text:
\section{Introduction}

Explanations of happenings in one's life, \textit{causal explanations}, are an important topic of study in social, psychological, economic, and behavioral sciences.
For example, psychologists have analyzed people's causal explanatory style~\cite{peterson1988pessimistic} and  found strong negative relationships with  depression, passivity, and hostility, as well as positive relationships with life satisfaction, quality of life, and length of life~\cite{scheier1989dispositional,carver1987optimism,peterson1988pessimistic}.

\begin{figure}
\centering
  \includegraphics[width=3.0in]{Causal_Explanation_Figure_ver4.png}
  \caption{A casual relation characterizes the connection between two discourse arguments, one of which is the \textit{causal explanation}. }
  \label{fig:causal_relation_parsing}
\end{figure}

To help understand the significance of causal explanations, consider how they are applied to measuring optimism (and its converse, pessimism)~\cite{peterson1988pessimistic}.
For example, in ``My parser failed \textit{because I always have bugs.}'', the emphasized text span is considered a causal explanation which indicates pessimistic personality -- a negative event where the author believes the cause is pervasive.
However, in ``My parser failed \textit{because I barely worked on the code.}'', the explanation would be considered a signal of optimistic personality -- a negative event for which the cause is believed to be short-lived.

Language-based models which can detect causal explanations from everyday social media language can be used for more than automating optimism detection.
Language-based assessments would enable other large-scale downstream tasks: tracking prevailing causal beliefs (e.g., about climate change or autism), better extracting process knowledge from non-fiction (e.g., gravity causes objects to move toward one another), or detecting attribution of blame or praise in product or service reviews (``I loved this restaurant because the fish was cooked to perfection'').

In this paper, we introduce causal explanation analysis and its subtasks of detecting the presence of causality (\textit{causality prediction}) and identifying explanatory phrases (\textit{causal explanation identification}). There are many challenges to achieving these task. First, the ungrammatical texts in social media incur poor syntactic parsing results which drastically affect the performance of  discourse relation parsing pipelines \footnote{Off-the-shelf Penn Discourse Treebank (PDTB) end-to-end parsers perform poorly on our Facebook causal prediction dataset (see Table~\ref{tab:causality})}. Many causal relations are \textit{implicit} and do not contain any discourse markers (e.g., `because').  Further, \textit{Explicit} causal relations are also more difficult in social media due to the abundance of abbreviations and variations of discourse connectives (e.g., `cuz' and `bcuz').

Prevailing approaches for social media analyses, utilizing traditional linear models or bag of words models (e.g., SVM trained with n-gram, part-of-speech (POS) tags, or lexicon-based features) alone do not seem appropriate for this task since they simply cannot segment the text into meaningful discourse units or discourse arguments \footnote{Each discourse relation theory uses a different term for minimal discourse text spans: `Elementary Discourse Unit (EDU)' in RST and `Discourse Argument' in PDTB. We will call it `Discourse Argument' in this paper, since we adapted the PDTB text segmentation method.} such as clauses or sentences rather than random consecutive token sequences or specific word tokens.
Even when the discourse units are clear, parsers may still fail to accurately identify discourse relations since the content of social media is quite different than that of newswire which is typically used for discourse parsing.

In order to overcome these difficulties of discourse relation parsing in social media, we simplify and minimize the use of syntactic parsing results and capture relations between discourse arguments, and investigate the use of a recursive neural network model (RNN).
Recent work has shown that RNNs are effective for utilizing discourse structures for their downstream tasks \cite{ji2017neural,bhatia2015better,wieting2015towards, paulus2014global}, but they have yet to be directly used for discourse relation prediction in social media.
We evaluated our model by comparing it to off-the-shelf end-to-end discourse relation parsers and traditional models. We found that the SVM and random forest classifiers work better than the LSTM classifier for the causality detection, while the LSTM classifier outperforms other models for identifying causal explanation.

The contributions of this work include: (1) the proposal of models for both (a) causality prediction and (b) causal explanation identification, (2) the extensive evaluation of a variety of models from social media classification models and discourse relation parsers to RNN-based application models, demonstrating that feature-based models work best for causality prediction while RNNs are superior for the more difficult task of causal explanation identification,  (3) performance analysis on architectural differences of the pipeline and the classifier structures, (4) exploration of the applications of causal explanation to downstream tasks, and (5) release of a novel, anonymized causality Facebook dataset along with our causality prediction and causal explanation identification models.

\section{Related Work}
Identifying causal explanations in documents can be viewed as discourse relation parsing.
The Penn Discourse Treebank (PDTB) \cite{prasad2007penn} has a `Cause' and `Pragmatic Cause' discourse type under a general `Contingency' class and Rhetorical Structure Theory (RST) \cite{mann1987rhetorical} has a `Relations of Cause'. In most cases, the development of discourse parsers has taken place \textit{in-domain}, where researchers have used the existing annotations of discourse arguments in newswire text (e.g. Wall Street Journal) from the discourse treebank and focused on exploring different features and optimizing various types of models for predicting relations \cite{pitler2009automatic,park2012improving,zhou2010predicting}.
In order to further develop automated systems, researchers have proposed end-to-end discourse relation parsers, building models which are trained and evaluated on the annotated PDTB and RST Discourse Treebank (RST DT).
These corpora consist of documents from Wall Street Journal (WSJ) which are much more well-organized and grammatical than social media texts \cite{biran:15,lin2014pdtb,ji2014representation,feng2014linear}.

Only a few works have attempted to parse discourse relations for out-of-domain problems such as text categorizations on social media texts; Ji and Bhatia used models which are pretrained with RST DT for building discourse structures from movie reviews, and Son adapted the PDTB discourse relation parsing approach for capturing counterfactual conditionals from tweets \cite{bhatia2015better,ji2017neural,son2017recognizing}.
These works had substantial differences to what propose in this paper.
First, Ji and Bhatia used a pretrained model (not fully optimal for some parts of the given task) in their pipeline; Ji's model performed worse than the baseline on the categorization of legislative bills, which is thought to be due to legislative discourse structures differing from those of the training set (WSJ corpus). Bhatia also used a pretrained model finding that utilizing discourse relation features did not boost accuracy~\cite{bhatia2015better,ji2017neural}.
Both Bhatia and Son used manual schemes which may limit the coverage of certain types of positive samples-- Bhatia used a hand-crafted schema for weighting discourse structures for the neural network model and Son manually developed seven surface forms of counterfactual thinking for the rule-based system~\cite{bhatia2015better,son2017recognizing}.
We use social-media-specific features from pretrained models which are directly trained on tweets and we avoid any hand-crafted rules except for those included in the existing discourse argument extraction techniques.

The automated systems for discourse relation parsing involve multiple subtasks from segmenting the whole text into discourse arguments to classifying discourse relations between the arguments.
Past research has found that different types of models and features yield varying performance for each subtask.
Some have optimized models for discourse relation classification (i.e.~given a document indicating if the relation existing) without discourse argument parsing using models such as Naive-Bayes or SVMs, achieve relatively stronger accuracies but a simpler task than that associated with discourse arguments~\cite{park2012improving,zhou2010predicting,pitler2009automatic}.
Researchers who, instead, tried to build the end-to-end parsing pipelines considered a wider range of approaches including sequence models and RNNs~\cite{biran:15,feng2014linear,ji2014representation,li2014recursive}. Particularly, when they tried to utilize the discourse structures for out-domain applications, they used RNN-based models and found that those models are advantageous for their downstream tasks \cite{bhatia2015better,ji2017neural}.

In our case, for identifying causal explanations from social media using discourse structure, we build an RNN-based model for its structural effectiveness in this task (see details in section \ref{sec:rnn_model}). However, we also note that simpler models such as SVMs and logistic regression obtained the state-of-the-art performances for text categorization tasks in social media \cite{lynn2017human,mohammad2013nrc}, so we build relatively simple models with different properties for each stage of the full pipeline of our parser.
\iffalse
\begin{itemize}
    \item Discourse Relation Parsing: Yangfeng Ji ACL 2017, Yangfeng Ji RST parser, Bhatia Movie Review RST, Ziheng Lin PDTB tagger, Son ACL 2017
    \item Psychology: Optimistic Causal Explanation

\end{itemize}
\fi
\section{Methods}
We build our model based on PDTB-style discourse relation parsing since PDTB has a relatively simpler text segmentation method;\footnote{RST parsing builds fully hierarchical discourse tree structures out of the whole span of target text which highly depends on syntactic parsing and exact matching of elementary discourse units which are extremely hard to obtain from social media texts} for explicit discourse relations, it finds the presence of discourse connectives within a document and extracts discourse arguments which parametrize the connective while for implicit relations, it considers all adjacent sentences as candidate discourse arguments.
\subsection{Dataset}

We created our own causal explanation dataset by collecting 3,268 random Facebook status update messages. Three well-trained annotators manually labeled whether or not each message contains the causal explanation and obtained 1,598 causality messages with substantial agreement ($\kappa=0.61$). We used the majority vote for our gold standard. Then, on each causality message, annotators identified which text spans are causal explanations.

For each task, we used 80\% of the dataset for training our model and 10\% for tuning the hyperparameters of our models. Finally, we evaluated all of our models on the remaining 10\% (Table~\ref{tab:causality_dataset} and Table~\ref{tab:causality_tweet_da}). For causal explanation detection task, we extracted discourse arguments using our parser and selected discourse arguments which most cover the annotated causal explanation text span as our gold standard.

\begin{table}
\centering
\begin{tabular}{|l|l|l|l|}
\hline \bf Dataset & Causality & Non-Causal & Total   \\ \hline
Training & 1,284 & 1,330  & 2,614  \\
Validation & 150  & 177 & 327  \\
Test & 164 & 163 & 327 \\ \hline
Total & 1,598 & 1,670 & 3,268 \\
\hline
\end{tabular}
\caption{\label{tab:causality_dataset} Number of messages containing causality or not in our dataset. }
\end{table}

\begin{table}
\centering
\begin{tabular}{|l|l|l|l|}
\hline \bf Causality messages & CE DA & Total DA   \\ \hline
Training & 1,278   & 5,606  \\
Validation & 160 & 652  \\
Test & 160 & 757 \\ \hline
Total & 1,598 & 7,015 \\
\hline
\end{tabular}
\caption{\label{tab:causality_tweet_da} The number of discourse arguments in causality messages. Across 1,598 total causality messages, we found 7,015 discourse arguments (Total DA) and the one which covers annotated causal explanation are used as causal explanation discourse arguments (CE DA)}
\end{table}

\subsection{Model}
We build two types of models. First, we develop feature-based models which utilize features of the successful models in social media analysis and causal relation discourse parsing. Then, we build a recursive neural network model which uses distributed representation of discourse arguments as this approach can even capture latent properties of causal relations which may exist between distant discourse arguments. We specifically selected bidirectional LSTM since the model with the discourse distributional structure built in this form outperformed the traditional models in similar NLP downstream tasks \cite{ji2017neural}.

\paragraph{Discourse Argument Extraction}
As the first step of our pipeline, we use Tweebo parser \cite{kong2014dependency} to extract syntactic features from messages. Then, we demarcate sentences using punctuation (`,') tag and periods. Among those sentences, we find discourse connectives defined in PDTB annotation along with a Tweet POS tag for conjunction words which can also be a discourse marker. In order to decide whether these connectives are really discourse connectives (e.g., I went home, \textbf{but} he stayed) as opposed to simple connections of two words (I like apple \textbf{and} banana) we see if verb phrases \footnote{minimal discourse unit is verb phrases with very few exceptions \cite{prasad2007penn}} exist before and after the connective by using dependency parsing results. Although discourse connective disambiguation is a complicated task which can be much improved by syntactic features \cite{pitler2009using}, we try to minimize effects of syntactic parsing and simplify it since it is highly error-prone in social media. Finally, according to visual inspection, emojis (`E' tag) are crucial for discourse relation in social media so we take them as separate discourse arguments (e.g.,in ``My test result... :('' the sad feeling is caused by the test result, but it cannot be captured by plain word tokens).
\iffalse
\todo{Figure for explanatory style in tweet (discourse units)}
\todo{We just detect VPs (which is minimal discourse unit of PDTB}
\fi

\begin{figure*}
  \centering
  \includegraphics[width=4.8in]{LSTM_Classifier_Figure4.png}
  \caption{LSTM classifier for causality detection and explanation identification}
  \label{fig:LSTM_classifier}
\end{figure*}

\paragraph{Feature Based Models}

We trained a linear SVM, an rbf SVM, and a random forest with N-gram, charater N-gram, and tweet POS tags, sentiment tags, average word lengths and word counts from each message as they have a pivotal role in the models for many NLP downstream tasks in social media \cite{mohammad2013nrc,lynn2017human}. In addition to these features, we also extracted \textit{First-Last, First3}  features and \textit{Word Pairs} from every adjacent pair of discourse arguments since these features were most helpful for causal relation prediction \cite{pitler2009automatic}. \textit{First-Last, First3} features are first and last word and first three words of two discourse arguments of the relation, and \textit{Word Pairs} are the cross product of words of those discourse arguments. These two features enable our model to capture interaction between two discourse arguments. \cite{pitler2009automatic} reported that these two features along with verbs, modality, context, and polarity (which can be captured by N-grams, sentiment tags and POS tags in our previous features) obtained the best performance for predicting Contingency class to which causality belongs.

\iffalse
We use the models that PDTB and RST researchers and NLP reserachers found helpful (Word Pairs, First-Last, N-gram, Char N-gram, POS)
Cite: pitler (Word Pairs, First-Last3), Mohammad Saif (N-gram, Char N-gram, POS)
\fi
\paragraph{\label{sec:rnn_model} Recursive Neural Network Model}
We load the GLOVE word embedding \cite{pennington2014glove} trained in Twitter \footnote{\url{http://nlp.stanford.edu/data/glove.twitter.27B.zip}} for each token of extracted discourse arguments from messages. For the distributional representation of discourse arguments, we run a Word-level LSTM on the words' embeddings within each discourse argument and concatenate last hidden state vectors of forward LSTM ($\overrightarrow{h}$) and backward LSTM ($\overleftarrow{h}$) which is suggested by \cite{ji2017neural} ($DA = [\overrightarrow{h};\overleftarrow{h}]$). Then, we feed the sequence of the vector representation of discourse arguments to the Discourse-argument-level LSTM (DA-level LSTM) to make a final prediction with log softmax function. With this structure, the model can learn the representation of interaction of tokens inside each discourse argument, then capture discourse relations across all of the discourse arguments in each message (Figure \ref{fig:LSTM_classifier}). In order to prevent the overfitting, we added a dropout layer between the Word-level LSTM and the DA-level LSTM layer.

\paragraph{Architectural Variants}
We also explore subsets of the full RNN architecture, specifically with one of the two LSTM layers removed.
In the first model variant, we directly input all word embeddings of a whole message to a BiLSTM layer and make prediction (\textbf{Word LSTM}) without the help of the distributional vector representations of discourse arguments. In the second model variant, we take the average of all word embeddings of each discourse argument ($DA_k=\frac{1}{N_k} \sum_{i=1}^{N_k}W_{i}$), and use them as inputs to a BiLSTM layer (\textbf{DA AVG LSTM})  as the average vector of embeddings were quite effective for representing the whole sequence  \cite{ji2017neural,wieting2015towards}.
As with the full architectures, for CP both of these variants ends with a many-to-one classification per message, while the CEI model ends with a sequence of classifications.

\subsection{Experiment}
\paragraph{Feature Based Model}
We explored three types of models (RBF SVM, Linear SVM, and Random Forest Classifier) which have previously been shown empirically useful for the language analysis in social media. We filtered out low frequency \textbf{Word Pairs} features as they tend to be noisy and sparse \cite{pitler2009automatic}. Then, we conducted univariate feature selection to restrict all remaining features to those showing at least a small relationship with the outcome. Specifically, we keep all features passing a family-wise error rate of $\alpha = 60$ with the given outcome. After comparing the performance of the optimized version of each model, we also conducted a feature ablation test on the best model in order to see how much each feature contributes to the causality prediction.

\paragraph{Neural Network Model}
We used bidirectional LSTMs for causality classification and causal explanation identification since the discourse arguments for causal explanation can show up either before and after the effected events or results and we want our model to be optimized for both cases. However, there is a risk of overfitting due to the dataset which is relatively small for the high complexity of the model, so we added a dropout layer (p=0.3) between the Word-level LSTM and the DA-level LSTM.

For tuning our model, we explore the dimensionality of word vector and LSTM hidden state vectors of discourse arguments of 25, 50, 100, and 200 as pretrained GLOVE vectors were trained in this setting. For optimization, we used Stochastic Gradient Descent (SGD) and Adam \cite{kingma2014adam} with learning rates 0.01 and 0.001.

We ignore missing word embeddings because our dataset is quite small for retraining new word embeddings. However, if embeddings are extracted as separate discourse arguments, we used the average of all vectors of all discourse arguments in that message. Average embeddings have performed well for representing text sequences in other tasks~\cite{wieting2015towards}.

\paragraph{Model Evaluation}
We first use state-of-the-art PDTB taggers for our baseline \cite{lin2014pdtb,biran:15} for the evaluation of the causality prediction of our models (\cite{biran:15} requires sentences extracted from the text as its input, so we used our parser to extract sentences from the message). Then, we compare how models work for each task and disassembled them to inspect how each part of the models can affect their final prediction performances. We conducted McNemar's test to determine whether the performance differences are statistically significant at $p < .05$.
\iffalse
\todo{compare the different characteristic (behaviors, patterns they can capture, pros and cons of each, so I can evaluate my hypothesis with the experiment results}
Pre-trained GLOVE word embeddings from messages (details follow...)

\paragraph{Discourse Argument Representation}
We used a representation of discourse units which worked well for text categorization using RST parsing (Ji ACL 2017).

\paragraph{Neural Network Model}
We feed them into our bidirectional LSTM classifier to capture relations between each adjacent discourse units

\fi
\section{Results}
We investigated various models for both causality detection and explanation identification. Based on their performances on the task, we analyzed the relationships between the types of models and the tasks, and scrutinized further for the best performing models. For performance analysis, we reported weighted F1 of classes.

\subsection{Causality Prediction}
\begin{table}
\centering
\begin{tabular}{|l|l|l|l|}
\hline \bf Model & F1  \\ \hline
\cite{biran:15} & 0.434 \\
\cite{lin2014pdtb} & 0.638\\ \hline
Linear SVM & \textbf{0.791}  \\
RBF SVM & \textbf{0.777} \\
Random Forest & 0.771\\ \hline
LSTM & 0.758\\

\hline
\end{tabular}
\caption{\label{tab:causality} Causality prediction performance across different predictive models. Bold indicates significant improvement over the LSTM
\end{table}

\begin{table}
\centering
\begin{tabular}{|l|l|l|l|}
\hline \bf Model & F1 \\ \hline
All & 0.791 \\
\hline
- First-Last, First3 & 0.788 \\
- Word Pairs & 0.787 \\
- POS tags & 0.734\\
- (Char + Word) N-grams & 0.769 \\
- Sentiment tags & 0.791 \\
\hline
\end{tabular}
\caption{\label{tab:lsvm-feat}Feature ablation test of Linear SVM for causality prediction}
\end{table}

In order to classify whether a message contains causal relation, we compared off-the-shelf PDTB parsers, linear SVM, RBF SVM, Random forest and LSTM classifiers. The off-the-shelf parsers achieved the lowest accuracies (\cite{biran:15} and \cite{lin2014pdtb} in Table \ref{tab:causality}). This result can be expected since 1) these models were trained with news articles and 2) they are trained for all possible discourse relations in addition to causal relations (e.g., contrast, condition, etc). Among our suggested models, SVM and random forest classifier performed better than LSTM and, in the general trend, the more complex the models were, the worse they performed. This suggests that the models with more direct and simpler learning methods with features might classify the causality messages better than the ones more optimized for capturing distributional information or non-linear relationships of features.

\paragraph{Causality Classifier Analysis}
Table \ref{tab:lsvm-feat} shows the results of a feature ablation test to see how each feature contributes to causality classification performance of the linear SVM classifier.
POS tags caused the largest drop in F1.
We suspect POS tags played a unique role because discourse connectives can have various surface forms (e.g., because, cuz, bcuz, etc) but still the same POS tag `P'.
Also POS tags can capture the occurrences of modal verbs, a feature previously found to be very useful for detecting similar discourse relations~\cite{pitler2009automatic}. N-gram features caused 0.022 F1 drop while sentiment tags did not affect the model when removed. Unlike the previous work where \textit{First-Last, First3} and \textit{Word pairs} tended to gain a large F1 increase for multiclass discourse relation prediction, in our case, they did not affect the prediction performance compared to other feature types such as POS tags or N-grams.

\subsection{Causal Explanation Identification}
\begin{table}
\centering
\begin{tabular}{|l|l|l|l|}
\hline \bf Model & Prec & Rec & F1  \\ \hline
Linear SVM & 0.773 & 0.727 & 0.743\\
RBF SVM & 0.739 & 0.771 & 0.749 \\
Random Forest & 0.747 & 0.790 & 0.746 \\
\hline
LSTM & \textbf{0.851} & \textbf{0.858} & \textbf{0.853} \\
\hline
\end{tabular}
\caption{\label{tab:ce} Causal explanation identification performance. Bold indicates significant imrpovement over next best model ($p < .05$)}
\end{table}

In this task, the model identifies causal explanations given the discourse arguments of the causality message. We explored over the same models as those we used for causality (sans the output layer), and found the almost opposite trend of performances (see Table \ref{tab:ce}). The Linear SVM obtained lowest F1 while the LSTM model made the best identification performance. As opposed to the simple binary classification of the causality messages, in order to detect causal explanation, it is more beneficial to consider the relation across discourse arguments of the whole message and implicit distributional representation due to the implicit causal relations between two distant arguments.

\subsection{Architectural Variants}

\begin{table}
\centering
\begin{tabular}{|l|l|l|l|}
\hline \bf Model & CP (F1) & CEI (F1) \\
\hline
Full LSTM & 0.758 & 0.853\\
\hline
DA AVG LSTM & 0.685 & 0.818 \\
Word LSTM & 0.694 & 0.792 \\
\hline
\end{tabular}

\caption{\label{tab:lstm-variant} The effect of Word-level LSTM (Word LSTM) and discourse argument LSTM (DA AVG LSTM) for causality prediction (CP) and causal explanation identification (CEI). Note that, as described in methods, there are architectural differences for CP and CEI models with the same names, most notably that the output layer is always a single classification for CP and a sequence of classifications for CEI. }
\end{table}

For causality prediction, we experimented with only word tokens in the whole message without help of Word-level LSTM layer (\textbf{Word LSTM}), and F1 dropped by 0.064  (CP in Table \ref{tab:lstm-variant}). Also, when we used the average of the sequence of word embeddings of each discourse argument as an input to the DA-level LSTM and it caused F1 drop of 0.073. This suggests that the information gained from both the interaction of words in and in between discourse arguments help when the model utilizes the distributional representation of the texts.

For causal explanation identification, in order to test how the LSTM classifier works without its capability of capturing the relations between discourse arguments, we removed DA-level LSTM layer and ran the LSTM directly on the word embedding sequence for each discourse argument for classifying whether the argument is causal explanation, and the model had 0.061 F1 drop (\textbf{Word LSTM} in CEI in Table \ref{tab:lstm-variant}). Also, when we ran DA-level LSTM on the average vectors of the word sequences of each discourse argument of messages, F1 decreased to 0.818. This follows the similar pattern observed from other types of models performances (i.e., SVMs and Random Forest classifiers) that the models with higher complexity for capturing the interaction of discourse arguments tend to identify causal explanation with the higher accuracies.

For CEI task, we found that when the model ran on the sequence representation of discourse argument (\textbf{DA AVG LSTM}), its performance was higher than the plain sequence of word embeddings (\textbf{Word LSTM}). Finally, in both subtasks, when the models ran on both Word-level and DA-Level (\textbf{Full LSTM}), they obtained the highest performance.

\subsection{Complete Pipeline}

\begin{table}
\centering
\begin{tabular}{|l|l|l|l|}
\hline \bf Model & Prec & Rec & F1 \\
\hline
CP + CEI$_{causal}$ & \bf 0.864 & \bf 0.877 & \textbf{0.868} \\
CP + CEI$_{all}$ & 0.842 & 0.864 & 0.848 \\
CEI$_{causal}$ Only & 0.847 & 0.788 & 0.810 \\
CEI$_{all}$ Only & 0.836 & 0.848 & 0.842 \\
\hline
\end{tabular}
\caption{\label{tab:pipeline} The effect of Linear SVM Cauality model (CP) within our pipeline. CEI$_{all}$: LSTM CEI models trained on all messages; CEI$_{causal}$: LSTM CEI models trained only on causality messages (CEI$_{causal}$); CP + CEI$_{all|causal}$: the combination of Linear SVM and each LSTM model. Bold: significant ($p < .05$) increase in F1 over the next best model, suggesting the two-step approach worked best. }
\end{table}

Evaluations thus far zeroed-in on each subtask of causal explanation analysis (i.e.~CEI only focused on data already identified to contain causal explanations). Here, we seek to evaluate the complete pipeline of CP and CEI, starting from all of test data (those or without causality) and evaluating the final accuracy of CEI predictions.
This is intended to evaluate CEI performance under an applied setting where one does not already know whether a document has a causal explanation.

There are several approaches we could take to perform CEI starting from unannotated data.
We could simply run CEI prediction by itself (\textbf{CEI Only}) or the pipeline of CP first and then only run CEI on documents predicted as causal (\textbf{CP + CEI}).
Further, the CEI model could be trained only on those documents annotated causal (as was done in the previous experiments) or on all training documents including many that are not causal.

Table \ref{tab:pipeline} show results varying the pipeline and how CEI was trained. Though all setups performed decent ($F1 > 0.81$) we see that the pipelined approach, first predicting causality (with the linear SVM) and then predicting causal explanations only for those with marked causal (CP + CEI$_{causal}$) yielded the strongest results.
This also utilized the CEI model only trained on those annotated causal.
Besides performance, an added benefit from this two step approach is that the CP step is less computational intensive of the CEI step and approximately 2/3 of documents will never need the CEI step applied.

\paragraph{Limitations.} We had an inevitable limitation on the size of our dataset, since there is no other causality dataset over social media and the annotation required an intensive iterative process. This might have limited performances of more complex models, but considering the processing time and the computation load, the combination of the linear model and the RNN-based model of our pipeline obtained both the high performance and efficiency for the practical applications to downstream tasks. In other words, it's possible the linear model will not perform as well if the training size is increased substantially. However, a linear model could still be used to do a first-pass, computationally efficient labeling, in order to shortlist social media posts for further labeling from an LSTM or more complex model.

\section{Exploration}

Here, we explore the use of causal explanation analysis for downstream tasks.
First we look at the relationship between use of causal explanation and one's demographics: age and gender. Then, we consider their use in sentiment analysis for extracting the causes of polarity ratings. Research involving human subjects was approved by the University of Pennsylvania Institutional Review Board.

\paragraph*{Demographic differences.}

We first explored variance in number of causality posts by demographics.
To do this, we used self-authored posts from a random 300 consenting-users of the MyPersonality dataset~\cite{kosinski2013private}.
For each user we calculate a $cp\_ratio$, defined as the number of causality predicted posts divided by their total number of posts, indicating the percentage of their posts which include a causal explanation.
We then correlated this ratio with real-valued age using Pearson correlation and looked the differences by dichotomous gender using Cohen's \textit{d} (the difference in standardized means; only binary gender was available). We found significant ($p < .05$) moderate-sized associations for both, indicating both older individuals and females were likely to use more causal explanations.

\paragraph*{Causality in Sentiment Analysis}
We explored the application of causality explanation identification for sentiment analysis using the Yelp polarity dataset~\cite{zhang2015character}. We randomly selected 10,000 of both positive and negative reviews and ran our complete pipeline on them to extract the causal explanations from the reviews. We then analyzed the ngrams from (a) causal explanation and (b) all other discourse arguments testing for associations with polarity. We used the a Bayesian interpretation of the log odds ratio using an informative dirichlet prior defined by~\newcite{monroe2008fightin}. We found difference in the top ngrams depending on whether the argument the ngram originated from was a causal explanation or not  (see Table \ref{tab:yelp}).  Top ngrams for causal explanations included more content words (e.g.~`rude', `overpriced', `slow') suggesting analyzing causal explanations within reviews can better target the \textit{reasons} for the negative review.

\begin{table}
\centering
\begin{tabular}{|l|c|c|}
\hline & \textbf{CE} & \textbf{Non-CE}\\
 & Top Ngrams & Top Ngrams \\
\hline
1 & worst & not \\
2 & was &  no \\
3 & not &  "   \\
4 & the worst & asked  \\
5 & horrible & she \\
6 & rude & told  \\
7 & bad  & said  \\
8 & overpriced  & minutes  \\
9 & over & ? \\
10 & slow &  me  \\
\hline
\end{tabular}
\caption{\label{tab:yelp} Top words most associated with negative reviews from within causal explanations (CE) and outside of causal explanation (Non-CE). }
\end{table}

\section{Conclusion}
We developed a pipeline for causal explanation analysis over social media text, including both causality prediction and causal explanation identification. We  examined a variety of model types and RNN architectures for each part of the pipeline, finding an SVM best for causality prediction and a hierarchy of BiLSTMs for causal explanation identification, suggesting the later task relies more heavily on sequential information.
In fact, we found replacing either layer of the hierarchical LSTM architecture (the word-level or the DA-level) with a an equivalent ``bag of features'' approach resulted in reduced accuracy.
Results of our whole pipeline of causal explanation analysis were found quite strong, achieving an $F1=0.868$ at identifying discourse arguments that are causal explanations.

Finally, we demonstrated use of our models in applications, finding associations between demographics and rate of mentioning causal explanations, as well as showing differences in the top words predictive of negative ratings in Yelp reviews.
Utilization of discourse structure in social media analysis has been a largely untapped area of exploration, perhaps due to its perceived difficulty.
We hope the strong results of causal explanation identification here leads to the integration of more syntax and deeper semantics into social media analyses and ultimately enables new applications beyond the current state of the art.

\section*{Acknowledgments}
This work was supported, in part, by a grant from the Templeton Religion Trust (ID \#TRT0048). The funders had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript. We also thank Laura Smith, Yiyi Chen, Greta Jawel and Vanessa Hernandez for their work in identifying causal explanations.

\iffalse
\todo{consider moving to discussion or conclusion (as applications):}
More broadly, causal explanations is studied importantly in social science (philosophy, social science, psychology)
for a long time. In philosophy, ... In social science, ... . In a broader view, causality analysis has been a significant basic principle for numerous types of scientific inferences for both social science (e.g., how policies affect economy) and natural science (e.g., factors of climate change, chemical reactions, etc). In this paper, we focus on causality detected in human language and recognizing causal explanation which is crucial for psychological analysis.

\begin{itemize}
    \item Causal Relation in general
    \begin{itemize}
        \item Science: chemistry: chemical reaction, medical: virus, bacteria of diesease, earth science: climate change
        \item NLP Tasks
        \begin{itemize}
            \item Question Answering: Cause \& Effect
            \item Sentiment Analysis: Why negative or positive? What factors contribute to the sentiment towards targets (e.g., stock prices, movies, products, restaurants)
        \end{itemize}
        \item \textbf{Psychology: Optimism that can be useful for the analysis personality, mental health, physical health, resiliency, etc }
    \end{itemize}
    \item Discourse Relation Parsing: General NLP models for predicting for all existing classes of discourse relation have low performances. We focus on causal relation which is very important and widely applicable to many downstream NLP tasks and a variety of domains (as we saw above)
    \item Applications of Causal Relation
\end{itemize}
\fi

redpajama_set_name: RedPajamaArXiv


Moving day is fun for the kids but a little rough on you…like most days really. The only thing that will get you through it, is the knowledge that you are going somewhere lovely…and that there's a glass of wine or calming cup of tea at the end of it. So, we've come up with a list of ways that you can reduce the madness that is moving. We still recommend the wine of course.
Before you do anything else - pack a box of everything the family needs to make their first night a joy Things like include sheets, towels, cutlery, toiletries, multiple packets of cleaning wipes, snacks, a kettle and above all of course the TV remote and phone chargers. You'll feel more at home and there'll be fewer tears. Use the guide at the bottom of this article for some essentials.
The temptation to just start throwing things into boxes is great. But if you don't want to find yourself searching for a school uniform at 3am, we suggest you list the contents of every box, and even colour code the boxes. Taking photos is also not bad idea.
Sliding clothes straight into bags or boxes can save hours at the other end. Cut a small hole at the bottom of a bin bag and place the clothes inside with the hangers towards the bottom of the bag. Once the hanger hooks are through the hole at the bottom of the bag, tie and secure them with a rubber band or string.
This will save room in expensive packing boxes.
It will be tempting to leave stuff in, but you'll wind up picking up things that have dropped out for weeks afterwards.
Cotton balls are genius and will cushion your powdered cosmetics and one almighty stop mess.
Take that used toilet roll and wrap cords around it so they don't get knotted up. If you don't there is 100% chance they will end up in a big ball.
A little bit of plastic could save you from wiping shampoo and conditioner off your hairdryer and anything else in your bathroom box. Don't risk it.
Opened containers of juice and food do not travel at all well.
Keep your dinner set safe, and lose the need for bubble wrap.
If you put towels under heavy furniture you will save the floorboards and tiles from scratches.
If you are taking the TV and pictures off the walls, keep all the screws and hooks in sandwich bags.
Quite simply don't do it. No one has yet invented unbreakable packing boxes or removalists.
Put something like a rubber band over the front door lock, so that whilst everyone is going in and out, the door stays open…one in ten people lock themselves out on moving day, don't let it be you!
Most importantly, remember you are about to start a new chapter in your life and once all the bruises have gone down and you can actually move around the rooms without boxes, the move will be totally worth it.


WASHINGTON (AP) -- Hillary Clinton says she will visit flood-damaged Louisiana when "the presence of a political campaign will not disrupt the response."
In a statement Monday, the Democratic presidential nominee called the floods a crisis in need of a national response. She noted that she had asked supporters to contribute to the Red Cross to help recovery efforts for the more than 100,000 people affected by the floods.
Clinton added that she wants to make sure there is a focus on Zika prevention, so that the mosquitoes that carry the virus don't spread to Louisiana.
Republican nominee Donald Trump visited Louisiana Friday and President Barack Obama is expected Tuesday. Heavy flooding this month killed at least 13 people and displaced thousands more after water engulfed their homes.


Unfortunately currently from oracle's site oracle-instantclient is only available in RPM and ZIP format. So Debian and Ubuntu users will have to install it either by downloading zip and unarchiving somewhere or using alien (deb rpm install) tool.
I've been thinking for a long time. How computers and involved technology copy God's creation. This kind of thoughts poped up in my mind right after I became a believer. As I'm having a strong IT background I tend to view thinks in world via the prism of my IT knowledge. If I have to learn a new science my mind tend to compare how this translates to my previous knowledge obtained in IT. Probably some other people out there has the same kind of thinking? I'm not sure if this is a geek thinking or it is usual and people from other fields of science tend to also understand the world by using accommodated knowledge in field of profession they practice. Anyways since my days I believed in Jesus Christ, I started to also to compare my so far knowledge with what I've red in Holy Bible and in the book of The Living of Saints (which btw is unknown to most protestant world). It is very interesting that if you deeply look into how all Information Technology knowledge is organized you can see how Computers resembles the visible God's creation. In reality I came to realization how Moden Man self-deceives himself. We think with every new modern technology we achieved something new revolutionory which didn't existed before. But is it really true? Lets take some technology like Microsoft Active Directory (using LDAP) for example. LDAP structures data in a tree form where each branch could have a number of sub branches (variables). In reality it appears LDAP is not new it a translation of previous already existent knowledge in universe served in a different kind of form. Let me give some other examples, lets pick up the Internet, we claim its a new invention and from human point of view it is. But if we look on it via the prism of existing created world. It is just a interconnection between "BIG DATA" in real world it is absolutely the same latest researches already know all in world is data and all data in world is interconnected. So obviously the internet is another copying of the wonderful things God created in material and for those who can accept (the spiritual world) world. Many who are hard-core atheists will argue that we copy things in the world but all the material world is just a co-incidence. But having in mind that the world is so perfectly tuned "for living beings to exist" it is near to impossible that all this life and perfection emerged by random. The tree structure model is existing everywhere in OS and programming. We can see it in hiearchy of a file system, we can see it in hashes (arrays) in programming and all this just copies the over-simplified model of a real tree (which we know well from Biology is innemous times more complex). Probably the future of computing is in Biotechnologies and people's attempts to copy how living organism works. We know from well from science-fiction and cyberpunk the future should be mostly in Bio-technologies and computer as we know it but even this high-tech next generation technology will be based on existent things. Meaning man doesn't invent something so different he does copy a model and then modify the model according to environment or just makes a combination of a number of models to achieve a next one. Sorry for the rant post but I'm thinking on this for quite a while and I thought i should spit it here and interested to hear what people think and what are the arguments for or against my thesis?
I remember GNU / Linux, 11 years from now, times when ALSA was not standardly shipped with Linux.
Back then ALSA still lacked good support for many SoundCards and was still a "baby project".
In that time what we used to have sound on Linux was OSS – Open Sound System. OSS emerged right after the first ever Linux sound system VoxWare (formerly known as the Linux Sound Driver).
Back in those days OSS was used for multimedia support on both GNU / Linux and BSD based free OSes. It was few years later when I heard and used ALSA for a fist time and it wasn't really a love from first sigth.
One can easily find out by the name ALSA it is a system especially built for the Linux kernel and that's one of the reasosn why *BSD systems has their custom separate sound system.
There is plenty of reasons why OSS was substituted with ALSA. Main reason was its commercial like license, OSS wasn't completely "open source" GPLed (free software), there was resctions on use of OSS for commercial goals.
With its emerge ALSA started to push away OSS slowly. Somewhere in 2003, alsa has officially entered the Linux kernel source and until 2005 it was the default standard for all GNU / Linux operating systems.
As of time of writting ALSA has become the only sound system to have support for multiple sound card devices for Linux.
My experiences with ALSA, however ain't so nice if I take a look in my past experiences.
Since the very beginning of using ALSA, I had plenty of troubles with configuring properly my sound card not to mention, even after configuring it the MIDI support was not there.
Besides all the troubles main problems were stemming from the many applications still written to use OSS as sound system and hence with those sound was impossible with ALSA.The most problematic thing about apps written with OSS in mind was all of them tried to stream sound via /dev/dsp (OSS Digital Sound Processor), since alsa did not used /dev/dsp those programs was soundless.
On the other hand OSS was creating issues as well, one severe problem with OSS was the inability to stream multiple sounds simultaneously, because each sound stream required to pass voice through /dev/dsp and usually there was only one /dev/dsp.
and the proceeding irritation that used to annoy us in the early GNU / Linux days had of course some raw workarounds hacks but generally the workaround did not fix problems always.
ALSA has modular structure and this imposes a great problem nowdays. The modular architecture is generally a good idea, however the way this was implemented within ALSA is far away from clear and easy to understand by the end user and therefore makes it very unintuitive and obscure.
Alsa misses simplicity which somehow was partially in the days of OSS. Thinking over the general situation with Linux multimedia nowdays, I believe it was exactly ALSA Project responsible for the so delayed mass Desktop Linux adoption.
Many long year standing Linux users had certainly had the alsa troubles during new system installs (correct me if I'm wrong).
The only fix to multiple soundcard initialization problems was to download alsa source and compile from source and hence made it hard and discouraging for people giving Linux a try.
Hope some ALSA developers will read this post work on changing the crazy structure of ALSA over complexity. ALSA needs automate way to solve issues with itself, the configuration should be more trivial and unified if Linux has to become more attractive for Desktop adoption.
Anyways, after the few words of history and indicating my pesonal observations on ALSA. I will proceed and explain few things on how ALSA can be configured to support and play nice with OSS dependant programs as well some basic explanations on common incompatibility between esd and pulseaudio and how this can be fixed;.
The Linux sound situation becomes even more messy when ESD enters the scene. Many of the novice new Linux users certainly don't remember (Enlightened Sound Daemon) . ESD historically preceded PulseAudio . Hence it will be good to mention ESD was used for few years in GNOME and in around 2006-2007 it was substituted by PulseAudio.
Many applications, however who was ported or written for Linux especially (the proprietary ported ones) was already built to work with ESD and even though newer GNOME releases was fully using pulseaudio, this (non free software apps and games) were still depending on ESD.
The package for almost all Linux distributions which does the esd emulation via pulse is pulseaudio-esound-compat . In latest Fedora Linux pulseaudio-esound-compat is installed by default.
padsp – is a PulseAudio OSS Wrapper.
Another common problem with sound on Linux is when running windows applications (running windows games with wine).
Quite often sound fails to work since wine tries to directly communicate with alsa and fails because alsa sound channel is taken by pulseaudio.
This script was reported by many people as fix to problems with wine games failing to play sounds and music, anyhow I personally prefer using the stop / start pulseaudio method.
Listing all table names from a MySQL database is a very easy and trivial task that every sql or system administrator out there is aware of.
However excluding certain table names from a whole list of tables belonging to a database is not that commonly used and therefore I believe many people have no clue how to do it when they have to.
Today for one of my sql backup scripts it was necessary that certain tables from a database to be excluded from the whole list of tables for a database I'm backupping.
However as my desire was to exclude certain tables from the list (preferrably with a certain SQL query) I had to ask around in irc.freenode.net for some hints on a ways to achieve my exclude table goals.
I was adviced by some people in #mysql that what I need to achieve my goal is the information_schema mysql structure, which is available since MySQL version 5.0.
It seems using the information_schema is very easy and is not much different from your normal queries syntax used to do trivial operations in the mysql server.
If you wonder just like I did what is mysql's information_schema go and use the information_schema database (which I believe is a virtual database that is stored in the system memory).
I used the limit clause in order to prevent being overfilled with data, where I could still see the table fields name to get general and few lines of the table to get an idea what kind of information the TABLES table contains.
That will also give you a few hints about the exact way the MySQL works and comprehends it's contained data structures.
In short information_schema virtual database and it's existing tables provides a very thorough information and if you're an SQL admin you certainly want to look over it every now and then.
A bit of playing with it lead me to a command which is actually a good substitute for the normal SHOW TABLES; mysql command.
Of course this example can easily be adopted to a MySQL backup script which requires the exclusion of certain tables from a backed up database.
Now this little bash one-liner can easily be customized to a backup script to create backups of a certain databases with a certain tables (e.g. with excluded number of tables) from the backup.
It's seriously a pity that by default the mysqldump command does not have an option for a certain tables exclude while making a database dump.
I've saw the mysqldump exclude option, being suggested somewhere online as a future feature of mysqldump, I've also seen it being reported in the mysql.com's bug database, I truly hope in the upcoming releases we will see the exclude option to appear as a possible mysqldump argument.


text:
Q: vueJs routing components adds resolve: You may need an appropriate loader to handle this file type I added dynamic components to the route。
component: resolve => require([`${item.component}`], resolve)

Console warning:

You may need an appropriate loader to handle this file type.
If I delete, it does not appear
component: resolve => require([`${item.component}`], resolve)

But that is not what I want, I do not want this warning

A: component: resolve => require([`${item.component}.vue`], resolve)

String variables are spliced into a string. But there's no problem with a single page entrance.

redpajama_set_name: RedPajamaStackExchange


Available with 16, 24, 32, 40 or 48 mono mic/line input channels, the new APB DynaSonics ProDesk-4 includes four stereo line-only input channels, each selectable from two different input sources and controlled from 100mm faders with stereo EQ and access to all aux buses. All mono input channels also include internally illuminated Mute and PFL switches, 6-segment channel metering, 48-volt phantom power switch with LED, mic/line, pad and polarity reverse switches.
Input channels are similar to those used on APB's ProRack mixers, while the cosmetics are identical to Spectra consoles. At the heart of ProDesk-4 is APB's wide-bandwidth circuitry, most noticeable in the mic pre's, EQ and summing amps. Six aux sends are switchable pre/post-fader in pairs and pre/post-EQ as a group; two of the auxes are switchable between mono and stereo operation. The 4-band EQ has sweepable mid-frequencies and an illuminated EQ "on" switch, as well as variable (20 Hz to 400 Hz) highpass filters. Outputs include four analog subgroups, four matrix outputs, and left/right, center and mono (subwoofer) outs. All include XLR balanced output connectors, TRS pre-fader inserts and TRS-balanced bus inputs.
Other ProDesk features include a digital USB stereo input and record output, remote muting of all primary outputs using a single Euro connector, and the use of mother/daughterboard construction for easy servicing. A plug-in power supply ensures reliability; an optional redundant power supply can be added to all but the 16 mono input chassis systems.
For more information, visit APB at InfoComm Booth #C4769 or on the Web at www.apb-dynasonics.com. For additional new SR product news, visit the Briefing Room.


text:
Scribes allegedly attacked by goons of illegal traders
By Correspondent - 8 Aug 2021 3:45 AM GMT
AT Photo
Dhubri, Aug 8: Dhubri based senior journalist Aminur Ahmed and Video Journalist Santosh Roy, was allegedly kept in confinement at Dhubri Free India Port by goons of illegal traders of PDS rice and attacked recently, when the scribes went on the spot to cover the news.
Ahmed stated that they got information that few persons are operating illegal trade of PDS Rice from Dhubri Port (Jahaj Ghat) and accordingly both of them went to the spot for covering the news on Tuesday evening at about 6:30 PM. He stated that on such PDS Rice from Dhubri District, which are meant for poor people traded illegally to South Salmara Mankachar District and then sent to Bangladesh. "I have spoken about it in my media houses several times in the past and about such illegal trade, but it is continuing after the SOP restrictions are implemented", Ahmed said.
He added that alleged illegal traders are using a government river port, after the crowds are less in the town in the evening due to SOP implementations. "Trucks filled with illegally traded PDS rice enter the port, the rice bags are unloaded and then reloaded in boats and then exported for Bangladesh. And all these things are done taking advantage of darkness, inside a locked govt port at the time of night curfew, when streets are least crowded".
However, when the scribes entered the port, few antisocial persons armed with sharp weapons reached the spot and confined senior journalists inside the port snatched their mobile phones, later they released Roy and tried to kill Ahmed, stating that Ahmed has been vocal about such trade, informed the video journalist who was present at the spot.
Ahmed stated that the goons have expressed that they were instructed by one Uttam Kumar Saha, to kill journalists who were regularly broadcasting news about illegal trade of PDS commodities and for naming Saha in such trades. Meanwhile, after escaping Roy informed other journalists of Dhubri town area and subsequently Police and an Addl. DC reached on the spot and rescued Ahmed. In this regard Dhubri Police have also registered an FIR against Saha and his good Under Section 147/325/307/382/506 of IPC, for attempting to murder Ahmed.
The journalist fraternity of Dhubri has also submitted a memorandum to the DC demanding security to the journalist and for proper investigation in the entire episode of illegal trade of PDS commodities in the district.
"Attack on any journalist is an attack on the entire democratic value of the country. The fourth pillar of democracy needs to be protected", said Ashiqure Rahman, District President of Assam Union of Working Journalists.
"This is not the first time a journalist is attacked in Dhubri for raising his voice against illegal activities and for the masses. But it has to be the last. The journalist needs to be protected specially and or else it will hamper the society", said Rajib Sarmah, General Secretary of Dhubri Press Club.
Many politicians, educationalists, senior citizens and individuals have condemned the entire event and have spoken in favour of Ahmed and demanded immediate inquiry into the entire matter and arrest the accused persons immediately.

redpajama_set_name: RedPajamaCommonCrawl


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Harry C. Johnson & Son Inc.
About Harry C Johnson Inc.
Harry C Johnson & Son Inc. was founded in 1906 in Washington DC by Mr. Harry C. Johnson . This family-owned furniture repair shop has been serving the Maryland, Northern Virginia and Washington DC areas for over 100 years.
Formerly located in Bethesda, Maryland, this long time furniture shop has been located in old town Gaithersburg for more than 30 years. Current director J Monterosa joined the company over three decades ago and under his direction our compnay continues to thrive and grow. His expertise and high work ethics will help carry the Johnson name well into the future.
Quality workmanship and long lasting relationships has always been the vision of Harry C Johnson. Today that vision continues evidenced by customer testimonials and repeat Business.
Ask about our pick up, delivery and setup options.
I would like to thank your company for the beautiful job done in connection with the repairs on my two pieces, the work was completed in a quick, efficient and professional manner..


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text:
Q: content inside popup not display in IE 9 content inside div is not displaying in Ie9.Its display in all other browsers?

    x

     **There will be Delay with shipping this week.**

   $(document).ready(function () {
       $('#element_to_pop_up2').bPopup({
              speed: 650,
              transition: 'slideIn'
       });
    });

A: See my fiddle http://jsfiddle.net/rYLu8/1/ where I've litterally copied your code, and copied the code from the JS (as couldn't be served from GitHub directly)
$(document).ready(function () {
   $('#element_to_pop_up2').bPopup({
          speed: 650,
          transition: 'slideIn'
   });
});

It seems to "work" for me on FF, Chrome and IE9 - however I wouldn't describe it as "working properly" in any situation, as it seems to only center the content once you resize the window, which doesn't make sense.
In conclusion, I'm guessing there's something wrong with your code, but I'd seriously recommend looking at something a bit better (and without some juvenile tagline) such as http://twitter.github.io/bootstrap/javascript.html#modals or http://jqueryui.com/dialog/, they'll be easier to use, more reliable across browsers, and they have a huge community of people you can ask if you ever get stuck.


text:
\section{Introduction}
It is well known \cite{szegedy2014intriguing} that modern classification models are vulnerable to adversarial attacks. There were many attempts \cite{43405,35184,Liu_2018_ECCV} to construct empirically robust classifiers, but all of them are proven to be vulnerable to some new more powerful attacks \cite{10.1145/3128572.3140444,pmlr-v80-athalye18a}
. As a result, a lot of certification methods have been introduced \cite{wong,li2020provable,alfarra2021deformrs}
that provide provably correct robustness certificates. Among this variety of certification methods the randomized smoothing remains one of the most effective and feasible approaches which is easy to implement and to parallel. Randomized smoothing \cite{cohen,li,lecuyer} can be applied to data and classifiers of any nature since it does not use any information about the internal structure of the classifier. It is also easily scalable to large networks unlike other methods. All these benefits lead to the fact that the randomized smoothing approach is currently one of the most popular and powerful solutions to robustness certification tasks.

Consequently, there are a lot of works devoted to the randomized smoothing. While most of them concern a setting of robustness within $l_p$- or $l_{\infty}$-balls, some generalize the approach to cover parameterized transformations \cite{fischer,li2020provable}. These parameters usually add up to each other when composing transformations, which allows to smooth classifiers over them. Yet there are some other types of perturbations such that their parameters do not add up but multiply to each other (e.g. a volume change of audio signals or a gamma correction of images). Though it is possible to reduce this case to the additive one by a logarithm (like it is done in \cite{fischer}), we find that alternative distributions (like Rayleigh) may result in better certificates. Thus we propose a new approach of smoothing directly over multiplicative parameters. We prove the theoretical guarantees for classifiers smoothed with Rayleigh distribution and compare them with those obtained via other smoothing distributions for the gamma correction perturbation. Our experiments show that the Rayleigh smoothing cannot be consistently outperformed by the others and thus may be more suitable for certain values of the perturbation parameter.

The main contributions of this work are:
\begin{itemize}
	\item Direct generalization of the randomized smoothing method to the case of multiplicative parameters;
	\item Practical comparison of the proposed method and other smoothing techniques for the gamma correction perturbation, which shows superiority of our method for some factors;
	\item Construction of the first certifiably robust classifiers for the gamma correction perturbation.
\end{itemize}

\section{Background}

The randomized smoothing is a method of replacement of an original classifier $f$ with its ``smoothed'' version ${g(x) = \arg\max\limits_{c}{\mathbb{P}_{\beta}(f\circ\psi_{\beta}(x)=c)}}$ that returns the class the base classifier $f$ is most likely to return when an instance is perturbed by some randomly distributed composable transformation $\psi$. A right choice of the distribution $\beta$ allows to predict a robustness certificate for a given input $x$ for this smoothed classifier $g$ using its confidence in its own answer. Here a robustness certificate is a range of transformations such that they do not change the predicted class $g(x)$ when applied to the given input $x$. One expects the new (smoothed) classifier to be reasonably good if the base one is sufficiently robust.

This technique has only two major drawbacks. The first one is that the transformation we want out classifier to be robust against has to be composable. It is a strong constraint since even theoretically composable transformations lose this property in practice due to rounding and interpolation errors. The other one is that generally we are unable to calculate the smoothed classifier $g$ directly and instead have to use approximations. Since we cannot calculate the actual prediction of $g$, we use the Monte-Carlo approximation algorithm with $n$ samples to obtain the result with an arbitrary high level of confidence. The introduced parameter $n$ controls the trade-off between the approximation accuracy and the evaluation time on inference. Though it is possible to smooth a default undefended base classifier $f$, experiments show that the best results can be achieved by smoothing of already (empirically) robust base classifier $f$. Thus during training of the base classifier $f$ we use the same type of augmentation we expect the smoothed classifier $g$ to be robust against. For a more detailed description of the randomized smoothing method see \cite{cohen,li,lecuyer}.

\section{Generalization of smoothing for multiplicative parameters}

Let us extend the notion of composable transformations in case of multiplicative parameters.

\begin{definition}
	A parameterized map ${\psi_{\delta} : X \rightarrow X,} \; {\delta\in \mathcal{B}\subset\mathbb{R}^n}$ is called multiplicatively composable if
	\begin{equation}(\psi_{\delta}\circ\psi_{\theta}) (x) = \psi_{(\delta\cdot \theta)} (x), \; \forall x\in X,\; \forall \delta,\theta\in \mathcal{B},
	\end{equation}
	where $\delta\cdot \theta\in\mathcal{B}$ means the element-wise multiplication of vectors.
\end{definition}

Usually multiplicative parameters are positive, thus one needs probability distributions with positive supports for randomized smoothing. Yet unlike the additive case an identity transformation corresponds to the parameter value equal to 1 rather than 0. So a conventional exponential distribution \cite{li2020provable} is unsuitable here (we want to concentrate probability mass at 1). Thus we propose a Rayleigh distribution for these needs.

\begin{definition}
	A random variable $\zeta$ has a Rayleigh distribution with the scale parameter $\sigma > 0$ $(\zeta\sim Rayleigh(\sigma))$ if its probability density function (PDF) has a form \begin{equation}
	p_{\zeta}(z) = \sigma^{-2}z e^{-z^2/(2\sigma^2)}, z \geq 0.
	\end{equation}
\end{definition}

For classifiers smoothed with the Rayleigh distribution the following robustness guarantee can be obtained:

\begin{theorem}\label{main}
	Let ${x\in {\mathbb{R}^m}}$, ${f:\mathbb{R}^m \rightarrow Y}$ be a classifier, ${\psi_{\beta}:\mathbb{R}^m \rightarrow \mathbb{R}^m}$ be a multiplicatively composable  transformation for ${\beta\sim Rayleigh(\sigma)}$ and ${g(x) = \arg\max\limits_{c}{\mathbb{P}_{\beta}(f\circ\psi_{\beta}(x)=c)}}$. Denote
	$${p_A = \mathbb{P}_{\beta}(f\circ \psi_{\beta}(x)=c_A)},\;\;
	{p_B = \max\limits_{c_B \not= c_A}{ \mathbb{P}_{\beta}(f\circ\psi_{\beta}(x)}=c_B)}.
	$$
	If
	$$p_A \geq \underline{p_A} > \overline{p_B}\geq p_B,$$
	then $g\circ\psi_{\gamma}(x)=c_A$ for all $\gamma$ satisfying $\gamma_1 < \gamma < \gamma_2$, where $\gamma_1, \gamma_2$ are the only solutions of the following equations
	\begin{equation}
	F(\gamma_1^{-1}F^{-1}(\overline{p_B})) + F(\gamma_1^{-1}F^{-1}(1-\underline{p_A}))=1,
	\end{equation}
	\begin{equation}
	F(\gamma_2^{-1}F^{-1}(\underline{p_A})) + F(\gamma_2^{-1}F^{-1}(1-\overline{p_B}))=1,
	\end{equation}
	and $F(z) = 1 - e^{-z^2/(2\sigma^2)}$ is the CDF of $\beta$.\end{theorem}

A proof is similar to the one provided in \cite{fischer}, though it is also possible to obtain the same result using the constrained adversarial certification framework \cite{zhang} or TSS framework \cite{li2020provable}. All variants are given in Appendix \ref{A},\ref{B}. Notice that the certificate bounds $\gamma_1, \gamma_2$ can be easily found numerically (e.g. by a simple bisection method). But it is also possible to use a trivial estimation $\overline{p_B}=1-\underline{p_A}$ for the probability of the top two class. In that case the equations can be solved analytically:
\begin{equation}
\gamma_1 = \frac{F^{-1}(1-\underline{p_A})}{F^{-1}(\frac{1}{2})}=\sqrt{\frac{\log\underline{p_A}}{\log\frac{1}{2}}},
\end{equation}
\begin{equation}
\gamma_2 = \frac{F^{-1}(\underline{p_A})}{F^{-1}(\frac{1}{2})}=\sqrt{\frac{\log(1-\underline{p_A})}{\log\frac{1}{2}}}.
\end{equation}

The only parameter to be chosen for the Rayleigh distribution is the scale $\sigma$. It seems reasonable to choose it in such a way that either the median or the mean of the random value equals one (value multiplied by 1 stays the same). We try both variants and find that the median equal to 1 results in better certificates (See Appendix \ref{C} for comparison details). Thus hereafter the scale is equal to $\frac{1}{\sqrt{2\ln{(2)}}}$.

The above theorem can be generalized in case of transformations with multiple multiplicative parameters.

\begin{theorem}
	Let $x\in {\mathbb{R}^m}$, $f:\mathbb{R}^m \rightarrow Y$ be a classifier, ${\psi_{\beta}:\mathbb{R}^m \rightarrow \mathbb{R}^m}$, ${\beta=(\beta_1, ..., \beta_n)^{T}}$ be a multiplicatively composable  transformation for independent and identically distributed random variables $\beta_i\sim Rayleigh(\sigma)$ and ${g(x) = \arg\max\limits_{c}{\mathbb{P}_{\beta}(f\circ\psi_{\beta}(x)=c)}}$.
	Denote $${p_A = \mathbb{P}_{\beta}(f\circ \psi_{\beta}(x)=c_A)},\;\; {p_B = \max\limits_{c_B \not= c_A}{ \mathbb{P}_{\beta}(f\circ\psi_{\beta}(x)}=c_B)}.$$ If
	$$p_A \geq \underline{p_A} > \overline{p_B}\geq p_B,$$
	then $g\circ\psi_{\gamma}(x)=c_A$ for all $\gamma\in\Omega$, where $\Omega$ is a region defined by the following inequality

	\begin{multline}

	\mathbb{P}((\gamma_1^{2}-1)\beta_1^2 +...+ (\gamma_n^{2}-1)\beta_n^2 \leq r) \\>\mathbb{P}((\gamma_1^{2}-1)\beta_1^2 +...+ (\gamma_n^{2}-1)\beta_n^2 \geq \theta),

	\end{multline}
	where $r$ and $\theta$ are the only solutions of the following equations
	\begin{equation}
	\mathbb{P}((1-\gamma_1^{-2})\beta_1^2 +...+ (1-\gamma_n^{-2})\beta_n^2 \leq r)=\underline{p_A},
	\end{equation}
	\begin{equation}
	\mathbb{P}((1-\gamma_1^{-2})\beta_1^2 +...+ (1-\gamma_n^{-2})\beta_n^2 \geq \theta)=\overline{p_B}.
	\end{equation}
\end{theorem}

A proof is analogous to the one for a single parameter case and can be found in Appendix \ref{A}.

Usually we cannot evaluate the probabilities $p_A, p_B$. Thus we use the Clopper-Pearson \cite{clopper} bounds $p_B \leq \overline{p_B} < \underline{p_A} \leq p_A$ that can be calculated with an arbitrary high confidence probability $1-\alpha$. In case $\underline{p_A} \leq \frac{1}{2}$ the classifier $g$ abstains from answering (i.e. returns the ``abstain'' answer).

We present $\gamma_1, \gamma_2$ values for some $\underline{p_A}, \overline{p_B}$ in Table \ref{gammas} to show the certificates that can possibly be achieved.
\begin{table}[t]
\centering
\begin{tabular}{@{}rrrr@{}}
    \toprule
	$\underline{p_A}$ & $\overline{p_B}$ & $\gamma_1$ & $\gamma_2$\\ \midrule
        0.600 & 0.400 & 0.86 & 1.15 \\
		 & 0.200 & 0.71 & 1.33 \\
		0.700 & 0.300 & 0.72 & 1.32 \\
		 & 0.100 & 0.54 &  1.56 \\
		0.800 & 0.200 & 0.57 & 1.52 \\
		0.900 & 0.100 & 0.39 & 1.82 \\
		0.990 & 0.010 & 0.12 & 2.58 \\
		0.999 & 0.001 & 0.04 & 3.16 \\ \bottomrule
\end{tabular}
\caption{The calculated robustness certificates $(\gamma_1, \gamma_2)$ for the top two class probabilities $\underline{p_A}, \overline{p_B}$.}
\label{gammas}
\end{table}

\begin{figure*}[b]

\subcaptionbox{$\gamma = 0.5$}{\includegraphics[width=0.33\textwidth]{3}}
\subcaptionbox{$\gamma = 1$}{\includegraphics[width=0.33\textwidth]{2}}
\subcaptionbox{$\gamma = 2$}{\includegraphics[width=0.33\textwidth]{1}}

\caption{Demonstration of the gamma correction transformation on a frog image.}
\label{frog}
\end{figure*}

\section{Experiments}

There are a lot of multiplicatively composable transformations. We choose the gamma correction of images for our experiments due to its importance and simplicity.

Gamma correction is a very popular operation in image processing. There are a lot of works on how it is used for image enhancement \cite{VELUCHAMY2019329} and medical image processing \cite{Karuppanagounder,AGARWAL2017509}.
Yet to the best of our knowledge no works on certified robustness to gamma correction attacks have been released. Thus we are eager to both fill this gap and try out our new smoothing technique on it.

If we consider an input image in the RGB-format as a tensor $x$ with entries in $[0,1]$, then the gamma correction $G_{\gamma}$ with the gamma factor $\gamma$ simply raises $x$ to the power $\gamma$ in the element-wise manner: $G_{\gamma}(x)=x^{\gamma}$.
For a human eye it looks like change of brightness (Fig.\ref{frog}).

Obviously, this transformation is multiplicatively composable:
\begin{equation}
G_{\beta}\circ G_{\gamma}(x)=(x^{\gamma})^{\beta}=x^{\gamma\cdot\beta}=G_{\gamma\cdot\beta}(x).
\end{equation}
But it should be noticed that in reality a gamma corrected image is likely to be converted to some image format with colour channel of limited width afterwards. In that case some information is lost and the resulting transformation is no longer composable. Thus we have two settings: 1) ``idealized'' --- when the colour channel is so wide that the conversion error is negligible in comparison with the rounding error of a machine; 2) ``realistic'' --- when the conversion error is significant and cannot be ignored.

Other works usually consider idealized setting where the only type of error we encounter is an interpolation error. Following their lead we conduct most of our experiments in this setting though we also show how to deal with conversion errors.

\subsection{Idealized setting}

As it is mentioned before the multiplicative parameters $\beta, \gamma$ can be converted into the additive ones $a = \log_c{\gamma}$, $b =\log_c{\beta}$ via logarithm: \begin{equation}
\gamma\cdot\beta = c^{\log_c{\gamma}}\cdot c^{\log_c{\beta}} = c^a \cdot c^b = c^{a+b},
\end{equation}
where $c$ is some fixed base. Thus one can smooth over these new parameters with the standard Gaussian distribution to obtain certificates on the original ones. The certified accuracy for a factor $\gamma$ is the proportion of correctly classified test images whose certificates contain the value $\gamma$. All certificates are obtained with the mistake probability $\alpha=0.001$.

\begin{figure*}[t]
\subcaptionbox{ResNet-20 on CIFAR-10}{\includegraphics[width=0.45\textwidth]{cifar10_gamma_cropped}}
\hspace{1.2 cm}
\subcaptionbox{ResNet-50 on ImageNet}{\includegraphics[width=0.45\textwidth]{imagenet_gamma_cropped}}

\caption{Comparison of theoretically predicted certificates with empirically obtained ones for the gamma correction perturbation.}
\label{comparison}
\end{figure*}

We compare the theoretical certificates for the smoothed classifier with empirical ones for: a) the same smoothed classifier (real smoothed classifier), b) the base architecture without augmentations on training or smoothing (undefended classifier), and c) the base architecture with augmentations on training but without smoothing (augmented classifier) (Fig.\ref{comparison}a). Numbers in brackets show values at 1, that is benign or clean accuracy: the portion of unperturbed images that were correctly classified.

The empirical certificates are obtained via evaluations of classifiers' predictions on perturbed images. For every image $x$ starting from gamma factor $\gamma_0=1$ we evaluate classifier's predictions on $G_{\gamma_k}(x)$ for every $\gamma_k=\gamma_{k-1}+\epsilon$ with a step $\epsilon=0.01$. If the classifier predicts a wrong label on the input $G_{\gamma_k}(x)$ for some $\gamma_k$, we assume that it is robust only in the interval $[1, \gamma_{k-1}]$ and stop the certification procedure. The same procedure is used to find the left end of a certification interval. We use $\epsilon=0.1$ searching for the right ends of certificate intervals for some classifiers (you can spot them by the less smooth right parts of corresponding graphs) due to computational complexity of this process. The resulting certificate can be over optimistic but still it provides a good approximation of the actual robustness. One can see that the real accuracy is bounded from below by the theoretically guaranteed one, which is expected.

We also observe that the smoothed classifier has almost the same actual robustness as the just augmented one. Thus it can be concluded that smoothing provides theoretical certificates at a cost of extra time on inference but does not significantly impact the actual robustness of the augmented base classifier.

The randomized smoothing method is easily scalable to deep models and large datasets. We repeat certification experiments for ResNet-50 on ImageNet \cite{inproceedings} (Fig.\ref{comparison}b) but do not approximate actual robustness due to computational complexity of this process.

\begin{figure*}[b!]
\includegraphics[width=0.33\textwidth]{cifar10_gamma_uniform_rescaled.png}
\includegraphics[width=0.33\textwidth]{cifar10_gamma_gauss_rescaled.png}
\includegraphics[width=0.33\textwidth]{cifar10_gamma_laplace_rescaled.png}

\caption{Comparison of Rayleigh smoothing with uniform, Gaussian and Laplace smoothing for ResNet-110 on CIFAR-10 for the gamma correction perturbation. The base is equal to $e$ everywhere, the mean equals 0 for all alternate distributions and we only change the length of support (uniform), standard deviation (Gaussian) and scale (Laplace) parameters.}
\label{distributions}
\end{figure*}

In order to compare Rayleigh smoothing with other smoothing distributions we train several smoothed ResNet-110 classifiers on CIFAR-10 dataset with uniform, Gaussian and Laplace distributions following TSS protocol (including consistency regularization \cite{NEURIPS2020_77330e13}). We also use the TSS certification protocol with $\alpha = 0.001, n = 100000$ and 500 images randomly picked from the test set for certification. The results are presented at Fig.\ref{distributions}. To ease comparison the factor axes have been re-scaled for factors smaller than 1 to represent uniform multiplicative scale and a dotted line has been used for Rayleigh smoothing.

One can see that the Rayleigh smoothing outperforms all competitors for small gamma factors. Apparently asymmetric nature of the Rayleigh distribution allows to obtain much better certificates for small perturbation values than it would be possible with any conventional symmetric distribution. See Appendix \ref{E} for more detailed analysis of this phenomenon.

\subsection{Realistic setting}

Our study shows that if we convert gamma corrected images to some format with 8 bits per colour channel (RGB true colour), then the conversion error is usually too big to be ignored. Thus we need to make the base classifier $f$ robust in some $l_2$-ball with a radius larger then the conversion error for the most images. For these needs Fischer et al. \cite{fischer} propose to initially smooth the base classifier with the additive Gaussian noise to make it robust in $l_2$-norm. This new classifier can then be used as a base one for smoothing over transformation parameters. Thus instead of $n$ samples for the Monte-Carlo simulation algorithm we use $n_{\varepsilon}$ samples to simulate the smoothing in $l_2$-norm and $n_{\gamma}$ samples to simulate the smoothing over the gamma correction. Since we need $n_{\varepsilon}$ samples for every gamma correction sample, the resulting number of samples is equal to $n_{\varepsilon}\cdot n_{\gamma}$.

There are two types of guarantees that can be obtained by this method: 1) a distributional guarantee, when the conversion error is pre-calculated on the training dataset; 2) an individual guarantee, when we calculate the conversion error for each input at inference time.

For a given input image $x$ and the gamma correction transformation $G$ we define
\begin{equation}
\varepsilon(\beta,\gamma, x):= G_{\beta}\circ G_{\gamma}(x)-G_{\beta\cdot\gamma}(x)
\end{equation}
as the conversion error for gamma factors $\beta,\gamma$. In the idealized setting we assume $\varepsilon(\beta,\gamma,x)=0$ for all inputs. But now we need to estimate this value with some upper bound $E$.

In case of distributional guarantees the conversion error can be estimated with $E$ such that
\begin{equation}
q_E=\mathbb{P}_{x\sim D, \beta\sim Rayleigh}\left(\max\limits_{\gamma\in\Gamma}||\varepsilon(\beta,\gamma,x)||_2 \leq E\right),
\end{equation}
where $D$ is the data distribution, $\Gamma$ is a fixed interval of possible attacks and $1 - q_E$ is a desirable error rate.

The conversion errors $E$ are estimated for intervals $\Gamma$ of gamma factors $\Gamma_1 = [0.86, 1.15]$ and $\Gamma_2 = [0.71, 1.33]$ with the same error rate $q_E=0.9$ as $E_1 = 0.18$ and $E_2 = 0.22$ respectively. These intervals are chosen randomly but in such a way that they could be certificate intervals obtained with the Rayleigh smoothing for some input images, i.e. there exist such $\underline{p_A}, \overline{p_B}$ that the corresponding certificates obtained via Theorem \ref{main} equal to $\Gamma_1$ or $\Gamma_2$. Then we smooth the base classifier with the Gaussian noise with a deviation $\sigma=0.25$, that is found experimentally to deliver the best results for our base models and datasets.

The resulting mistake probability $\rho$ of that smoothed classifier on a ball $B_E(x)$ for a given $x$ can be estimated as the sum of mistake probabilities on all steps:
\begin{equation}
\rho \leq \alpha + 1 - q_E + \alpha_E,
\end{equation}
where $1-\alpha$ is the confidence of the base classifier, $q_E$ is the probabilistic guarantee for $E$ and $1-\alpha_E$ is the confidence with which $E$ is obtained. In all our experiments we set $\alpha_E = 0.01$.

\begin{figure}[t]
	\begin{center}
		\includegraphics[width = 0.45\textwidth]{real_cifar_cropped}
	\end{center}
	\caption{Certified accuracy of smoothed ResNet-20 classifiers in the idealized (IS) and the realistic (RS) settings on CIFAR-10 for the gamma correction perturbation.}
	\label{real_cifar}
\end{figure}

This smoothed classifier is then smoothed with the Rayleigh distributed gamma correction. For the certification procedure the probabilities $\underline{p_A},\overline{p_B}$ are adjusted by $\rho$:
\begin{equation} \underline{p_A}^{\prime}=\underline{p_A} - \rho,\; \overline{p_B}^{\prime} = \overline{p_B} + \rho
\end{equation}
such that we take into consideration the mistake probability of the base classifier we have estimated previously. In order to preserve the correctness of conversion error estimations the obtained certificates are clipped to the selected attack intervals $\Gamma$.

We provide distributional guarantees for smoothed classifiers on CIFAR-10 for different numbers of samples $n_{\varepsilon},n_{\gamma}$, mistake probabilities $\alpha$ and attack intervals $\Gamma$ (Fig. \ref{real_cifar}). A significant drop in accuracy can be seen when we switch from the idealized setting to the realistic one. It happens because we are not able to preserve the same number of samples for smoothing over the gamma parameter (due to the computational complexity) as well as because our base classifier has to be robust in $l_2$-norm.

\begin{figure*}[t]

\subcaptionbox{$\gamma = 0.5$}{\includegraphics[width=0.33\textwidth]{4}}
\subcaptionbox{$\gamma = 1$}{\includegraphics[width=0.33\textwidth]{5}}
\subcaptionbox{$\gamma = 2$}{\includegraphics[width=0.33\textwidth]{6}}

\caption{Demonstration of the scaling interpolation error on a frog image.}
\label{frog_scaling}
\end{figure*}

One can see that the smaller attack interval allows the lower estimation $E$ (which is expected) and thus the higher accuracy can be obtained. It should also be noticed that a greater mistake probability $\alpha$ increases the accuracy due to fewer ``abstain'' answers by both smoothed classifiers, but at the same time a greater $\alpha$ decreases the resulting confidence with which certificates are obtained.

The achieved certification results can be found in Appendix \ref{D}.

\subsection{Experiments with scaling interpolation error}

The other type of distortion we experiment with is a scaling interpolation error. By that we mean a transformation $S_{r}(x)$ which scales the image $x$ by factor $r$ followed by re-scaling to the original size. For a human eye it looks like a loss of clarity (Fig. \ref{frog_scaling}). Note that this transformation is not the same as scaling (e.g. like here \cite{li2020provable}) where there is no re-scaling to the original size and black padding is used instead. The resulting interpolation error is a perturbation we want our classifier to be robust against. It is clear that in the idealized setting scale factors multiply to each other when we scale an image several times and in that case the interpolation error does not exist. Thus this transformation is multiplicatively composable. But in reality we face non-zero interpolation errors that cannot be ignored. Though it is possible to handle this problem via double smoothing like it is done with the gamma correction, we find that the average interpolation error on CIFAR-10 images is too great to be handled via Gaussian smoothing with satisfactory accuracy. Fischer et al. \cite{fischer} encounter the same problem dealing with rotations. They propose several techniques to reduce interpolation errors (preprocessing with vignetting and low-pass filter, noise partitioning). Some of these techniques can be applied to our case but we find that the problem persists even for high-resolution images (e.g. ImageNet) that is not the case for rotations.

It should be noticed that down-scaling presents a greater challenge than up-scaling since the latter is theoretically invertible and results in smaller interpolation errors. Therefore it seems reasonable to use non-symmetric (with respect to multiplication) distributions for smoothing like the Rayleigh one, which has a distinct bias towards smaller values rather than bigger ones. One can expect such distributions to provide better certification results than the standard Gaussian and other symmetric distributions for perturbations whose symmetric parameter values do not cause the same level of distortion.

Of course one could use the TSS framework \cite{li2020provable} to deal with the interpolation errors but this approach does not require any smoothing distribution and thus it does not allow to exploit the asymmetrical nature of scaling distortions via asymmetrical probability distributions.

\section{Further research}

There are some recent works revealing the limitations of the standard Gaussian smoothing \cite{hayes,zhang}, especially in case of multidimensional perturbations with $l_p$-ball certificates for large $p$. It seems reasonable to search for alternative smoothing distributions not only among those suitable for additive parameters but also among distributions of multiplicative parameters. Indeed, it is interesting to find a good family of distributions for smoothing over multiplicative parameters, so that one could vary the variance like in additive case with uniform, Gaussian and Laplace distributions. Moreover, an example of the scaling perturbation shows that we may be more interested in asymmetric distributions which are able to exploit asymmetrical nature of certain transformations. There is a chance that it will allow to avoid the pitfalls and limitations of conventional smoothing distributions and achieve better results in the robustness certification tasks.

Also this technique can be applied to other types of perturbations and even to other types of data (audio signals, video, etc.). Therefore further experiments can include construction of certifiably robust audio and video classifiers based on the Rayleigh smoothing and comparison of them with those obtained via other methods.

\section{Related work}

In this section we want to discuss similar works in certification theory and point out some key differences between them and the present paper. Usually randomized smoothing is applied to classifiers having $R^d$ as their domain, resulting in the threat model when the signal is directly fed into the model without interpolation, rounding or clipping (like in \cite{li2020provable}). In real life it is unlikely that we will be able to perturb the original signal itself. Usually we have only a digital record of the signal (e.g. an image) and cannot access the original. Thus this setting seems unrealistic and we call it "idealized" in the paper. We go further and propose to consider "realistic" attacks that take into account limitations of digital image processing (limited channel width and bounds on minimal and maximal pixel values). Notice that gamma correction preserves pixel values within $[0,1]^3$ RGB-area  (unlike brightness or contrast changes) and thus can be easily certified against in "realistic" setting via above mentioned double smoothing technique.

The realistic setting results in a big certified accuracy drop, but we believe that it is a much more adequate way to simulate real-world attacks.

\section{Conclusion}

This work proposes a novel approach of randomized smoothing directly over multiplicative parameters of input transformations. We prove certification theorems and provide experimental comparison of the proposed method with conventional smoothing distributions for gamma correction transformations and show that it cannot be consistently outperformed by the latter. We also propose to consider realistic attacks which include additional conversion errors caused by the limited color channel width. As a result, the first certifiably robust image classifiers in the idealized and realistic settings are constructed for the gamma correction perturbation.

\bibliographystyle{named}


text:
Q: Hyper-V: Assigning VMs a public IP from a subnet We are running a Hyper-V server on our Windows Server 2008 R2. I have ordered a subnet because we would like to provide each of your VM's with a public IP address.
My data center provider (Hetzner AG from germany) writes the following about it:

Problems with virtualization
With this type of IP/subnet allocation, it is not possible to use a
  "bridged" setup, as with such a setup several MAC addresses appear.
  VPS (linux virtual servers, Xen, vmware, etc) must use a so-called
  "Routed" setup (VMware: "host-only networking"). With an additional
  subnet the host system or dom0 must be configured with an IP address
  from the subnet which is then used as a gatewar for the VPS. The
  (additional) address of the host system must therefore be configured
  in the VPS in each case as a gateway. An exception to this rule is
  "openvz", which does not require a gateway. On the host system or dom0
  "ip_forward" must be activated for each virtualization:

Well, what does this mean for me now, how do I have to configure hyper V?
Thanks for your help!

A: What they're saying is that if you have multiple different clients each with their own VM in your setup, using a subnet in bridged mode allows me to "see" other clients' MAC's on my virtual NIC.  This is a bit of a security thing, in that ideally, you'll want to isolate each client completely so that there's no risk of one client causing "harm" to another of your clients, which you'd likely wind up being responsible for.
The person at your hosting facility is simply reminding you that if you do what you want to do, you may be at risk of getting yourself into trouble.
I'm not sure it's possible to do host-only networking with public IPs on Hyper-V (or VMWare for that matter).  What I believe you should be doing instead is assigning all these IP's to the host, and then creating appropriate rules for NAT etc for each IP to point to the private network of each client's VM(s).
Now, if this is a Server for one organization we're talking about, and you don't have a need, security-wise, to isolate each service, then you may not care about this.  Personally, I'd still isolate them, so that if one gets compromised, an attacker can't discover new machines just by looking at ARP.

A: It's saying that you'll need a router between your new subnet and their network to address their requirement of having 1 MAC address:1 Switchport.  I'm not sure that Hyper-V provides an option for this natively, but you could setup RRAS on the host or buy a hardware router to put in between the host and the network.

A: In this context, this explanation of how to set it up in Windows Server 2012 might help:
http://www.wiki.hetzner.de/index.php/Windows_Server_2012_Subnet/en
(provided by the same provider mentioned in the initial question)
I don't know if this works for Win Server 2008, but I could't see a reason why it shouldn't.