \mathrm { a t t a c k } }$ . This is different from Madry et al. (2017), where the authors train and test with the same ϵattack.
+
+For Fashion MNIST experiments, we randomly sample $\epsilon _ { \mathrm { a t t a c k } }$ from $\mathcal { N } ( 0 , \delta )$ , take the absolute value and truncate it to $[ 0 , 2 \delta ]$ , where $\delta \ : = \ : 8$ or 25. For CIFAR-10 experiments, we follow the same procedure but fix $\delta = 8$ .
+
+Feature Squeezing For implementing the feature squeezing defense, we reduce the number of colors to 8 on Fashion MNIST, and use 32 colors for CIFAR-10. The numbers are chosen to make sure color reduction will not lead to significant deterioration of image quality. After color depth reduction, we apply a $2 \times 2$ median filter with reflective paddings, since it is reported in $\mathrm { X u }$ et al. (2017b) to be most effective for preventing CW attacks.
+
+Models We use ResNet (62-layer) and VGG (16-layer) as classifiers. In our experiments, normally trained networks have the same architectures as adversarially trained networks. Since the images of Fashion MNIST contain roughly one quarter values of those of CIFAR-10, we use a smaller network for classifying Fashion MNIST. More specifically, we reduce the number of feature maps for Fashion MNIST to 1/4 while keeping the same depths. In practive, VGG is more robust than ResNet due to using of dropout layers. The network architecture details are described in Appendix C. For the PixelCNN generative model, we adopted the implementation of $\mathrm { P i x e l C N N + + }$ (Salimans et al., 2017), but modified the output from mixture of logistic distributions to softmax. The feature maps are also reduced to 1/4 for training PixelCNN on Fashion MNIST.
+
+Adaptive Threshold We chose the adaptive threshold discussed in Section 4.2 using validation data. We set the threshold at the lowest value which did not decrease the performance of the strongest adversary. For Fashion MNIST, the threshold of bits per dimension was set to 1.8, and for CIFAR-10 the number was 3.2. As a reference, the mean value of bits per dimension for Fashion MNIST test images is 2.7 and for CIFAR-10 is 3.0. However, we admit that using a validation set to choose the best threshold makes the adaptive version of PixelDefend not strictly attack-agnostic.
+
+# APPENDIX C IMAGE CLASSIFIER ARCHITECTURES∗
+
+C.1 RESNET CLASSIFIER FOR CIFAR-10 & FASHION MNIST   
+
+<table><tr><td rowspan=1 colspan=1>NAME</td><td rowspan=1 colspan=2>CONFIGURATION</td></tr><tr><td rowspan=1 colspan=1>Initial Layer</td><td rowspan=1 colspan=2>conv (filter size: 3 × 3,feature maps: 16 (4), stride size: 1 × 1)</td></tr><tr><td rowspan=1 colspan=1>Residual Block 1</td><td rowspan=1 colspan=1>batch normalization &amp; leaky reluconv (filter size: 3 × 3, feature maps: 16 (4), stride size: 1 × 1)batch normalization &amp;leaky reluconv (filter size: 3 × 3, feature maps: 16 (4), stride size: 1 × 1)residual addition</td><td rowspan=1 colspan=1>×10 times</td></tr><tr><td rowspan=2 colspan=1>Residual Block 2</td><td rowspan=1 colspan=2>batch normalization&amp; leaky reluconv (filter size: 3 × 3, feature maps: 32 (8), stride size: 2 × 2)batch normalization&amp;leakyreluconv (filter size: 3 × 3, feature maps: 32 (8), stride size: 1 × 1)average pooling &amp; padding &amp; residual addition</td></tr><tr><td rowspan=1 colspan=1>batch normalization &amp; leaky reluconv (filter size: 3 × 3, feature maps: 32(8), stride size: 1 × 1)batch normalization&amp;leaky reluconv (filter size: 3 × 3, feature maps: 32(8), stride size: 1 × 1)residual addition</td><td rowspan=1 colspan=1>×9 times</td></tr><tr><td rowspan=2 colspan=1>Residual Block 3</td><td rowspan=1 colspan=2>batch normalization&amp; leaky reluconv (filter size: 3 × 3, feature maps: 64 (16),stride size: 2 × 2)batch normalization &amp; leaky reluconv (filter size: 3 × 3,feature maps: 64(16), stride size:1 × 1)average pooling &amp; padding &amp; residual addition</td></tr><tr><td rowspan=1 colspan=1>batch normalization &amp; leaky reluconv(filter size: 3 × 3,feature maps: 64(16), stride size:1 × 1)batch normalization &amp; leaky reluconv(filter size: 3 × 3,feature maps: 64(16), stride size:1 × 1)residual addition</td><td rowspan=1 colspan=1>×9 times</td></tr><tr><td rowspan=1 colspan=1>Pooling Layer</td><td rowspan=1 colspan=2>batch normalization &amp; leaky relu &amp; average pooling</td></tr><tr><td rowspan=1 colspan=1>Output Layer</td><td rowspan=1 colspan=2>fc_10 &amp; softmax</td></tr></table>
+
+# C.2 VGG CLASSIFIER FOR CIFAR-10 & FASHION MNIST
+
+<table><tr><td rowspan=1 colspan=1>NAME</td><td rowspan=1 colspan=2>CONFIGURATION</td></tr><tr><td rowspan=2 colspan=1>Feature Block 1</td><td rowspan=1 colspan=1>conv (filter size: 3 × 3, feature maps: 16 (4), stride size: 1 × 1)batch normalization &amp; relu</td><td rowspan=1 colspan=1>×2 times</td></tr><tr><td rowspan=1 colspan=2>max pooling (stride size: 2 × 2)</td></tr><tr><td rowspan=2 colspan=1>Feature Block 2</td><td rowspan=1 colspan=1>conv (filter size: 3 × 3, feature maps: 128 (32), stride size: 1 × 1)batch normalization &amp; relu</td><td rowspan=1 colspan=1>×2 times</td></tr><tr><td rowspan=1 colspan=1>max pooling (stride size:2 × 2)</td><td rowspan=1 colspan=1></td></tr><tr><td rowspan=2 colspan=1>Feature Block 3</td><td rowspan=1 colspan=1>conv (filter size: 3 × 3, feature maps: 512 (128), stride size: 1 × 1)batch normalization &amp; relu</td><td rowspan=1 colspan=1>×3 times</td></tr><tr><td rowspan=1 colspan=2>max pooling (stride size: 2 × 2)</td></tr><tr><td rowspan=2 colspan=1>Feature Block 4</td><td rowspan=1 colspan=1>conv (filter size: 3 × 3, feature maps: 512 (128), stride size: 1 × 1)batch normalization &amp; relu</td><td rowspan=1 colspan=1>×3 times</td></tr><tr><td rowspan=1 colspan=2>max pooling (stride size:2× 2) &amp; flatten</td></tr><tr><td rowspan=1 colspan=1>Classifier Block</td><td rowspan=1 colspan=2>dropout &amp; fc_512(128)&amp; reludropout&amp;fc_1O&amp; softmax</td></tr></table>
+
+# APPENDIX D SAMPLED IMAGES FROM PIXELCNN
+
+# D.1 FASHION MNIST
+
+![](images/a52c9a517dca2176580fff1389c7f949da2bad0c291e18c24545fefdf06d727d.jpg)  
+Figure 8: True and generated images from Fashion MNIST. The upper part shows true images sampled from the dataset while the bottom shows generated images from PixelCNN.
+
+D.2 CIFAR-10
+
+![](images/31c1a6c112e70b99ae7152ef1092a77c22d51e7cabee2a609f7352a85edd4c55.jpg)  
+Figure 9: True and generated images from CIFAR-10. The upper part shows true images sampled from the dataset while the bottom part shows generated images from PixelCNN.
+
+# APPENDIX E SAMPLED PURIFIED IMAGES FROM PIXELDEFEND
+
+# E.1 FASHION MNIST
+
+![](images/45672d7a01c00ca3bc643f6903a64a18eec8b52de727d98e8a3c13f4bba2bbb2.jpg)  
+Figure 10: The upper part shows adversarial images generated from FGSM attack while the bottom part shows corresponding purified images after PixelDefend. Here $\epsilon _ { \mathrm { a t t a c k } } = 2 5 $ and $\epsilon _ { \mathrm { d e f e n d } } = 3 2 $ .
+
+# E.2 CIFAR-10
+
+![](images/5a6af051392cefa48293cd2d5b943cf4e7dfb45eaebf9508ebc2ad16fec1b77e.jpg)  
+Figure 11: The upper part shows adversarial images generated from FGSM attack while the bottom part shows corresponding purified images by PixelDefend. Here $\epsilon _ { \mathrm { a t t a c k } } = 8$ and $\epsilon _ { \mathrm { d e f e n d } } = 1 6$ .

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+        "text": "PIXELDEFEND: LEVERAGING GENERATIVE MODELS TO UNDERSTAND AND DEFEND AGAINST ADVERSARIAL EXAMPLES ",
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+        "text": "Yang Song   \nStanford University   \nyangsong@cs.stanford.edu ",
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+        "text": "Taesup Kim Université de Montréal taesup.kim@umontreal.ca ",
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+        "text": "Sebastian Nowozin Microsoft Research nowozin@microsoft.com ",
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+        "text": "Stefano Ermon Stanford University ermon@cs.stanford.edu ",
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+        "text": "Nate Kushman   \nMicrosoft Research   \nnkushman@microsoft.com ",
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+        "type": "text",
+        "text": "ABSTRACT ",
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+        "text": "Adversarial perturbations of normal images are usually imperceptible to humans, but they can seriously confuse state-of-the-art machine learning models. What makes them so special in the eyes of image classifiers? In this paper, we show empirically that adversarial examples mainly lie in the low probability regions of the training distribution, regardless of attack types and targeted models. Using statistical hypothesis testing, we find that modern neural density models are surprisingly good at detecting imperceptible image perturbations. Based on this discovery, we devised PixelDefend, a new approach that purifies a maliciously perturbed image by moving it back towards the distribution seen in the training data. The purified image is then run through an unmodified classifier, making our method agnostic to both the classifier and the attacking method. As a result, PixelDefend can be used to protect already deployed models and be combined with other model-specific defenses. Experiments show that our method greatly improves resilience across a wide variety of state-of-the-art attacking methods, increasing accuracy on the strongest attack from $63 \\%$ to $84 \\%$ for Fashion MNIST and from $32 \\%$ to $70 \\%$ for CIFAR-10. ",
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+        "text": "1 INTRODUCTION ",
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+        "text": "Recent work has shown that small, carefully chosen modifications to the inputs of a neural network classifier can cause the model to give incorrect labels (Szegedy et al., 2013; Goodfellow et al., 2014). This weakness of neural network models is particularly surprising because the modifications required are often imperceptible, or barely perceptible, to humans. As deep neural networks are being deployed in safety-critical applications such as self-driving cars (Amodei et al., 2016), it becomes increasingly important to develop techniques to handle these kinds of inputs. ",
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+        "text": "Rethinking adversarial examples The existence of such adversarial examples seems quite surprising. A neural network classifier can get super-human performance (He et al., 2015) on clean test images, but will give embarrassingly wrong predictions on the same set of images if some imperceptible noise is added. What makes this noise so special to deep neural networks? ",
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+        "text": "In this paper, we propose and empirically evaluate the following hypothesis: Even though they have very small deviations from clean images, adversarial examples largely lie in the low probability regions of the distribution that generated the data used to train the model. Therefore, they fool classifiers mainly due to covariate shift. This is analogous to training models on MNIST (LeCun et al., 1998) but testing them on Street View House Numbers (Netzer et al., 2011). ",
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+        "text": "To study this hypothesis, we first need to estimate the probability density of the underlying training distribution. To this end, we leverage recent developments in generative models. Specifically, we choose a PixelCNN (van den Oord et al., 2016b) model for its state-of-the-art performance in modeling image distributions (van den Oord et al., 2016a; Salimans et al., 2017) and tractability of evaluating the data likelihood. In the first part of the paper, we show that a well-trained PixelCNN generative model is very sensitive to adversarial inputs, typically giving them several orders of magnitude lower likelihoods compared to those of training and test images. ",
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+    {
+        "type": "text",
+        "text": "Detecting adversarial examples An important step towards handling adversarial images is the ability to detect them. In order to catch any kind of threat, existing work has utilized confidence estimates from Bayesian neural networks (BNNs) or dropout (Li & Gal, 2017; Feinman et al., 2017). However, if their model is misspecified, the uncertainty estimates can be affected by covariate shift (Shimodaira, 2000). This is problematic in an adversarial setting, since the attacker might be able to make use of the inductive bias from the misspecified classifier to bypass the detection. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Protection against the strongest adversary requires a pessimistic perspective—our assumption is that the classifier cannot give reliable predictions for any input outside of the training distribution. Therefore, instead of relying on label uncertainties given by the classifier, we leverage statistical hypothesis testing to detect any input not drawn from the same distribution as training images. ",
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+    {
+        "type": "text",
+        "text": "Specifically, we first compute the probabilities of all training images under the generative model. Afterwards, for a novel input we compute the probability density at the input and evaluate its rank (in ascending order) among the density values of all training examples. Next, the rank can be used as a test statistic and gives us a $p$ -value for whether or not the image was drawn from the training distribution. This method is general and practical and we show that the $p$ -value enables us to detect adversarial images across a large number of different attacking methods with high probability, even when they differ from clean images by only a few pixel values. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Purifying adversarial examples Since adversarial examples are generated from clean images by adding imperceptible perturbations, it is possible to decontaminate them by searching for more probable images within a small distance of the original ones. By limiting the $L ^ { \\infty }$ distance1, this image purification procedure generates only imperceptible modifications to the original input, so that the true labels of the purified images remain the same. The resulting purified images have higher probability under the training distribution, so we can expect that a classifier trained on the clean images will have more reliable predictions on the purified images. Moreover, for inputs which are not corrupted by adversarial perturbations the purified results remain in a high density region. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "We use this intuition to build PixelDefend, an image purification procedure which requires no knowledge of the attack nor the targeted classifier. PixelDefend approximates the training distribution using a PixelCNN model. The constrained optimization problem of finding the highest probability image within an $\\epsilon$ -ball of the original is computationally intractable, however, so we approximate it using a greedy decoding procedure. Since PixelDefend does not change the classification model, it can be combined with other adversarial defense techniques, including adversarial training (Goodfellow et al., 2014), to provide synergistic improvements. We show experimentally that PixelDefend performs exceptionally well in practice, leading to state-of-the art results against a large number of attacks, especially when combined with adversarial training. ",
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+    {
+        "type": "text",
+        "text": "Contributions Our main contributions are as follows: ",
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+    },
+    {
+        "type": "text",
+        "text": "• We show that generative models can be used for detecting adversarially perturbed images and observe that most adversarial examples lie in low probability regions.   \n• We introduce a novel family of methods for defending against adversarial attacks based on the idea of purification.   \nWe show that a defensive technique from this family, PixelDefend, can achieve state-of-theart results on a large number of attacking techniques, improving the accuracy against the strongest adversary on the CIFAR-10 dataset from $32 \\%$ to $70 \\%$ . ",
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+    {
+        "type": "text",
+        "text": "2 BACKGROUND ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "2.1 ATTACKING METHODS ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "Given a test image $\\mathbf { X }$ , an attacking method tries to find a small perturbation $\\pmb { \\Delta }$ with $\\| \\pmb { \\Delta } \\| _ { \\infty } \\le \\epsilon _ { \\mathrm { a t t a c k } }$ such that a classifier $f$ gives different predictions on $\\mathbf { X } ^ { a d v } \\triangleq \\mathbf { X } + \\Delta$ and $\\mathbf { X }$ . Here colors in the image are represented by integers from 0 to 255. Each attack method is controlled by a configurable $\\epsilon _ { \\mathrm { a t t a c k } }$ parameter which sets the maximum perturbation allowed for each pixel in integer increments on the color scale. We only consider white-box attacks in this paper, i.e., the attack methods can get access to weights of the classifier. In the following, we give an introduction to all the attacking methods used in our experiments. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Random perturbation (RAND) Random perturbation is arguably the weakest attacking method, and we include it as the simplest baseline. Formally, the randomly perturbed image is given by ",
+        "bbox": [
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+    {
+        "type": "equation",
+        "img_path": "images/88fdf1bf5bcf98d728cc80ce5bc2b68fb4c7b43eebce351fdeedee5d1897ea18.jpg",
+        "text": "$$\n{ \\bf X } ^ { a d v } = { \\bf X } + \\mathcal { U } ( - \\lfloor \\epsilon _ { \\mathrm { a t t a c k } } \\rfloor , \\lfloor \\epsilon _ { \\mathrm { a t t a c k } } \\rfloor ) ,\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "where $\\textstyle { \\mathcal { U } } ( a , b )$ denotes an element-wise uniform distribution of integers from $[ a , b ]$ ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Fast gradient sign method (FGSM) Goodfellow et al. (2014) proposed the generation of malicious perturbations in the direction of the loss gradient $\\nabla _ { \\mathbf { X } } L ( \\mathbf { X } , y )$ , where $L ( \\mathbf { X } , y )$ is the loss function used to train the model. The adversarial examples are computed by ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/6f13081640a25abc19110d9211bb705b09252d85e338fbffcf427a41bff2970b.jpg",
+        "text": "$$\n{ \\bf X } ^ { a d v } = { \\bf X } + \\epsilon _ { \\mathrm { a t t a c k } } \\mathrm { s i g n } ( \\nabla _ { \\bf X } L ( { \\bf X } , y ) ) .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Basic iterative method (BIM) Kurakin et al. (2016) tested a simple variant of the fast gradient sign method by applying it multiple times with a smaller step size. Formally, the adversarial examples are computed as ",
+        "bbox": [
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+    {
+        "type": "equation",
+        "img_path": "images/dfccee158267e02117a63f21ef53e9c14a46fff49fb267761d98ce7d93057429.jpg",
+        "text": "$$\n{ \\bf X } _ { 0 } ^ { a d v } = { \\bf X } , { \\bf X } _ { n + 1 } ^ { a d v } = \\mathrm { C l i p } _ { \\bf x } ^ { \\epsilon _ { \\mathrm { a t a c k } } } \\left\\{ { \\bf X } _ { n } ^ { a d v } + \\alpha \\mathrm { s i g n } ( \\nabla _ { \\bf X } L ( { \\bf X } _ { n } ^ { a d v } , y ) ) \\right\\} ,\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "where $\\mathrm { C l i p } _ { \\mathbf { X } } ^ { \\epsilon _ { \\mathrm { a t t a c k } } }$ means we clip the resulting image to be within the $\\epsilon _ { \\mathrm { a t t a c k } }$ -ball of $\\mathbf { X }$ . Following Kurakin et al. (2016), we set $\\alpha = 1$ and the number of iterations to be $\\left\\lfloor \\operatorname* { m i n } ( \\epsilon _ { \\mathrm { a t t a c k } } + 4 , 1 . 2 5 \\epsilon _ { \\mathrm { a t t a c k } } ) \\right\\rfloor$ This method is also called Projected Gradient Descent (PGD) in Madry et al. (2017). ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "DeepFool DeepFool (Moosavi-Dezfooli et al., 2016) works by iteratively linearizing the decision boundary and finding the closest adversarial examples with geometric formulas. However, compared to FGSM and BIM, this method is much slower in practice. We clip the resulting image so that its perturbation is no larger than $\\epsilon _ { \\mathrm { a t t a c k } }$ . ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Carlini-Wagner (CW) Carlini & Wagner (2017b) proposed an efficient optimization objective for iteratively finding the adversarial examples with the smallest perturbations. As with DeepFool, we clip the output image to make sure the perturbations are limited by $\\epsilon _ { \\mathrm { a t t a c k } }$ . ",
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+        "page_idx": 2
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+    {
+        "type": "text",
+        "text": "2.2 DEFENSE METHODS ",
+        "text_level": 1,
+        "bbox": [
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+        "type": "text",
+        "text": "Current defense methods generally fall into two classes. They either (1) change the network architecture or training procedure to make it more robust, or (2) modify adversarial examples to reduce their harm. In this paper, we take the following defense methods into comparison. ",
+        "bbox": [
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+        "page_idx": 2
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+    {
+        "type": "text",
+        "text": "Adversarial training This defense works by generating adversarial examples on-the-fly during training and including them into the training set. FGSM adversarial examples are the most commonly used ones for adversarial training, since they are fast to generate and easy to train. Although training with higher-order adversarial examples (e.g., BIM) has witnessed some success in small datasets (Madry et al., 2017), other work has reported failure in larger ones (Kurakin et al., 2016). We consider both variants in our work. ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Label smoothing In contrast to adversarial training, label smoothing (Warde-Farley & Goodfellow, 2016) is agnostic to the attack method. It converts one-hot labels to soft targets, where the correct class has value $1 - \\epsilon$ while the other (wrong) classes have value $\\epsilon / ( N - \\bar { 1 } )$ . Here $\\epsilon$ is a small constant and $N$ is the number of classes. When the classifier is re-trained on these soft targets rather than the one-hot labels it is significantly more robust to adversarial examples. This method was originally devised to achieve a similar effect as defensive distillation (Papernot et al., 2016c), and their performance is comparable. We didn’t compare to defensive distillation since it is more computationally expensive. ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Feature squeezing Feature squeezing $\\mathrm { { X u } }$ et al., 2017a) is both attack-agnostic and modelagnostic. Given any input image, it first reduces the color range from [0, 255] to a smaller value, and then smooths the image with a median filter. The resulting image is then passed to a classifier for predictions. Since this technique does not depend on attacking methods and classifiers, it can be combined with other defensive methods such as adversarial training, similar to PixelDefend. ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "2.3 EXPERIMENT METHODOLOGIES ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Datasets Two datasets are used in our experiments: Fashion MNIST (Xiao et al., 2017) and CIFAR-10 (Krizhevsky et al.). Fashion MNIST was designed as a more difficult, but drop-in replacement for MNIST (LeCun et al., 1998). Thus it shares all of MNIST’s characteristics, i.e., 60, 000 training examples and 10, 000 test examples where each example is a $2 8 \\times 2 8$ gray-scale image associated with a label from 1 of 10 classes. CIFAR-10 is another dataset that is also broadly used for image classification tasks. It consists of 60, 000 examples, where 50, 000 are used for training and 10, 000 for testing, and each sample is a $3 2 \\times 3 2$ color image associated with 1 of 10 classes. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Models We examine two state-of-the-art deep neural network image classifiers: ResNet (He et al., 2016) and VGG (Simonyan & Zisserman, 2014). The architectures are described in Appendix C. ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "PixelCNN The PixelCNN (van den Oord et al., 2016b; Salimans et al., 2017) is a generative model with tractable likelihood especially designed for images. The model defines the joint distribution over all pixels by factorizing it into a product of conditional distributions. ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/3b33f33024299e7b1171a53e7a0175d8984971cab0926b5c2aaafba8cb286486.jpg",
+        "text": "$$\np _ { \\mathrm { C N N } } ( \\mathbf { X } ) = \\prod _ { i } p _ { \\mathrm { C N N } } ( x _ { i } | x _ { 1 : ( i - 1 ) } ) .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "The pixel dependencies are in raster scan order (row by row and column by column within each row). We train the PixelCNN model for each dataset using only clean (not perturbed) image samples. In Appendix D, we provide clean sample images from the datasets as well as generated image samples from PixelCNN (see Figure 8 and Figure 9). ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "As a convenient representation of $p _ { \\mathrm { C N N } } ( \\mathbf { X } )$ for images, we also use the concept of bits per dimension, which is defined as $\\mathrm { B P D } ( \\mathbf { X } ) \\triangleq - \\log p _ { \\mathrm { C N N } } ( \\mathbf { X } ) / ( I \\times J \\times K \\times \\log 2 )$ for an image of resolution $I \\times J$ and $K$ channels. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "3 DETECTING ADVERSARIAL EXAMPLES ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Adversarial images are defined with respect to a specific classifier. Intuitively, a maliciously perturbed image that causes one network to give a highly confident incorrect prediction might not fool another network. However, recent work (Papernot et al., 2016a; Liu et al., 2016; Tramèr et al., 2017) has shown that adversarial images can transfer across different classifiers. This indicates that there are some intrinsic properties of adversarial examples that are independent of classifiers. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "One possibility is that, compared to normal training and test images, adversarial examples have much lower probability densities under the image distribution. As a result, classifiers do not have enough training instances to get familiarized with this part of the input space. The resulting prediction task suffers from covariate shift, and since all of the classifiers are trained on the same dataset, this covariate shift will affect all of them similarly and will likely lead to misclassifications. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "To empirically verify this hypothesis, we train a PixelCNN model on the CIFAR-10 (Krizhevsky & Hinton, 2009) dataset and use its log-likelihood as an approximation to the true underlying probability density. The adversarial examples are generated with respect to a ResNet (He et al., 2016), which gets $92 \\%$ accuracy on the test images. We generate adversarial examples from RAND, FGSM, BIM, DeepFool and CW methods with $\\epsilon _ { \\mathrm { a t t a c k } } = 8$ . Note that as shown in Figure 1, the resulting adversarial perterbations are barely perceptible to humans. ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "image",
+        "img_path": "images/f08adab06239984a46d4b77ee6fabaca7d40d3ac138ce08f629e31bcb656bf1d.jpg",
+        "image_caption": [
+            "Figure 1: An image sampled from the CIFAR-10 test dataset and various adversarial examples generated from it. The text above shows the attacking method while the text below shows the predicted label of the ResNet. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "image",
+        "img_path": "images/7e219f80073784145f4f5b8e698056f47d0e1bc7934c8580b00b3f76bfe5f436.jpg",
+        "image_caption": [
+            "Figure 2: (a) Likelihoods of different perturbed images with $\\epsilon _ { \\mathrm { a t t a c k } } = 8$ . (b) Test errors of a ResNet on different adversarial examples. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "However, the distribution of log-likelihoods show considerable difference between perturbed images and clean images. As summarized in Figure 2, even a $3 \\%$ perturbation can lead to systematic decrease of log-likelihoods. Note that the PixelCNN model has no information about the attacking methods for producing those adversarial examples, and no information about the ResNet model either. ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "We can see from Figure 3(b) that random perturbations also push the images outside of the training distribution, even though they do not have the same adverse effect on accuracy. We believe this is due to an inductive bias that is shared by many neural network models but not inherent to all models, as discussed further in Appendix A. ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Besides qualitative analysis, the log-likelihoods from PixelCNN also provide a quantitative measure for detecting adversarial examples. Combined with permutation test (Efron & Tibshirani, 1994), we can provide a uncertainty value for each input about whether it comes from the training distribution or not. Specifically, let the input $\\mathbf { X } ^ { \\prime } \\overset { \\mathrm { i . i . d . } } { \\sim } q ( \\mathbf { X } )$ and training images $\\mathbf { X } _ { 1 } , \\cdots , \\mathbf { X } _ { N } \\overset { \\mathrm { i . i . d . } } { \\sim } p ( \\mathbf { X } )$ . The null hypothesis is $H _ { 0 } : p ( \\mathbf { X } ) = { \\\\overset { \\cdot } { q } } ( \\mathbf { X } )$ while the alternative is $H _ { 1 } : p ( \\mathbf { X } ) \\neq q ( \\mathbf { X } )$ . We first compute the probabilities give by a PixelCNN for $\\mathbf { X } ^ { \\prime }$ and $\\mathbf { X } _ { 1 } , \\cdots , \\mathbf { X } _ { N }$ , then use the rank of $p _ { \\mathrm { C N N } } ( \\mathbf { X } ^ { \\prime } )$ in $\\{ p _ { \\mathrm { C N N } } ( \\mathbf { X } _ { 1 } ) , \\cdots , \\bar { p } _ { \\mathrm { C N N } } ( \\mathbf { \\bar { X } } _ { N } ) \\}$ as our test statistic: ",
+        "bbox": [
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+    },
+    {
+        "type": "equation",
+        "img_path": "images/2049830d3501c2e478d4bdf8a93d601dfa95693dd378e6ef8d40e73e59e984ff.jpg",
+        "text": "$$\nT = T ( \\mathbf { X } ^ { \\prime } ; \\mathbf { X } _ { 1 } , \\cdots , \\mathbf { X } _ { N } ) \\triangleq \\sum _ { i = 1 } ^ { N } \\mathbb { I } [ p _ { \\mathrm { C N N } } ( \\mathbf { X } _ { i } ) \\leq p _ { \\mathrm { C N N } } ( \\mathbf { X } ^ { \\prime } ) ] .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Here $\\mathbb { I } [ \\cdot ]$ is the indicator function, which equals 1 when the condition inside brackets is true and otherwise equals 0. Let $T _ { i } = T ( { \\bf X } _ { i } ; { \\bf X } _ { 1 } , \\cdot \\cdot \\cdot , { \\bf X } _ { i - 1 } , { \\bf X } ^ { \\prime } , { \\bf X } _ { i + 1 } , \\cdot \\cdot \\cdot , { \\bf X } _ { N } )$ . According to the permutation principle, $T _ { i }$ has the same distribution as $T$ under the null hypothesis $H _ { 0 }$ . We can therefore compute the $p$ -value exactly by ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/d4909f1290510dfbd7e44d742cecb6cbc7df50dd9a9e3452a3e2b0ae5b43a064.jpg",
+        "text": "$$\np = \\frac { 1 } { N + 1 } \\left( \\sum _ { i = 1 } ^ { N } \\mathbb { I } [ T _ { i } \\leq T ] + 1 \\right) = \\frac { T + 1 } { N + 1 } = \\frac { 1 } { N + 1 } \\left( \\sum _ { i = 1 } ^ { N } \\mathbb { I } [ p _ { \\mathrm { C N N } } ( \\mathbf { X } _ { i } ) \\leq p _ { \\mathrm { C N N } } ( \\mathbf { X } ^ { \\prime } ) ] + 1 \\right) .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "image",
+        "img_path": "images/841fb2606539c7e4c79789b143754a98d8357f97e4cec03ca413c7385fcfd8ef.jpg",
+        "image_caption": [
+            "Figure 3: The distribution of $p$ -values under the PixelCNN generative model. The inputs are more outside of the training distribution if their $p$ -value distribution has a larger deviation from uniform. Here “clean” means clean test images. From definition, the $p$ -values of clean training images have a uniform distribution. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "image",
+        "img_path": "images/84c36078978cc3206f8239f356b8f2bdca0f5fa11b8e53dcc42b5f50501efca7.jpg",
+        "image_caption": [
+            "Figure 4: An example of how purification works. The above row shows an image from CIFAR10 test set and various attacking images generated from it. The bottom row shows corresponding purified images. The text below each image is the predicted label given by our ResNet. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "For CIFAR-10, we provide histograms of $p$ -values for different adversarial examples in Figure 3 and ROC curves of using $p$ -values for detection in Figure 6(a). Note that in the ideal case, the $p$ -value distribution of clean test images should be uniform. The method works especially well for attacks producing larger perturbations, such as RAND, FGSM, and BIM. For DeepFool and CW adversarial examples, we can also observe significant deviations from uniform. As shown in Figure 3(a), the $p$ - value distribution of test images are almost uniform, indicating good generalization of the PixelCNN model. ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "4 PURIFYING IMAGES WITH PIXELDEFEND ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "In many circumstances, simply detecting adversarial images is not sufficient. It is often critical to be able to correctly classify images despite such adversarial modifications. In this section we introduce PixelDefend, a specific instance of a new family of defense methods that significantly improves the state-of-the-art performance on advanced attacks, while simultaneously performing well against all other attacks. ",
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+    },
+    {
+        "type": "text",
+        "text": "Algorithm 1 PixelDefend ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "Input: Image X, Defense parameter $\\epsilon _ { \\mathrm { d e f e n d } }$ , Pre-trained PixelCNN model $p _ { \\mathrm { C N N } }$   \nOutput: Purified Image $\\mathbf { X } ^ { * }$   \n1: $\\bar { \\mathbf { X } } ^ { * }  \\mathbf { X }$   \n2: for each row $i$ do   \n3: for each column $j$ do   \n4: for each channel $k$ do   \n5: $x \\gets \\mathbf { X } [ i , j , k ]$   \n6: Set feasible range $R \\gets [ \\mathrm { m a x } ( x - \\epsilon _ { \\mathrm { d e f e n d } } , 0 ) , \\mathrm { m i n } ( x + \\epsilon _ { \\mathrm { d e f e n d } } , 2 5 5 ) ]$   \n7: Compute the 256-way softmax $p _ { \\mathrm { C N N } } ( \\mathbf { X } ^ { * } )$ .   \n8: Update $\\mathbf { X } ^ { * } [ i , j , k ] \\gets \\mathrm { a r g } \\operatorname* { m a x } _ { z \\in R } p _ { \\mathrm { C N N } } [ i , j , k , z ]$   \n9: end for   \n10: end for   \n11: end for ",
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+        "img_path": "images/9cd8d3d299ff4ff7eed038f5178135903d715fc07f90ba51f32ac6907fe381b5.jpg",
+        "image_caption": [
+            "Figure 5: The bits-per-dimension distributions of purified images from FGSM adversarial examples. We tested two purification methods, L-BFGS-B and greedy decoding, the latter of which is used in PixelDefend. A good purification method should give images that have lower bits per dimension compared to FGSM images and ideally similar bits per dimension compared to clean ones. "
+        ],
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+    {
+        "type": "text",
+        "text": "4.1 RETURNING IMAGES TO THE TRAINING DISTRIBUTION ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "The basic idea behind PixelDefend is to purify input images, by making small changes to them in order to move them back towards the training distribution, i.e., move the images towards a highprobability region. We then classify the purified image using any existing classifier. As the example in Figure 4 shows, the purified images can usually be classified correctly. ",
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+    {
+        "type": "text",
+        "text": "Formally, we have training image distribution $p ( \\mathbf { X } )$ , and input image $\\mathbf { X }$ of resolution $I \\times J$ with $\\mathbf { X } [ i , j , k ]$ the pixel at location $( i , j )$ and channel $k \\in \\{ 1 , \\cdots , C \\}$ . We wish to find an image $\\mathbf { X } ^ { * }$ that maximizes $p ( \\mathbf { X } )$ subject to the constraint that $\\mathbf { X } ^ { * }$ is within the $\\epsilon _ { \\mathrm { d e f e n d } }$ -ball of $\\mathbf { X }$ : ",
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+    },
+    {
+        "type": "equation",
+        "img_path": "images/e19de716d9c37c602609b11f571210c98f9c20f8a628a3a71c82302093a0bda4.jpg",
+        "text": "$$\n\\begin{array} { c } { \\displaystyle { \\operatorname* { m a x } _ { \\mathbf { X } ^ { * } } p ( \\mathbf { X } ^ { * } ) } } \\\\ { \\mathrm { s . t . } \\quad \\left\\| \\mathbf { X } ^ { * } - \\mathbf { X } \\right\\| _ { \\infty } \\leq \\epsilon _ { \\mathrm { d e f e n d } } . } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Here $\\epsilon _ { \\mathrm { d e f e n d } }$ reflects a trade-off, since large $\\epsilon _ { \\mathrm { d e f e n d } }$ may change the meaning of $\\mathbf { X }$ while small ϵdefend may not be sufficient for returning $\\mathbf { X }$ to the correct distribution. In practice, we choose ϵdefend to be some value that overestimates $\\epsilon _ { \\mathrm { a t t a c k } }$ but still keeps high accuracies on clean images. As in Section 3, we approximate $p ( \\mathbf { X } )$ with the PixelCNN distribution $p _ { \\mathrm { C N N } } ( \\mathbf { X } )$ , which is trained on the same training set as the classifier. ",
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+    },
+    {
+        "type": "text",
+        "text": "However, exact constrained optimization of $p _ { \\mathrm { C N N } } ( \\mathbf { X } )$ is computationally intractable. Surprisingly, even gradient-based optimization faces great difficulty on that problem. We found that one advanced methods in gradient-based constrained optimization, L-BFGS-B (Byrd et al., 1995) (we use the scipy implementation based on Zhu et al. (1997)), actually decreases $p _ { \\mathrm { C N N } } ( \\mathbf { X } )$ for most random initializations within the $\\epsilon _ { \\mathrm { d e f e n d } }$ -ball. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "For efficient optimization, we instead use a greedy technique described in Algorithm 1, which is similar to the greedy decoding process typically used in sequence-to-sequence models (Sutskever et al., 2014). The method is similar to generating images from PixelCNN, with the additional constraint that the generated image should be within an $\\epsilon _ { \\mathrm { d e f e n d } }$ -ball of a perturbed image. As an autoregressive model, PixelCNN is slow in image generation. Nonetheless, by caching redundant calculation, Ramachandran et al. (2017) proposes a very fast generation algorithm for PixelCNN. In our experiments, adoption of Ramachandran et al. (2017)’s method greatly increases the speed of PixelDefend. For CIFAR-10 images, PixelDefend on average processes 3.6 images per second on one NVIDIA TITAN Xp GPU. ",
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+    },
+    {
+        "type": "image",
+        "img_path": "images/9541592e4ea68f210ba02c6038cd122b98e392a4788d54a8ad822f5b222e631c.jpg",
+        "image_caption": [
+            "Figure 6: ROC curves showing the efficacy of using $p$ -values as scores to detect adversarial examples. For computing the ROC, we assign negative labels to training images and positive labels to adversarial images (or clean test images). (a) Original adversarial examples. (b) Purified adversarial examples after PixelDefend. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "To show the effectiveness of this greedy method compared to L-BFGS-B, we take the first 10 images from CIFAR-10 test set, attack them by FGSM with $\\epsilon _ { \\mathrm { a t t a c k } } = 8$ , and purify them with L-BFGS-B and PixelDefend respectively. We used random start points for L-BFGS-B and repeated 100 times for each image. As depicted in Figure 5, most L-BFGS-B attempts failed at minimizing the bits per dimension of FGSM adversarial examples. Because of the rugged gradient landscape of PixelCNN, L-BFGS-B even results in images that have lower probabilities. In contrast, PixelDefend works much better in increasing the probabilities of purified images, although their probabilities are still lower compared to clean ones. ",
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+    {
+        "type": "text",
+        "text": "In Figure 6 and Figure 7, we empirically show that after PixelDefend, purified images are more likely to be drawn from the training distribution. Specifically, Figure 6 shows that the detecting power of $p$ -values greatly decreases for purified images. For DeepFool and CW examples, purification makes them barely distinguishable from normal samples of the data distribution. This is also manifested by Figure 7, as the $p$ -value distributions of purified examples are closer to uniform. Visually, purified images indeed look much cleaner than adversarially perturbed ones. In Appendix E, we provide sampled purified images from Fashion MNIST and CIFAR-10. ",
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+    {
+        "type": "text",
+        "text": "4.2 ADAPTIVE PIXELDEFEND ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "One concern with the approach of purifying images is what happens when we purify a clean image. More generally, we will never know $\\epsilon _ { \\mathrm { a t t a c k } }$ and if we set $\\epsilon _ { \\mathrm { d e f e n d } }$ too large for a given attack, then we will modify all images to become the mode image, which would mostly result in misclassifications. One way to avoid this problem is to tune $\\epsilon _ { \\mathrm { d e f e n d } }$ adaptively based on the probability of the input image under the generative model. In this way, images that already have high probability under the training distribution would have a very low $\\epsilon _ { \\mathrm { d e f e n d } }$ preventing significant modification, while low probability images would have a high $\\epsilon _ { \\mathrm { d e f e n d } }$ thus allowing significant modifications. We implemented a very simple thresholding version of this, which sets $\\epsilon _ { \\mathrm { d e f e n d } }$ to zero if the input image probability is below a threshold value, and otherwise leaves it fixed at a manually chosen setting. In practice, we set this threshold based on knowledge of the set of possible attacks, so strictly speaking, the adaptive version of our technique is no longer attack-agnostic. ",
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+    },
+    {
+        "type": "image",
+        "img_path": "images/d8ac12f892812f533f4b6b1ae8304e24aeb92180d4e2963b938837985ba1b75c.jpg",
+        "image_caption": [
+            "Figure 7: The distributions of $p$ -values under the PixelCNN model after PixelDefend purification. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+    },
+    {
+        "type": "table",
+        "img_path": "images/1c77371270d5ac9326269f2a703c1a451eeaad2d30c814c7772dbdfd40390bd4.jpg",
+        "table_caption": [
+            "Table 1: Fashion MNIST $( \\epsilon _ { \\mathrm { a t t a c k } } = 8 / 2 5 $ , $\\epsilon _ { \\mathrm { d e f e n d } } = 3 2$ ) "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>NETWORK</td><td>TRAINING TECHNIQUE</td><td>CLEAN</td><td>RAND</td><td>FGSM</td><td>BIM</td><td>DEEP FOOL</td><td>Cw</td><td>STRONGEST ATTACK</td></tr><tr><td>ResNet</td><td>Normal</td><td>93/93</td><td>89/71</td><td>38/24</td><td>00/00</td><td>06/06</td><td>20/01</td><td>00/00</td></tr><tr><td>VGG</td><td>Normal</td><td>92/92</td><td>91/87</td><td>73/58</td><td>36/08</td><td>49/14</td><td>43/23</td><td>36/08</td></tr><tr><td rowspan=\"5\">ResNet</td><td>Adversarial FGSM</td><td>93/93</td><td>92/89</td><td>85/85</td><td>51/00</td><td>63/07</td><td>67/21</td><td>51/00</td></tr><tr><td>Adversarial BIM</td><td>92/91</td><td>92/91</td><td>84/79</td><td>76/63</td><td>82/72</td><td>81/70</td><td>76/63</td></tr><tr><td>Label Smoothing</td><td>93/93</td><td>91/76</td><td>73/45</td><td>16/00</td><td>29/06</td><td>33/14</td><td>16/00</td></tr><tr><td>Feature Squeezing</td><td>84/84</td><td>84/70</td><td>70/28</td><td>56/25</td><td>83/83</td><td>83/83</td><td>56/25</td></tr><tr><td>Adversarial FGSM + Feature Squeezing</td><td>88/88</td><td>87/82</td><td>80/77</td><td>70/46</td><td>86/82</td><td>84/85</td><td>70/46</td></tr><tr><td>ResNet</td><td>Normal +PixelDefend</td><td>88/88</td><td>88/89</td><td>85/74</td><td>83/76</td><td>87/87</td><td>87/87</td><td>83/74</td></tr><tr><td>VGG</td><td>Normal+PixelDefend</td><td>89/89</td><td>89/89</td><td>87/82</td><td>85/83</td><td>88/88</td><td>88/88</td><td>85/82</td></tr><tr><td rowspan=\"2\">ResNet</td><td>Adversarial FGSM +PixelDefend</td><td>90/89</td><td>91/90</td><td>88/82</td><td>85/76</td><td>90/88</td><td>89/88</td><td>85/76</td></tr><tr><td>Adversarial FGSM +Adaptive PixelDefend</td><td>91/91</td><td>91/91</td><td>88/88</td><td>85/84</td><td>89/90</td><td>89/84</td><td>85/84</td></tr></table>",
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+    {
+        "type": "text",
+        "text": "4.3 PIXELDEFEND RESULTS ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "We carried out a comprehensive set of experiments to test various defenses versus attacks. Detailed information on experimental settings is provided in Appendix B. All experimental results are summarized in Tab. 1 and Tab. 2. In the upper part of the tables, we show how the various baseline defenses fare against each of the attacks, while in the lower part of the tables we show how our PixelDefend technique works. Each table cell contains accuracies on adversarial examples generated with different $\\epsilon _ { \\mathrm { a t t a c k } }$ . More specifically, for Fashion MNIST (Tab. 1), we tried $\\epsilon _ { \\mathrm { a t t a c k } } = 8$ and 25. The cells in Tab. 1 is formated as $x / y$ , where $x$ denotes the accuracy $( \\% )$ on images attacked with $\\epsilon _ { \\mathrm { a t t a c k } } = 8$ , while $y$ denotes the accuracy when $\\epsilon _ { \\mathrm { a t t a c k } } = 2 5 $ . For CIFAR-10 (Tab. 2), we tried $\\epsilon _ { \\mathrm { a t t a c k } } = 2$ , 8, and 16, and the cells are formated in a similar way. We use the same $\\epsilon _ { \\mathrm { d e f e n d } }$ for different ϵattack’s to show that PixelDefend is insensitive to ϵattack. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "From the tables we observe that adversarial training successfully defends against the basic FGSM attack, but cannot defend against the more advanced ones. This is expected, as training on simple adversarial examples does not guarantee robustness to more complicated attacking techniques. Consistent with Madry et al. (2017), adversarial training with BIM examples is more successful at preventing a wider spectrum of attacks. For example, it improves the accuracy on strongest attack from $2 \\%$ to $32 \\%$ on CIFAR-10 when $\\epsilon _ { \\mathrm { a t t a c k } } = 8$ . But the numbers are still not ideal even with respect to BIM attack itself. As in Tab. 2, it only gets $6 \\%$ on BIM and $8 \\%$ on CW when $\\epsilon _ { \\mathrm { a t t a c k } } = 1 6$ . We also observe that label smoothing, which learns smoothed predictions so that the gradient $\\nabla _ { \\mathbf { X } } L ( \\mathbf { X } , y )$ becomes very small, is only effective against simple FGSM attack. Model-agnostic methods, such as feature squeezing, can be combined with other defenses for strengthened performance. We observe that combining it with adversarial training indeed makes it more robust. Actually, Tab. 1 and Tab. 2 show that feature squeezing combined with adversarial training dominates using feature squeezing along in all settings. It also gets good performance on DeepFool and CW attacks. However, for iterative attacks with larger perturbations, i.e., BIM, feature squeezing performs poorly. On CIFAR-10, it only gets $2 \\%$ and $0 \\%$ accuracy on BIM with $\\epsilon _ { \\mathrm { a t t a c k } } = 8$ and 16 respectively. ",
+        "bbox": [
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+    },
+    {
+        "type": "table",
+        "img_path": "images/10883ba06ebf9f7119b414bf4ec59fdcb548a9e4683e717608d83610503cd77e.jpg",
+        "table_caption": [
+            "Table 2: CIFAR-10 $\\mathcal { C } _ { \\mathrm { a t t a c k } } = 2 / 8 / 1 6$ , $\\epsilon _ { \\mathrm { d e f e n d } } = 1 6$ ) "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>NETWORK</td><td>TRAINING TECHNIQUE</td><td>CLEAN</td><td>RAND</td><td>FGSM</td><td>BIM</td><td>DEEP FOOL</td><td>CW</td><td>STRONGEST ATTACK</td></tr><tr><td>ResNet</td><td>Normal</td><td>92/92/92</td><td>92/87/76</td><td>33/15/11</td><td>10/00/00</td><td>12/06/06</td><td>07/00/00</td><td>07/00/00</td></tr><tr><td>VGG</td><td>Normal</td><td>89/89/89</td><td>89/88/80</td><td>60/46/30</td><td>44/02/00</td><td>57/25/11</td><td>37/00/00</td><td>37/00/00</td></tr><tr><td rowspan=\"5\">ResNet</td><td>Adversarial FGSM</td><td>91/91/91</td><td>90/88/84</td><td>88/91/91</td><td>24/07/00</td><td>45/00/00</td><td>20/00/07</td><td>20/00/00</td></tr><tr><td>Adversarial BIM</td><td>87/87/87</td><td>87/87/86</td><td>80/52/34</td><td>74/32/06</td><td>79/48/25</td><td>76/42/08</td><td>74/32/06</td></tr><tr><td>Label Smoothing</td><td>92/92/92</td><td>91/88/77</td><td>73/54/28</td><td>59/08/01</td><td>56/20/10</td><td>30/02/02</td><td>30/02/01</td></tr><tr><td>Feature Squeezing</td><td>84/84/84</td><td>83/82/76</td><td>31/20/18</td><td>13/00/00</td><td>75/75/75</td><td>78/78/78</td><td>13/00/00</td></tr><tr><td>Adversarial FGSM + Feature Squeezing</td><td>86/86/86</td><td>85/84/81</td><td>73/67/55</td><td>55/02/00</td><td>85/85/85</td><td>83/83/83</td><td>55/02/00</td></tr><tr><td>ResNet</td><td>Normal+PixelDefend</td><td>85/85/88</td><td>82/83/84</td><td>73/46/24</td><td>71/46/25</td><td>80/80/80</td><td>78/78/78</td><td>71/46/24</td></tr><tr><td>VGG</td><td>Normal+PixelDefend</td><td>82/82/82</td><td>82/82/84</td><td>80/62/52</td><td>80/61/48</td><td>81/76/76</td><td>81/79/79</td><td>80/61/48</td></tr><tr><td rowspan=\"2\">ResNet</td><td>Adversarial FGSM +PixelDefend</td><td>88/88/86</td><td>86/86/87</td><td>81/68/67</td><td>81/69/56</td><td>85/85/85</td><td>84/84/84</td><td>81/69/56</td></tr><tr><td>Adversarial FGSM +Adaptive PixelDefend</td><td>90/90/90</td><td>86/87/87</td><td>81/70/67</td><td>81/70/56</td><td>82/81/82</td><td>81/80/81</td><td>81/70/56</td></tr></table>",
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+    {
+        "type": "text",
+        "text": "",
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+    {
+        "type": "text",
+        "text": "PixelDefend, our model-agnostic and attack-agnostic method, performs well on different classifiers (ResNet and VGG) and different attacks without modification. In addition, we can see that augmenting basic adversarial training with PixelDefend can sometimes double the accuracies. We hypothesize that the purified images from PixelDefend are still not perfect, and adversarially trained networks have more toleration for perturbations. This also corroborates the plausibility and benefit of combining PixelDefend with other defenses. ",
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+    {
+        "type": "text",
+        "text": "Furthermore, PixelDefend can simultaneously obtain accuracy above $70 \\%$ for all other attacking techniques, while ensuring that performance on clean images only declines slightly. Models with PixelDefend consistently outperform other methods with respect to the strongest attack. On Fashion MNIST, PixelDefend methods improve the accuracy on strongest attack from $76 \\%$ to $8 5 \\%$ and $63 \\%$ to $84 \\%$ . On CIFAR-10, the improvements are even more significant, i.e., from $74 \\%$ to $81 \\%$ , $32 \\%$ to $70 \\%$ and $6 \\%$ to $56 \\%$ , for $\\epsilon _ { \\mathrm { a t t a c k } } = 2$ , 8, and 16 respectively. In a security-critical scenario, the weakest part of a system determines the overall reliability. Therefore, the outstanding performance of PixelDefend on the strongest attack makes it a valuable and useful addition for improving AI security. ",
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+    {
+        "type": "text",
+        "text": "4.4 END-TO-END ATTACK OF PIXELDEFEND ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "A natural question that arises is whether we can generate a new class of adversarial examples targeted specifically at the combined PixelDefend architecture of first purifying the image and then using an existing classifier to predict the label of the purified image. We have three pieces of empirical evidence to believe that such adversarial examples are hard to find in general. First, we attempted to apply the iterative BIM attack to an end-to-end differentiable version of PixelDefend generated by unrolling the PixelCNN purification process. However we found the resulting network was too deep and led to problems with vanishing gradients (Bengio et al., 1994), resulting in adversarial images that were identical to the original images. Moreover, attacking the whole system is very time consuming. Empirically, it took about 10 hours to generate 100 attacking images with one TITAN $\\mathrm { X p }$ GPU which failed to fool PixelDefend. Secondly, we found the optimization problem in Eq. (4.1) was not amenable to gradient descent, as indicated in Figure 5. This makes gradient-based attacks especially difficult. Last but not least, the generative model and classifier are trained separately and have independent parameters. Therefore, the perturbation direction that leads to higher probability images has a smaller correlation with the perturbation direction that results in misclassification. Accordingly, it is harder to find adversarial examples that can fool both of them together. However, we will open source our codes and look forward to any possible attack from the community. ",
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+    {
+        "type": "text",
+        "text": "5 RELATED WORK ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Most recent work on detecting adversarial examples focuses on adding an outlier class detection module to the classifier, such as Grosse et al. (2017), Gong et al. (2017) and Metzen et al. (2017). Those methods require the classification model to be changed, and are thus not model-agnostic. Feinman et al. (2017) also presents a detection method based on kernel density estimation and Bayesian neural network uncertainty. However, Carlini & Wagner (2017a) shows that all those methods can be bypassed. ",
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+        "page_idx": 10
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+    {
+        "type": "text",
+        "text": "Grosse et al. (2017) also studied the distribution of adversarial examples from a statistical testing perspective. They reported the same discovery that adversarial examples are outside of the training distribution. However, our work is different from theirs in several important aspects. First, the kernel-based two-sample test used in their paper needs a large number of suspicious inputs, while our method only requires one data point. Second, they mainly tested on first-order methods such as FGSM and JSMA (Papernot et al., 2016b). We show the efficacy of PixelCNN on a wider range of attacking methods (see Figure 3), including both first-order and iterative methods. Third, we further demonstrate that random perturbed inputs are also outside of the training distribution. ",
+        "bbox": [
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+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Some other work has focused on modifying the classifier architecture to increase its robustness, e.g., Gu & Rigazio (2014), Cisse et al. (2017) and Nayebi & Ganguli (2017). Although they have witnessed some success, such modifications of models might limit their representative power and are also not model-agnostic. ",
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+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Our basic idea of moving points to higher-density regions is also present in other machine learning methods not specifically designed for handling adversarial data; for example, the manifold denoising method of Hein & Maier (2007), the direct density gradient estimation of Sasaki et al. (2014), and the denoising autoencoders of Vincent et al. (2008) all move data points from low to high-density regions. In the future some of these methods could be adapted to amortize the purification process directly, that is, to learn a purification network. ",
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+    {
+        "type": "text",
+        "text": "6 CONCLUSION ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "In this work, we discovered that state-of-the-art neural density models, e.g., PixelCNN, can detect small perturbations with high sensitivity. This sensitivity broadly exists for a large number of perturbations generated with different methods. An interesting fact is that PixelCNN is only sensitive in one direction—it is relatively easy to detect perturbations that lead to lower probabilities rather than higher probabilities. ",
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+    },
+    {
+        "type": "text",
+        "text": "Based on the sensitivity of PixelCNN, we utilized statistical hypothesis testing to verify that adversarial examples lie outside of the training distribution. With the permutation test, we give exact $p$ -values which can be used as a uncertainty measure for detecting outlier perturbations. ",
+        "bbox": [
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+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Furthermore, we make use of the sensitivity of generative models to explore the idea of purifying adversarial examples. We propose the PixelDefend algorithm, and experimentally show that returning adversarial examples to high probability regions of the training distribution can significantly decrease their damage to classifiers. Different from many other defensive techniques, PixelDefend is model-agnostic and attack-agnostic, which means it can be combined with other defenses to improve robustness without modifying the classification model. As a result PixelDefend is a practical and effective defense against adversarial inputs. ",
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+    {
+        "type": "text",
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+    {
+        "type": "text",
+        "text": "APPENDIX A ON RANDOM PERTURBATIONS ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "One may observe from Figure 3(b) that random perturbations have very low $p$ -values, and thus also live outside of the high density area. Although many classifiers are robust to random noise, it is not a property granted by the dataset. The fact is that robustness to random noise could be from model inductive bias, and there exist classifiers which have high generalization performance on clean images, but can be attacked by small random perturbations. ",
+        "bbox": [
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+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "It is easy to construct a concrete classifier that are susceptible to random perturbations. Our ResNet on CIFAR-10 gets $9 2 . 0 \\%$ accuracy on the test set and $8 7 . 3 \\%$ on randomly perturbed test images with $\\epsilon _ { \\mathrm { a t t a c k } } = 8$ . According to our PixelCNN, 175 of 10000 test images have a bits per dimension (BPD) larger than 4.5, while the number for random images is 9874. Therefore, we can define a new classifier ",
+        "bbox": [
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+        "page_idx": 14
+    },
+    {
+        "type": "equation",
+        "img_path": "images/bfbc7526fd64d72b16291bfbfeaca7dbb00235893de16bf7f6b3b068a196976d.jpg",
+        "text": "$$\n\\mathrm { R e s N e t ^ { \\prime } ( X ) } \\triangleq \\left\\{ \\begin{array} { l l } { \\mathrm { R e s N e t ( X ) } , } & { \\mathrm { B P D } ( \\mathbf { X } ) < 4 . 5 } \\\\ { \\mathrm { r a n d o m \\ l a b e l } , } & { \\mathrm { B P D } ( \\mathbf { X } ) \\geq 4 . 5 } \\end{array} \\right. ,\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "which will get roughly $9 2 \\% \\times 9 8 2 5 / 1 0 0 0 0 + 1 0 \\% \\times 1 7 5 / 1 0 0 0 0 \\approx 9 0 . 6 \\%$ accuracy on the test set, while only $8 7 . 3 \\% \\times 1 2 6 / 1 0 0 0 0 + 1 0 \\% \\times 9 8 7 4 / 1 0 0 0 0 \\approx 1 1 . 0 \\%$ accuracy on the randomly perturbed images. This classifier has comparable generalization performance to the original ResNet, but will give incorrect labels to most randomly perturbed images. ",
+        "bbox": [
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+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "APPENDIX B EXPERIMENTAL SETTINGS ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            404,
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+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "Adversarial Training We have tested adversarial training with both FGSM and BIM examples. During training, we take special care of the label leaking problem as noted in Kurakin et al. (2016)— we use the predicted labels of the model to generate adversarial examples, instead of using the true labels. This prevents the adversarially trained network to perform better on adversarial examples than clean images by simply retrieving ground-truth labels. Following Kurakin et al. (2016), we also sample $\\epsilon _ { \\mathrm { a t t a c k } }$ from a truncated Gaussian distribution for generating FGSM or BIM adversarial examples, so that the adversarially trained network won’t overfit to any specific $\\epsilon _ { \\mathrm { a t t a c k } }$ . This is different from Madry et al. (2017), where the authors train and test with the same ϵattack. ",
+        "bbox": [
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+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "For Fashion MNIST experiments, we randomly sample $\\epsilon _ { \\mathrm { a t t a c k } }$ from $\\mathcal { N } ( 0 , \\delta )$ , take the absolute value and truncate it to $[ 0 , 2 \\delta ]$ , where $\\delta \\ : = \\ : 8$ or 25. For CIFAR-10 experiments, we follow the same procedure but fix $\\delta = 8$ . ",
+        "bbox": [
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+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "Feature Squeezing For implementing the feature squeezing defense, we reduce the number of colors to 8 on Fashion MNIST, and use 32 colors for CIFAR-10. The numbers are chosen to make sure color reduction will not lead to significant deterioration of image quality. After color depth reduction, we apply a $2 \\times 2$ median filter with reflective paddings, since it is reported in $\\mathrm { X u }$ et al. (2017b) to be most effective for preventing CW attacks. ",
+        "bbox": [
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+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "Models We use ResNet (62-layer) and VGG (16-layer) as classifiers. In our experiments, normally trained networks have the same architectures as adversarially trained networks. Since the images of Fashion MNIST contain roughly one quarter values of those of CIFAR-10, we use a smaller network for classifying Fashion MNIST. More specifically, we reduce the number of feature maps for Fashion MNIST to 1/4 while keeping the same depths. In practive, VGG is more robust than ResNet due to using of dropout layers. The network architecture details are described in Appendix C. For the PixelCNN generative model, we adopted the implementation of $\\mathrm { P i x e l C N N + + }$ (Salimans et al., 2017), but modified the output from mixture of logistic distributions to softmax. The feature maps are also reduced to 1/4 for training PixelCNN on Fashion MNIST. ",
+        "bbox": [
+            173,
+            695,
+            825,
+            821
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "Adaptive Threshold We chose the adaptive threshold discussed in Section 4.2 using validation data. We set the threshold at the lowest value which did not decrease the performance of the strongest adversary. For Fashion MNIST, the threshold of bits per dimension was set to 1.8, and for CIFAR-10 the number was 3.2. As a reference, the mean value of bits per dimension for Fashion MNIST test images is 2.7 and for CIFAR-10 is 3.0. However, we admit that using a validation set to choose the best threshold makes the adaptive version of PixelDefend not strictly attack-agnostic. ",
+        "bbox": [
+            174,
+            837,
+            825,
+            921
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "APPENDIX C IMAGE CLASSIFIER ARCHITECTURES∗ ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            102,
+            612,
+            118
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "table",
+        "img_path": "images/9c46a54f1aedca3ac9e82f6b3f47bfdd25d25147c12ea0b2a44ada33913ed4d5.jpg",
+        "table_caption": [
+            "C.1 RESNET CLASSIFIER FOR CIFAR-10 & FASHION MNIST "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=1 colspan=1>NAME</td><td rowspan=1 colspan=2>CONFIGURATION</td></tr><tr><td rowspan=1 colspan=1>Initial Layer</td><td rowspan=1 colspan=2>conv (filter size: 3 × 3,feature maps: 16 (4), stride size: 1 × 1)</td></tr><tr><td rowspan=1 colspan=1>Residual Block 1</td><td rowspan=1 colspan=1>batch normalization &amp; leaky reluconv (filter size: 3 × 3, feature maps: 16 (4), stride size: 1 × 1)batch normalization &amp;leaky reluconv (filter size: 3 × 3, feature maps: 16 (4), stride size: 1 × 1)residual addition</td><td rowspan=1 colspan=1>×10 times</td></tr><tr><td rowspan=2 colspan=1>Residual Block 2</td><td rowspan=1 colspan=2>batch normalization&amp; leaky reluconv (filter size: 3 × 3, feature maps: 32 (8), stride size: 2 × 2)batch normalization&amp;leakyreluconv (filter size: 3 × 3, feature maps: 32 (8), stride size: 1 × 1)average pooling &amp; padding &amp; residual addition</td></tr><tr><td rowspan=1 colspan=1>batch normalization &amp; leaky reluconv (filter size: 3 × 3, feature maps: 32(8), stride size: 1 × 1)batch normalization&amp;leaky reluconv (filter size: 3 × 3, feature maps: 32(8), stride size: 1 × 1)residual addition</td><td rowspan=1 colspan=1>×9 times</td></tr><tr><td rowspan=2 colspan=1>Residual Block 3</td><td rowspan=1 colspan=2>batch normalization&amp; leaky reluconv (filter size: 3 × 3, feature maps: 64 (16),stride size: 2 × 2)batch normalization &amp; leaky reluconv (filter size: 3 × 3,feature maps: 64(16), stride size:1 × 1)average pooling &amp; padding &amp; residual addition</td></tr><tr><td rowspan=1 colspan=1>batch normalization &amp; leaky reluconv(filter size: 3 × 3,feature maps: 64(16), stride size:1 × 1)batch normalization &amp; leaky reluconv(filter size: 3 × 3,feature maps: 64(16), stride size:1 × 1)residual addition</td><td rowspan=1 colspan=1>×9 times</td></tr><tr><td rowspan=1 colspan=1>Pooling Layer</td><td rowspan=1 colspan=2>batch normalization &amp; leaky relu &amp; average pooling</td></tr><tr><td rowspan=1 colspan=1>Output Layer</td><td rowspan=1 colspan=2>fc_10 &amp; softmax</td></tr></table>",
+        "bbox": [
+            173,
+            165,
+            838,
+            582
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "text",
+        "text": "C.2 VGG CLASSIFIER FOR CIFAR-10 & FASHION MNIST ",
+        "text_level": 1,
+        "bbox": [
+            173,
+            606,
+            593,
+            621
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "table",
+        "img_path": "images/8f148f1fc6d4a0711f26ea19c7ba43946462fff3fc79810797c6cd151e362b32.jpg",
+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=1 colspan=1>NAME</td><td rowspan=1 colspan=2>CONFIGURATION</td></tr><tr><td rowspan=2 colspan=1>Feature Block 1</td><td rowspan=1 colspan=1>conv (filter size: 3 × 3, feature maps: 16 (4), stride size: 1 × 1)batch normalization &amp; relu</td><td rowspan=1 colspan=1>×2 times</td></tr><tr><td rowspan=1 colspan=2>max pooling (stride size: 2 × 2)</td></tr><tr><td rowspan=2 colspan=1>Feature Block 2</td><td rowspan=1 colspan=1>conv (filter size: 3 × 3, feature maps: 128 (32), stride size: 1 × 1)batch normalization &amp; relu</td><td rowspan=1 colspan=1>×2 times</td></tr><tr><td rowspan=1 colspan=1>max pooling (stride size:2 × 2)</td><td rowspan=1 colspan=1></td></tr><tr><td rowspan=2 colspan=1>Feature Block 3</td><td rowspan=1 colspan=1>conv (filter size: 3 × 3, feature maps: 512 (128), stride size: 1 × 1)batch normalization &amp; relu</td><td rowspan=1 colspan=1>×3 times</td></tr><tr><td rowspan=1 colspan=2>max pooling (stride size: 2 × 2)</td></tr><tr><td rowspan=2 colspan=1>Feature Block 4</td><td rowspan=1 colspan=1>conv (filter size: 3 × 3, feature maps: 512 (128), stride size: 1 × 1)batch normalization &amp; relu</td><td rowspan=1 colspan=1>×3 times</td></tr><tr><td rowspan=1 colspan=2>max pooling (stride size:2× 2) &amp; flatten</td></tr><tr><td rowspan=1 colspan=1>Classifier Block</td><td rowspan=1 colspan=2>dropout &amp; fc_512(128)&amp; reludropout&amp;fc_1O&amp; softmax</td></tr></table>",
+        "bbox": [
+            173,
+            637,
+            839,
+            856
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "text",
+        "text": "APPENDIX D SAMPLED IMAGES FROM PIXELCNN ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            101,
+            609,
+            118
+        ],
+        "page_idx": 16
+    },
+    {
+        "type": "text",
+        "text": "D.1 FASHION MNIST ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            132,
+            341,
+            148
+        ],
+        "page_idx": 16
+    },
+    {
+        "type": "image",
+        "img_path": "images/a52c9a517dca2176580fff1389c7f949da2bad0c291e18c24545fefdf06d727d.jpg",
+        "image_caption": [
+            "Figure 8: True and generated images from Fashion MNIST. The upper part shows true images sampled from the dataset while the bottom shows generated images from PixelCNN. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            173,
+            256,
+            825,
+            765
+        ],
+        "page_idx": 16
+    },
+    {
+        "type": "text",
+        "text": "D.2 CIFAR-10 ",
+        "bbox": [
+            173,
+            103,
+            294,
+            117
+        ],
+        "page_idx": 17
+    },
+    {
+        "type": "image",
+        "img_path": "images/31c1a6c112e70b99ae7152ef1092a77c22d51e7cabee2a609f7352a85edd4c55.jpg",
+        "image_caption": [
+            "Figure 9: True and generated images from CIFAR-10. The upper part shows true images sampled from the dataset while the bottom part shows generated images from PixelCNN. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            173,
+            242,
+            825,
+            747
+        ],
+        "page_idx": 17
+    },
+    {
+        "type": "text",
+        "text": "APPENDIX E SAMPLED PURIFIED IMAGES FROM PIXELDEFEND ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            101,
+            714,
+            118
+        ],
+        "page_idx": 18
+    },
+    {
+        "type": "text",
+        "text": "E.1 FASHION MNIST ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            132,
+            339,
+            148
+        ],
+        "page_idx": 18
+    },
+    {
+        "type": "image",
+        "img_path": "images/45672d7a01c00ca3bc643f6903a64a18eec8b52de727d98e8a3c13f4bba2bbb2.jpg",
+        "image_caption": [
+            "Figure 10: The upper part shows adversarial images generated from FGSM attack while the bottom part shows corresponding purified images after PixelDefend. Here $\\epsilon _ { \\mathrm { a t t a c k } } = 2 5 $ and $\\epsilon _ { \\mathrm { d e f e n d } } = 3 2 $ . "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            173,
+            256,
+            825,
+            762
+        ],
+        "page_idx": 18
+    },
+    {
+        "type": "text",
+        "text": "E.2 CIFAR-10 ",
+        "text_level": 1,
+        "bbox": [
+            173,
+            103,
+            294,
+            117
+        ],
+        "page_idx": 19
+    },
+    {
+        "type": "image",
+        "img_path": "images/5a6af051392cefa48293cd2d5b943cf4e7dfb45eaebf9508ebc2ad16fec1b77e.jpg",
+        "image_caption": [
+            "Figure 11: The upper part shows adversarial images generated from FGSM attack while the bottom part shows corresponding purified images by PixelDefend. Here $\\epsilon _ { \\mathrm { a t t a c k } } = 8$ and $\\epsilon _ { \\mathrm { d e f e n d } } = 1 6$ . "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            173,
+            242,
+            825,
+            747
+        ],
+        "page_idx": 19
+    }
+]