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+# Adaptive Conformal Inference Under Distribution Shift
+
+# Isaac Gibbs
+
+Emmanuel J. Candès
+
+Department of Statistics Stanford University igibbs@stanford.edu
+
+Department of Statistics Department of Mathematics Stanford University candes@stanford.edu
+
+# Abstract
+
+We develop methods for forming prediction sets in an online setting where the data generating distribution is allowed to vary over time in an unknown fashion. Our framework builds on ideas from conformal inference to provide a general wrapper that can be combined with any black box method that produces point predictions of the unseen label or estimated quantiles of its distribution. While previous conformal inference methods rely on the assumption that the data points are exchangeable, our adaptive approach provably achieves the desired coverage frequency over long-time intervals irrespective of the true data generating process. We accomplish this by modelling the distribution shift as a learning problem in a single parameter whose optimal value is varying over time and must be continuously re-estimated. We test our method, adaptive conformal inference, on two real world datasets and find that its predictions are robust to visible and significant distribution shifts.
+
+# 1 Introduction
+
+Machine learning algorithms are increasingly being employed in high stakes decision making processes. For instance, deep neural networks are currently being used in self-driving cars to detect nearby objects $\pmb { \mathbb { D } }$ and parole decisions are being made with the assistance of complex models that combine over a hundred features $\mathbb { I I }$ . As the popularity of black box methods and the cost of making wrong decisions grow it is crucial that we develop tools to quantify the uncertainty of their predictions.
+
+In this paper we develop methods for constructing prediction sets that are guaranteed to contain the target label with high probability. We focus specifically on an online learning setting in which we observe covariate-response pairs $\{ ( X _ { t } , Y _ { t } ) \} _ { t \in \mathbb { N } } \subseteq \mathbb { R } ^ { d } \times \mathbf { \bar { \mathbb { R } } }$ in a sequential fashion. At each time step $t \in \mathbb { N }$ we are tasked with using the previously observed data $\{ ( X _ { r } , Y _ { r } ) \} _ { 1 \leq r \leq t - 1 }$ along with the new covariates, $X _ { t }$ , to form a prediction set $\hat { C } _ { t }$ for $Y _ { t }$ . Then, given a target coverage level $\alpha \in ( 0 , 1 )$ our generic goal is to guarantee that $Y _ { t }$ belongs to $\hat { C } _ { t }$ at least $1 0 0 ( 1 - \alpha ) \%$ of the time.
+
+Perhaps the most powerful and flexible tools for solving this problem come from conformal inference [see e.g. 34, 16, 32, 22, 31, 15, 3] . This framework provides a generic methodology for transforming the outputs of any black box prediction algorithm into a prediction set. The generality of this approach has facilitated the development of a large suite of conformal methods, each specialized to a specific prediction problem of interest [e.g. 30, 11, 23, 8, 24, 21]. With only minor exceptions all of these algorithms share the same common guarantee that if the training and test data are exchangeable, then the prediction set has valid marginal coverage $\mathbb { P } ( Y _ { t } \in \hat { C } _ { t } ) = \bar { 1 } - \alpha$ .
+
+While exchangeability is a common assumption, there are many real-world applications in which we do not expect the marginal distribution of $( X _ { t } , Y _ { t } )$ to be stationary. For example, in finance and economics market behaviour can shift drastically in response to new legislation or major world events. Alternatively, the distribution of $( X _ { t } , Y _ { t } )$ may change as we deploy our prediction model in new environments. This paper develops adaptive conformal inference (ACI), a method for forming prediction sets that are robust to changes in the marginal distribution of the data. Our approach is both simple, in that it requires only the tracking of a single parameter that models the shift, and general as it can be combined with any modern machine learning algorithm that produces point predictions or estimated quantiles for the response. We show that over long time intervals ACI achieves the target coverage frequency without any assumptions on the data-generating distribution. Moreover, when the distribution shift is small and the prediction algorithm takes a certain simple form we show that ACI will additionally obtain approximate marginal coverage at most time steps.
+
+# 1.1 Conformal inference
+
+Suppose we are given a fitted regression model for predicting the value of $Y$ from $X$ . Let $y$ be a candidate value for $Y _ { t }$ . To determine if $y$ is a reasonable estimate of $Y _ { t }$ , we define a conformity score $S ( X , Y )$ that measures how well the value $y$ conforms with the predictions of our fitted model. For example, if our regression model produces point predictions ${ \hat { \mu } } ( X )$ then we could use a conformity score that measures the distance between $\hat { \mu } ( X _ { t } )$ and $y$ . One such example is
+
+$$
+S ( X _ { t } , y ) = | \hat { \mu } ( X _ { t } ) - y | .
+$$
+
+Alternatively, suppose our regression model outputs estimates ${ \hat { q } } ( X ; p )$ of the $p$ th quantile of the distribution of $Y | X$ . Then, we could use the method of conformal quantile regression (CQR) $\left[ \left[ 2 8 \right] \right]$ which examines the signed distance between $y$ and fitted upper and lower quantiles through the score
+
+$$
+S ( X _ { t } , y ) = \operatorname* { m a x } \{ \hat { q } ( X _ { t } ; \alpha / 2 ) - y , y - \hat { q } ( X _ { t } ; 1 - \alpha / 2 ) \} .
+$$
+
+Regardless of what conformity score is chosen the key issue is to determine how small $S ( X _ { t } , y )$ should be in order to accept $y$ as a reasonable prediction for $Y _ { t }$ . Assume we have a calibration set $\mathcal { D } _ { \mathrm { c a l } } \subseteq \{ ( X _ { r } , Y _ { r } ) \} _ { 1 \leq r \leq t - 1 }$ that is different from the data that was used to fit the regression model. Using this calibration set we define the fitted quantiles of the conformity scores to be
+
+$$
+\hat { Q } ( p ) : = \operatorname* { i n f } \left\{ s : \left( \frac 1 { | \mathscr { D } _ { \mathrm { c a l } } | } \sum _ { ( X _ { r } , Y _ { r } ) \in \mathscr { D } _ { \mathrm { c a l } } } \mathbb { 1 } _ { \{ S ( X _ { r } , Y _ { r } ) \leq s \} } \right) \geq p \right\} ,
+$$
+
+and say that $y$ is a reasonable prediction for $Y _ { t }$ if $S ( X _ { t } , y ) \leq \hat { Q } ( 1 - \alpha )$ .
+
+The crucial observation is that if the data $\mathcal { D } _ { \operatorname { c a l } } \cup \{ ( X _ { t } , Y _ { t } ) \}$ are exchangeable and we break ties uniformly at random then the rank of $S ( X _ { t } , Y _ { t } )$ amongst the points $\{ S ( X _ { r } , Y _ { r } ) \} _ { ( X _ { r } , Y _ { r } ) \in \mathcal { D } _ { \mathrm { c a l } } } \cup$ $\{ S ( X _ { t } , Y _ { t } ) \}$ will be uniform. Therefore,
+
+$$
+\mathbb { P } ( S ( X _ { t } , Y _ { t } ) \le \hat { Q } ( 1 - \alpha ) ) = \frac { \lceil | \mathcal { D } _ { \mathrm { c a l } } | ( 1 - \alpha ) \rceil } { | \mathcal { D } _ { \mathrm { c a l } } | + 1 } .
+$$
+
+Thus, defining our prediction set to be $\hat { C } _ { t } : = \{ y : S ( X _ { t } , y ) \leq \hat { Q } ( 1 - \alpha ) \}$ gives the marginal coverage guarantee
+
+$$
+\mathbb { P } ( Y _ { t } \in \hat { C } _ { t } ) = \mathbb { P } ( S ( X _ { t } , Y _ { t } ) \leq \hat { Q } ( 1 - \alpha ) ) = \frac { \lceil | \mathcal { D } _ { \mathrm { c a l } } | ( 1 - \alpha ) \rceil } { | \mathcal { D } _ { \mathrm { c a l } } | + 1 } .
+$$
+
+By introducing additional randomization this generic procedure can be altered slightly to produce a set $\hat { C } _ { t }$ that satisfies the exact marginal coverage guarantee $\mathbb { P } ( Y _ { t } \in \hat { C } _ { t } ) = 1 - \alpha$ [34]. For the purposes of this paper this adjustment is not critical and so we omit the details here. Additionally, we remark that the method outlined above is often referred to as split or inductive conformal inference $[ 1 2 7 , 1 3 4 , | 2 6 |$ . This refers to the fact that we have split the observed data between a training set used to fit the regression model and a withheld calibration set. The adaptive conformal inference method developed in this article can also be easily adjusted to work with full conformal inference in which data splitting is avoided at the cost of greater computational resources $\pmb { \mathbb { B 4 } }$ .
+
+# 2 Adapting conformal inference to distribution shifts
+
+Up until this point we have been working with a single score function $S ( \cdot )$ and quantile function ${ \hat { Q } } ( \cdot )$ In the general case where the distribution of the data is shifting over time both these functions should
+
+be regularly re-estimated to align with the most recent observations. Therefore, we assume that at each time $t$ we are given a fitted score function $S _ { t } ( \cdot )$ and corresponding quantile function $\hat { Q } _ { t } ( \cdot )$ . We define the realized miscoverage rate of the prediction set $\hat { C } _ { t } ( \alpha ) : = \{ y : S _ { t } ( X _ { t } , y ) \leq \hat { Q } _ { t } ( 1 - \alpha ) \}$ as
+
+$$
+M _ { t } ( \alpha ) : = \mathbb { P } ( S _ { t } ( X _ { t } , Y _ { t } ) > \hat { Q } _ { t } ( 1 - \alpha ) ) ,
+$$
+
+where the probability is over the test point $( X _ { t } , Y _ { t } )$ as well as the data used to fit $S _ { t } ( \cdot )$ and $\hat { Q } _ { t } ( \cdot )$
+
+Now, since the distribution generating the data is non-stationary we do not expect $M _ { t } ( \alpha )$ to be equal, or even close to, $\alpha$ . Even so, we can still postulate that if the conformity scores used to fit $\hat { Q } _ { t } ( \cdot )$ cover the bulk of the distribution of $S _ { t } ( X _ { t } , Y _ { t } )$ then there may be an alternative value $\alpha _ { t } ^ { * } \in [ 0 , 1 ]$ such that $M _ { t } ( \alpha _ { t } ^ { * } ) \cong \alpha$ . More rigorously, assume that with probability one, $\hat { Q } _ { t } ( \cdot )$ is continuous, non-decreasing and such that $\hat { Q } _ { t } ( 0 ) = - \infty$ and $\hat { Q } _ { t } ( 1 ) = \infty$ . This does not hold for the split conformal quantile functions defined in $( 1 )$ , but in the case where there are no ties amongst the conformity scores we can adjust our definition to guarantee this by smoothing over the jump discontinuities in ${ \hat { Q } } ( \cdot )$ . Then, $M _ { t } ( \cdot )$ will be non-decreasing on $[ 0 , 1 ]$ with $M _ { t } ( 0 ) = 0$ and $M _ { t } ( 1 ) = 1$ and so we may define
+
+$$
+\alpha _ { t } ^ { \ast } : = \operatorname* { s u p } \{ \beta \in [ 0 , 1 ] : M _ { t } ( \beta ) \leq \alpha \} .
+$$
+
+Moreover, if we additionally assume that
+
+$$
+\mathbb { P } ( S _ { t } ( X _ { t } , Y _ { t } ) = \hat { Q } _ { t } ( 1 - \alpha _ { t } ^ { * } ) ) = 0 ,
+$$
+
+then we will have that $M _ { t } ( \alpha _ { t } ^ { * } ) = \alpha$ . So, in particular we find that by correctly calibrating the argument to $\hat { Q } _ { t } ( \cdot )$ we can achieve either approximate or exact marginal coverage.
+
+To perform this calibration we will use a simple online update. This update proceeds by examining the empirical miscoverage frequency of the previous prediction sets and then decreasing (resp. increasing) our estimate of $\alpha _ { t } ^ { * }$ if the prediction sets were historically under-covering (resp. over-covering) $Y _ { t }$ . In particular, let $\alpha _ { 1 }$ denote our initial estimate (in our experiments we will choose $\alpha _ { 1 } = \alpha$ ). Recursively define the sequence of miscoverage events
+
+$$
+\begin{array} { r } { \mathsf { e r r } _ { t } : = \Bigl \{ \displaystyle 1 , \mathrm { ~ i f ~ } Y _ { t } \notin \hat { C } _ { t } ( \alpha _ { t } ) , \qquad \mathrm { w h e r e ~ } \hat { C } _ { t } ( \alpha _ { t } ) : = \{ y : S _ { t } ( X _ { t } , y ) \leq \hat { Q } _ { t } ( 1 - \alpha _ { t } ) \} . } \end{array}
+$$
+
+Then, fixing a step size parameter $\gamma > 0$ we consider the simple online update
+
+$$
+\alpha _ { t + 1 } : = \alpha _ { t } + \gamma ( \alpha - \mathrm { e r r } _ { t } ) .
+$$
+
+We refer to this algorithm as adaptive conformal inference. Here, $\operatorname { e r r } _ { t }$ plays the role of our estimate of the historical miscoverage frequency. A natural alternative to this is the update
+
+$$
+\alpha _ { t + 1 } = \alpha _ { t } + \gamma \left( \alpha - \sum _ { s = 1 } ^ { t } w _ { s } \mathrm { e r r } _ { s } \right) ,
+$$
+
+where the ap $\{ w _ { s } \} _ { 1 \leq s \leq t } \subseteq [ 0 , 1 ]$ is a sequence of increasing weights with y evaluating the recent empirical miscov $\textstyle \sum _ { s = 1 } ^ { t } w _ { s } = 1$ . This update hasy when deciding whether or not to lower or raise $\alpha _ { t }$ . In practice, we find that $\textcircled{2}$ and $\textcircled{3}$ produce almost identical results. For example, in Section ${ \bf A } . 3$ in the Appendix we show some sample trajectories for $\alpha _ { t }$ obtained using the update $\textcircled { 3 }$ with
+
+$$
+w _ { s } : = \frac { 0 . 9 5 ^ { t - s } } { \sum _ { s ^ { \prime } = 1 } ^ { t } 0 . 9 5 ^ { t - s ^ { \prime } } } .
+$$
+
+We find that these trajectories are very similar to those produced by $( 2 )$ . The main difference is that the trajectories obtained with $\textcircled { 3 }$ are smoother with less local variation in $\alpha _ { t }$ . In the remainder of this article we will focus on $( 2 )$ for simplicity.
+
+# 2.1 Choosing the step size
+
+The choice of $\gamma$ gives a tradeoff between adaptability and stability. While raising the value of $\gamma$ will make the method more adaptive to observed distribution shifts, it will also induce greater volatility in the value of $\alpha _ { t }$ . In practice, large fluctuations in $\alpha _ { t }$ may be undesirable as it allows the method to oscillate between outputting small conservative and large anti-conservative prediction sets.
+
+In Theorem $\left| \overline { { 4 . 2 } } \right|$ we give an upper bound on $( M _ { t } ( \alpha _ { t } ) - \alpha ) ^ { 2 }$ that is optimized by choosing $\gamma$ proportional to $\sqrt { \left| \alpha _ { t + 1 } ^ { * } - \alpha _ { t } ^ { * } \right| }$ . While not directly applicable in practice, this result supports the intuition that in environments with greater distributional shift the algorithm needs to be more adapatable and thus $\gamma$ should be chosen to be larger. In our experiments we will take $\gamma = 0 . 0 0 5$ . This value was chosen because it was found to give relatively stable trajectories for $\alpha _ { t }$ while still being sufficiently large as to allow $\alpha _ { t }$ to adapt to observed shifts. In agreement with the general principles outlined above we found that larger values of $\gamma$ also successfully protect against distribution shifts, while taking $\gamma$ to be too small causes adaptive conformal inference to perform similar to non-adaptive methods that hold $\alpha _ { t } = \alpha$ constant across time.
+
+# 2.2 Real data example: predicting market volatility
+
+We apply ACI to the prediction of market volatility. Let $\{ P _ { t } \} _ { 1 \leq t \leq T }$ denote a sequence of daily open prices for a stock. For all $t \geq 2$ , define the return $R _ { t } : = ( \bar { P } _ { t } - \bar { P } _ { t - 1 } ) / P _ { t - 1 }$ and realized volatility $\dot { V } _ { t } = R _ { t } ^ { 2 }$ . Our goal is to use the previously observed returns $X _ { t } : = \{ R _ { s } \} _ { 1 \leq s \leq t - 1 }$ to form prediction sets for $Y _ { t } : = V _ { t }$ . More sophisticated financial models might augment $X _ { t }$ with additional market covariates (available to the analyst at time $t - 1$ ). As the primary purpose of this section is to illustrate adaptive conformal inference we work with only a simple prediction method.
+
+We start off by forming point predictions using a GARCH(1,1) model $\mathbb { \lVert }$ . This method assumes that $R _ { t } = \sigma _ { t } \epsilon _ { t }$ with $\epsilon _ { 2 } , \dots , \epsilon _ { T }$ taken to be i.i.d. $\mathcal { N } ( 0 , 1 )$ and $\sigma _ { t }$ satisfying the recursive update
+
+$$
+\sigma _ { t } ^ { 2 } = \omega + \tau V _ { t - 1 } + \beta \sigma _ { t - 1 } ^ { 2 } .
+$$
+
+This is a common approach used for forecasting volatility in economics. In practice, shifting market dynamics can cause the predictions of this model to become inaccurate over large time periods. Thus, when forming point predictions we fit the model using only the last 1250 trading days (i.e. approximately 5 years) of market data. More precisely, for all times $t > 1 2 5 0$ we fit the coefficients $\hat { \omega } _ { t } , ~ \hat { \tau } _ { t } , \hat { \beta } _ { t }$ as well as the sequence of variances $\{ \hat { \sigma } _ { s } ^ { t } \} _ { 1 \leq s \leq t - 1 }$ using only the data $\{ R _ { r } \} _ { t - 1 2 5 0 \leq r < t }$ . Then, our point prediction for the realized volatility at time $t$ is
+
+$$
+( \hat { \sigma } _ { t } ^ { t } ) ^ { 2 } : = \hat { \omega } _ { t } + \hat { \tau } _ { t } V _ { t - 1 } + \hat { \beta } _ { t } ( \hat { \sigma } _ { t - 1 } ^ { t } ) ^ { 2 } .
+$$
+
+To form prediction intervals we define the sequence of conformity scores
+
+$$
+S _ { t } : = \frac { | V _ { t } - ( \hat { \sigma } _ { t } ^ { t } ) ^ { 2 } | } { ( \hat { \sigma } _ { t } ^ { t } ) ^ { 2 } }
+$$
+
+and the corresponding quantile function
+
+$$
+\hat { Q } _ { t } ( p ) : = \operatorname* { i n f } \left\{ x : \frac { 1 } { 1 2 5 0 } \sum _ { r = t - 1 2 5 0 } ^ { t - 1 } \mathbb { 1 } _ { S _ { r } \leq x } \geq p \right\} .
+$$
+
+Then, our prediction set at time $t$ is
+
+$$
+\hat { C } _ { t } ( \alpha _ { t } ) : = \left\{ v : \frac { | v - ( \hat { \sigma } _ { t } ^ { t } ) ^ { 2 } | } { ( \hat { \sigma } _ { t } ^ { t } ) ^ { 2 } } \leq \hat { Q } _ { t } ( 1 - \alpha _ { t } ) \right\} ,
+$$
+
+where $\left\{ \alpha _ { t } \right\}$ is initialized with $\alpha _ { 1 2 5 0 } = \alpha = 0 . 1$ and then updated recursively as in $\bigoplus$ .
+
+We compare this algorithm to a non-adaptive alternative that takes $\alpha _ { t } = \alpha$ fixed. To measure the performance of these methods across time we examine their local coverage frequencies defined as the average coverage rate over the most recent two years, i.e.
+
+$$
+\mathrm { l o c a l C o v } _ { t } : = 1 - { \frac { 1 } { 5 0 0 } } \sum _ { r = t - 2 5 0 + 1 } ^ { t + 2 5 0 } \mathrm { e r r } _ { r } .
+$$
+
+If the methods perform well then we expect the local coverage frequency to stay near the target value $1 - \alpha$ across all time points.
+
+![](images/c65e1feee99d5c67ae7fcb7de068af025556a955d04f9e89d028fdc3e4e06f15.jpg)  
+Figure 1: Local coverage frequencies for adaptive conformal (blue), a non-adaptive method that holds $\alpha _ { t } = \alpha$ fixed (red), and an i.i.d. Bernoulli(0.1) sequence (grey) for the prediction of stock market volatility. The coloured dotted lines mark the average coverage obtained across all time points, while the black line indicates the target level of $1 - \alpha = 0 . 9$ .
+
+Daily open prices were obtained from publicly available datasets published by The Wall Street Journal. The realized local coverage frequencies for the non-adaptive and adaptive conformal methods on four different stocks are shown in Figure $1 .$ These stocks were selected out of a total of 12 stocks that we examined because they showed a clear failure of the non-adaptive method. Adaptive conformal inference was found to perform well in all cases (see Figure 9 in the appendix).
+
+As a visual comparator, the grey curves show the moving average 1500 Pt+250r=t250+1 Ir for sequences $\left\{ I _ { t } \right\} _ { 1 \leq t \leq T }$ that are i.i.d. Bernoulli(0.1). We see that the local coverage frequencies obtained by adaptive conformal inference (blue lines) always stay within the variation that would be expected from an i.i.d. Bernoulli sequence. On the other hand, the non-adaptive method undergoes large excursions away from the target level of $1 - \alpha = 0 . 9$ (red lines). For example, in the bottom right panel we can see that the non-adaptive method fails to cover the realized volatility of Fannie Mae during the 2008 financial crisis, while the adaptive method is robust to this event (see Figure 4 in the Appendix for a plot of the price of Fannie Mae over this time period).
+
+# 3 Related Work
+
+Prior work on conformal inference has considered two different types of distribution shift $\mathbb { \lVert 3 3 \rVert \lVert 0 \rVert }$ . In both cases the focus was on environments in which the calibration data is drawn i.i.d. from a single distribution $P _ { 0 }$ , while the test point comes from a second distribution $P _ { 1 }$ . In this setting Tibshirani et al. $\mathbb { \lVert 3 3 \rVert }$ showed that valid prediction sets can be obtained by re-weighting the calibration data using the likelihood ratio between $P _ { 1 }$ and $P _ { 0 }$ . However, this requires the conditional distribution of $Y | X$ to be constant between training and testing and the likelihood ratio $P _ { 1 } ( X ) / P _ { 0 } ( X )$ to be either known or very accurately estimated. On the other hand, Cauchois et al. $\mathbb { m }$ develop methods for forming prediction sets that are valid whenever $P _ { 1 }$ and $P _ { 0 }$ are close in $f$ -divergence. Similar to our work, they show that if $D _ { f } ( P _ { 1 } | | P _ { 0 } ) \leq \rho$ then there exists a conservative value $\alpha _ { \rho } \in ( 0 , 1 )$ such that
+
+$$
+M ( \alpha _ { \rho } ) : = \mathbb { P } ( S ( X _ { t } , Y _ { t } ) > \hat { Q } ( 1 - \alpha _ { \rho } ) ) \leq \alpha .
+$$
+
+The difference between our approach and theirs is twofold. First, while they fix a single conservative value $\alpha _ { \rho }$ our methods aim to estimate the optimal choice $\alpha ^ { * }$ satisfying $M ( \alpha ^ { * } ) = \alpha$ . This is not possible in the setting of $\mathbb { m }$ as they do not observe any data from which the size of the distribution shift can be estimated. Second, while they consider only one training and one testing distribution we work in a fully online setting in which the distribution is allowed to shift continuously over time.
+
+# 4 Coverage guarantees
+
+# 4.1 Distribution-free results
+
+In this section we outline the theoretical coverage guarantees of adaptive conformal inference. We will assume throughout that with probability one $\alpha _ { 1 } \in [ 0 , 1 ]$ and $\hat { Q } _ { t }$ is non-decreasing with $\hat { Q } _ { t } ( x ) = - \infty$ for all $x < 0$ and $\hat { Q } _ { t } ( x ) = \infty$ for all $x > 1$ . Our first result shows that over long time intervals adaptive conformal inference obtains the correct coverage frequency irrespective of any assumptions on the data-generating distribution.
+
+Lemma 4.1 With probability one we have that $\forall t \in \mathbb { N }$ , $\alpha _ { t } \in [ - \gamma , 1 + \gamma ]$ .
+
+Proof: Assume by contradiction that with positive probability $\{ \alpha _ { t } \} _ { t \in \mathbb { N } }$ is such that $\operatorname* { i n f } _ { t } \alpha _ { t } < - \gamma$ (the case where $\operatorname* { s u p } _ { t } \alpha _ { t } > 1 + \gamma$ is identical). Note that $\begin{array} { r } { \operatorname* { s u p } _ { t } | \alpha _ { t + 1 } - \alpha _ { t } | = \operatorname* { s u p } _ { t } \gamma | \alpha - \mathrm { e r r } _ { t } | < \gamma . } \end{array}$ . Thus, with positive probability we may find $t \in \mathbb { N }$ such that $\alpha _ { t } < 0$ and $\alpha _ { t + 1 } < \alpha _ { t }$ . However,
+
+$$
+\alpha _ { t } < 0 \implies \hat { Q } _ { t } ( 1 - \alpha _ { t } ) = \infty \implies \mathrm { e r r } _ { t } = 0 \implies \alpha _ { t + 1 } = \alpha _ { t } + \gamma ( \alpha - \mathrm { e r r } _ { t } ) \geq \alpha _ { t }
+$$
+
+and thus $\mathbb { P } ( \exists t$ such that $\alpha _ { t + 1 } < \alpha _ { t } < 0 ) = 0$ . We have reached a contradiction.
+
+Proposition 4.1 With probability one we have that for all $T \in \mathbb { N }$ ,
+
+$$
+\left| \frac { 1 } { T } \sum _ { t = 1 } ^ { T } e r r _ { t } - \alpha \right| \leq \frac { \operatorname* { m a x } \{ \alpha _ { 1 } , 1 - \alpha _ { 1 } \} + \gamma } { T \gamma } .
+$$
+
+$\begin{array} { r } { \operatorname* { l i m } _ { T \to \infty } \frac { 1 } { T } \sum _ { t = 1 } ^ { T } e r r _ { t } \stackrel { a . s . } { = } \alpha } \end{array}$
+
+Proof: By expanding the recursion defined in $( 2 )$ and applying Lemma 4.1 we find that
+
+$$
+[ - \gamma , 1 + \gamma ] \ni \alpha _ { T + 1 } = \alpha _ { 1 } + \sum _ { t = 1 } ^ { T } \gamma ( \alpha - \mathrm { e r r } _ { t } ) .
+$$
+
+Rearranging this gives the result.
+
+Proposition $\boxed { 4 . 1 }$ puts no constraints on the data generating distribution. One may immediately ask whether these results can be improved by making mild assumptions on the distribution shifts. We argue that without assumptions on the quality of the initialization the answer to this question is negative. To understand this, consider a setting in which there is a single fixed optimal target $\alpha ^ { * } \in [ 0 , 1 ]$ and assume that
+
+$$
+M _ { t } ( p ) = M ( p ) = { \left\{ \begin{array} { l l } { \alpha + { \frac { 1 - \alpha } { 1 - \alpha ^ { * } } } ( p - \alpha ^ { * } ) , { \mathrm { i f ~ } } p > \alpha ^ { * } , } \\ { \alpha + { \frac { \alpha } { \alpha ^ { * } } } ( p - \alpha ^ { * } ) { \mathrm { i f ~ } } p \leq \alpha ^ { * } . } \end{array} \right. } ~ .
+$$
+
+Suppose additionally that $\mathbb { E } [ \mathrm { e r r } _ { t } | \alpha _ { t } ] = M ( \alpha _ { t } ) . ^ { 1 }$ In order to simplify the calculations consider the noiseless update $\alpha _ { t + 1 } = \alpha _ { t } + \gamma ( \alpha - M ( \alpha _ { t } ) ) \equiv \alpha _ { t } + \gamma ( \alpha - \mathbb { E } [ \operatorname { e r r } _ { t } | \alpha _ { t } ] )$ . Intuitively, the noiseless update can be viewed as the average case behaviour of $\textstyle \operatorname* { m i n } \{ { \frac { 1 - \alpha ^ { * } } { 1 - \alpha } } , { \frac { \alpha ^ { * } } { \alpha } } \}$ there exists a constant $\textstyle c \in \{ { \frac { 1 - \alpha } { 1 - \alpha ^ { * } } } , { \frac { \alpha } { \alpha ^ { * } } } \}$ $\textcircled { 2 }$ such that for all . Now, for any initialization $t$ , $M ( \alpha _ { t } ) - \alpha = c ( \alpha _ { t } - \alpha ^ { * } )$ $\alpha _ { 1 }$ and any $\gamma \leq$ So, we have that
+
+$$
+\begin{array} { r } { \mathbb { E } [ \operatorname { e r r } _ { t } ] - \alpha = c \mathbb { E } [ \alpha _ { t } - \alpha ^ { * } ] = c \mathbb { E } [ \alpha _ { t - 1 } + \gamma ( \alpha - M _ { t - 1 } ( \alpha _ { t - 1 } ) ) - \alpha ^ { * } ] = c ( 1 - c \gamma ) \mathbb { E } [ \alpha _ { t - 1 } - \alpha ^ { * } ] . } \end{array}
+$$
+
+Repeating this calculation recursively gives that
+
+$$
+\begin{array} { r } { \mathbb { E } [ \mathsf { e r r } _ { t } ] - \alpha = c ( 1 - c \gamma ) ^ { t - 1 } \mathbb { E } [ \alpha _ { 1 } - \alpha ^ { * } ] = c ( 1 - c \gamma ) ^ { t - 1 } ( \alpha _ { 1 } - \alpha ^ { * } ) , } \end{array}
+$$
+
+and thus,
+
+$$
+\left| \frac { 1 } { T } \sum _ { t = 1 } ^ { T } \mathbb { E } [ \mathsf { e r r } _ { t } ] - \alpha \right| = \frac { 1 - ( 1 - c \gamma ) ^ { T } } { T \gamma } | \alpha _ { 1 } - \alpha ^ { * } | .
+$$
+
+The comparison of this bound to $( 5 )$ is self-evident. The main difference is that we have replaced $\operatorname* { m a x } \{ 1 - \alpha _ { 1 } , \alpha _ { 1 } \}$ with $| \alpha _ { 1 } - \alpha ^ { * } |$ . This arises from the fact that $\alpha ^ { * } \in ( 0 , 1 )$ is arbitrary and thus $\operatorname* { m a x } \{ 1 - \alpha _ { 1 } , \alpha _ { 1 } \}$ is the best possible upper bound on $| \alpha _ { 1 } - \alpha ^ { * } |$ . So, we view Proposition $^ { 4 . 1 }$ as both an agnostic guarantee that shows that our method gives the correct long-term empirical coverage frequency irrespective of the true data generating process, and as an approximately tight bound on the worst-case behaviour immediately after initialization.
+
+# 4.2 Performance in a hidden Markov model
+
+Although we believe Proposition $4 . 1$ is an approximately tight characterization of the behaviour after initialization, we can still ask whether better bounds can be obtained for large time steps. In this section we answer this question positively by showing that if $\alpha _ { 1 }$ is initialized appropriately and the distribution shift is small, then tighter coverage guarantees can be given. In order to obtain useful results we will make some simplifying assumptions about the data generating process. While we do not expect these assumptions to hold exactly in any real-world setting, we do consider our results to be representative of the true behaviour of adaptive conformal inference and we expect similar results to hold under alternative models.
+
+# 4.2.1 Setting
+
+We model the data as coming from a hidden Markov model. In particular, we let $\{ A _ { t } \} _ { t \in \mathbb { N } } \subseteq { \mathcal { A } }$ denote the underlying Markov chain for the environment and we assume that conditional on $\{ A _ { t } \} _ { t \in \mathbb { N } }$ $\{ ( X _ { t } , Y _ { t } ) \} _ { t \in \mathbb { N } }$ is an independent sequence with $( X _ { t } , Y _ { t } ) \sim P _ { A _ { t } }$ for some collection of distributions $\{ P _ { a } : a \in { \mathcal { A } } \}$ . In order to simplify our calculations, we assume additionally that the estimated quantile function $\hat { Q } _ { t } ( \cdot )$ and score function $S _ { t } ( \cdot )$ do not depend on $t$ and we denote them by ${ \hat { Q } } ( \cdot )$ and $\bar { S } ( \cdot )$ . This occurs for example in the split conformal setting with fixed training and calibration sets.
+
+In this setting, $\{ ( \alpha _ { t } , A _ { t } ) \} _ { t \in \mathbb { N } }$ forms a Markov chain on $[ - \gamma , 1 + \gamma ] \times \mathcal { A }$ . We assume that this chain has a unique stationary distribution $\pi$ and that $( \alpha _ { 1 } , A _ { 1 } ) \sim \pi$ . This implies that $\left( \alpha _ { t } , A _ { t } , \mathbf { e r r } _ { t } \right)$ is a stationary process and thus will greatly simplify our characterization of the behaviour of $\mathrm { e r r } _ { t }$ . While there is little doubt that the theory can be extended, recall our that main goal is to get useful and simple results. That said, what we really have in mind here is that $\{ A _ { t } \} _ { t \in \mathbb { N } }$ is sufficiently wellbehaved to guarantee that $( \alpha _ { t } , A _ { t } )$ has a limiting stationary distribution. In Section ${ \bf A } . 5$ we give an example where this is indeed provably the case. Lastly, the assumption that $( \alpha _ { 1 } , \overline { { A _ { 1 } } } ) \sim \overline { { \pi } }$ is essentially equivalent to assuming that we have been running the algorithm for long enough to exit the initialization phase described in Section 4.1.
+
+# 4.2.2 Large deviation bound for the errors
+
+Our first have that $\operatorname { e r r } _ { t }$ orrect aveand since value. More precisely, by is stationary it follows that $\boxed { 4 . 1 }$ weus, ${ \mathrm { l i m } } _ { T \to \infty } T ^ { - 1 } \sum _ { t = 1 } ^ { T } { \mathrm { e r r } } _ { t } \ { \overset { a . s . } { = } } \alpha$ $\operatorname { e r r } _ { t }$ $\mathbb { E } [ \mathsf { e r r } _ { t } ] = \alpha$ to understand the deviation of T 1 PTt= from $\alpha$ we simply need to characterize the dependence structure of $\{ \mathrm { e r r } _ { t } \} _ { t \in \mathbb { N } }$ .
+
+We accomplish this in Theorem $^ { 4 . 1 , }$ which gives a large deviation bound on $\begin{array} { r } { | T ^ { - 1 } \sum _ { t = 1 } ^ { T } \mathrm { e r r } _ { t } - \alpha | } \end{array}$ The idea behind this result is to decompose the dependence in into two parts. First, there is dependence due to the fact that $\alpha _ { t }$ is a function of $\{ \mathrm { e r r } _ { r } \} _ { 1 \leq r \leq t - 1 }$ . In Section $\boxed { \mathbf { A } . 7 }$ in the Appendix we argue that this dependence induces a negative correlation and thus the errors concentrate around their expectation at a rate no slower than that of an i.i.d. Bernoulli sequence. This gives rise to the first term in $( 6 )$ , which is what would be obtained by applying Hoeffding’s inequality to an i.i.d. sequence. Second, there is dependence due to the fact that $A _ { t }$ depends on $A _ { t - 1 }$ . More specifically, consider a setting in which the distribution of $Y | X$ has more variability in some states than others. The goal of adaptive conformal inference is to adapt to the level of variability and thus return larger prediction sets in states where the distribution of $Y | X$ is more spread. However, this algorithm is not perfect and as a result there may be some states $a \in { \mathcal { A } }$ in which $\mathbb { E } [ \mathbf { e r r } _ { t } | A _ { t } = a ]$ is biased away from $\alpha$ . Furthermore, if the environment tends to spend long stretches of time in more variable (or less variable) states this will induce a positive dependence in the errors and cause T 1 PTt=1 to deviate from $\alpha$ . To control this dependence we use a Bernstein inequality for Markov chains to bound $\begin{array} { r } { | T ^ { - 1 } \sum _ { t = 1 } ^ { T } \mathbb { E } [ \mathrm { e r r } _ { t } | A _ { t } ] - \alpha | } \end{array}$ . This gives rise to the second term in $( 6 )$
+
+Theorem 4.1 Assume that $\{ A _ { t } \} _ { t \in \mathbb { N } }$ has non-zero absolute spectral gap $1 - \eta > 0$ . Let
+
+$$
+B : = \operatorname* { s u p } _ { a \in { \mathcal { A } } } | \mathbb { E } [ \operatorname { e r r } _ { t } | A _ { t } = a ] - \alpha | \quad a n d \quad \sigma _ { B } ^ { 2 } : = \mathbb { E } [ ( \mathbb { E } [ \operatorname { e r r } _ { t } | A _ { t } ] - \alpha ) ^ { 2 } ] .
+$$
+
+Then,
+
+$$
+\mathbb { P } \left( \left| \frac { 1 } { T } \sum _ { t = 1 } ^ { T } \mathrm { e r r } _ { t } - \alpha \right| \geq \epsilon \right) \leq 2 \exp \left( - \frac { T \epsilon ^ { 2 } } { 8 } \right) + 2 \exp \left( - \frac { T ( 1 - \eta ) \epsilon ^ { 2 } } { 8 ( 1 + \eta ) \sigma _ { B } ^ { 2 } + 2 0 B \epsilon } \right) .
+$$
+
+A formal proof of this result can be found in Section $\boxed { \mathbf { A . 7 } }$ The quality of this concentration inequality will depend critically on the size of the bias terms $B$ and $\sigma _ { B } ^ { 2 }$ . Before proceeding, it is important that we emphasize that the definitions of $B$ and $\sigma _ { B } ^ { 2 }$ are independent of the choice of $t$ owing to the fact that $\left( \alpha _ { t } , A _ { t } , \mathbf { e r r } _ { t } \right)$ is assumed stationary. Now, to understand these quantities, let
+
+$$
+M ( p | a ) : = \mathbb { P } ( S ( X _ { t } , Y _ { t } ) > \hat { Q } ( 1 - p ) | A _ { t } = a )
+$$
+
+denote the realized miscoverage level in state $a \in { \mathcal { A } }$ obtained by the quantile $\hat { Q } ( 1 - p )$ . Assume that $M ( p | a )$ is continuous. This will happen for example when ${ \hat { Q } } ( \cdot )$ is continuous and ${ \cal S } ( X _ { t } , Y _ { t } ) | A _ { t } = a$ is continuously distributed. Then, there exists an optimal value $\alpha _ { a } ^ { * }$ such that $M ( \alpha _ { a } ^ { * } | a ) = \alpha$ . Lemma A.4 in the Appendix shows that if in addition $M ( \cdot | a )$ admits a second order Taylor expansion, then
+
+$$
+B \leq C \left( \gamma + \gamma ^ { - 1 } \operatorname* { s u p } _ { a \in \mathcal { A } } \operatorname* { s u p } _ { k \in \mathbb { N } } \mathbb { E } [ \left| \alpha _ { A _ { t + 1 } } ^ { * } - \alpha _ { A _ { t } } ^ { * } \right| \left| A _ { t + k } = a \right| \right) \quad \mathrm { a n d } \quad \sigma _ { B } ^ { 2 } \leq B ^ { 2 } .
+$$
+
+Here, the constant $C$ will depend on how much $M ( \cdot | a )$ differs from the ideal case in which ${ \hat { Q } } ( \cdot )$ is the true quantile function for ${ \cal S } ( X _ { t } , Y _ { t } ) | A _ { t } = a$ . In this case we would have that $M ( \cdot | a )$ is the linear function ${ \bf \bar { \cal M } } ( p | a ) = p$ , $\forall p \in [ 0 , 1 ]$ and $C \le 2$ .
+
+We remark that the term $\mathbb { E } [ | \alpha _ { A _ { t + 1 } } ^ { * } - \alpha _ { A _ { t } } ^ { * } | \big | A _ { t + k } = a ]$ can be seen as a quantitative measurement of the size of the distribution shift in terms of the change in the critical value $\alpha _ { a } ^ { * }$ . Thus, we interpret these results as showing that if the distribution shift is small and $\forall a \in { \mathcal { A } }$ , ${ \hat { Q } } ( \cdot )$ gives reasonable coverage of the distribution of ${ \cal S } ( X _ { t } , Y _ { t } ) | A _ { t } = a$ , then $T ^ { - 1 } \sum _ { t = 1 } ^ { T } \mathrm { e r r } _ { t }$ 2 A ·will concentrate well around .
+
+# 4.2.3 Achieving approximate marginal coverage
+
+Theorem $\boxed { 4 . 1 }$ bounds the distance between the average miscoverage rate and the target level over long stretches of time. On the other hand, it provides no information about the marginal coverage frequency at a single time step. The following result shows that if the distribution shift is small, the realized marginal coverage rate $M ( \alpha _ { t } | A _ { t } )$ will be close to $\alpha$ on average.
+
+Theorem 4.2 Assume that there exists a constant $L > 0$ such that for all $a \in { \mathcal { A } }$ and all $\alpha _ { 1 } , \alpha _ { 2 } \in \mathbb { R } ,$ ,
+
+$$
+| M ( \alpha _ { 2 } | a ) - M ( \alpha _ { 1 } | a ) | \leq L | \alpha _ { 2 } - \alpha _ { 1 } | .
+$$
+
+Assume additionally that for all $a \in { \mathcal { A } }$ there exists $\alpha _ { a } ^ { * } \in ( 0 , 1 )$ such that $M ( \alpha _ { a } ^ { * } | a ) = \alpha .$ . Then,
+
+$$
+\mathbb { E } [ ( M ( \alpha _ { t } \vert A _ { t } ) - \alpha ) ^ { 2 } ] \leq \frac { L ( 1 + \gamma ) } { \gamma } \mathbb { E } [ \vert \alpha _ { A _ { t + 1 } } ^ { \ast } - \alpha _ { A _ { t } } ^ { \ast } \vert ] + \frac { L } { 2 } \gamma .
+$$
+
+Once again we emphasize that $\textcircled{7}$ holds for any choice of $t$ owing to the fact that $\left( \alpha _ { t } , A _ { t } , \mathbf { e r r } _ { t } \right)$ is assumed stationary and thus the quantities appearing in the bound are invariant across $t$ . Proof of this result can be found in Section $\mathbf { \bar { A . 8 } }$ of the Appendix. We remark that the right-hand side of $\textcircled { 7 }$ is minimized by choosing $\gamma = ( 2 \mathbb { E } [ | \overline { { \alpha _ { A _ { t + 1 } } ^ { * } } } - \alpha _ { A _ { t } } ^ { * } | ] ) ^ { 1 / 2 }$ , which gives the inequality
+
+$$
+\mathbb { E } [ ( M ( \alpha _ { t } | A _ { t } ) - \alpha ) ^ { 2 } ] \leq L ( \sqrt { 2 } + 1 ) \sqrt { \mathbb { E } [ | \alpha _ { A _ { t + 1 } } ^ { * } - \alpha _ { A _ { t } } ^ { * } | ] } .
+$$
+
+As above we have that in the ideal case ${ \hat { Q } } ( \cdot )$ is a perfect estimate of the quantiles of ${ \cal S } ( X _ { t } , Y _ { t } ) | A _ { t } = a$ and thus $M ( p | a ) = p$ and $L = 1$ . Moreover, we once again have the interpretation that $\mathbb { E } [ | \alpha _ { A _ { t + 1 } } ^ { * } -$ $\alpha _ { A _ { t } } ^ { * } | ]$ is a quantitative measurement of the distribution shift. Thus, this result can be interpreted as bounding the average difference between the realized and target marginal coverage in terms of the size of the underlying distribution shift. Finally, note that the choice $\gamma = ( 2 \mathbb { E } [ | \alpha _ { A _ { t + 1 } } ^ { * } - \alpha _ { A _ { t } } ^ { * } | ] ) ^ { 1 / 2 }$ formalizes our intuition that $\gamma$ should be chosen to be larger in domains with greater distribution shift, while not being so large as to cause $\alpha _ { t }$ to be overly volatile.
+
+# 5 Impact of $S _ { t } ( \cdot )$ on the performance
+
+The performance of all conformal inference methods depends heavily on the design of the conformity score. Previous work has shown how carefully chosen scores or even explicit optimization of the interval width can be used to obtain smaller prediction sets [e.g. 28, 29, 20, 12]. Adaptive conformal inference can work with any conformity score $S _ { t } ( \cdot )$ and quantile function $\hat { Q } _ { t } ( \cdot )$ and thus can be directly combined with other improvements in conformal inference to obtain shorter intervals. One important caveat here is that the lengths of conformal prediction sets depend directly on the quality of the fitted regression model. Thus, to obtain smaller intervals one should re-fit the model at each time step using the most recent data to build the most accurate predictions. This is exactly what we have done in our experiments in Sections $2 . 2$ and 6.
+
+![](images/3a880b81d462e11cd1f20f8e8ab2d3c46e21a0e890896e590d96462c030ea136.jpg)  
+Figure 2: Local coverage frequencies for adaptive conformal (blue), a non-adaptive method that holds $\alpha _ { t } = \alpha$ fixed (red), and an i.i.d. Bernoulli(0.1) sequence (grey) for the prediction of stock market volatility with conformity score $\tilde { S } _ { t }$ . The coloured dotted lines mark the average coverage obtained across all time points, while the black line indicates the target level of $1 - \alpha = 0 . 9$ .
+
+In addition to this, the choice of $S _ { t } ( \cdot )$ can also have a direct effect on the coverage properties of adaptive conformal inference. Theorems $\boxed { 4 . 1 }$ and $4 . 2$ show that the performance of adaptive conformal inference is controlled by the size of the shift in the optimal parameter $\alpha _ { t } ^ { * }$ across time. Moreover, $\alpha _ { t } ^ { * }$ itself is in one-to-one correspondence with the $1 - \alpha$ quantile of $S _ { t } ( X _ { t } , Y _ { t } )$ . Thus, the coverage properties of adaptive conformal inference depend on how close $S _ { t } ( X _ { t } , Y _ { t } )$ is to being stationary.
+
+For a simple example illustrating the impact of this dependence, note that in Section $\boxed { 2 . 2 }$ we formed prediction sets using the conformity score
+
+$$
+S _ { t } : = \frac { | V _ { t } - \hat { \sigma } _ { t } ^ { 2 } | } { \hat { \sigma } _ { t } ^ { 2 } } .
+$$
+
+An a priori reasonable alternative to this is the unnormalized score
+
+$$
+\tilde { S } _ { t } : = | V _ { t } - \hat { \sigma } _ { t } ^ { 2 } | .
+$$
+
+However, after a more careful examination it becomes unsurprising that normalization by $\hat { \sigma } _ { t } ^ { 2 }$ is critical for obtaining an approximately stationary conformity score and thus $\tilde { S } _ { t }$ leads to much worse coverage properties. Figure $2$ shows the local coverage frequency (see $( 4 )$ ) of adaptive conformal inference using $\tilde { S } _ { t }$ . In comparison to Figure $^ 1$ the coverage now undergoes much wider swings away from the target level of 0.9. This issue can be partially mitigated by choosing a larger value of $\gamma$ that gives greater adaptivity to the algorithm.
+
+# 6 Real data example: election night predictions
+
+During the 2020 US presidential election The Washington Post used conformalized quantile regression (CQR) (see $\mathbb { \underline { { \left( \mathrm { 1 } \right) } } }$ and Section $\boxed { 1 . 1 }$ to produce county level predictions of the vote total on election night [13]. Here we replicate the core elements of this method using both fixed and adaptive quantiles.
+
+![](images/c0d7449ad604c682f9a8bf69819c9ade50744b6d0f32c4c16d3e878e7ead701f.jpg)  
+Figure 3: Local coverage frequencies of adaptive conformal (blue), a non-adaptive method that holds $\alpha _ { t } = \alpha$ fixed (red), and an i.i.d. Bernoulli(0.1) sequence (grey) for county-level election predictions. Coloured dotted lines show the average coverage across all time points, while the black line indicates the target coverage level of $1 - \alpha = 0 . 9$ .
+
+To make the setting precise, let $\{ Y _ { t } \} _ { 1 \leq t \leq T }$ denote the number of votes cast for presidential candidate Joe Biden in the 2020 election in each of approximately $T = 3 0 0 0$ counties in the United States. Let $X _ { t }$ denote a set of demographic covariates associated to the tth county. In our experiment $X _ { t }$ will include information on the make-up of the county population by ethnicity, age, sex, median income and education (see Section $\boxed { \overline { { \mathrm { A } . 6 . 1 } } }$ for details). On election night county vote totals were observed as soon as the vote count was completed. If the order in which vote counts completed was uniformly random $\{ ( X _ { t } , Y _ { t } ) \} _ { 1 \leq t \leq T }$ would be an exchangeable sequence on which we could run standard conformal inference methods. In reality, larger urban counties tend to report results later than smaller rural counties and counties on the east coast of the US report earlier than those on the west coast. Thus, the distribution of $( X _ { t } , Y _ { t } )$ can be viewed as drifting throughout election night.
+
+We apply CQR to predict the county-level vote totals (see Section A.6.2 for details). To replicate the east to west coast bias observed on election night we order the counties by their time zone with eastern time counties appearing first and Hawaiian counties appearing last. Within each time zone counties are ordered uniformly at random. Figure $3$ shows the realized local coverage frequency over the most recent 300 counties (see $\textcircled{4}$ ) for the non-adaptive and adaptive conformal methods. We find that the non-adaptive method fails to maintain the desired $9 0 \%$ coverage level, incurring large troughs in its coverage frequency during time zone changes. On the other hand, the adaptive method maintains approximate $9 0 \hat { \% }$ coverage across all time points with deviations in its local coverage level comparable to what is observed in Bernoulli sequences.
+
+# 7 Discussion
+
+There are still many open problems in this area. The methods we develop are specific to cases where $Y _ { t }$ is revealed at each time point. However, there are many settings in which we receive the response in a delayed fashion or in large batches. In addition, our theoretical results in Section $\boxed { 4 . 2 }$ are limited to a single model for the data generating distribution and the special case where the quantile function $\hat { Q } _ { t } ( \cdot )$ is fixed across time. It would be interesting to determine if similar results can be obtained in settings where $\hat { Q } _ { t } ( \cdot )$ is fit in an online fashion on the most recent data. Another potential area for improvement is in the choice of the step size $\gamma$ . In Section $\boxed { 2 . 1 }$ we give some heuristic guidelines for choosing $\gamma$ based on the size of the distribution shift in the environment. Ideally however we would like to be able to determine $\gamma$ adaptively without prior knowledge. Finally, our experimental results are limited to just two domains. Additional work is needed to determine if our methods can successfully protect against a wider variety of real-world distribution shifts.
+
+# 8 Acknowledgements
+
+E.C. was supported by Office of Naval Research grant N00014-20-12157, by the National Science Foundation grants OAC 1934578 and DMS 2032014, by the Army Research Office (ARO) under grant W911NF-17-1-0304, and by the Simons Foundation under award 814641. We thank John Cherian for valuable discussions related to Presidential Election Night 2020 and Lihua Lei for helpful comments on the connection to online learning.
+
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+[30] Yaniv Romano, Matteo Sesia, and Emmanuel Candes. Classification with valid and adaptive coverage. In H. Larochelle, M. Ranzato, R. Hadsell, M. F. Balcan, and H. Lin, editors, Advances in Neural Information Processing Systems, volume 33, pages 3581–3591. Curran Associates, Inc., 2020. URL https://proceedings.neurips.cc/paper/2020/ file/244edd7e85dc81602b7615cd705545f5-Paper.pdf.   
+[31] Mauricio Sadinle, Jing Lei, and Larry Wasserman. Least ambiguous set-valued classifiers with bounded error levels. Journal of the American Statistical Association, 114(525):223–234, 2019. doi: 10.1080/01621459.2017.1395341. URL https://doi.org/10.1080/01621459.2017. 1395341.   
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+[33] Ryan J Tibshirani, Rina Foygel Barber, Emmanuel Candes, and Aaditya Ramdas. Conformal prediction under covariate shift. In H. Wallach, H. Larochelle, A. Beygelzimer, F. d'Alché-Buc, E. Fox, and R. Garnett, editors, Advances in Neural Information Processing Systems, volume 32. Curran Associates, Inc., 2019. URL https://proceedings.neurips.cc/paper/2019/ file/8fb21ee7a2207526da55a679f0332de2-Paper.pdf.   
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+    {
+        "type": "text",
+        "text": "Adaptive Conformal Inference Under Distribution Shift ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Isaac Gibbs ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 0
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+        "type": "text",
+        "text": "Emmanuel J. Candès ",
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Department of Statistics Stanford University igibbs@stanford.edu ",
+        "bbox": [
+            272,
+            237,
+            436,
+            277
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Department of Statistics Department of Mathematics Stanford University candes@stanford.edu ",
+        "bbox": [
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+        "page_idx": 0
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+    {
+        "type": "text",
+        "text": "Abstract ",
+        "text_level": 1,
+        "bbox": [
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+            535,
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "We develop methods for forming prediction sets in an online setting where the data generating distribution is allowed to vary over time in an unknown fashion. Our framework builds on ideas from conformal inference to provide a general wrapper that can be combined with any black box method that produces point predictions of the unseen label or estimated quantiles of its distribution. While previous conformal inference methods rely on the assumption that the data points are exchangeable, our adaptive approach provably achieves the desired coverage frequency over long-time intervals irrespective of the true data generating process. We accomplish this by modelling the distribution shift as a learning problem in a single parameter whose optimal value is varying over time and must be continuously re-estimated. We test our method, adaptive conformal inference, on two real world datasets and find that its predictions are robust to visible and significant distribution shifts. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "1 Introduction ",
+        "text_level": 1,
+        "bbox": [
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+            550,
+            310,
+            568
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Machine learning algorithms are increasingly being employed in high stakes decision making processes. For instance, deep neural networks are currently being used in self-driving cars to detect nearby objects $\\pmb { \\mathbb { D } }$ and parole decisions are being made with the assistance of complex models that combine over a hundred features $\\mathbb { I I }$ . As the popularity of black box methods and the cost of making wrong decisions grow it is crucial that we develop tools to quantify the uncertainty of their predictions. ",
+        "bbox": [
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+            582,
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "In this paper we develop methods for constructing prediction sets that are guaranteed to contain the target label with high probability. We focus specifically on an online learning setting in which we observe covariate-response pairs $\\{ ( X _ { t } , Y _ { t } ) \\} _ { t \\in \\mathbb { N } } \\subseteq \\mathbb { R } ^ { d } \\times \\mathbf { \\bar { \\mathbb { R } } }$ in a sequential fashion. At each time step $t \\in \\mathbb { N }$ we are tasked with using the previously observed data $\\{ ( X _ { r } , Y _ { r } ) \\} _ { 1 \\leq r \\leq t - 1 }$ along with the new covariates, $X _ { t }$ , to form a prediction set $\\hat { C } _ { t }$ for $Y _ { t }$ . Then, given a target coverage level $\\alpha \\in ( 0 , 1 )$ our generic goal is to guarantee that $Y _ { t }$ belongs to $\\hat { C } _ { t }$ at least $1 0 0 ( 1 - \\alpha ) \\%$ of the time. ",
+        "bbox": [
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Perhaps the most powerful and flexible tools for solving this problem come from conformal inference [see e.g. 34, 16, 32, 22, 31, 15, 3] . This framework provides a generic methodology for transforming the outputs of any black box prediction algorithm into a prediction set. The generality of this approach has facilitated the development of a large suite of conformal methods, each specialized to a specific prediction problem of interest [e.g. 30, 11, 23, 8, 24, 21]. With only minor exceptions all of these algorithms share the same common guarantee that if the training and test data are exchangeable, then the prediction set has valid marginal coverage $\\mathbb { P } ( Y _ { t } \\in \\hat { C } _ { t } ) = \\bar { 1 } - \\alpha$ . ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "While exchangeability is a common assumption, there are many real-world applications in which we do not expect the marginal distribution of $( X _ { t } , Y _ { t } )$ to be stationary. For example, in finance and economics market behaviour can shift drastically in response to new legislation or major world events. Alternatively, the distribution of $( X _ { t } , Y _ { t } )$ may change as we deploy our prediction model in new environments. This paper develops adaptive conformal inference (ACI), a method for forming prediction sets that are robust to changes in the marginal distribution of the data. Our approach is both simple, in that it requires only the tracking of a single parameter that models the shift, and general as it can be combined with any modern machine learning algorithm that produces point predictions or estimated quantiles for the response. We show that over long time intervals ACI achieves the target coverage frequency without any assumptions on the data-generating distribution. Moreover, when the distribution shift is small and the prediction algorithm takes a certain simple form we show that ACI will additionally obtain approximate marginal coverage at most time steps. ",
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+    {
+        "type": "text",
+        "text": "1.1 Conformal inference ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 1
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+    {
+        "type": "text",
+        "text": "Suppose we are given a fitted regression model for predicting the value of $Y$ from $X$ . Let $y$ be a candidate value for $Y _ { t }$ . To determine if $y$ is a reasonable estimate of $Y _ { t }$ , we define a conformity score $S ( X , Y )$ that measures how well the value $y$ conforms with the predictions of our fitted model. For example, if our regression model produces point predictions ${ \\hat { \\mu } } ( X )$ then we could use a conformity score that measures the distance between $\\hat { \\mu } ( X _ { t } )$ and $y$ . One such example is ",
+        "bbox": [
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+        "type": "equation",
+        "img_path": "images/35dbdf0bb86d318c315e5e5be304c7b27b3032312f00bcd0c84a2e89efb39b25.jpg",
+        "text": "$$\nS ( X _ { t } , y ) = | \\hat { \\mu } ( X _ { t } ) - y | .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Alternatively, suppose our regression model outputs estimates ${ \\hat { q } } ( X ; p )$ of the $p$ th quantile of the distribution of $Y | X$ . Then, we could use the method of conformal quantile regression (CQR) $\\left[ \\left[ 2 8 \\right] \\right]$ which examines the signed distance between $y$ and fitted upper and lower quantiles through the score ",
+        "bbox": [
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+        "type": "equation",
+        "img_path": "images/e42f3b0535d32f4a86f02075dbea58f1611c76988936ff3266c2ee4b3bfac2b2.jpg",
+        "text": "$$\nS ( X _ { t } , y ) = \\operatorname* { m a x } \\{ \\hat { q } ( X _ { t } ; \\alpha / 2 ) - y , y - \\hat { q } ( X _ { t } ; 1 - \\alpha / 2 ) \\} .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Regardless of what conformity score is chosen the key issue is to determine how small $S ( X _ { t } , y )$ should be in order to accept $y$ as a reasonable prediction for $Y _ { t }$ . Assume we have a calibration set $\\mathcal { D } _ { \\mathrm { c a l } } \\subseteq \\{ ( X _ { r } , Y _ { r } ) \\} _ { 1 \\leq r \\leq t - 1 }$ that is different from the data that was used to fit the regression model. Using this calibration set we define the fitted quantiles of the conformity scores to be ",
+        "bbox": [
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+        "img_path": "images/dd2b887fa806a5a2517ebcdbd9c10d8d8aef04cb01f852e1b30fb308b5bd244c.jpg",
+        "text": "$$\n\\hat { Q } ( p ) : = \\operatorname* { i n f } \\left\\{ s : \\left( \\frac 1 { | \\mathscr { D } _ { \\mathrm { c a l } } | } \\sum _ { ( X _ { r } , Y _ { r } ) \\in \\mathscr { D } _ { \\mathrm { c a l } } } \\mathbb { 1 } _ { \\{ S ( X _ { r } , Y _ { r } ) \\leq s \\} } \\right) \\geq p \\right\\} ,\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 1
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+    {
+        "type": "text",
+        "text": "and say that $y$ is a reasonable prediction for $Y _ { t }$ if $S ( X _ { t } , y ) \\leq \\hat { Q } ( 1 - \\alpha )$ . ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "The crucial observation is that if the data $\\mathcal { D } _ { \\operatorname { c a l } } \\cup \\{ ( X _ { t } , Y _ { t } ) \\}$ are exchangeable and we break ties uniformly at random then the rank of $S ( X _ { t } , Y _ { t } )$ amongst the points $\\{ S ( X _ { r } , Y _ { r } ) \\} _ { ( X _ { r } , Y _ { r } ) \\in \\mathcal { D } _ { \\mathrm { c a l } } } \\cup$ $\\{ S ( X _ { t } , Y _ { t } ) \\}$ will be uniform. Therefore, ",
+        "bbox": [
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+        "page_idx": 1
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+    {
+        "type": "equation",
+        "img_path": "images/1fec0089858b69f6565ff21de5a70e6745e787bc76bc57bab06a561b7e3926dd.jpg",
+        "text": "$$\n\\mathbb { P } ( S ( X _ { t } , Y _ { t } ) \\le \\hat { Q } ( 1 - \\alpha ) ) = \\frac { \\lceil | \\mathcal { D } _ { \\mathrm { c a l } } | ( 1 - \\alpha ) \\rceil } { | \\mathcal { D } _ { \\mathrm { c a l } } | + 1 } .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Thus, defining our prediction set to be $\\hat { C } _ { t } : = \\{ y : S ( X _ { t } , y ) \\leq \\hat { Q } ( 1 - \\alpha ) \\}$ gives the marginal coverage guarantee ",
+        "bbox": [
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+        "page_idx": 1
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+    {
+        "type": "equation",
+        "img_path": "images/c6122cf1be94086549d158acdfbe4b1000bd2ae942ef0809a95caae9f53172b2.jpg",
+        "text": "$$\n\\mathbb { P } ( Y _ { t } \\in \\hat { C } _ { t } ) = \\mathbb { P } ( S ( X _ { t } , Y _ { t } ) \\leq \\hat { Q } ( 1 - \\alpha ) ) = \\frac { \\lceil | \\mathcal { D } _ { \\mathrm { c a l } } | ( 1 - \\alpha ) \\rceil } { | \\mathcal { D } _ { \\mathrm { c a l } } | + 1 } .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "By introducing additional randomization this generic procedure can be altered slightly to produce a set $\\hat { C } _ { t }$ that satisfies the exact marginal coverage guarantee $\\mathbb { P } ( Y _ { t } \\in \\hat { C } _ { t } ) = 1 - \\alpha$ [34]. For the purposes of this paper this adjustment is not critical and so we omit the details here. Additionally, we remark that the method outlined above is often referred to as split or inductive conformal inference $[ 1 2 7 , 1 3 4 , | 2 6 |$ . This refers to the fact that we have split the observed data between a training set used to fit the regression model and a withheld calibration set. The adaptive conformal inference method developed in this article can also be easily adjusted to work with full conformal inference in which data splitting is avoided at the cost of greater computational resources $\\pmb { \\mathbb { B 4 } }$ . ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "2 Adapting conformal inference to distribution shifts ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Up until this point we have been working with a single score function $S ( \\cdot )$ and quantile function ${ \\hat { Q } } ( \\cdot )$ In the general case where the distribution of the data is shifting over time both these functions should ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "be regularly re-estimated to align with the most recent observations. Therefore, we assume that at each time $t$ we are given a fitted score function $S _ { t } ( \\cdot )$ and corresponding quantile function $\\hat { Q } _ { t } ( \\cdot )$ . We define the realized miscoverage rate of the prediction set $\\hat { C } _ { t } ( \\alpha ) : = \\{ y : S _ { t } ( X _ { t } , y ) \\leq \\hat { Q } _ { t } ( 1 - \\alpha ) \\}$ as ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/a2328ce53ec690cc59ea238922a95185e5b3dc38f3f526c481249051b278dbfe.jpg",
+        "text": "$$\nM _ { t } ( \\alpha ) : = \\mathbb { P } ( S _ { t } ( X _ { t } , Y _ { t } ) > \\hat { Q } _ { t } ( 1 - \\alpha ) ) ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            367,
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "where the probability is over the test point $( X _ { t } , Y _ { t } )$ as well as the data used to fit $S _ { t } ( \\cdot )$ and $\\hat { Q } _ { t } ( \\cdot )$ ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Now, since the distribution generating the data is non-stationary we do not expect $M _ { t } ( \\alpha )$ to be equal, or even close to, $\\alpha$ . Even so, we can still postulate that if the conformity scores used to fit $\\hat { Q } _ { t } ( \\cdot )$ cover the bulk of the distribution of $S _ { t } ( X _ { t } , Y _ { t } )$ then there may be an alternative value $\\alpha _ { t } ^ { * } \\in [ 0 , 1 ]$ such that $M _ { t } ( \\alpha _ { t } ^ { * } ) \\cong \\alpha$ . More rigorously, assume that with probability one, $\\hat { Q } _ { t } ( \\cdot )$ is continuous, non-decreasing and such that $\\hat { Q } _ { t } ( 0 ) = - \\infty$ and $\\hat { Q } _ { t } ( 1 ) = \\infty$ . This does not hold for the split conformal quantile functions defined in $( 1 )$ , but in the case where there are no ties amongst the conformity scores we can adjust our definition to guarantee this by smoothing over the jump discontinuities in ${ \\hat { Q } } ( \\cdot )$ . Then, $M _ { t } ( \\cdot )$ will be non-decreasing on $[ 0 , 1 ]$ with $M _ { t } ( 0 ) = 0$ and $M _ { t } ( 1 ) = 1$ and so we may define ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/8f13a9f93b0d9778c0015c17425fc170037ec067c85dad55259aa961e9d405f7.jpg",
+        "text": "$$\n\\alpha _ { t } ^ { \\ast } : = \\operatorname* { s u p } \\{ \\beta \\in [ 0 , 1 ] : M _ { t } ( \\beta ) \\leq \\alpha \\} .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Moreover, if we additionally assume that ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/bf8f875d83dc9ab34b16a30fc097995ab96a677061daefadb1dc074d425180a4.jpg",
+        "text": "$$\n\\mathbb { P } ( S _ { t } ( X _ { t } , Y _ { t } ) = \\hat { Q } _ { t } ( 1 - \\alpha _ { t } ^ { * } ) ) = 0 ,\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "then we will have that $M _ { t } ( \\alpha _ { t } ^ { * } ) = \\alpha$ . So, in particular we find that by correctly calibrating the argument to $\\hat { Q } _ { t } ( \\cdot )$ we can achieve either approximate or exact marginal coverage. ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "To perform this calibration we will use a simple online update. This update proceeds by examining the empirical miscoverage frequency of the previous prediction sets and then decreasing (resp. increasing) our estimate of $\\alpha _ { t } ^ { * }$ if the prediction sets were historically under-covering (resp. over-covering) $Y _ { t }$ . In particular, let $\\alpha _ { 1 }$ denote our initial estimate (in our experiments we will choose $\\alpha _ { 1 } = \\alpha$ ). Recursively define the sequence of miscoverage events ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/20180bf5597f8b25bd6ae48b9e047693e73c84dc45829fe3f100449bd69578de.jpg",
+        "text": "$$\n\\begin{array} { r } { \\mathsf { e r r } _ { t } : = \\Bigl \\{ \\displaystyle 1 , \\mathrm { ~ i f ~ } Y _ { t } \\notin \\hat { C } _ { t } ( \\alpha _ { t } ) , \\qquad \\mathrm { w h e r e ~ } \\hat { C } _ { t } ( \\alpha _ { t } ) : = \\{ y : S _ { t } ( X _ { t } , y ) \\leq \\hat { Q } _ { t } ( 1 - \\alpha _ { t } ) \\} . } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Then, fixing a step size parameter $\\gamma > 0$ we consider the simple online update ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/9b039b90e1dc96247282522271ecf4a2c6991b6e950be60eda1fd0fb48c25b17.jpg",
+        "text": "$$\n\\alpha _ { t + 1 } : = \\alpha _ { t } + \\gamma ( \\alpha - \\mathrm { e r r } _ { t } ) .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "We refer to this algorithm as adaptive conformal inference. Here, $\\operatorname { e r r } _ { t }$ plays the role of our estimate of the historical miscoverage frequency. A natural alternative to this is the update ",
+        "bbox": [
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+        "page_idx": 2
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+    {
+        "type": "equation",
+        "img_path": "images/a8ea41cb9338dabd0e0b0a7bc933c267c1a05f1b8a2a10984d45e01f698e965c.jpg",
+        "text": "$$\n\\alpha _ { t + 1 } = \\alpha _ { t } + \\gamma \\left( \\alpha - \\sum _ { s = 1 } ^ { t } w _ { s } \\mathrm { e r r } _ { s } \\right) ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            379,
+            628,
+            617,
+            672
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "where the ap $\\{ w _ { s } \\} _ { 1 \\leq s \\leq t } \\subseteq [ 0 , 1 ]$ is a sequence of increasing weights with y evaluating the recent empirical miscov $\\textstyle \\sum _ { s = 1 } ^ { t } w _ { s } = 1$ . This update hasy when deciding whether or not to lower or raise $\\alpha _ { t }$ . In practice, we find that $\\textcircled{2}$ and $\\textcircled{3}$ produce almost identical results. For example, in Section ${ \\bf A } . 3$ in the Appendix we show some sample trajectories for $\\alpha _ { t }$ obtained using the update $\\textcircled { 3 }$ with ",
+        "bbox": [
+            173,
+            679,
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/52f134113e049bf36291eb0e4504ad9f16b1345d0563bd328e090451e8da0fe8.jpg",
+        "text": "$$\nw _ { s } : = \\frac { 0 . 9 5 ^ { t - s } } { \\sum _ { s ^ { \\prime } = 1 } ^ { t } 0 . 9 5 ^ { t - s ^ { \\prime } } } .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+            756,
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "We find that these trajectories are very similar to those produced by $( 2 )$ . The main difference is that the trajectories obtained with $\\textcircled { 3 }$ are smoother with less local variation in $\\alpha _ { t }$ . In the remainder of this article we will focus on $( 2 )$ for simplicity. ",
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "2.1 Choosing the step size ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            857,
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+            872
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "The choice of $\\gamma$ gives a tradeoff between adaptability and stability. While raising the value of $\\gamma$ will make the method more adaptive to observed distribution shifts, it will also induce greater volatility in the value of $\\alpha _ { t }$ . In practice, large fluctuations in $\\alpha _ { t }$ may be undesirable as it allows the method to oscillate between outputting small conservative and large anti-conservative prediction sets. ",
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "In Theorem $\\left| \\overline { { 4 . 2 } } \\right|$ we give an upper bound on $( M _ { t } ( \\alpha _ { t } ) - \\alpha ) ^ { 2 }$ that is optimized by choosing $\\gamma$ proportional to $\\sqrt { \\left| \\alpha _ { t + 1 } ^ { * } - \\alpha _ { t } ^ { * } \\right| }$ . While not directly applicable in practice, this result supports the intuition that in environments with greater distributional shift the algorithm needs to be more adapatable and thus $\\gamma$ should be chosen to be larger. In our experiments we will take $\\gamma = 0 . 0 0 5$ . This value was chosen because it was found to give relatively stable trajectories for $\\alpha _ { t }$ while still being sufficiently large as to allow $\\alpha _ { t }$ to adapt to observed shifts. In agreement with the general principles outlined above we found that larger values of $\\gamma$ also successfully protect against distribution shifts, while taking $\\gamma$ to be too small causes adaptive conformal inference to perform similar to non-adaptive methods that hold $\\alpha _ { t } = \\alpha$ constant across time. ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "2.2 Real data example: predicting market volatility ",
+        "text_level": 1,
+        "bbox": [
+            173,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "We apply ACI to the prediction of market volatility. Let $\\{ P _ { t } \\} _ { 1 \\leq t \\leq T }$ denote a sequence of daily open prices for a stock. For all $t \\geq 2$ , define the return $R _ { t } : = ( \\bar { P } _ { t } - \\bar { P } _ { t - 1 } ) / P _ { t - 1 }$ and realized volatility $\\dot { V } _ { t } = R _ { t } ^ { 2 }$ . Our goal is to use the previously observed returns $X _ { t } : = \\{ R _ { s } \\} _ { 1 \\leq s \\leq t - 1 }$ to form prediction sets for $Y _ { t } : = V _ { t }$ . More sophisticated financial models might augment $X _ { t }$ with additional market covariates (available to the analyst at time $t - 1$ ). As the primary purpose of this section is to illustrate adaptive conformal inference we work with only a simple prediction method. ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "We start off by forming point predictions using a GARCH(1,1) model $\\mathbb { \\lVert }$ . This method assumes that $R _ { t } = \\sigma _ { t } \\epsilon _ { t }$ with $\\epsilon _ { 2 } , \\dots , \\epsilon _ { T }$ taken to be i.i.d. $\\mathcal { N } ( 0 , 1 )$ and $\\sigma _ { t }$ satisfying the recursive update ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/0e2f8b2bb7d97a12fed249c2ec7366a0380ca81587e4c711b94dbaf05c278fda.jpg",
+        "text": "$$\n\\sigma _ { t } ^ { 2 } = \\omega + \\tau V _ { t - 1 } + \\beta \\sigma _ { t - 1 } ^ { 2 } .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+            420,
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+            439
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "This is a common approach used for forecasting volatility in economics. In practice, shifting market dynamics can cause the predictions of this model to become inaccurate over large time periods. Thus, when forming point predictions we fit the model using only the last 1250 trading days (i.e. approximately 5 years) of market data. More precisely, for all times $t > 1 2 5 0$ we fit the coefficients $\\hat { \\omega } _ { t } , ~ \\hat { \\tau } _ { t } , \\hat { \\beta } _ { t }$ as well as the sequence of variances $\\{ \\hat { \\sigma } _ { s } ^ { t } \\} _ { 1 \\leq s \\leq t - 1 }$ using only the data $\\{ R _ { r } \\} _ { t - 1 2 5 0 \\leq r < t }$ . Then, our point prediction for the realized volatility at time $t$ is ",
+        "bbox": [
+            173,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/f7f2097e044f8279e1f7ebe77f6bd02e92672dd29dee2672cdc3905c0f84f122.jpg",
+        "text": "$$\n( \\hat { \\sigma } _ { t } ^ { t } ) ^ { 2 } : = \\hat { \\omega } _ { t } + \\hat { \\tau } _ { t } V _ { t - 1 } + \\hat { \\beta } _ { t } ( \\hat { \\sigma } _ { t - 1 } ^ { t } ) ^ { 2 } .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            379,
+            537,
+            617,
+            558
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "To form prediction intervals we define the sequence of conformity scores ",
+        "bbox": [
+            173,
+            564,
+            651,
+            579
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/32ac0f0fb08eaa2b52776d92925fb267f05f243f25b8555992aa08b90fd01889.jpg",
+        "text": "$$\nS _ { t } : = \\frac { | V _ { t } - ( \\hat { \\sigma } _ { t } ^ { t } ) ^ { 2 } | } { ( \\hat { \\sigma } _ { t } ^ { t } ) ^ { 2 } }\n$$",
+        "text_format": "latex",
+        "bbox": [
+            434,
+            585,
+            563,
+            621
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "and the corresponding quantile function ",
+        "bbox": [
+            174,
+            627,
+            436,
+            642
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/e3ab8304eb9e66bd7d8f7f4b5fafa9b04b90c7c1df9fd9de89d3dd0ad99d9469.jpg",
+        "text": "$$\n\\hat { Q } _ { t } ( p ) : = \\operatorname* { i n f } \\left\\{ x : \\frac { 1 } { 1 2 5 0 } \\sum _ { r = t - 1 2 5 0 } ^ { t - 1 } \\mathbb { 1 } _ { S _ { r } \\leq x } \\geq p \\right\\} .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            338,
+            647,
+            658,
+            693
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Then, our prediction set at time $t$ is ",
+        "bbox": [
+            174,
+            699,
+            406,
+            713
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/79b92091c57e2386738f3839d3610ea5edf76f5b0730f539be7b6fea39f99e63.jpg",
+        "text": "$$\n\\hat { C } _ { t } ( \\alpha _ { t } ) : = \\left\\{ v : \\frac { | v - ( \\hat { \\sigma } _ { t } ^ { t } ) ^ { 2 } | } { ( \\hat { \\sigma } _ { t } ^ { t } ) ^ { 2 } } \\leq \\hat { Q } _ { t } ( 1 - \\alpha _ { t } ) \\right\\} ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            346,
+            720,
+            650,
+            756
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "where $\\left\\{ \\alpha _ { t } \\right\\}$ is initialized with $\\alpha _ { 1 2 5 0 } = \\alpha = 0 . 1$ and then updated recursively as in $\\bigoplus$ . ",
+        "bbox": [
+            174,
+            762,
+            738,
+            779
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "We compare this algorithm to a non-adaptive alternative that takes $\\alpha _ { t } = \\alpha$ fixed. To measure the performance of these methods across time we examine their local coverage frequencies defined as the average coverage rate over the most recent two years, i.e. ",
+        "bbox": [
+            174,
+            784,
+            825,
+            825
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/426bb35fdb6d5c98a0bf00d70c82275e5900b91df16dfff98edef60d4138974b.jpg",
+        "text": "$$\n\\mathrm { l o c a l C o v } _ { t } : = 1 - { \\frac { 1 } { 5 0 0 } } \\sum _ { r = t - 2 5 0 + 1 } ^ { t + 2 5 0 } \\mathrm { e r r } _ { r } .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            369,
+            832,
+            627,
+            876
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "If the methods perform well then we expect the local coverage frequency to stay near the target value $1 - \\alpha$ across all time points. ",
+        "bbox": [
+            173,
+            883,
+            823,
+            911
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "image",
+        "img_path": "images/c65e1feee99d5c67ae7fcb7de068af025556a955d04f9e89d028fdc3e4e06f15.jpg",
+        "image_caption": [
+            "Figure 1: Local coverage frequencies for adaptive conformal (blue), a non-adaptive method that holds $\\alpha _ { t } = \\alpha$ fixed (red), and an i.i.d. Bernoulli(0.1) sequence (grey) for the prediction of stock market volatility. The coloured dotted lines mark the average coverage obtained across all time points, while the black line indicates the target level of $1 - \\alpha = 0 . 9$ . "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            187,
+            87,
+            807,
+            344
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Daily open prices were obtained from publicly available datasets published by The Wall Street Journal. The realized local coverage frequencies for the non-adaptive and adaptive conformal methods on four different stocks are shown in Figure $1 .$ These stocks were selected out of a total of 12 stocks that we examined because they showed a clear failure of the non-adaptive method. Adaptive conformal inference was found to perform well in all cases (see Figure 9 in the appendix). ",
+        "bbox": [
+            174,
+            438,
+            825,
+            508
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "As a visual comparator, the grey curves show the moving average 1500 Pt+250r=t\u0000250+1 Ir for sequences $\\left\\{ I _ { t } \\right\\} _ { 1 \\leq t \\leq T }$ that are i.i.d. Bernoulli(0.1). We see that the local coverage frequencies obtained by adaptive conformal inference (blue lines) always stay within the variation that would be expected from an i.i.d. Bernoulli sequence. On the other hand, the non-adaptive method undergoes large excursions away from the target level of $1 - \\alpha = 0 . 9$ (red lines). For example, in the bottom right panel we can see that the non-adaptive method fails to cover the realized volatility of Fannie Mae during the 2008 financial crisis, while the adaptive method is robust to this event (see Figure 4 in the Appendix for a plot of the price of Fannie Mae over this time period). ",
+        "bbox": [
+            173,
+            515,
+            825,
+            630
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "3 Related Work ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            648,
+            321,
+            666
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Prior work on conformal inference has considered two different types of distribution shift $\\mathbb { \\lVert 3 3 \\rVert \\lVert 0 \\rVert }$ . In both cases the focus was on environments in which the calibration data is drawn i.i.d. from a single distribution $P _ { 0 }$ , while the test point comes from a second distribution $P _ { 1 }$ . In this setting Tibshirani et al. $\\mathbb { \\lVert 3 3 \\rVert }$ showed that valid prediction sets can be obtained by re-weighting the calibration data using the likelihood ratio between $P _ { 1 }$ and $P _ { 0 }$ . However, this requires the conditional distribution of $Y | X$ to be constant between training and testing and the likelihood ratio $P _ { 1 } ( X ) / P _ { 0 } ( X )$ to be either known or very accurately estimated. On the other hand, Cauchois et al. $\\mathbb { m }$ develop methods for forming prediction sets that are valid whenever $P _ { 1 }$ and $P _ { 0 }$ are close in $f$ -divergence. Similar to our work, they show that if $D _ { f } ( P _ { 1 } | | P _ { 0 } ) \\leq \\rho$ then there exists a conservative value $\\alpha _ { \\rho } \\in ( 0 , 1 )$ such that ",
+        "bbox": [
+            173,
+            680,
+            825,
+            808
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/2d2e01a4d415fbe60ff1f7fc642344f3d1f7732d232bc5b59abb28861a10c48d.jpg",
+        "text": "$$\nM ( \\alpha _ { \\rho } ) : = \\mathbb { P } ( S ( X _ { t } , Y _ { t } ) > \\hat { Q } ( 1 - \\alpha _ { \\rho } ) ) \\leq \\alpha .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            351,
+            814,
+            647,
+            834
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "The difference between our approach and theirs is twofold. First, while they fix a single conservative value $\\alpha _ { \\rho }$ our methods aim to estimate the optimal choice $\\alpha ^ { * }$ satisfying $M ( \\alpha ^ { * } ) = \\alpha$ . This is not possible in the setting of $\\mathbb { m }$ as they do not observe any data from which the size of the distribution shift can be estimated. Second, while they consider only one training and one testing distribution we work in a fully online setting in which the distribution is allowed to shift continuously over time. ",
+        "bbox": [
+            174,
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+            825,
+            911
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "4 Coverage guarantees ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            89,
+            382,
+            107
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "4.1 Distribution-free results ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            119,
+            380,
+            135
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "In this section we outline the theoretical coverage guarantees of adaptive conformal inference. We will assume throughout that with probability one $\\alpha _ { 1 } \\in [ 0 , 1 ]$ and $\\hat { Q } _ { t }$ is non-decreasing with $\\hat { Q } _ { t } ( x ) = - \\infty$ for all $x < 0$ and $\\hat { Q } _ { t } ( x ) = \\infty$ for all $x > 1$ . Our first result shows that over long time intervals adaptive conformal inference obtains the correct coverage frequency irrespective of any assumptions on the data-generating distribution. ",
+        "bbox": [
+            173,
+            143,
+            826,
+            219
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Lemma 4.1 With probability one we have that $\\forall t \\in \\mathbb { N }$ , $\\alpha _ { t } \\in [ - \\gamma , 1 + \\gamma ]$ . ",
+        "bbox": [
+            173,
+            228,
+            656,
+            244
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Proof: Assume by contradiction that with positive probability $\\{ \\alpha _ { t } \\} _ { t \\in \\mathbb { N } }$ is such that $\\operatorname* { i n f } _ { t } \\alpha _ { t } < - \\gamma$ (the case where $\\operatorname* { s u p } _ { t } \\alpha _ { t } > 1 + \\gamma$ is identical). Note that $\\begin{array} { r } { \\operatorname* { s u p } _ { t } | \\alpha _ { t + 1 } - \\alpha _ { t } | = \\operatorname* { s u p } _ { t } \\gamma | \\alpha - \\mathrm { e r r } _ { t } | < \\gamma . } \\end{array}$ . Thus, with positive probability we may find $t \\in \\mathbb { N }$ such that $\\alpha _ { t } < 0$ and $\\alpha _ { t + 1 } < \\alpha _ { t }$ . However, ",
+        "bbox": [
+            173,
+            256,
+            828,
+            299
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "equation",
+        "img_path": "images/08b37ca69b7f3e21fd8876f1444e9ff8fa69d2ab8215a3b29056453f7ea9d868.jpg",
+        "text": "$$\n\\alpha _ { t } < 0 \\implies \\hat { Q } _ { t } ( 1 - \\alpha _ { t } ) = \\infty \\implies \\mathrm { e r r } _ { t } = 0 \\implies \\alpha _ { t + 1 } = \\alpha _ { t } + \\gamma ( \\alpha - \\mathrm { e r r } _ { t } ) \\geq \\alpha _ { t }\n$$",
+        "text_format": "latex",
+        "bbox": [
+            220,
+            301,
+            776,
+            319
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "and thus $\\mathbb { P } ( \\exists t$ such that $\\alpha _ { t + 1 } < \\alpha _ { t } < 0 ) = 0$ . We have reached a contradiction. ",
+        "bbox": [
+            176,
+            320,
+            694,
+            335
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Proposition 4.1 With probability one we have that for all $T \\in \\mathbb { N }$ , ",
+        "bbox": [
+            173,
+            344,
+            606,
+            359
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "equation",
+        "img_path": "images/c5537b0819284f0f30c9fd40d2c4dfb823967687f66674275c1314e79e254c83.jpg",
+        "text": "$$\n\\left| \\frac { 1 } { T } \\sum _ { t = 1 } ^ { T } e r r _ { t } - \\alpha \\right| \\leq \\frac { \\operatorname* { m a x } \\{ \\alpha _ { 1 } , 1 - \\alpha _ { 1 } \\} + \\gamma } { T \\gamma } .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            351,
+            361,
+            647,
+            404
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "$\\begin{array} { r } { \\operatorname* { l i m } _ { T \\to \\infty } \\frac { 1 } { T } \\sum _ { t = 1 } ^ { T } e r r _ { t } \\stackrel { a . s . } { = } \\alpha } \\end{array}$ ",
+        "bbox": [
+            176,
+            405,
+            455,
+            424
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Proof: By expanding the recursion defined in $( 2 )$ and applying Lemma 4.1 we find that ",
+        "bbox": [
+            171,
+            434,
+            754,
+            452
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "equation",
+        "img_path": "images/37ceeaff7b8909dc42ec596a213e7add047f17fdebe2f3adc66baf32b28765aa.jpg",
+        "text": "$$\n[ - \\gamma , 1 + \\gamma ] \\ni \\alpha _ { T + 1 } = \\alpha _ { 1 } + \\sum _ { t = 1 } ^ { T } \\gamma ( \\alpha - \\mathrm { e r r } _ { t } ) .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            346,
+            452,
+            650,
+            494
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Rearranging this gives the result. ",
+        "bbox": [
+            174,
+            494,
+            390,
+            508
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Proposition $\\boxed { 4 . 1 }$ puts no constraints on the data generating distribution. One may immediately ask whether these results can be improved by making mild assumptions on the distribution shifts. We argue that without assumptions on the quality of the initialization the answer to this question is negative. To understand this, consider a setting in which there is a single fixed optimal target $\\alpha ^ { * } \\in [ 0 , 1 ]$ and assume that ",
+        "bbox": [
+            173,
+            515,
+            825,
+            585
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "equation",
+        "img_path": "images/ea8156067e3367201a0d3345bea700cdc5341689b042442319cbfe2179483672.jpg",
+        "text": "$$\nM _ { t } ( p ) = M ( p ) = { \\left\\{ \\begin{array} { l l } { \\alpha + { \\frac { 1 - \\alpha } { 1 - \\alpha ^ { * } } } ( p - \\alpha ^ { * } ) , { \\mathrm { i f ~ } } p > \\alpha ^ { * } , } \\\\ { \\alpha + { \\frac { \\alpha } { \\alpha ^ { * } } } ( p - \\alpha ^ { * } ) { \\mathrm { i f ~ } } p \\leq \\alpha ^ { * } . } \\end{array} \\right. } ~ .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            313,
+            587,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Suppose additionally that $\\mathbb { E } [ \\mathrm { e r r } _ { t } | \\alpha _ { t } ] = M ( \\alpha _ { t } ) . ^ { 1 }$ In order to simplify the calculations consider the noiseless update $\\alpha _ { t + 1 } = \\alpha _ { t } + \\gamma ( \\alpha - M ( \\alpha _ { t } ) ) \\equiv \\alpha _ { t } + \\gamma ( \\alpha - \\mathbb { E } [ \\operatorname { e r r } _ { t } | \\alpha _ { t } ] )$ . Intuitively, the noiseless update can be viewed as the average case behaviour of $\\textstyle \\operatorname* { m i n } \\{ { \\frac { 1 - \\alpha ^ { * } } { 1 - \\alpha } } , { \\frac { \\alpha ^ { * } } { \\alpha } } \\}$ there exists a constant $\\textstyle c \\in \\{ { \\frac { 1 - \\alpha } { 1 - \\alpha ^ { * } } } , { \\frac { \\alpha } { \\alpha ^ { * } } } \\}$ $\\textcircled { 2 }$ such that for all . Now, for any initialization $t$ , $M ( \\alpha _ { t } ) - \\alpha = c ( \\alpha _ { t } - \\alpha ^ { * } )$ $\\alpha _ { 1 }$ and any $\\gamma \\leq$ So, we have that ",
+        "bbox": [
+            173,
+            625,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "equation",
+        "img_path": "images/27bdc545f9ae15ca273446a05c4cca4f7041727cca72f5e9fe906d84fb9b5329.jpg",
+        "text": "$$\n\\begin{array} { r } { \\mathbb { E } [ \\operatorname { e r r } _ { t } ] - \\alpha = c \\mathbb { E } [ \\alpha _ { t } - \\alpha ^ { * } ] = c \\mathbb { E } [ \\alpha _ { t - 1 } + \\gamma ( \\alpha - M _ { t - 1 } ( \\alpha _ { t - 1 } ) ) - \\alpha ^ { * } ] = c ( 1 - c \\gamma ) \\mathbb { E } [ \\alpha _ { t - 1 } - \\alpha ^ { * } ] . } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            176,
+            700,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Repeating this calculation recursively gives that ",
+        "bbox": [
+            176,
+            718,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "equation",
+        "img_path": "images/e286f4409eeaffbf2df5b5f0e145eee6f275c017bfd9e70979d012b32f2b2447.jpg",
+        "text": "$$\n\\begin{array} { r } { \\mathbb { E } [ \\mathsf { e r r } _ { t } ] - \\alpha = c ( 1 - c \\gamma ) ^ { t - 1 } \\mathbb { E } [ \\alpha _ { 1 } - \\alpha ^ { * } ] = c ( 1 - c \\gamma ) ^ { t - 1 } ( \\alpha _ { 1 } - \\alpha ^ { * } ) , } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            276,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "and thus, ",
+        "bbox": [
+            173,
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "equation",
+        "img_path": "images/1512d6d4a6e9dfc7202ce72aefd1e37b4c3808c7c5e2631962532a07aaec5e12.jpg",
+        "text": "$$\n\\left| \\frac { 1 } { T } \\sum _ { t = 1 } ^ { T } \\mathbb { E } [ \\mathsf { e r r } _ { t } ] - \\alpha \\right| = \\frac { 1 - ( 1 - c \\gamma ) ^ { T } } { T \\gamma } | \\alpha _ { 1 } - \\alpha ^ { * } | .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "The comparison of this bound to $( 5 )$ is self-evident. The main difference is that we have replaced $\\operatorname* { m a x } \\{ 1 - \\alpha _ { 1 } , \\alpha _ { 1 } \\}$ with $| \\alpha _ { 1 } - \\alpha ^ { * } |$ . This arises from the fact that $\\alpha ^ { * } \\in ( 0 , 1 )$ is arbitrary and thus $\\operatorname* { m a x } \\{ 1 - \\alpha _ { 1 } , \\alpha _ { 1 } \\}$ is the best possible upper bound on $| \\alpha _ { 1 } - \\alpha ^ { * } |$ . So, we view Proposition $^ { 4 . 1 }$ as both an agnostic guarantee that shows that our method gives the correct long-term empirical coverage frequency irrespective of the true data generating process, and as an approximately tight bound on the worst-case behaviour immediately after initialization. ",
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "4.2 Performance in a hidden Markov model ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            90,
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Although we believe Proposition $4 . 1$ is an approximately tight characterization of the behaviour after initialization, we can still ask whether better bounds can be obtained for large time steps. In this section we answer this question positively by showing that if $\\alpha _ { 1 }$ is initialized appropriately and the distribution shift is small, then tighter coverage guarantees can be given. In order to obtain useful results we will make some simplifying assumptions about the data generating process. While we do not expect these assumptions to hold exactly in any real-world setting, we do consider our results to be representative of the true behaviour of adaptive conformal inference and we expect similar results to hold under alternative models. ",
+        "bbox": [
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "4.2.1 Setting ",
+        "text_level": 1,
+        "bbox": [
+            174,
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+            256
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "We model the data as coming from a hidden Markov model. In particular, we let $\\{ A _ { t } \\} _ { t \\in \\mathbb { N } } \\subseteq { \\mathcal { A } }$ denote the underlying Markov chain for the environment and we assume that conditional on $\\{ A _ { t } \\} _ { t \\in \\mathbb { N } }$ $\\{ ( X _ { t } , Y _ { t } ) \\} _ { t \\in \\mathbb { N } }$ is an independent sequence with $( X _ { t } , Y _ { t } ) \\sim P _ { A _ { t } }$ for some collection of distributions $\\{ P _ { a } : a \\in { \\mathcal { A } } \\}$ . In order to simplify our calculations, we assume additionally that the estimated quantile function $\\hat { Q } _ { t } ( \\cdot )$ and score function $S _ { t } ( \\cdot )$ do not depend on $t$ and we denote them by ${ \\hat { Q } } ( \\cdot )$ and $\\bar { S } ( \\cdot )$ . This occurs for example in the split conformal setting with fixed training and calibration sets. ",
+        "bbox": [
+            173,
+            263,
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "In this setting, $\\{ ( \\alpha _ { t } , A _ { t } ) \\} _ { t \\in \\mathbb { N } }$ forms a Markov chain on $[ - \\gamma , 1 + \\gamma ] \\times \\mathcal { A }$ . We assume that this chain has a unique stationary distribution $\\pi$ and that $( \\alpha _ { 1 } , A _ { 1 } ) \\sim \\pi$ . This implies that $\\left( \\alpha _ { t } , A _ { t } , \\mathbf { e r r } _ { t } \\right)$ is a stationary process and thus will greatly simplify our characterization of the behaviour of $\\mathrm { e r r } _ { t }$ . While there is little doubt that the theory can be extended, recall our that main goal is to get useful and simple results. That said, what we really have in mind here is that $\\{ A _ { t } \\} _ { t \\in \\mathbb { N } }$ is sufficiently wellbehaved to guarantee that $( \\alpha _ { t } , A _ { t } )$ has a limiting stationary distribution. In Section ${ \\bf A } . 5$ we give an example where this is indeed provably the case. Lastly, the assumption that $( \\alpha _ { 1 } , \\overline { { A _ { 1 } } } ) \\sim \\overline { { \\pi } }$ is essentially equivalent to assuming that we have been running the algorithm for long enough to exit the initialization phase described in Section 4.1. ",
+        "bbox": [
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "4.2.2 Large deviation bound for the errors ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            494,
+            480,
+            511
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Our first have that $\\operatorname { e r r } _ { t }$ orrect aveand since value. More precisely, by is stationary it follows that $\\boxed { 4 . 1 }$ weus, ${ \\mathrm { l i m } } _ { T \\to \\infty } T ^ { - 1 } \\sum _ { t = 1 } ^ { T } { \\mathrm { e r r } } _ { t } \\ { \\overset { a . s . } { = } } \\alpha$ $\\operatorname { e r r } _ { t }$ $\\mathbb { E } [ \\mathsf { e r r } _ { t } ] = \\alpha$ to understand the deviation of T \u00001 PTt= from $\\alpha$ we simply need to characterize the dependence structure of $\\{ \\mathrm { e r r } _ { t } \\} _ { t \\in \\mathbb { N } }$ . ",
+        "bbox": [
+            173,
+            518,
+            826,
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "We accomplish this in Theorem $^ { 4 . 1 , }$ which gives a large deviation bound on $\\begin{array} { r } { | T ^ { - 1 } \\sum _ { t = 1 } ^ { T } \\mathrm { e r r } _ { t } - \\alpha | } \\end{array}$ The idea behind this result is to decompose the dependence in into two parts. First, there is dependence due to the fact that $\\alpha _ { t }$ is a function of $\\{ \\mathrm { e r r } _ { r } \\} _ { 1 \\leq r \\leq t - 1 }$ . In Section $\\boxed { \\mathbf { A } . 7 }$ in the Appendix we argue that this dependence induces a negative correlation and thus the errors concentrate around their expectation at a rate no slower than that of an i.i.d. Bernoulli sequence. This gives rise to the first term in $( 6 )$ , which is what would be obtained by applying Hoeffding’s inequality to an i.i.d. sequence. Second, there is dependence due to the fact that $A _ { t }$ depends on $A _ { t - 1 }$ . More specifically, consider a setting in which the distribution of $Y | X$ has more variability in some states than others. The goal of adaptive conformal inference is to adapt to the level of variability and thus return larger prediction sets in states where the distribution of $Y | X$ is more spread. However, this algorithm is not perfect and as a result there may be some states $a \\in { \\mathcal { A } }$ in which $\\mathbb { E } [ \\mathbf { e r r } _ { t } | A _ { t } = a ]$ is biased away from $\\alpha$ . Furthermore, if the environment tends to spend long stretches of time in more variable (or less variable) states this will induce a positive dependence in the errors and cause T \u00001 PTt=1 to deviate from $\\alpha$ . To control this dependence we use a Bernstein inequality for Markov chains to bound $\\begin{array} { r } { | T ^ { - 1 } \\sum _ { t = 1 } ^ { T } \\mathbb { E } [ \\mathrm { e r r } _ { t } | A _ { t } ] - \\alpha | } \\end{array}$ . This gives rise to the second term in $( 6 )$ ",
+        "bbox": [
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Theorem 4.1 Assume that $\\{ A _ { t } \\} _ { t \\in \\mathbb { N } }$ has non-zero absolute spectral gap $1 - \\eta > 0$ . Let ",
+        "bbox": [
+            171,
+            811,
+            748,
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "equation",
+        "img_path": "images/72a53d3376fd872ad6920dffe1d5f34c64921d3623add0a960e5deb3cc20d40d.jpg",
+        "text": "$$\nB : = \\operatorname* { s u p } _ { a \\in { \\mathcal { A } } } | \\mathbb { E } [ \\operatorname { e r r } _ { t } | A _ { t } = a ] - \\alpha | \\quad a n d \\quad \\sigma _ { B } ^ { 2 } : = \\mathbb { E } [ ( \\mathbb { E } [ \\operatorname { e r r } _ { t } | A _ { t } ] - \\alpha ) ^ { 2 } ] .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            271,
+            829,
+            722,
+            856
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Then, ",
+        "bbox": [
+            173,
+            858,
+            210,
+            871
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "equation",
+        "img_path": "images/4f418237109cce3ccc2245942d1ece16d07f348e7f20a6337f999b35ad45f399.jpg",
+        "text": "$$\n\\mathbb { P } \\left( \\left| \\frac { 1 } { T } \\sum _ { t = 1 } ^ { T } \\mathrm { e r r } _ { t } - \\alpha \\right| \\geq \\epsilon \\right) \\leq 2 \\exp \\left( - \\frac { T \\epsilon ^ { 2 } } { 8 } \\right) + 2 \\exp \\left( - \\frac { T ( 1 - \\eta ) \\epsilon ^ { 2 } } { 8 ( 1 + \\eta ) \\sigma _ { B } ^ { 2 } + 2 0 B \\epsilon } \\right) .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            223,
+            872,
+            772,
+            916
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "A formal proof of this result can be found in Section $\\boxed { \\mathbf { A . 7 } }$ The quality of this concentration inequality will depend critically on the size of the bias terms $B$ and $\\sigma _ { B } ^ { 2 }$ . Before proceeding, it is important that we emphasize that the definitions of $B$ and $\\sigma _ { B } ^ { 2 }$ are independent of the choice of $t$ owing to the fact that $\\left( \\alpha _ { t } , A _ { t } , \\mathbf { e r r } _ { t } \\right)$ is assumed stationary. Now, to understand these quantities, let ",
+        "bbox": [
+            173,
+            90,
+            825,
+            147
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "equation",
+        "img_path": "images/f7cf3d9e27d332cefdd05a2b713efccb1d0cf478472f7e4962522b325a459286.jpg",
+        "text": "$$\nM ( p | a ) : = \\mathbb { P } ( S ( X _ { t } , Y _ { t } ) > \\hat { Q } ( 1 - p ) | A _ { t } = a )\n$$",
+        "text_format": "latex",
+        "bbox": [
+            346,
+            152,
+            651,
+            172
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "denote the realized miscoverage level in state $a \\in { \\mathcal { A } }$ obtained by the quantile $\\hat { Q } ( 1 - p )$ . Assume that $M ( p | a )$ is continuous. This will happen for example when ${ \\hat { Q } } ( \\cdot )$ is continuous and ${ \\cal S } ( X _ { t } , Y _ { t } ) | A _ { t } = a$ is continuously distributed. Then, there exists an optimal value $\\alpha _ { a } ^ { * }$ such that $M ( \\alpha _ { a } ^ { * } | a ) = \\alpha$ . Lemma A.4 in the Appendix shows that if in addition $M ( \\cdot | a )$ admits a second order Taylor expansion, then ",
+        "bbox": [
+            173,
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "equation",
+        "img_path": "images/3731c424c09c8e0b2954227bcf5dc31e8af14fdc3b88d1d2d8e7a99bace1b6d0.jpg",
+        "text": "$$\nB \\leq C \\left( \\gamma + \\gamma ^ { - 1 } \\operatorname* { s u p } _ { a \\in \\mathcal { A } } \\operatorname* { s u p } _ { k \\in \\mathbb { N } } \\mathbb { E } [ \\left| \\alpha _ { A _ { t + 1 } } ^ { * } - \\alpha _ { A _ { t } } ^ { * } \\right| \\left| A _ { t + k } = a \\right| \\right) \\quad \\mathrm { a n d } \\quad \\sigma _ { B } ^ { 2 } \\leq B ^ { 2 } .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            245,
+            244,
+            751,
+            280
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Here, the constant $C$ will depend on how much $M ( \\cdot | a )$ differs from the ideal case in which ${ \\hat { Q } } ( \\cdot )$ is the true quantile function for ${ \\cal S } ( X _ { t } , Y _ { t } ) | A _ { t } = a$ . In this case we would have that $M ( \\cdot | a )$ is the linear function ${ \\bf \\bar { \\cal M } } ( p | a ) = p$ , $\\forall p \\in [ 0 , 1 ]$ and $C \\le 2$ . ",
+        "bbox": [
+            173,
+            287,
+            825,
+            332
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "We remark that the term $\\mathbb { E } [ | \\alpha _ { A _ { t + 1 } } ^ { * } - \\alpha _ { A _ { t } } ^ { * } | \\big | A _ { t + k } = a ]$ can be seen as a quantitative measurement of the size of the distribution shift in terms of the change in the critical value $\\alpha _ { a } ^ { * }$ . Thus, we interpret these results as showing that if the distribution shift is small and $\\forall a \\in { \\mathcal { A } }$ , ${ \\hat { Q } } ( \\cdot )$ gives reasonable coverage of the distribution of ${ \\cal S } ( X _ { t } , Y _ { t } ) | A _ { t } = a$ , then $T ^ { - 1 } \\sum _ { t = 1 } ^ { T } \\mathrm { e r r } _ { t }$ 2 A ·will concentrate well around . ",
+        "bbox": [
+            173,
+            337,
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "4.2.3 Achieving approximate marginal coverage ",
+        "text_level": 1,
+        "bbox": [
+            173,
+            429,
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Theorem $\\boxed { 4 . 1 }$ bounds the distance between the average miscoverage rate and the target level over long stretches of time. On the other hand, it provides no information about the marginal coverage frequency at a single time step. The following result shows that if the distribution shift is small, the realized marginal coverage rate $M ( \\alpha _ { t } | A _ { t } )$ will be close to $\\alpha$ on average. ",
+        "bbox": [
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Theorem 4.2 Assume that there exists a constant $L > 0$ such that for all $a \\in { \\mathcal { A } }$ and all $\\alpha _ { 1 } , \\alpha _ { 2 } \\in \\mathbb { R } ,$ , ",
+        "bbox": [
+            173,
+            518,
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "equation",
+        "img_path": "images/d719211f908a5b5ebe6d3a69d988982edd2b02bf5c9b4a8a2db87720378c2a33.jpg",
+        "text": "$$\n| M ( \\alpha _ { 2 } | a ) - M ( \\alpha _ { 1 } | a ) | \\leq L | \\alpha _ { 2 } - \\alpha _ { 1 } | .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            370,
+            540,
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Assume additionally that for all $a \\in { \\mathcal { A } }$ there exists $\\alpha _ { a } ^ { * } \\in ( 0 , 1 )$ such that $M ( \\alpha _ { a } ^ { * } | a ) = \\alpha .$ . Then, ",
+        "bbox": [
+            173,
+            561,
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "equation",
+        "img_path": "images/8c06b7befde5238365ee25ee63a810e19dee4b4984204c3c3975a82e646a6daf.jpg",
+        "text": "$$\n\\mathbb { E } [ ( M ( \\alpha _ { t } \\vert A _ { t } ) - \\alpha ) ^ { 2 } ] \\leq \\frac { L ( 1 + \\gamma ) } { \\gamma } \\mathbb { E } [ \\vert \\alpha _ { A _ { t + 1 } } ^ { \\ast } - \\alpha _ { A _ { t } } ^ { \\ast } \\vert ] + \\frac { L } { 2 } \\gamma .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            302,
+            582,
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+            616
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Once again we emphasize that $\\textcircled{7}$ holds for any choice of $t$ owing to the fact that $\\left( \\alpha _ { t } , A _ { t } , \\mathbf { e r r } _ { t } \\right)$ is assumed stationary and thus the quantities appearing in the bound are invariant across $t$ . Proof of this result can be found in Section $\\mathbf { \\bar { A . 8 } }$ of the Appendix. We remark that the right-hand side of $\\textcircled { 7 }$ is minimized by choosing $\\gamma = ( 2 \\mathbb { E } [ | \\overline { { \\alpha _ { A _ { t + 1 } } ^ { * } } } - \\alpha _ { A _ { t } } ^ { * } | ] ) ^ { 1 / 2 }$ , which gives the inequality ",
+        "bbox": [
+            173,
+            627,
+            826,
+            688
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "equation",
+        "img_path": "images/e2f98d209c6888dcca3a153c57bea23aaab0499e0211f9be432638529cec93b9.jpg",
+        "text": "$$\n\\mathbb { E } [ ( M ( \\alpha _ { t } | A _ { t } ) - \\alpha ) ^ { 2 } ] \\leq L ( \\sqrt { 2 } + 1 ) \\sqrt { \\mathbb { E } [ | \\alpha _ { A _ { t + 1 } } ^ { * } - \\alpha _ { A _ { t } } ^ { * } | ] } .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            310,
+            694,
+            686,
+            722
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "As above we have that in the ideal case ${ \\hat { Q } } ( \\cdot )$ is a perfect estimate of the quantiles of ${ \\cal S } ( X _ { t } , Y _ { t } ) | A _ { t } = a$ and thus $M ( p | a ) = p$ and $L = 1$ . Moreover, we once again have the interpretation that $\\mathbb { E } [ | \\alpha _ { A _ { t + 1 } } ^ { * } -$ $\\alpha _ { A _ { t } } ^ { * } | ]$ is a quantitative measurement of the distribution shift. Thus, this result can be interpreted as bounding the average difference between the realized and target marginal coverage in terms of the size of the underlying distribution shift. Finally, note that the choice $\\gamma = ( 2 \\mathbb { E } [ | \\alpha _ { A _ { t + 1 } } ^ { * } - \\alpha _ { A _ { t } } ^ { * } | ] ) ^ { 1 / 2 }$ formalizes our intuition that $\\gamma$ should be chosen to be larger in domains with greater distribution shift, while not being so large as to cause $\\alpha _ { t }$ to be overly volatile. ",
+        "bbox": [
+            173,
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "5 Impact of $S _ { t } ( \\cdot )$ on the performance ",
+        "text_level": 1,
+        "bbox": [
+            173,
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "The performance of all conformal inference methods depends heavily on the design of the conformity score. Previous work has shown how carefully chosen scores or even explicit optimization of the interval width can be used to obtain smaller prediction sets [e.g. 28, 29, 20, 12]. Adaptive conformal inference can work with any conformity score $S _ { t } ( \\cdot )$ and quantile function $\\hat { Q } _ { t } ( \\cdot )$ and thus can be directly combined with other improvements in conformal inference to obtain shorter intervals. One important caveat here is that the lengths of conformal prediction sets depend directly on the quality of the fitted regression model. Thus, to obtain smaller intervals one should re-fit the model at each time step using the most recent data to build the most accurate predictions. This is exactly what we have done in our experiments in Sections $2 . 2$ and 6. ",
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+        "type": "image",
+        "img_path": "images/3a880b81d462e11cd1f20f8e8ab2d3c46e21a0e890896e590d96462c030ea136.jpg",
+        "image_caption": [
+            "Figure 2: Local coverage frequencies for adaptive conformal (blue), a non-adaptive method that holds $\\alpha _ { t } = \\alpha$ fixed (red), and an i.i.d. Bernoulli(0.1) sequence (grey) for the prediction of stock market volatility with conformity score $\\tilde { S } _ { t }$ . The coloured dotted lines mark the average coverage obtained across all time points, while the black line indicates the target level of $1 - \\alpha = 0 . 9$ . "
+        ],
+        "image_footnote": [],
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+    {
+        "type": "text",
+        "text": "In addition to this, the choice of $S _ { t } ( \\cdot )$ can also have a direct effect on the coverage properties of adaptive conformal inference. Theorems $\\boxed { 4 . 1 }$ and $4 . 2$ show that the performance of adaptive conformal inference is controlled by the size of the shift in the optimal parameter $\\alpha _ { t } ^ { * }$ across time. Moreover, $\\alpha _ { t } ^ { * }$ itself is in one-to-one correspondence with the $1 - \\alpha$ quantile of $S _ { t } ( X _ { t } , Y _ { t } )$ . Thus, the coverage properties of adaptive conformal inference depend on how close $S _ { t } ( X _ { t } , Y _ { t } )$ is to being stationary. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "For a simple example illustrating the impact of this dependence, note that in Section $\\boxed { 2 . 2 }$ we formed prediction sets using the conformity score ",
+        "bbox": [
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+    {
+        "type": "equation",
+        "img_path": "images/86d02443f3a6a5ef64d172834783b068202c788027becc99e87b378469a2356c.jpg",
+        "text": "$$\nS _ { t } : = \\frac { | V _ { t } - \\hat { \\sigma } _ { t } ^ { 2 } | } { \\hat { \\sigma } _ { t } ^ { 2 } } .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 8
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+    {
+        "type": "text",
+        "text": "An a priori reasonable alternative to this is the unnormalized score ",
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "equation",
+        "img_path": "images/a783dad874d8b4b59525940c9a7f2e6c9039106b3a191673bb38c625021b7d71.jpg",
+        "text": "$$\n\\tilde { S } _ { t } : = | V _ { t } - \\hat { \\sigma } _ { t } ^ { 2 } | .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            442,
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "However, after a more careful examination it becomes unsurprising that normalization by $\\hat { \\sigma } _ { t } ^ { 2 }$ is critical for obtaining an approximately stationary conformity score and thus $\\tilde { S } _ { t }$ leads to much worse coverage properties. Figure $2$ shows the local coverage frequency (see $( 4 )$ ) of adaptive conformal inference using $\\tilde { S } _ { t }$ . In comparison to Figure $^ 1$ the coverage now undergoes much wider swings away from the target level of 0.9. This issue can be partially mitigated by choosing a larger value of $\\gamma$ that gives greater adaptivity to the algorithm. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "6 Real data example: election night predictions ",
+        "text_level": 1,
+        "bbox": [
+            173,
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+        "page_idx": 8
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+    {
+        "type": "text",
+        "text": "During the 2020 US presidential election The Washington Post used conformalized quantile regression (CQR) (see $\\mathbb { \\underline { { \\left( \\mathrm { 1 } \\right) } } }$ and Section $\\boxed { 1 . 1 }$ to produce county level predictions of the vote total on election night [13]. Here we replicate the core elements of this method using both fixed and adaptive quantiles. ",
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+    {
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+        "img_path": "images/c0d7449ad604c682f9a8bf69819c9ade50744b6d0f32c4c16d3e878e7ead701f.jpg",
+        "image_caption": [
+            "Figure 3: Local coverage frequencies of adaptive conformal (blue), a non-adaptive method that holds $\\alpha _ { t } = \\alpha$ fixed (red), and an i.i.d. Bernoulli(0.1) sequence (grey) for county-level election predictions. Coloured dotted lines show the average coverage across all time points, while the black line indicates the target coverage level of $1 - \\alpha = 0 . 9$ . "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            197,
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+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "To make the setting precise, let $\\{ Y _ { t } \\} _ { 1 \\leq t \\leq T }$ denote the number of votes cast for presidential candidate Joe Biden in the 2020 election in each of approximately $T = 3 0 0 0$ counties in the United States. Let $X _ { t }$ denote a set of demographic covariates associated to the tth county. In our experiment $X _ { t }$ will include information on the make-up of the county population by ethnicity, age, sex, median income and education (see Section $\\boxed { \\overline { { \\mathrm { A } . 6 . 1 } } }$ for details). On election night county vote totals were observed as soon as the vote count was completed. If the order in which vote counts completed was uniformly random $\\{ ( X _ { t } , Y _ { t } ) \\} _ { 1 \\leq t \\leq T }$ would be an exchangeable sequence on which we could run standard conformal inference methods. In reality, larger urban counties tend to report results later than smaller rural counties and counties on the east coast of the US report earlier than those on the west coast. Thus, the distribution of $( X _ { t } , Y _ { t } )$ can be viewed as drifting throughout election night. ",
+        "bbox": [
+            173,
+            431,
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+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "We apply CQR to predict the county-level vote totals (see Section A.6.2 for details). To replicate the east to west coast bias observed on election night we order the counties by their time zone with eastern time counties appearing first and Hawaiian counties appearing last. Within each time zone counties are ordered uniformly at random. Figure $3$ shows the realized local coverage frequency over the most recent 300 counties (see $\\textcircled{4}$ ) for the non-adaptive and adaptive conformal methods. We find that the non-adaptive method fails to maintain the desired $9 0 \\%$ coverage level, incurring large troughs in its coverage frequency during time zone changes. On the other hand, the adaptive method maintains approximate $9 0 \\hat { \\% }$ coverage across all time points with deviations in its local coverage level comparable to what is observed in Bernoulli sequences. ",
+        "bbox": [
+            173,
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+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "7 Discussion ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            722,
+            292,
+            738
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "There are still many open problems in this area. The methods we develop are specific to cases where $Y _ { t }$ is revealed at each time point. However, there are many settings in which we receive the response in a delayed fashion or in large batches. In addition, our theoretical results in Section $\\boxed { 4 . 2 }$ are limited to a single model for the data generating distribution and the special case where the quantile function $\\hat { Q } _ { t } ( \\cdot )$ is fixed across time. It would be interesting to determine if similar results can be obtained in settings where $\\hat { Q } _ { t } ( \\cdot )$ is fit in an online fashion on the most recent data. Another potential area for improvement is in the choice of the step size $\\gamma$ . In Section $\\boxed { 2 . 1 }$ we give some heuristic guidelines for choosing $\\gamma$ based on the size of the distribution shift in the environment. Ideally however we would like to be able to determine $\\gamma$ adaptively without prior knowledge. Finally, our experimental results are limited to just two domains. Additional work is needed to determine if our methods can successfully protect against a wider variety of real-world distribution shifts. ",
+        "bbox": [
+            173,
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+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "8 Acknowledgements ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            89,
+            367,
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+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "E.C. was supported by Office of Naval Research grant N00014-20-12157, by the National Science Foundation grants OAC 1934578 and DMS 2032014, by the Army Research Office (ARO) under grant W911NF-17-1-0304, and by the Simons Foundation under award 814641. We thank John Cherian for valuable discussions related to Presidential Election Night 2020 and Lihua Lei for helpful comments on the connection to online learning. ",
+        "bbox": [
+            174,
+            119,
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+    },
+    {
+        "type": "text",
+        "text": "References ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            208,
+            267,
+            224
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
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parse/train/BJlVeyHFwH/BJlVeyHFwH.md

@@ -0,0 +1,613 @@
+# ON THE INVERTIBILITY OF INVERTIBLE NEURAL NETWORKS
+
+Anonymous authors Paper under double-blind review
+
+# ABSTRACT
+
+Guarantees in deep learning are hard to achieve due to the interplay of flexible modeling schemes and complex tasks. Invertible neural networks (INNs), however, provide several mathematical guarantees by design, such as the ability to approximate non-linear diffeomorphisms. One less studied advantage of INNs is that they enable the design of bi-Lipschitz functions. This property has been used implicitly by various works to design generative models, memory-saving gradient computation, regularize classifiers, and solve inverse problems.
+
+In this work, we study Lipschitz constants of invertible architectures in order to investigate guarantees on stability of their inverse and forward mapping. Our analysis reveals that commonly-used INN building blocks can easily become noninvertible, leading to questionable “exact” log likelihood computations and training difficulties. We make use of numerical analysis tools to diagnose non-invertibility in practice. Finally, based on our theoretical analysis, we show how to guarantee numerical invertibility for one of the most common INN architectures.
+
+# 1 INTRODUCTION
+
+Invertible neural networks (INNs) have become a standard building block in the deep learning toolkit. Invertibility is useful for training generative models with exact likelihoods (Dinh et al., 2014; 2017; Kingma & Dhariwal, 2018; Kingma et al., 2016; Behrmann et al., 2019; Chen et al., 2019), increasing posterior flexibility in VAEs (Rezende & Mohamed, 2015; Tomczak & Welling, 2016; Papamakarios et al., 2017), learning transition operators in MCMC samplers (Song et al., 2017; Levy et al., 2017), computing memory-efficient gradients (Gomez et al., 2017; Donahue & Simonyan, 2019), allowing for bi-directional training (Grover et al., 2018), solving inverse problems (Ardizzone et al., 2019) and analysing adversarial robustness (Jacobsen et al., 2019).
+
+The application space of INNs is rapidly growing and many approaches for constructing invertible architectures have been proposed. A common way to construct invertible networks is to use triangular coupling layers (Dinh et al., 2014; 2017; Kingma & Dhariwal, 2018), where dimension partitioning is interleaved with ResNet-type computation. Another approach is to use various forms of masked convolutions, generalizing the dimension partitioning approach of coupling layers (Song et al., 2019; Hoogeboom et al., 2019). To avoid dimension partitioning altogether, multiple approaches based on efficiently estimating the log-determinant of the Jacobian, necessary for applying the change of variable formula, have been proposed to allow for free-form Jacobian structure (Grathwohl et al., 2019; Behrmann et al., 2019; Chen et al., 2019).
+
+From a mathematical perspective, invertible architectures enable several unique guarantees like:
+
+• Enabling flexible approximation of non-linear diffeomorphisms (Rezende & Mohamed, 2015; Dinh et al., 2017; Kingma & Dhariwal, 2018; Chen et al., 2019)   
+Memory-saving gradient computation (Gomez et al., 2017; Donahue & Simonyan, 2019)   
+Fast analytical invertibility (Dinh et al., 2014)   
+Guaranteed preservation of mutual information and exact access to invariants of deep networks (Jacobsen et al., 2018; 2019).
+
+Despite the increased interest in invertible neural networks, little attention has been paid to guarantees on their numerical invertibility. Specifically, this means analyzing their ability to learn bi-Lipschitz neural networks, i.e. Lipschitz continuous neural networks with a bound on the Lipschitz constant of the forward and inverse mapping.
+
+While the stability analysis of neural networks has received significant attention e.g. due to adversarial examples (Szegedy et al., 2013), the focus here is only on bounding Lipschitz constants of the forward mapping. However, bounding the Lipschitz constant of the inverse mapping is of major interest, e.g. when reconstructing inputs from noisy or imprecise features. In fact, analytical invertibility as provided by some invertible architectures does not necessarily imply numerical invertibility in practice.
+
+In this paper, we first discuss the relevance of controlling the bi-Lipschitz bounds of invertible networks. Afterwards we analyze Lipschitz bounds of commonly used invertible neural network building blocks. Our contributions are:
+
+• We argue for forward and inverse stability analysis as a unified viewpoint on invertible network (non-)invertibility. To this end, we derive Lipschitz bounds of commonly-used invertible building blocks for their forward and inverse maps.   
+We numerically monitor and detect (non-)invertibility for different practical tasks such as classification and generative modeling. We show how this overlooked issue with non-invertibility can lead to questionable claims when computing exact likelihoods with the change-of-variable formula.   
+• Finally, we study spectral normalization as a stabilizer for one of the most commonly-used family of INN architectures, namely additive coupling blocks.
+
+# 2 BACKGROUND AND MOTIVATION
+
+Invertible neural networks are bijective functions with a parametrized forward mapping $F _ { \theta } : \mathbb { R } ^ { d }  \mathbb { R } ^ { d }$ with $F _ { \theta } : x \mapsto z$ , where $\theta \in \mathbb { R } ^ { p }$ defines the parameter vector. Additionally, they define an inverse mapping $F _ { \theta } ^ { - 1 } : \mathbb { R } ^ { d }  \mathbb { R } ^ { d }$ with $F _ { \theta } ^ { - 1 } : z \mapsto { \dot { x } }$ . This inverse can be given in closed-form (analytical inverse, e.g. Dinh et al. (2017); Kingma & Dhariwal (2018)) or approximated numerically (numerical inverse, e.g. Behrmann et al. (2019); Song et al. (2019)).
+
+Before we discuss building blocks of invertible networks, we provide some background and motivation for studying forward and inverse stability.
+
+Definition 1 (Lipschitz and bi-Lipschitz continuity). A function $F : ( \mathbb { R } ^ { d _ { 1 } } , \| \cdot \| ) \to ( \mathbb { R } ^ { d _ { 2 } } , \| \cdot \| )$ is called Lipschitz continuous if there exists a constant $L = : \operatorname { L i p } ( F )$ such that
+
+$$
+\| F ( x _ { 1 } ) - F ( x _ { 2 } ) \| \leq L \| x _ { 1 } - x _ { 2 } \| , \quad \forall x _ { 1 } , x _ { 2 } \in \mathbb R ^ { d _ { 1 } } .
+$$
+
+If an inverse $F ^ { - 1 } : ( \mathbb { R } ^ { d _ { 2 } } , \lVert \cdot \rVert ) \to ( \mathbb { R } ^ { d _ { 1 } } , \lVert \cdot \rVert )$ and a constant $L ^ { * } = : \operatorname { L i p } ( F ^ { - 1 } )$ exists such that
+
+$$
+\| F ^ { - 1 } ( y _ { 1 } ) - F ^ { - 1 } ( y _ { 2 } ) \| \leq L ^ { * } \| y _ { 1 } - y _ { 2 } \| , \quad \forall y _ { 1 } , y _ { 2 } \in \mathbb { R } ^ { d _ { 2 } } ,
+$$
+
+then $F$ is called bi-Lipschitz continuous.
+
+Remark 2. We focus on invertible functions $F : ( \mathbb { R } ^ { d } , \| \cdot \| _ { 2 } ) \to ( \mathbb { R } ^ { d } , \| \cdot \| _ { 2 } ) ,$ , i.e. functions where the domain and co-domain are of the same dimensionality $d$ and the norm is given by the euclidian norm.
+
+Lemma 3. (Rademacher (Federer, 1969, Theorem 3.1.6))   
+If $F : \mathbb { R } ^ { d }  \mathbb { R } ^ { d }$ is a locally Lipschitz continuous function (i.e. functions whose restriction to a neighborhood around any point is Lipschitz), then $F$ is differentiable almost everywhere. Moreover, $i f F$ is Lipschitz continuous, then
+
+$$
+\mathrm { L i p } ( F ) = \operatorname* { s u p } _ { x \in \mathbb { R } ^ { d } } \| J _ { F } ( x ) \| _ { 2 } ,
+$$
+
+where $J _ { F } ( x )$ is the Jacobian matrix of $F$ at $x$ and $\| J _ { F } ( x ) \| _ { 2 }$ denotes its spectral norm.
+
+Lipschitz bounds on the forward mapping are of crucial importance in several areas, including in adversarial example research (Szegedy et al., 2013), to avoid exploding gradients, or the training of Wasserstein GANs (Anil et al., 2019). The stability of the inverse, however, can have a similar impact. For instance, having a Lipschitz bound on the inverse may avoid vanishing gradients during training.
+
+Given that deep-learning computations are carried out with limited precision, imprecision is always introduced in both the forward and backward passes, i.e., $z ^ { \delta } = { \bf { \dot { F } } } ( x ) + \delta$ and $\hat { x } ^ { \delta } = F ^ { - 1 } ( z ^ { \check { \delta } } )$ . Instability in either pass will aggravate this problem, and essentially make the invertible network numerically non-invertible. To summarize, this problem occurs in the following situations:
+
+• Numerical reconstruction of $x$ , where features $z ^ { \delta }$ are inexact due to limited precision (e.g. when computations are executed in single precision as common on modern hardware).   
+Reconstruction based on imprecise measurements from physical devices (e.g. when using invertible networks for inverse problems (Ardizzone et al., 2019)).   
+• Numerical re-computation of intermediate activations of the neural network to allow for memory-efficient backpropagation (Gomez et al., 2017).
+
+Furthermore, some computations are performed via numerical approximation, which in turn adds another source of imprecision that might be aggravated via instability. Examples include:
+
+• Numerical forward computation, as in Neural ODEs (Chen et al., 2018) (numerical solver is used to approximate dynamic of ODE).   
+Numerical inverse computation, e.g. via fixed-point iterations as in i-ResNets (Behrmann et al., 2019) or MintNet (Song et al., 2019) or via ODE-solvers for the backward dynamics as in Neural ODEs (Chen et al., 2018).
+
+As an example of why bi-Lipschitz continuity is critical for numerical stability in invertible functions, let’s consider the simple mappings $F _ { 1 } ( x ) \stackrel { \bullet } { = } \log ( x ) , F _ { 1 } ^ { - 1 } ( z ) = \exp ( z )$ , and $F _ { 2 } ( x ) = x , F _ { 2 } ^ { - 1 } ( z ) =$ $z$ . Though both functions tend to infinity when $x  \infty$ , $F _ { 1 }$ is much less stable. Consider the introduction of numerical imprecision as $z ^ { \delta } = F _ { 1 } ( x ) + \delta$ where $\delta$ denotes the introduced imprecision. Then this imprecision is magnified in the inverse pass as:
+
+$$
+| | F _ { 1 } ^ { - 1 } ( z ) - F _ { 1 } ^ { - 1 } ( z ^ { \delta } ) | | _ { 2 } ^ { 2 } \approx | | \delta \frac { \partial F _ { 1 } ^ { - 1 } ( z ^ { \delta } ) } { \partial z ^ { \delta } } | | _ { 2 } ^ { 2 } = | | \delta \exp ( z ^ { \delta } ) | | _ { 2 } ^ { 2 } .
+$$
+
+A similar example can be constructed for both the forward and backward passes, which speaks to the importance of bi-Lipschitz continuity. For an additional discussion on the connection of Lipschitz constants and numerical errors, we refer to Appendix $\mathbf { B }$ .
+
+# 3 STABILITY OF INVERTIBLE NEURAL NETWORKS
+
+# 3.1 LIPSCHITZ BOUNDS FOR BUILDING BLOCKS OF INVERTIBLE NETWORKS
+
+Research on invertible networks has produced a large variety of architectural building blocks. Yet, the focus of prior work was on obtaining flexible architectures while maintaining invertibility guarantees. Here, we build on the work in (Behrmann et al., 2019), where bi-Lipschitz bounds were proven for invertible ResNets, by deriving Lipschitz bounds on the forward and inverse mapping of common building blocks. Together with an overview of common invertible building blocks, we provide our main results in Table 1. We chose these particular model classes in order to cover both coupling-based approaches and free-form approaches like Neural ODE (Chen et al., 2018) and i-ResNets (Behrmann et al., 2019). The derivations of the bounds are given in Appendix A. Note that the bounds provide the worst-case stability and serve mainly as a guideline for future designs of invertible building blocks.
+
+# 3.2 CONTROLLING STABILITY OF BUILDING BLOCKS
+
+As shown in Table 1, there are many factors that influence the stability of INNs. Of particular importance are the Lipschitz constants $\operatorname { L i p } ( g )$ of the sub-network $g$ for i-ResNets (Behrmann et al., 2019) and affine coupling blocks (Dinh et al., 2014), and $\mathrm { L i p } ( s ) , \mathrm { L i p } ( t )$ for additive coupling blocks (Dinh et al., 2017). Whereas computing the Lipschitz constants of neural networks is NP-hard (Virmaux & Scaman, 2018), there is a simple data-independent upper bound:
+
+$$
+\mathrm { L i p } ( g ) \leq \prod _ { i = 1 } ^ { L } \| A _ { i } \| _ { 2 } , \quad \mathrm { f o r } \quad g ( x ) = A _ { L } \circ \phi \circ A _ { L - 1 } \circ \cdot \cdot \circ A _ { 2 } \circ \phi \circ A _ { 1 } ,
+$$
+
+Table 1: Lipschitz bounds on building blocks of invertible neural networks. The second column shows the operations of the forward mapping and the last two columns show bounds on the Lipschitz constant of the forward and inverse mapping. $M$ in the row for the forward mapping of an affine block is defined as ${ \cal M } = \operatorname* { m a x } ( | a | , | b | ) \cdot \bar { c } _ { g ^ { \prime } } \cdot \bar { \mathrm { L i p } } ( s ) + \mathrm { L i p } ( t )$ . Furthermore, $M ^ { * }$ for the inverse of an affine block is $M ^ { * } = \operatorname* { m a x } ( | a ^ { * } | , | b ^ { * } | ) \cdot c _ { ( \frac { 1 } { g } ) ^ { \prime } } \cdot \mathrm { L i p } ( s ) + c _ { \left( \frac { 1 } { g } \right) ^ { \prime } } \cdot \mathrm { L i p } ( s ) \cdot c _ { t } + c _ { \frac { 1 } { g } } \cdot \mathrm { L i p } ( t ) .$ . Note that the bounds of the affine blocks hold only locally. Derivations of the bounds are given in Appendix A.   
+
+<table><tr><td>Building Block</td><td>Forward Operation</td><td>Lipschitz Forward</td><td>Lipschitz Inverse</td></tr><tr><td>Additive Coupling Block (Dinh et al., 2014)</td><td>F(x)1=x1 F(x)1=x1+g(x1）</td><td>≤1+Lip(g)</td><td>≤1+Lip(g)</td></tr><tr><td>Affine Coupling Block (Dinh et al., 2017)</td><td>F(x)1=x1 F(x)1=x1①g(s(𝑥1))+t(𝑥1) g()≠0</td><td>≤max(1,cg)+M local for x ∈ [a,b]d g(x）≤cg</td><td>≤max(1,c1)+M* local for y ∈[a*,b*]d g (≤</td></tr><tr><td>Invertible Residual Layer (Behrmann et al.,2019)</td><td>F(x)=x+g(x) Lip(g)&lt;1</td><td>≤1+Lip(g)</td><td>≤1-Lip(@) 1</td></tr><tr><td>Neural ODE (Chen et al.,2018)</td><td>d(t）=F(x(t）,t） dt t∈[0,T]</td><td>≤eLip(F).t</td><td>≤eLip(F).t</td></tr><tr><td>Invertible Downsampling ／Squeeze (Dinh et al., 2017)</td><td>F(x)=Px P permutation</td><td>=1</td><td>=1</td></tr><tr><td>Diagonal Scaling (Dinh et al., 2014) ActNorm</td><td>F(x)=Dx D diagonal</td><td>= maxi|Diil</td><td>miniDii</td></tr><tr><td>(Kingma &amp; Dhariwal,2018) Invertible 1×1 Convolution (Kingma &amp; Dhariwal,2018)</td><td>D≠0 F(x)=PL(U+diag(s))=:W P permutation,L lower-triangular U upper-triangular,s ∈Rd</td><td>≤|W|2</td><td>≤ |W-1|2</td></tr></table>
+
+where $A _ { i }$ are linear layers, $\| \cdot \| _ { 2 }$ is the spectral norm and $\phi$ a contractive activation function $( \mathrm { L i p } ( \phi ) \leq 1 )$ ). The above bound was used by (Behrmann et al., 2019) in conjunction with spectral normalization (Miyato et al., 2018; Gouk et al., 2018) to ensure a contractive residual block $g$ . In particular, this employs a normalization via:
+
+$$
+\tilde { A } = \kappa \frac { A } { \hat { \sigma } _ { 1 } } , \quad \mathrm { w i t h } \quad \hat { \sigma } _ { 1 } \approx \sigma _ { 1 } = \| A \| _ { 2 } \quad ( \mathrm { a p p r o x . \ v i a \ p o w e r { - } m e t h o d } ) ,
+$$
+
+where $\kappa > 0$ is a coefficient that sets the approximate upper bound on the spectral norm of each linear layer $A _ { i }$ . Thus, by setting an appropriate coefficient $\kappa$ depending on the targeted Lipschitz bound of the building block, this approach enables one to control both forward and inverse stability. Note that the above discussion can be generalized to other $\ell _ { p }$ -norms, see (Chen et al., 2019).
+
+However, this is not sufficient when using affine coupling blocks because their bound on the Lipschitz constant holds only locally. In particular, it depends on the regions of the inputs $x$ to the coupling block. While inputs to the first layer are usually bounded by the nature of the data, obtaining bounds for intermediate activations is less straightforward. One interesting avenue for future work could be local regularizers like gradient penalties (Gulrajani et al., 2017), where spectral normalization could be used post-hoc to certify stability.
+
+Lastly, we use ActNorm (Kingma & Dhariwal, 2018) in several architectures and avoid small diagonal terms which would yield large Lipschitz constants in the inverse (see Table 1) by adding a positive constant. Further stabilization could be achieved via bounding the scaling.
+
+# 4 NUMERICAL EXPERIMENTS
+
+In this section, we study numerical invertibility for several objectives and architecture settings. This section is structured by task:
+
+1. Classification: we show that INN classifiers can become non-invertible on CIFAR-10 and discuss consequences for memory-efficient backpropagation.
+
+2. Density estimation: we analyze the numerical invertibility of SOTA trained density models.
+
+3. Generative modeling: we study the stability of adversarially trained INN generators, discuss the consequences for likelihood evaluation, and stabilize an additive-coupling based INN generator using spectral normalization.
+
+4. Decorrelation: we perform an in-depth study of the effect of different architecture settings on a simple task, where both stable and unstable solutions are possible. Furthermore, we show that spectral normalization is effective at stabilizing additive-coupling based flows.
+
+We use the following measures to diagnose the numerical instability of invertible models:
+
+• Reconstruction error. We measure the $\ell _ { 2 }$ -distance between the input $x$ and its reconstruction, i.e. $| | x ^ { ( i ) } - F _ { \theta } ^ { - 1 } ( F _ { \theta } ( x ^ { ( i ) } ) ) | | _ { 2 }$ . • Conditioning of the Jacobian and max/min singular values. For forward stability, we are interested in the behavior of the Jacobian $J _ { F } ( x )$ , while for inverse stability we are interested in the Jacobian of the inverse mapping $J _ { F ^ { - 1 } } ( x )$ . We compute the singular values of the Jacobians using the SVD, which allows us to compute its condition number. 1
+
+While the reconstruction error allows us to quantitatively monitor non-invertibility even before reconstruction artifacts are perceptible, the linear approximation $J _ { F } ( x )$ and its singular values provide insights into unstable directions of the forward (very large singular values of $J _ { F } ( x ) )$ and inverse (very small singular values of $J _ { F } ( x ) )$ mapping. Both measures were also used in (Jacobsen et al., 2018), where an ill-conditioned inverse was observed.
+
+# 4.1 CLASSIFICATION WITH INVERTIBLE MODELS
+
+In this section, we show that when training an INN for classification, there exist both stable and unstable solutions. We compare a stable model that uses additive coupling and an unstable model that uses affine coupling and ActNorm both inside and between the blocks—for additional experimental details see Appendix C. These models achieve similar test accuracies of $9 0 . 2 \%$ and $9 0 . 5 \%$ , respectively (Figure 6, Appendix C); however, we note that the goal of this experiment is not to achieve SOTA accuracy on CIFAR-10. Rather, we aim to show that models trained for classification with reasonable accuracy, can vary greatly with respect to stability. To observe the differences in stability, we plot the reconstruction results in Figure 2.
+
+An important use-case of INNs is to enable memoryefficient training, by re-computing activations in the backward pass rather than storing them in memory during the forward pass (Gomez et al., 2017). This approach enables e.g. large-scale generative modeling (Donahue & Simonyan, 2019) and scaling segmentation networks to high-resolution medical images (Brugger et al., 2019). ¨
+
+![](images/dfb05d9964d1ce3f73bc4858a4a168d46174d6f5c4ef1a99b209d30870609c49.jpg)  
+Figure 1: Impact of stability on memory-efficient training. We measure the angle between the true gradient (using stored activations) and the memory-saving gradient (using recomputing activations). For all epochs after the dotted vertical line, the affine model had numerically infinite or nan gradients, while the additive model gives accurate memory-efficient gradients.
+
+Re-computing the activations, however, relies on a numerically precise inverse mapping. To better understand the effect of numerical errors, we perform an analysis similar to Gomez et al. (2017): we track the angle between the true and memory-saving gradients during training (Figure 1). As expected from the reconstruction results, we observe that the affine model yields gradients very different from the true gradient; in fact, after approximately 20 epochs of training, the memory-saving gradients of the affine model contain numerically infinite or nan values. Thus, it would not be possible to train the affine model successfully using memory-saving gradients.
+
+![](images/0484406ee1d4e5a9b1025b473a8fd88c55c9bfbc28a9043a1075941d4ce97ada.jpg)  
+Figure 2: Measuring stability of additive and affine INN classifiers on CIFAR-10. Right: we track the reconstruction errors for a fixed minibatch over the course of training. In each iteration (vertical slice) we plot the errors of each minibatch element. Orange points in the affine plot denote inf and nan values. c denotes the condition number of the Jacobian, $\sigma _ { m a x }$ and $\sigma _ { m i n }$ denote the max and min singular values. At the top, we show the original images $x$ and in each snapshot, the reconstructions $\bar { x } = F ^ { - 1 } ( F ( x ) )$ and reconstruction errors $R = | x - { \hat { x } } |$ .
+
+# 4.2 ANALYZING STATE-OF-THE-ART DENSITY MODELS
+
+![](images/5fe3592eeeaec0088a9d1e651d1cc8d01f62b4f9e1f8829bd826d2751b78bd61.jpg)  
+Figure 3: Crafting non-invertible inputs for a CIFAR-10 Glow model. For three different images, we show: 1) the original datapoint $x$ from which we start running PGD; 2) the crafted input $x ^ { \prime }$ that results from PGD; and 3) the reconstruction ${ \hat { x } } ^ { \prime } = F ^ { - 1 } ( F ( x ^ { \prime } ) )$ of the crafted input. In all cases, the reconstructions of adversarial inputs are heavily corrupted, indicating that the attacks were successful, and exposing non-invertibility in this Glow model.
+
+In this section, we analyze the invertibility of trained density models. In particular, we expose non-invertibility in models that otherwise appear stable by optimizing in input space to find examples that are poorly reconstructed by the model. Here, we take a trained Glow model (Kingma $\&$ Dhariwal, $2 0 1 8 ) ^ { 2 }$ and optimize the input using Projected Gradient Descent (PGD) (Madry et al., 2018). Our goal is to find a point $x ^ { \prime }$ in the domain of the invertible model such that the reconstruction $F ^ { - 1 } ( F ( x ^ { \prime } ) )$ differs from $x ^ { \prime }$ . In particular, we start with a datapoint $x$ and use PGD to find a perturbed example $x ^ { \prime }$ that has high reconstruction error via Eq. 3 (additional details in Appendix E).
+
+$$
+\underset { | | x ^ { \prime } - x | | _ { \infty } \leq \epsilon } { \arg \operatorname* { m a x } } | | x ^ { \prime } - F ^ { - 1 } ( F ( x ^ { \prime } ) ) | | _ { 2 } .
+$$
+
+As shown in Figure 3, this attack is effective for finding examples that are perceptually identical to test examples, yet induce large reconstruction errors. This attack can be understood as a worstcase invertibility diagnosis; however, we note that unsuccessful attacks can be due to algorithmic issues and thus do not necessarily imply stable invertible models. Also, it is not always clear how to get gradients that are able to exploit numerical instabilities, leaving room for improvement via gradient-free methods (e.g., the boundary attack from Brendel et al. (2018)).
+
+We present in Appendix E additional experiments using the PGD attack on the additive Glow model from (Kingma & Dhariwal, 2018) trained on CelebA, where we show that the model becomes non-invertible outside the valid input range of images. Furthermore, we analyze a trained residual flow model (Chen et al., 2019) on CIFAR-10, where we cannot find such dramatic non-invertible inputs, which is to be expected given the stability bounds of i-ResNet blocks (see Table 1).
+
+# 4.3 GENERATIVE MODELING WITH INVERTIBLE MODELS
+
+Generative models based on invertible networks have been predominantly trained using maximum likelihood estimation (MLE). Another viable approach is to train them adversarially (ADV), as done in Flow-GAN (Danihelka et al., 2017; Grover et al., 2018). Flow-GAN is appealing as it can result in a generator capable of producing high-quality samples (as in GANs), while also giving access to exact density estimates, which GANs lack. Prior work (Danihelka et al., 2017; Grover et al., 2018) has compared these two techniques for training flows (MLE vs ADV); the main conclusion of these studies was that training with MLE yields good likelihoods but relatively poor samples, while training with a GAN loss yields good samples but likelihoods orders of magnitude worse than MLE training.
+
+<table><tr><td></td><td>MLE Affine -</td><td>MLE Additive</td><td>ADV Additive Unstable</td><td>ADV Additive Stable</td></tr><tr><td>(Test) BPD</td><td>1.22</td><td>1.38</td><td>3.87e12*</td><td>709</td></tr><tr><td>FID</td><td>99.9</td><td>95.1</td><td>25.3</td><td>195</td></tr></table>
+
+Table 2: Comparison of bits-per-dimension (BPD) and sample quality via FID scores for MLE- and ADV-trained models. For reference, the FID of a untrained Flow is roughly 1500. Using the stable version with ADV, though improves BPD significantly, might come at a trade-off on FID.
+
+Here, we analyze in depth the effect of Flow-GAN training on numerical stability. We use networks with repeated additive coupling layers, and ActNorm between blocks. We examine two architectures, both with 3 levels, i.e., ‘squeeze’ between levels (additional details in Appendix D):
+
+• Stable: having a depth of 4 (i.e., 4 blocks per level) and spectral normalization applied to all the convolution layers, see section 3.2 for details.   
+• Unstable: having a depth of 16, ActNorm within coupling layer, and no spectral normalization applied to the convolutional layers.
+
+Table 2 shows that models trained with MLE objective can achieve good bits-per-dimenstion (BPD), but models trained with ADV can achieve better sample quality as measured by the Frechet Inception Distance (FID), a common measure of sample quality (Heusel et al., 2017; Lucic et al., 2018). This justifies why considering training INN with objective functions other than MLE is desired.
+
+Broken Flow-GAN. In Figure 4, we show that an INN trained only with adversarial loss can become non-invertible, depending on the architecture. We perform forward and inverse passes repeatedly on the same mini-batch. The unstable model shows visible reconstruction errors quickly. Table 3 shows that the unstable model has BPD orders of magnitude larger than the stable model.
+
+![](images/a2670486ac1f35a8d45909c8ee6dfce4a9bd47eb565e2348620a06ad00c573d6.jpg)  
+Figure 4: Reconstructions of images through multiple forward/inverse passes. First row in the ‘Recons’ subfigures are real-data, and each row after is one forward/inverse pass.
+
+<table><tr><td>Model</td><td>(Test) BPD</td><td>Max-SV</td><td>Min-SV</td><td>Cond-Num</td><td>Analytic LDJ</td></tr><tr><td>ADV (Stable)</td><td>709</td><td>1.99e3</td><td>0.6129</td><td>3.253e3</td><td>4734</td></tr><tr><td>ADV (Unstable)</td><td>3.87e12*</td><td>3.83e8</td><td>0.0309</td><td>1.239e10</td><td>9874</td></tr></table>
+
+Table 3: Stability analysis on two architectures trained adversarially (ADV), and with maximum likelihood (MLE). \*means this number is meaningless as the network is visibly non-invertible. LDJ denotes the log-determinant of the Jacobian, BPD denotes bits-per-dimension and SV singular value.
+
+Flow-GAN “Likelihood”. Typically likelihood is computed by a change of variables, which assumes invertibility. When a network is numerically non-invertible, the assumption breaks, and the computed value becomes some numerical approximation to the density. The model used in Figure 4 consists of additive coupling blocks, ActNorm, and squeezing operations. All these operations have data independent log-determinant Jacobian. Thus, obtaining a numerical value via the change of variable formula is straightforward. However, it is unclear what this numerical value represents, likely it cannot be trusted as true likelihood due to the lack of invertibility. In this case, we advocate for ensuring invertibility using the remedies discussed here to make sure BPD values are trustworthy. In terms of sample quality, however, we currently observe a tradeoff since the stable model yields higher FID scores than the unstable model (Table 2, includes also MLE-models for comparison). Yet, we believe that proper tuning e.g. of spectral normalization could remove this tradeoff.
+
+Lastly, one can adjust for the likelihood by adjusting the prior (see Appendix I). In summary, in this section we point out that Flow-GAN can become non-invertible, in which case the computed likelihood cannot be taken as ground truth likelihood (Grover et al., 2018). In sum, INN trained with MLE are stable, but the sample quality is worse than those trained with ADV. Yet, training with ADV loss might make INN non-invertible, which defeats the purpose of using INN in the first place. Hence, when training with alternative objectives, numerical stability is a crucial property to consider.
+
+# 4.4 DECORRELATION TASK
+
+As the last part of our empirical study, we use a simple decorrelation task to benchmark the stability of invertible models. In particular, we compute the correlation matrix $C \in \mathbb { R } ^ { d \times d }$ via
+
+$$
+C _ { j , l } ^ { \theta } = \frac { 1 } { N } \sum _ { i = 1 } ^ { N } \frac { \left( F _ { \theta } ( x ^ { ( i ) } ) _ { j } - \hat { \mu } _ { j } \right) \left( F _ { \theta } ( x ^ { ( i ) } ) _ { l } - \hat { \mu } _ { l } \right) } { \hat { \sigma } _ { j } \hat { \sigma } _ { l } } ,
+$$
+
+where $\hat { \mu } _ { j }$ is the estimated mean over output samples $F _ { \theta } ( x ^ { ( i ) } )$ and $\hat { \sigma } _ { j }$ the estimated standard deviation.   
+Then, we optimize the parameters $\theta$ to minimize the off-diagonal correlation, i.e.
+
+$$
+\operatorname* { m i n } _ { \theta } \| C ^ { \theta } - \mathrm { d i a g } ( C ^ { \theta } ) \| _ { F } ,
+$$
+
+where $\| \cdot \| _ { F }$ is the Frobenius-norm 3. Decorrelation objectives have been used in (Cogswell et al., 2015) to reduce overfitting and in (Cheung et al., 2014) to disentangle hidden activations.
+
+This objective serves as a good task for our purposes for two reasons:
+
+1. Decorrelation is a simpler objective than optimizing outputs $z = F _ { \theta } ( x )$ to follow a factorized Gaussian as in Normalizing Flows (Rezende & Mohamed, 2015). Furthermore, it will show that changing the objective to less standard tasks can lead to larger instabilities compared to using INNs on more common tasks such as density estimation.   
+2. Decorrelation allows multiple solutions using invertible mappings, where both stable and unstable transforms are equally valid for the given objective. See Appendix F for a motivation based on a simple 2D toy example.
+
+In summary, the decorrelation objective offers an environment to study which INN components steer the mapping towards stable or unstable solutions, that are equally plausible for the given task.
+
+In our experiments shown in Figure 5, we focus on coupling-based models like Glow (Kingma & Dhariwal, 2018), which are analytically invertible and thus allow to a simpler analysis compared to models relying on numerical inversion like i-ResNets (Behrmann et al., 2019). We evaluate the effects of different architectural choices on numerical stability, including additive vs. affine coupling layers, ActNorm, and architecture depth. For ActNorm we study two settings: 1) between coupling blocks, 2) inside blocks, i.e. as part of the function $g$ in additive blocks or $s$ and $t$ in affine blocks. Details on the architectures/ training schemes and extended results are provided in Appendix H.
+
+3We provide example PyTorch code for the decorrelation objective in Appendix G.
+
+![](images/5019444ab8fb70f6e314418fb7525817886728854b55e61bd8a622ce90618bb6.jpg)  
+Figure 5: Instability in affine and additive models. On the right, we track the reconstruction errors for a fixed minibatch over the course of training: in each iteration (vertical slice) we plot the errors of each minibatch element. Thus, we can observe the distribution of errors. Orange points in the affine plots denote inf and nan values. c denotes the condition number of the Jacobian, $\sigma _ { m a x }$ and $\sigma _ { m i n }$ denote the maximum and minimum singular values, respectively. At the top, we show the original images $x$ and in each snapshot, the reconstructions ${ \hat { x } } = { \dot { \operatorname { F } } } ^ { - 1 } ( F ( x ) )$ and reconstruction errors $R = | x - { \hat { x } } |$ . Note that the $\mathbf { X } ^ { - }$ and y-axes differ for different settings, as the models become unstable at different points during training.
+
+# 5 RELATED WORK
+
+Invertibility and stability of deep networks. The inversion from activations in standard neural networks to inputs has been studied in various works, e.g. via optimization in input space (Mahendran & Vedaldi, 2014). Linking invertibility and inverse stability for relu-networks was e.g. done in Behrmann et al. (2018). However, few works study the stability of INNs: Gomez et al. (2017) study the numerical errors in the gradient computation when using their memory-efficient backpropagation variant. Similarly to our empirical analysis, (Jacobsen et al., 2018) computed the SVD of the Jacobian of a trained i-RevNet and observed an ill-conditioned Jacobian. Lastly, the i-ResNet architecture (Behrmann et al., 2019) yields bi-Lipschitz bounds by design.
+
+On the other hand, the stability of neural networks has been of major interest due to the problem of exploding and vanishing gradients, and more recently due to adversarial examples (Szegedy et al., 2013) and training of Wasserstein GANs (Arjovsky et al., 2017). See e.g. (Anil et al., 2019) for a promising approach to learn flexible Lipschitz neural networks.
+
+Invertible building blocks. Besides the invertible building blocks we studied in Table 1, several other approaches were proposed. Most prominently, autogressive models like MAF (Papamakarios et al., 2017) or IAF (Kingma et al., 2016) provide invertible models that are not studied in our analysis. Furthermore, several newer coupling layers that require numerical inversion have been introduced (Jaini et al., 2019; Durkan et al., 2019). Besides the coupling-based approaches, multiple approaches (Chen et al., 2018; Behrmann et al., 2019; Chen et al., 2019; Song et al., 2019) use numerical inversion schemes, where the interplay of numerical errors due to stability and errors due to the numerical approximation of the inverse adds another dimension to the study of invertibility.
+
+Fixed-Point arithmetic and limited precision. Maclaurin et al. (2015); MacKay et al. (2018) implement invertible computation using fixed-point numbers, with specially-designed schemes to store information that is “lost” when bits are shifted due to multiplication/division, enabling exact invertibility at the cost of additional memory usage. As Gomez et al. (2017) point out, this approach allows exact numerical inversion when using additive coupling blocks independent of stability issues. However, our stability analysis aims for a broadly applicable methodology beyond the special case of additive coupling. Lastly, there may be connections to deep learning using limited precision, see e.g. (Gupta et al., 2015), which could provide more insights into our observed numerical errors.
+
+# 6 CONCLUSION
+
+Numerical instability is an important concern for the practical application of invertible models. If for instance analytical invertibility does not carry through to the numerical computation due to instabilities or numerical errors, the consequences can be arbitrarily severe. As shown in our experiments, this can impact memory-efficient backpropagation (Gomez et al., 2017) and thus significantly reduces the usability of invertible networks if not handled appropriately. Flow-GAN illustrates another application where non-invertibility poses a serious threat, as instabilities can strongly influence or even break likelihood-computation.
+
+In this paper, we shed light on the underlying causes of instability by deriving Lipschitz bounds on many of the atomic building blocks commonly used to construct INNs. From a practical standpoint, we used diagnostics to measure stability and provided an empirical framework to benchmark stability. Further, we have shown how to guarantee stability for one of the most common INN architectures. We hope that this will inspire future work to view numerical stability as a crucial axis in the design of new building blocks and architectures for invertible neural networks.
+
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+
+# A DERIVATIONS OF LIPSCHITZ BOUNDS
+
+The bounds for invertible ResNets are taken from (Behrmann et al., 2019). For Neural ODEs (Chen et al., 2018), one needs to consider a Lipschitz constant $\operatorname { L i p } ( F )$ that holds for all $t \in [ 0 , T ]$ , i.e.
+
+$$
+\| F ( t , x _ { 1 } ) - F ( t , x _ { 2 } ) \| _ { 2 } \leq \mathrm { { L i p } } ( F ) \| x _ { 1 } - x _ { 2 } \| _ { 2 } , \quad \mathrm { f o r ~ a l l } \quad t \in [ 0 , T ] .
+$$
+
+Then, the claimed bound is a standard result, see e.g. (Ascher, 2008, Theorem 2.3). Note that the inverse is given by $\begin{array} { r } { \frac { d y ( t ) } { d t } = - F ( y ( t ) , t ) } \end{array}$ , hence the same bound holds.
+
+In the subsequent subsections, we derive the bounds for coupling layers.
+
+# A.1 DERIVATION OF LIPSCHITZ BOUND FOR ADDITIVE COUPLING LAYERS
+
+Consider an additive coupling block defined as
+
+$$
+\begin{array} { l } { F ( x ) _ { I _ { 1 } } = x _ { I _ { 1 } } } \\ { F ( x ) _ { I _ { 2 } } = x _ { I _ { 2 } } + g ( x _ { I _ { 1 } } ) , } \end{array}
+$$
+
+where $I _ { 1 } , I _ { 2 }$ is a disjoint partition of indices $\{ 1 , . . . , d \}$ of the same cardinality, i.e. $\begin{array} { r } { | I _ { 1 } | = | I _ { 2 } | = \frac { d } { 2 } } \end{array}$ Further, $x _ { I _ { 1 } } , x _ { I _ { 2 } }$ correpsonds to the corresponding dimension of $x \in \mathbb { R } ^ { d }$ and $g : \mathbb { R } ^ { \frac { d } { 2 } }  \mathbb { R } ^ { \frac { d } { 2 } }$ . By Lemma 3, it is
+
+$$
+\mathrm { L i p } ( F ) = \operatorname* { s u p } _ { x \in \mathbb { R } ^ { d } } \| J _ { F } ( x ) \| _ { 2 } .
+$$
+
+Thus, in order to obtain a bound on the Lipschitz constant, it is helpful to look into the structure of the Jacobian. If the partitions $I _ { 1 }$ and $I _ { 2 }$ correspond to the first and last $\textstyle { \frac { d } { 2 } }$ indices, the Jacobian has a lower-block structure with an identity diagonal, i.e.
+
+$$
+J _ { F } ( x ) = \binom { I } { J _ { g } ( x ) } \binom { 0 } { I } .
+$$
+
+By using this structure, we can derive the following upper bound:
+
+$$
+\begin{array} { r l } { \| { \bf { J } } _ { 1 } ( { \bf { J } } _ { 2 } ^ { \mathrm { R } } ) - { \bf { A } } _ { 1 } ( { \bf { J } } _ { 1 } ^ { \mathrm { R } } ) \| ^ { 2 } } & { = } & { \kappa ^ { 2 } \kappa \| \sum _ { i = 1 } ^ { N } [ \kappa ( { \bf { J } } _ { i } ^ { \mathrm { R } } ) \kappa ^ { 2 } ] \| ^ { 2 } } \\ & { = } & { \kappa ^ { 2 } \kappa ^ { 2 } \kappa \| \sum _ { i = 1 } ^ { N } [ \kappa ( { \bf { J } } _ { i } ^ { \mathrm { R } } ) \kappa ^ { 2 } ] \| ^ { 2 } } \\ & { = } & { \kappa ^ { 2 } \kappa ^ { 3 } \kappa ^ { 4 } \kappa ^ { 4 } \kappa ^ { 4 } \kappa ^ { 4 } \kappa ^ { 4 } \kappa ^ { 4 } \kappa ^ { 4 } \kappa ^ { 4 } \kappa ^ { 4 } \kappa ^ { 4 } \kappa ^ { 4 } \kappa ^ { 4 } \kappa ^ { 4 } \kappa ^ { 4 } \kappa ^ { 4 } \kappa ^ { 4 } \kappa ^ { 4 } \kappa ^ { 4 } \kappa ^ { 4 } \kappa ^ { 4 } } \\ & { = } & { \kappa ^ { 4 } \kappa \| \sum _ { i = 1 } ^ { N } [ \kappa ( { \bf { J } } _ { i } ^ { \mathrm { R } } ) \kappa ^ { 2 } \kappa ^ { 2 } ] \| ^ { 2 } \kappa ^ { 4 } \| \kappa ^ { 4 } \kappa ^ { 4 } \kappa ^ { 4 } \kappa ^ { 4 } \kappa ^ { 4 } \kappa ^ { 4 } \kappa ^ { 4 } \kappa ^ { 4 } \kappa ^ { 4 } \kappa ^ { 4 } \kappa ^ { 4 } \kappa ^ { 4 } \kappa ^ { 4 } } \\ &  \quad \times \frac { \kappa ^ { 2 } \kappa ^ { 4 } \kappa ^ { 4 } }  2 \kappa ^ { 4 } \kappa ^ { 4 } \kappa ^ { 4 } \kappa ^ { 4 } \kappa ^ { 4 } \kappa ^ { 4 } \kappa ^ { 4 } \kappa ^ { 4 } \kappa ^ { 4 } \kappa ^ { 4 } \kappa ^ { 4 } \kappa ^ { 4 } \kappa ^ { 4 } \kappa ^ { 4 } \kappa ^ { 4 } \kappa ^ { 4 } \kappa ^ { 4 } \kappa ^ { 4 } \kappa ^ { 4 } \kappa ^ { 4 } \kappa ^ { 4 } \kappa ^ { 4 } \kappa ^ { 4 } \kappa ^ { 4 } \kappa ^ { 4 } \kappa ^ { 4 } \kappa ^ { 4 } \kappa ^ { 4 } \kappa ^ { 4 } \kappa ^ { 4 } \kappa ^ { 4 } \kappa  \end{array}
+$$
+
+Furthermore, the inverse of $F$ can be obtained via the simple algebraic transformation $( y : = F ( x ) )$ )
+
+$$
+\begin{array} { l } { { F ^ { - 1 } ( y ) _ { I _ { 1 } } = y _ { I _ { 1 } } } } \\ { { F ^ { - 1 } ( y ) _ { I _ { 2 } } = y _ { I _ { 2 } } - g ( y _ { I _ { 1 } } ) . } } \end{array}
+$$
+
+Since the only difference to the forward mapping is the minus sign, the Lipschitz bound for the inverse is the same as for the forward mapping.
+
+# A.2 DERIVATION OF LIPSCHITZ BOUND FOR AFFINE COUPLING LAYERS
+
+Since the structure of the forward and inverse mapping for affine coupling layers has some differences, we split the derivation of the Lipschitz bounds into two sections. First, we start with the forward mapping and then reuse several steps for the bounds on the inverse mapping.
+
+# A.2.1 DERIVATION FOR THE FORWARD MAPPING
+
+Consider an affine coupling block defined as
+
+$$
+\begin{array} { l } { F ( x ) _ { I _ { 1 } } = x _ { I _ { 1 } } } \\ { F ( x ) _ { I _ { 2 } } = x _ { I _ { 2 } } \odot g ( s ( x _ { I _ { 1 } } ) ) + t ( x _ { I _ { 1 } } ) , } \end{array}
+$$
+
+where $g ( \cdot ) \neq 0$ for all $X _ { I _ { 2 } }$ and $I _ { 1 } , I _ { 2 }$ as before. The Jacobian for this operation has the structure
+
+$$
+J _ { F } ( x ) = \left( \begin{array} { c c } { { I } } & { { 0 } } \\ { { D _ { I } ( x _ { I _ { 2 } } ) D _ { g ^ { \prime } } ( x _ { I _ { 1 } } ) J _ { s } ( x _ { I _ { 1 } } ) + J _ { t } ( x _ { I _ { 1 } } ) } } & { { D _ { g } ( s ( x _ { I _ { 1 } } ) ) } } \end{array} \right) ,
+$$
+
+where $D$ are following diagonal matrices
+
+$$
+\begin{array} { r l } & { \quad D _ { I } ( x _ { I _ { 2 } } ) = \mathrm { d i a g } \left( ( x _ { I _ { 2 } } ) _ { 1 } , \ldots , ( x _ { I _ { 2 } } ) _ { | I _ { 2 } | } \right) , } \\ & { \quad D _ { g ^ { \prime } } ( x _ { I _ { 1 } } ) = \mathrm { d i a g } \left( g ^ { \prime } ( s ( x _ { I _ { 2 } } ) _ { 1 } , \ldots , g ^ { \prime } ( s ( x _ { I _ { 2 } } ) _ { | I _ { 2 } | } ) \right) } \\ & { \quad D _ { g } ( s ( x _ { I _ { 1 } } ) ) = \mathrm { d i a g } \left( g ( s ( x _ { I _ { 2 } } ) _ { 1 } , \ldots , g ( s ( x _ { I _ { 2 } } ) _ { | I _ { 2 } | } ) \right) . } \end{array}
+$$
+
+Denote
+
+$$
+M ( x ) : = D _ { I } ( x _ { I _ { 2 } } ) D _ { g ^ { \prime } } ( x _ { I _ { 1 } } ) J _ { s } ( x _ { I _ { 1 } } ) + J _ { t } ( x _ { I _ { 1 } } ) .
+$$
+
+By using an analogous derivation as in equation 6 (up to the inequality sign), we get
+
+$$
+\begin{array} { l } { \displaystyle \mathrm { L i p } ( F ) ^ { 2 } \leq \underset { x \in \mathbb R ^ { d } } { \operatorname* { s u p } } \underset { | x ^ { * } | | _ { 2 } = 1 } { \operatorname* { s u p } } \| x _ { I _ { 1 } } ^ { * } \| _ { 2 } ^ { 2 } + \big ( \| D _ { g } ( s ( x _ { I _ { 1 } } ) ) x _ { I _ { 2 } } ^ { * } \| _ { 2 } + \| M ( x ) x _ { I _ { 1 } } ^ { * } \| _ { 2 } \big ) ^ { 2 } } \\ { = \underset { x \in \mathbb R ^ { d } } { \operatorname* { s u p } } \underset { | x | | _ { 1 } } { \operatorname* { m a x } } \big ( 1 , D _ { g } ( s ( x _ { I _ { 1 } } ) _ { i } ) \big ) ^ { 2 } + 2 \underset { i \in [ | I _ { 1 } | ] } { \operatorname* { m a x } } \big ( D _ { g } ( s ( x _ { I _ { 1 } } ) _ { i } ) \big ) \| M ( x ) \| _ { 2 } + \| M ( x ) \| _ { 2 } ^ { 2 } } \\ { \leq \underset { x \in \mathbb R ^ { d } } { \operatorname* { s u p } } \underset { | x | | _ { 1 } } { \operatorname* { m a x } } \big ( 1 , D _ { g } ( s ( x _ { I _ { 1 } } ) _ { i } ) \big ) ^ { 2 } + 2 \underset { i \in [ | I _ { 1 } | ] } { \operatorname* { m a x } } \big ( 1 , D _ { g } ( s ( x _ { I _ { 1 } } ) _ { i } ) \big ) \| M ( x ) \| _ { 2 } + \| M ( x ) \| _ { 2 } ^ { 2 } } \\ { = \underset { x \in \mathbb R ^ { d } } { \operatorname* { s u p } } \bigg ( \underset { i \in [ | I _ { 1 } | ] } { \operatorname* { m a x } } \big ( 1 , D _ { g } ( s ( x _ { I _ { 1 } } ) _ { i } ) \big ) + \| M ( x ) \| _ { 2 } \bigg ) ^ { 2 } } \\ { \Longleftrightarrow \underset { x \in \mathbb R ^ { d } } { \operatorname* { m a x } } \big ( 1 , D _ { g } ( s ( x _ { I _ { 1 } } ) _ { i } ) \big ) + \underset { x \in \mathbb R ^ { d } } { \operatorname* { s u p } } \| M ( x ) \| _ { 2 } . } \end{array}
+$$
+
+Next, we will look into the structure of $M ( x )$ to derive a more precise bound. Since inputs $x$ are assumed to be bounded as $x \in [ a , b ] ^ { d }$ , it holds
+
+$$
+\| D _ { I } ( x _ { I _ { 2 } } ) \| _ { 2 } \leq \operatorname* { m a x } ( | a | , | b | ) .
+$$
+
+Furthermore, let the derivative $g ^ { \prime }$ of the element-wise function $g$ be globally bounded by $c$ , i.e. $\begin{array} { r } { \operatorname* { s u p } _ { x \in \mathbb { R } } g ^ { \prime } ( x ) \leq c _ { g ^ { \prime } } } \end{array}$ . Then, it is
+
+$$
+\| D _ { g ^ { \prime } } ( x _ { I _ { 1 } } ) \| _ { 2 } \leq c _ { g ^ { \prime } } .
+$$
+
+In a similar manner as in section A.1, the spectral norm of the Jacobian of the scale-function $s$ and translation-function $t$ can be bounded by their Lipschitz constant, i.e.
+
+$$
+\begin{array} { r } { \| J _ { s } ( x _ { I _ { 1 } } ) \| _ { 2 } \leq \mathrm { L i p } ( s ) } \\ { \| J _ { t } ( x _ { I _ { 1 } } ) \| _ { 2 } \leq \mathrm { L i p } ( t ) . } \end{array}
+$$
+
+By using above bounds, we obtain
+
+$$
+\operatorname* { s u p } _ { x \in \mathbb { R } ^ { d } } \| M ( x ) \| _ { 2 } ^ { 2 } \leq \operatorname* { m a x } ( | a | , | b | ) \cdot c \cdot \mathrm { L i p } ( s ) + \mathrm { L i p } ( t ) .
+$$
+
+If we further assume, that the elementwise-function $g$ is globally upper bounded by $c _ { g }$ and we insert above bounds, we obtain
+
+$$
+\mathrm { L i p } ( F ) \leq \operatorname* { m a x } ( 1 , c _ { g } ) + \operatorname* { m a x } ( | a | , | b | ) \cdot c _ { g ^ { \prime } } \cdot \mathrm { L i p } ( s ) + \mathrm { L i p } ( t ) .
+$$
+
+A.2.2 DERIVATION FOR THE INVERSE MAPPING
+
+For the affine coupling block from section A.2.1, the inverse is defined as
+
+$$
+\begin{array} { r l } & { F ^ { - 1 } ( y ) _ { I _ { 1 } } = y _ { I _ { 1 } } } \\ & { F ^ { - 1 } ( y ) _ { I _ { 2 } } = \left( y _ { I _ { 2 } } - t ( x _ { I _ { 1 } } ) \right) \oslash g ( s ( y _ { I _ { 1 } } ) ) , } \end{array}
+$$
+
+where $g ( \cdot ) \neq 0$ for all $X _ { I _ { 2 } } , I _ { 1 } , I _ { 2 }$ as before and $\oslash$ denotes elementwise division. The Jacobian for this operation has the structure
+
+$$
+J _ { F } ( x ) = \left( \begin{array} { c c } { { I } } & { { 0 } } \\ { { M ^ { * } ( y ) } } & { { D _ { \frac { 1 } { g } } ( s ( x _ { I _ { 1 } } ) ) } } \end{array} \right) ,
+$$
+
+where $D _ { \frac { 1 } { g } } { \left( s \left( x _ { I _ { 1 } } \right) \right) }$ denotes a diagonal matrix, as before. Furthermore, $M *$ is defined as
+
+$\begin{array} { r } { { \cal M } ^ { * } ( y ) = D _ { I } ( y _ { I _ { 2 } } ) D _ { \left( { \frac { 1 } { g } } \right) ^ { \prime } } ( s ( y _ { I _ { 1 } } ) ) J _ { s } ( y _ { I _ { 1 } } ) - D _ { \left( { \frac { 1 } { g } } \right) ^ { \prime } } ( s ( y _ { I _ { 1 } } ) ) J _ { s } ( y _ { I _ { 1 } } ) D _ { I } ( t ( y _ { I _ { 1 } } ) ) - D _ { \frac { 1 } { g } } ( s ( y _ { I _ { 1 } } ) ) J _ { t } ( y _ { I _ { 1 } } ) , } \end{array}$ where $D _ { \left( { \frac { 1 } { g } } \right) ^ { \prime } } ( s ( x _ { I _ { 1 } } ) )$ also denotes a diagonal matrix. Using analogous arguments as in section A.2.1, we obtain the bound
+
+$$
+\mathrm { L i p } ( F ^ { - 1 } ) \leq \operatorname* { m a x } _ { i \in [ | I _ { 1 } | ] } ( 1 , D _ { \frac { 1 } { g } } ( s ( x _ { I _ { 1 } } ) _ { i } ) ) + \operatorname* { s u p } _ { x \in \mathbb { R } ^ { d } } \| M ^ { * } ( x ) \| _ { 2 } .
+$$
+
+Hence, we need to further bound the spectral norm of $M ^ { * }$ . First, assume that $\textstyle { \frac { 1 } { g } }$ , the derivative $\scriptstyle \left( { \frac { 1 } { g } } \right) ^ { \prime }$ and translation $t$ is globally upper bounded by $c _ { \frac { 1 } { g } } , c _ { \left( \frac { 1 } { g } \right) ^ { \prime } }$ and $c _ { t }$ respectively. Furthermore consider bounded inputs $y \in [ a ^ { * } , b ^ { * } ] ^ { d }$ . Then we obtain the bound
+
+$$
+\operatorname* { s u p } _ { x \in \mathbb { R } ^ { d } } \| M ^ { * } ( x ) \| _ { 2 } ^ { 2 } \leq \operatorname* { m a x } ( | a ^ { * } | , | b ^ { * } | ) \cdot c _ { \binom { 1 } { g } } \cdot \operatorname { L i p } ( s ) + c _ { \binom { 1 } { g } } \cdot \operatorname { L i p } ( s ) \cdot c _ { t } + c _ { \frac { 1 } { g } } \cdot \operatorname { L i p } ( t ) .
+$$
+
+Hence, we can bound the Lipschitz constant of the inverse of an affine block as
+
+$$
+\mathrm { L i p } ( F ^ { - 1 } ) \leq \operatorname* { m a x } _ { i \in [ [ I _ { 1 } ] ] } ( 1 , c _ { \frac { 1 } { g } } ) + \operatorname* { m a x } ( | a ^ { * } | , | b ^ { * } | ) \cdot c _ { ( \frac { 1 } { g } ) ^ { \prime } } \cdot \mathrm { L i p } ( s ) + c _ { ( \frac { 1 } { g } ) ^ { \prime } } \cdot \mathrm { L i p } ( s ) \cdot c _ { t } + c _ { \frac { 1 } { g } } \cdot \mathrm { L i p } ( t ) .
+$$
+
+# B NUMERICAL ERRORS AND LIPSCHITZ CONSTANTS
+
+In a general setting, connecting numerical errors e.g. due to floating point operations to Lipschitz constants of the underlying mapping in a quantitative manner is not straightforward. For example, numerical errors due to limited precision occurs when summing to floating point numbers. As discussed in (Gomez et al., 2017), this occurs in additive coupling layers and is one source of numerical errors we observe in our experiments.
+
+To formalize the connection to the Lipschitz constant, consider the following two mappings:
+
+$$
+\begin{array} { r l } & { F ( x ) = z , \quad \mathrm { ( a n a l y t i c a l \ e x a c t { c o m p u t a t i o n } ) } } \\ & { F _ { \delta } ( x ) = z + \delta = : z _ { \delta } , \quad \mathrm { ( f l o a t i n g \ p o i n t i n e x a c t { c o m p u t a t i o n } ) } } \end{array}
+$$
+
+In order to bound the error in the reconstruction due to the imprecision in the forward mapping, let $x _ { \delta _ { 1 } } = F ^ { - 1 } ( z _ { \delta } )$ . Now consider
+
+$$
+\| x - x _ { \delta _ { 1 } } \| _ { 2 } \le \mathrm { L i p } ( F ^ { - 1 } ) \| z - z _ { \delta } \| _ { 2 } = \mathrm { L i p } ( F ^ { - 1 } ) \| \delta \| _ { 2 } ,
+$$
+
+where the Lipschitz constant of the inverse is used to bound the influence of the numerical error in the forward mapping. However, similarly to the forward mapping, the inverse mapping can also be imprecise. Thus, we introduce
+
+$$
+F _ { \delta } ^ { - 1 } ( z _ { \delta } ) = x _ { \delta _ { 1 } } + \delta _ { 2 } : = x _ { \delta _ { 2 } }
+$$
+
+to formalize the numerical error in the inverse mapping. Hence, we obtain the bound
+
+$$
+\begin{array} { r l } & { \| \boldsymbol { x } - ( x _ { \delta _ { 1 } } + \delta _ { 2 } ) \| _ { 2 } \leq \| \boldsymbol { x } - x _ { \delta _ { 1 } } \| _ { 2 } + \| \delta _ { 2 } \| _ { 2 } } \\ & { \qquad \leq \mathrm { L i p } ( \boldsymbol { F } ^ { - 1 } ) \| \boldsymbol { z } - \boldsymbol { z } _ { \delta } \| _ { 2 } + \| \delta _ { 2 } \| _ { 2 } } \\ & { \qquad = \mathrm { L i p } ( \boldsymbol { F } ^ { - 1 } ) \| \delta \| _ { 2 } + \| \delta _ { 2 } \| _ { 2 } , } \end{array}
+$$
+
+where the numerical errors of the mapping are denoted via $\delta$ (forward) and $\delta _ { 2 }$ (inverse). While obtaining quantitative values for $\delta$ and $\delta _ { 2 }$ for a model as complex as deep neural networks is hard, above formalization still provides insights into a potential role of the inverse stability when reconstructing inputs.
+
+# C ADDITIONAL DETAILS FOR CLASSIFICATION EXPERIMENTS
+
+Both affine and additive coupling-based models have the same architecture, that consists of 3 levels, 16 blocks per level, and 128 hidden channels. Each level consists of a sequence of residual blocks that operate on the same dimensionality. Between levels, the input is spatially downsampled by $2 \times$ in both width and height, while the number of channels is increased by $4 \times$ . Each residual block consists of a chain of $3 \times 3$ , $1 \times 1$ , $3 \times 3$ convolutions, with ReLU activations in between. We trained on CIFAR-10 for 200 epochs, using SGD with Nesterov momentum 0.9 and weight decay 5e-4, with initial learning rate 0.01, decayed by a factor of 5 at epochs 60, 120, and 160. We found that using a smaller initial learning rate of 0.01 was important for training the INN classifier, as opposed to the standard initial learning rate of 0.1 used for ResNets (Zagoruyko & Komodakis, 2016). We used standard data augmentation (random cropping and horizontal flipping).
+
+![](images/2fe1a3649baa435844f4734a4f6d8df651e4bd1458c1d4ff731ce54e48fa327e.jpg)  
+Figure 6: Test classification accuracy on CIFAR-10.
+
+# D EXPERIMENTAL DETAILS FOR FLOW-GAN
+
+For the experiments in Section 4.3, the ADV models are trained with standard binary cross entropy loss and the hyperparameters in Table 4, whereas the MLE models are trained with learning rate 1e-3.   
+Table 4: Hyperparameters for ADV models   
+
+<table><tr><td>Variable</td><td>Values</td></tr><tr><td>batch size</td><td>64</td></tr><tr><td>learning rate (generator)</td><td>1e-5</td></tr><tr><td>learning rate (discriminator)</td><td>1e-4</td></tr><tr><td>weight decay (both)</td><td>0</td></tr><tr><td>optimizer</td><td>Adam(0.5, 0.99)</td></tr></table>
+
+# E EXTENDED RESULTS FOR CRAFTED, NON-INVERTIBLE INPUTS
+
+PGD Setup. To find non-invertible inputs for Glow and Residual Flows, we used PGD (Eq. 3) with $\epsilon = 0 . 1$ and step size 0.01. For the Glow model in Section 4.2, we consistently found inputs with severe reconstruction errors (as shown in Figure 3) in fewer than 10 PGD iterations. For the Residual Flow model analyzed in this section, we ran 200 iterations of PGD. In each iteration, the pixel values of the perturbed image were clipped to the valid input range that the respective model was trained on $( [ - 0 . 5 , 0 . 5 ]$ for Glow and [0, 1] for the Residual Flow).
+
+Crafted Inputs for Residual Flows. We also applied the PGD attack from Section 4.2 to a Residual Flow (Chen et al., 2019) pre-trained on CIFAR-10 (Figure 7).4 We find that, while there are visible differences between the crafted input $x ^ { \prime }$ and its reconstruction $\hat { x } ^ { \prime }$ , the reconstruction errors are less severe for the Residual Flow compared to Glow (analyzed in Section 4.2).
+
+![](images/8a1b23e08b9ef2e20c437c2b65fd4bca45906d8605f6bb773a8fd482a966624d.jpg)  
+Figure 7: Crafting non-invertible inputs for a CIFAR-10 Residual Flow model.
+
+Instability Outside the Range of Training Inputs. Here, we applied the PGD attack from Section 4.2 to a Glow model pre-trained on Celeb-A (Liu et al., 2015). 5 This model was trained on images normalized to the range $[ - 0 . 5 , 0 . 5 ]$ . While the PGD attack was not successful at finding adversarial inputs in $[ - 0 . 5 , 0 . 5 ]$ , it succeeded when the range was increased to $[ - 0 . 7 , 0 . 7 ]$ (by using $\epsilon = 0 . 2$ and not clipping the perturbed inputs to $[ - 0 . 5 , 0 . 5 ] )$ , yielding the example shown in Figure 8. Thus, we found that invertible models can become numerically non-invertible on out-of-distribution data.
+
+![](images/0049da560f3d5241c3f13f0c1561a603cb55362cd1b326f76e6615fee45db129.jpg)  
+Figure 8: Glow becomes non-invertible for inputs outside the training distribution. We use a SOTA Glow model trained on images normalized to $[ - 0 . 5 , 0 . 5 ]$ , and use PGD to find an adversarial input constrained to the larger range $[ - 0 . 7 , 0 . 7 ]$ . This attack succeeds in finding examples that induce dramatic reconstruction error.
+
+![](images/e5409efd8c2a7ea7138035bab2354724edcc5fc54d837ec1921c5bad083c8d54.jpg)  
+Figure 9: Toy example to motivate the decorrelation task. Left: standard normal data, 2. left: scaled by $D$ and rotated data by $R$ , 2. right: scaling and rotation backwards, low correlation but higher condition number, right: rotation backwards, low correlation and low condition number.
+
+To motivate the decorrelation task as simple toy environment for stability of invertible models consider the following task (and its visualization in Figure 9):
+
+• Consider input data $x$ that is distributed via a standard normal distribution, i.e. $x \sim \mathcal { N } ( 0 , I )$ (Figure 9 (left)).   
+• Assume the data is transformed by a rotation matrix $R$ and a diagonal matrix $D$ , i.e. $y = R D x$ (Figure 9 (2. from left)).   
+• Goal: decorrelate transformed data $y$ using an invertible mapping.
+
+Since correlation is independent of scale (scaling by standard deviation of the data), at least the two solutions $A _ { 2 } = D ^ { - 1 } R ^ { T }$ (Figure 9 (2. from right)) and $A _ { 1 } = R ^ { T }$ (Figure 9 (right)) are equally valid for the given decorrelation task. However, the conditioning of the mappings $A _ { 1 }$ and $A _ { 2 }$ can be largely different if the scaling matrix $D$ has a high condition number. Hence, this task both offers a stable solution, namely $A _ { 1 }$ , and a (potentially) unstable solution $A _ { 2 }$ .
+
+To conclude, decorrelation can allow multiple solutions with different stability. Hence, decorrelation is a natural simple task to study which solution the INN picks. Furthermore, guiding the network to a stable solution is a justified strategy for this task and it is not expected to harm performance.
+
+# G DECORRELATION EXAMPLE CODE
+
+Here we provide an example implementation of the decorrelation objective used in Section 4.4, that minimizes the norm of the off-diagonal entries in the correlation matrix.
+
+Listing 1: Example PyTorch code to implement the decorrelation loss used in our experiments.
+
+<table><tr><td>z = model(img)</td><td></td><td></td><td></td><td></td></tr><tr><td></td><td>z_flat = z.view(z.size(O)，-1)</td><td></td><td></td><td></td></tr><tr><td></td><td>z_flat = z_flat - z_flat.mean(dim=0)</td><td></td><td> Subtract mean</td><td></td></tr><tr><td></td><td></td><td>z_flat = z_flat / (z_flat.std(dim=O) + le-8) # Standardize</td><td></td><td></td></tr><tr><td></td><td>correlation = (torch.mm(z_flat.t()，z_flat)/ (z_flat.size(0)-1))</td><td></td><td></td><td></td></tr><tr><td></td><td>loss = torch.norm(correlation - torch.diag(torch.diagonal(correlation)))</td><td></td><td></td><td></td></tr><tr><td></td><td>optimizer.zero_grad()</td><td></td><td></td><td></td></tr><tr><td>loss.backward() optimizer.step()</td><td></td><td></td><td></td><td></td></tr></table>
+
+# H EXTENDED RESULTS FOR DECORRELATION
+
+Decorrelation Experiment Details Here we provide additional details on the model architectures and training schemes we used in our numerical experiments.
+
+For all the decorrelation experiments, we used a 3-level model with blocks of depth 16. We used Adam (Kingma & Ba, 2015) with fixed learning rate 1e-4 and no weight decay, and trained on mini-batches of size 64. The CIFAR-10 images were normalized to the range [-0.5, 0.5], and were dequantized with uniform noise in [0, 1e-6].
+
+Effect of Model Depth. Furthermore, we investigated the effect of network depth on stability since its an additional influence factor besides the selection of each invertible building block. Starting with a 3-level additive model with ActNorm, we vary the depth of the blocks between $\{ 4 , 1 6 , 3 2 \}$ and train with the decorrelation objective. The quantitative reconstruction errors and condition numbers of the Jacobians are shown in Figure 10. As expected, deeper architectures become unstable faster than shallow ones.
+
+![](images/a9712a6e50f9d99a72009394aa054d2829c4aba6941e5ccf64ced12fa2fb9a19.jpg)  
+Figure 10: Comparing stability of additive flows of different depths. These models all have ActNorm both between and inside the additive blocks.
+
+Loss Plots. Here we show that all the model variants investigated in the decorrelation experiments achieve their objective, i.e., the loss decreased enough for the correlation matrices to be diagonal.
+
+Evolution of Condition Numbers, Max & Min Singular Values for Decorrelation. Here we plot the condition numbers, maximum and minimum singular values during training for the decorrelation task. We include plots for all settings discussed in the main paper as well as the appendix.
+
+![](images/8246fdaaee39ea1b2766e9d20b29b93e13d309b35b16f74bb5230a3fb770bca9.jpg)  
+Figure 11: Additional settings for additive coupling, where: 1) ActNorm was applied only between blocks; and 2) ActNorm was applied only inside blocks. Both models become unstable and exhibit severe reconstruction artifacts.
+
+![](images/775832e2b93fb6447ecf9f96ceca73411d77dc064f08b1c3a5b41175f11b8cfe.jpg)  
+Figure 12: Additional setting for affine coupling, where ActNorm was applied only between blocks. Similarly to the other affine settings described in Section 4.4, this model becomes highly unstable rapidly during training.
+
+![](images/71ac2e99be346b4e0fa83cb839cb170d29f85b6fe2a554b5f7074709bc403353.jpg)  
+Figure 13: Decorrelation loss plots. These plots track the norm of the off-diagonal entries in the correlation matrix while training with the decorrelation objective.
+
+![](images/c8d21522a7b03a20b40abded9fe849d6c48da48eaf1748ee4f74990b4fc60f61.jpg)  
+Figure 14: Additive: ActNorm Between Blocks
+
+![](images/f066eb5452072ee8ecd8050d49acc55202b8a3250714bef50972760eb5d432aa.jpg)  
+Figure 15: Additive: ActNorm Both Inside and Between Blocks
+
+![](images/b0ecc4af73545855c6446c2b01fe10851fc2712b006fe77ff7e365d2a3070bc1.jpg)  
+Figure 16: Additive: ActNorm Inside Blocks
+
+![](images/b1a5254190d5d6d474392afdca78b25b213fd1ee58c958a2576a9767eff42aa7.jpg)  
+Figure 17: Additive: No ActNorm
+
+![](images/085f42bd97160d74ca2764db5e9ae10a7f200cf99f3c51756d8da69c80a02a0a.jpg)  
+Figure 18: Affine: No ActNorm
+
+![](images/74043fb0866a8110f988f84c8517069c0ab22f9163175848d22062f24e656013.jpg)  
+Figure 19: Affine: ActNorm Both Inside and Between Blocks
+
+![](images/cbfe7aa6f901958ea0911eba0a2ccd090e1adb2cde83938b6461a623a420cb90.jpg)  
+Figure 20: Affine: ActNorm Between Blocks
+
+# I REFITTING PRIOR IN FLOW-GAN
+
+In general, if the model is not optimized with forward KL, $D _ { K L } ( P _ { \mathrm { d a t a } } | | P _ { \theta } )$ , as in the case when optimizing with maximum likelihood, we cannot be sure ${ \cal F } ( x ) , \quad x \sim P _ { \mathrm { d a t a } }$ is best fitted with a standard Normal. Hence, a reasonable strategy, without changing anything in the learned network, is to refit the prior parameters. Here we simply optimize for maximum likelihood (as typically done for flow models) while only fitting a diagonal variance in prior. This can be interpreted as increasing the entropy of our model. In Kingma & Dhariwal (2018), they observed the opposite phenomenon that a model trained with maximum likelihood generates better samples after decreasing the entropy in the prior. See Figure 21 for samples after refitting the prior.
+
+![](images/4be6cb4fb59c7342bfccbc9d78835428e2a0f313fbc9bbc407f5fc7e108c9c14.jpg)  
+Figure 21: The row number corresponds to the number of epochs after refitting the prior variance. The unstable model fails to generate samples (i.e., outside of valid pixel values) after the prior is refitted, whereas even though the stable model degrades in sample quality, it is able to generate valid images.

parse/train/Bkxbrn0cYX/Bkxbrn0cYX.md

@@ -0,0 +1,360 @@
+# SELFLESS SEQUENTIAL LEARNING
+
+Rahaf Aljundi   
+KU Leuven   
+ESAT-PSI, Belgium   
+rahaf.aljundi@gmail.com   
+Tinne Tuytelaars   
+KU Leuven   
+ESAT-PSI, Belgium   
+tinne.tuytelaars@esat.kuleuven.be
+
+# ABSTRACT
+
+Sequential learning, also called lifelong learning, studies the problem of learning tasks in a sequence with access restricted to only the data of the current task. In this paper we look at a scenario with fixed model capacity, and postulate that the learning process should not be selfish, i.e. it should account for future tasks to be added and thus leave enough capacity for them. To achieve Selfless Sequential Learning we study different regularization strategies and activation functions. We find that imposing sparsity at the level of the representation (i.e. neuron activations) is more beneficial for sequential learning than encouraging parameter sparsity. In particular, we propose a novel regularizer, that encourages representation sparsity by means of neural inhibition. It results in few active neurons which in turn leaves more free neurons to be utilized by upcoming tasks. As neural inhibition over an entire layer can be too drastic, especially for complex tasks requiring strong representations, our regularizer only inhibits other neurons in a local neighbourhood, inspired by lateral inhibition processes in the brain. We combine our novel regularizer with state-of-the-art lifelong learning methods that penalize changes to important previously learned parts of the network. We show that our new regularizer leads to increased sparsity which translates in consistent performance improvement on diverse datasets.
+
+# 1 INTRODUCTION
+
+Sequential learning, also referred to as continual, incremental, or lifelong learning (LLL), studies the problem of learning a sequence of tasks, one at a time, without access to the training data of previous or future tasks. When learning a new task, a key challenge in this context is how to avoid catastrophic interference with the tasks learned previously (French, 1999; Li & Hoiem, 2016). Some methods exploit an additional episodic memory to store a small amount of previous tasks data to regularize future task learning (e.g. Lopez-Paz et al. (2017)). Others store previous tasks models and at test time, select one model or merge the models (Rusu et al., 2016; Aljundi et al., 2016; Lee et al., 2017). In contrast, in this work we are interested in the challenging situation of learning a sequence of tasks without access to any previous or future task data and restricted to a fixed model capacity, as also studied in Kirkpatrick et al. (2016); Aljundi et al. (2017); Fernando et al. (2017); Mallya & Lazebnik (2017); Serrà et al. (2018). This scenario not only has many practical benefits, including privacy and scalability, but also resembles more closely how the mammalian brain learns tasks over time.
+
+The mammalian brain is composed of billions of neurons. Yet at any given time, information is represented by only a few active neurons resulting in a sparsity of $9 0 { - } 9 5 \%$ (Lennie, 2003). In neural biology, lateral inhibition describes the process where an activated neuron reduces the activity of its weaker neighbors. This creates a powerful decorrelated and compact representation with minimum interference between different input patterns in the brain (Yu et al., 2014). This is in stark contrast with artificial neural networks, which typically learn dense representations that are highly entangled (Bengio et al., 2009). Such an entangled representation is quite sensitive to changes in the input patterns, in that it responds differently to input patterns with only small variations. French (1999) suggests that an overlapped internal representation plays a crucial role in catastrophic forgetting and reducing this overlap would result in a reduced interference. Cogswell et al. (2015) show that when the amount of overfitting in a neural network is reduced, the representation correlation is also reduced. As such, learning a disentangled representation is more powerful and less vulnerable to catastrophic interference. However, if the learned disentangled representation at a given task is not sparse, only little capacity is left for the learning of new tasks. This would in turn result in either an underfitting to the new tasks or again a forgetting of previous tasks. In contrast, a sparse and decorrelated representation would lead to a powerful representation and at the same time enough free neurons that can be changed without interference with the neural activations learned for the previous tasks.
+
+![](images/a017368bfb65afbb42bcb6dc21d6c13cc867ef1742a3c8d5f95e153027446ce3.jpg)  
+Figure 1: The difference between parameter sparsity (a) and representation sparsity (b) in a simple two tasks case. First layer indicates input patterns. Learning the first task utilizes parts indicated in red. Task 2 has different input patterns and uses parts shown in green. Orange indicates changed neurons activations as a result of the second task. In (a), when an example from the first task is encountered again, the activations of the first layer will not be affected by the changes, however, the second and later layer activations are changed. Such interference is largely reduced when imposing sparsity on the representation (b).
+
+In general, sparsity in neural networks can be thought of either in terms of the network parameters or in terms of the representation (i.e., the activations). In this paper we postulate, and confirm experimentally, that a sparse and decorrelated representation is preferable over parameter sparsity in a sequential learning scenario. There are two arguments for this: first, a sparse representation is less sensitive to new and different patterns (such as data from new tasks) and second, the training procedure of the new tasks can use the free neurons leading to less interference with the previous tasks, hence reducing forgetting. In contrast, when the effective parameters are spread among different neurons, changing the ineffective ones would change the function of their corresponding neurons and hence interfere with previous tasks (see also Figure 1). Based on these observations, we propose a new regularizer that exhibits a behavior similar to the lateral inhibition in biological neurons. The main idea of our regularizer is to penalize neurons that are active at the same time. This leads to more sparsity and a decorrelated representation. However, complex tasks may actually require multiple active neurons in a layer at the same time to learn a strong representation. Therefore, our regularizer, Sparse coding through Local Neural Inhibition and Discounting (SLNID), only penalizes neurons locally. Furthermore, we don’t want inhibition to affect previously learned tasks, even if later tasks use neurons from earlier tasks. An important component of SLNID is thus to discount inhibition from/to neurons which have high neuron importance – a new concept that we introduce in analogy to parameter importance (Kirkpatrick et al., 2016; Zenke et al., 2017; Aljundi et al., 2017). When combined with a state-of-the-art important parameters preservation method (Aljundi et al., 2017; Kirkpatrick et al., 2016), our proposed regularizer leads to sparse and decorrelated representations which improves the lifelong learning performance.
+
+Our contribution is threefold. First, we direct attention to Selfless Sequential Learning and study a diverse set of representation based regularizers, parameter based regularizers, as well as sparsity inducing activation functions to this end. These have not been studied extensively in the lifelong learning literature before. Second, we propose a novel regularizer, SLNID, which is inspired by lateral inhibition in the brain. Third, we show that our proposed regularizer consistently outperforms alternatives on three diverse datasets (Permuted MNIST, CIFAR, Tiny Imagenet) and we compare to and outperform state-of-the-art LLL approaches on an 8-task object classification challenge. SLNID can be applied to different regularization based LLL approaches, and we show experiments with MAS (Aljundi et al., 2017) and EWC (Kirkpatrick et al., 2016).
+
+In the following, we first discuss related approaches to LLL and different regularization criteria from a LLL perspective (Section 2). We proceed by introducing Selfless Sequential Learning and detailing our novel regularizer (Section 3). Section 4 describes our experimental evaluation, while Section 5 concludes the paper.
+
+# 2 RELATED WORK
+
+The goal in lifelong learning is to learn a sequence of tasks without catastrophic forgetting of previously learned ones (Thrun & Mitchell, 1995). One can identify different approaches to introducing lifelong learning in neural networks. Here, we focus on learning a sequence of tasks using a fixed model capacity, i.e. with a fixed architecture and fixed number of parameters. Under this setting, methods either follow a pseudo rehearsal approach, i.e. using the new task data to approximate the performance of the previous task (Li & Hoiem, 2016; Triki et al., 2017), or aim at identifying the important parameters used by the current set of tasks and penalizing changes to those parameters by new tasks (Kirkpatrick et al., 2016; Zenke et al., 2017; Aljundi et al., 2017; Chaudhry et al., 2018; Liu et al., 2018). To identify the important parameters for a given task, Elastic Weight Consolidation (Kirkpatrick et al., 2016) uses an approximation of the Fisher information matrix computed after training a given task. Liu et al. (2018) suggest a network reparameterization to obtain a better diagonal approximation of the Fisher Information matrix of the network parameters. Path Integral (Zenke et al., 2017) estimates the importance of the network parameters while learning a given task by accumulating the contribution of each parameter to the change in the loss. Chaudhry et al. (2018) suggest a KL-divergence based generalization of Elastic Weight Consolidation and Path Integral. Memory Aware Synapses (Aljundi et al., 2017) estimates the importance of the parameters in an online manner without supervision by measuring the sensitivity of the learned function to small perturbations on the parameters. This method is less sensitive to the data distribution shift, and a local version proposed by the authors resembles applying Hebb rule (Hebb, 2002) to consolidate the important parameters, making it more biologically plausible.
+
+A common drawback of all the above methods is that learning a task could utilize a good portion of the network capacity, leaving few "free" neurons to be adapted by the new task. This in turn leads to inferior performance on the newly learned tasks or forgetting the previously learned ones, as we will show in the experiments. Hence, we study the role of sparsity and representation decorrelation in sequential learning. This aspect has not received much attention in the literature yet. Very recently, (Serrà et al., 2018) proposed to overcome catastrophic forgetting through learned hard attention masks for each task with L1 regularization imposed on the accumulated hard attention masks. This comes closer to our approach although we study and propose a regularization scheme on the learned representation.
+
+The concept of reducing the representation overlap has been suggested before in early attempts towards overcoming catastrophic forgetting in neural networks (French, 1999). This has led to several methods with the goal of orthogonalizing the activations (French, 1992; 1994; Kruschke, 1992; 1993; Sloman & Rumelhart, 1992). However, these approaches are mainly designed for specific architectures and activation functions, which makes it hard to integrate them in recent neural network structures.
+
+The sparsification of neural networks has mostly been studied for compression. SVD decomposition can be applied to reduce the number of effective parameters (Xue et al., 2013). However, there is no guarantee that the training procedure converges to a low rank weight matrix. Other works iterate between pruning and retraining of a neural network as a post processing step (Liu et al., 2015; Sun et al., 2016; Aghasi et al., 2017; Louizos et al., 2017). While compressing a neural network by removing parameters leads to a sparser neural network, this does not necessarily lead to a sparser representation. Indeed, a weight vector can be highly sparse but spread among the different neurons. This reduces the effective size of a neural network, from a compression point of view, but it would not be beneficial for later tasks as most of the neurons are already occupied by the current set of tasks. In our experiments, we show the difference between using a sparse penalty on the representation versus applying it to the weights.
+
+# 3 SELFLESS SEQUENTIAL LEARNING
+
+One of the main challenges in single model sequential learning is to have capacity to learn new tasks and at the same time avoid catastrophic forgetting of previous tasks as a result of learning new tasks. In order to prevent catastrophic forgetting, importance weight based methods such as EWC (Kirkpatrick et al., 2016) or MAS (Aljundi et al., 2017) introduce an importance weight $\Omega _ { k }$ for each parameter $\theta _ { k }$ in the network. While these methods differ in how to estimate the important parameters, all of them penalize changes to important parameters when learning a new task $T _ { n }$ using $L _ { 2 }$ penalty:
+
+$$
+T _ { n } : \quad \operatorname* { m i n } _ { \theta } { \frac { 1 } { M } } \sum _ { m = 1 } ^ { M } { \mathcal { L } } ( y _ { m } , f ( x _ { m } , \theta ^ { n } ) ) + \lambda _ { \Omega } \sum _ { k } \Omega _ { k } ( \theta _ { k } ^ { n } - \theta _ { k } ^ { n - 1 } ) ^ { 2 }
+$$
+
+where $\theta ^ { n - 1 } = \{ \theta _ { k } ^ { n - 1 } \}$ are the optimal parameters learned so far, i.e. before the current task. $\{ x _ { m } \}$ is the set of $M$ training inputs, with $\{ f ( x _ { m } , \theta ^ { n } ) \}$ and $\{ y _ { m } \}$ the corresponding predicted and desired outputs, respectively. $\lambda _ { \Omega }$ is a trade-off parameter between the new task objective $\mathcal { L }$ and the changes on the important parameters, i.e. the amount of forgetting.
+
+In this work we introduce an additional regularizer $R _ { \mathrm { S S L } }$ which encourages sparsity in the activations $H _ { l } = \{ h _ { i } ^ { m } \}$ for each layer $l$ .
+
+$$
+T _ { n } : \quad \operatorname* { m i n } _ { \theta } { \frac { 1 } { M } } \sum _ { m = 1 } ^ { M } { \mathcal { L } } ( y _ { m } , f ( x _ { m } , \theta ^ { n } ) ) + \lambda _ { \Omega } \sum _ { k } \Omega _ { k } ( \theta _ { k } ^ { n } - \theta _ { k } ^ { n - 1 } ) ^ { 2 } + \lambda _ { \mathrm { S S L } } \sum _ { l } R _ { \mathrm { S S L } } ( H _ { l } )
+$$
+
+$\lambda _ { \mathrm { S S L } }$ and $\lambda _ { \Omega }$ are trade-off parameters that control the contribution of each term. When training the first task $\dot { n } = 1$ ), $\Omega _ { k } = 0$ .
+
+# 3.1 SPARSE CODING THROUGH NEURAL INHIBITION (SNI)
+
+Now we describe how we obtain a sparse and decorrelated representation. In the literature sparsity has been proposed by Glorot et al. (2011) to be combined with the rectifier activation function (ReLU) to control unbounded activations and to increase sparsity. They minimize the $L _ { 1 }$ norm of the activations (since minimizing the $L _ { 0 }$ norm is an NP hard problem). However, $L _ { 1 }$ norm imposes an equal penalty on all the active neurons leading to small activation magnitude across the network.
+
+Learning a decorrelated representation has been explored before with the goal of reducing overfitting. This is usually done by minimizing the Frobenius norm of the covariance matrix corrected by the diagonal, as in Cogswell et al. (2015) or Xiong et al. (2016). Such a penalty results in a decorrelated representation but with activations that are mostly close to a non zero mean value. We merge the two objectives of sparse and decorrelated representation resulting in the following objective:
+
+$$
+R _ { \mathrm { S N I } } ( H _ { l } ) = \frac { 1 } { M } \sum _ { i , j } \sum _ { m } h _ { i } ^ { m } h _ { j } ^ { m } , \quad i \neq j
+$$
+
+where we consider a hidden layer $l$ with activations $H _ { l } = \{ h _ { i } ^ { m } \}$ for a set of inputs $X = \{ x _ { m } \}$ and $i , j \in { 1 , . . , N }$ running over all $N$ neurons in the hidden layer. This formula differs from minimizing the Frobenius norm of the covariance matrix in two simple yet important aspects:
+
+(1) In the case of a ReLU activation function, used in most modern architectures, a neuron is active if its output is larger than zero, and zero otherwise. By assuming a close to zero mean of the activations, $\mu _ { i } \simeq 0 \forall i \in { 1 , . . , N }$ , we minimize the correlation between any two active neurons.
+
+(2) By evaluating the derivative of the presented regularizer w.r.t. the activation, we get:
+
+$$
+\frac { \partial { R _ { \mathrm { S N I } } ( H _ { l } ) } } { \partial { h _ { i } ^ { m } } } = \frac { 1 } { M } \sum _ { j \neq i } { h _ { j } ^ { m } }
+$$
+
+i.e., each active neuron receives a penalty from every other active neuron that corresponds to that other neuron’s activation magnitude. In other words, if a neuron fires, with a high activation value, for a given example, it will suppress firing of other neurons for that same example. Hence, this results in a decorrelated sparse representation.
+
+# 3.2 SPARSE CODING THROUGH LOCAL NEURAL INHIBITION (SLNI)
+
+The loss imposed by the SNI objective will only be zero when there is at most one active neuron per example. This seems to be too harsh for complex tasks that need a richer representation. Thus, we suggest to relax the objective by imposing a spatial weighting to the correlation penalty. In other words, an active neuron penalizes mostly its close neighbours and this effect vanishes for neurons further away. Instead of uniformly penalizing all the correlated neurons, we weight the correlation penalty between two neurons with locations $i$ and $j$ using a Gaussian weighting. This gives
+
+$$
+R _ { \mathrm { S L N I } } ( H _ { l } ) = \frac { 1 } { M } \sum _ { i , j } e ^ { - \frac { ( i - j ) ^ { 2 } } { 2 \sigma ^ { 2 } } } \sum _ { m } h _ { i } ^ { m } h _ { j } ^ { m } , \quad i \neq j
+$$
+
+As such, each active neuron inhibits its neighbours, introducing a locality in the network inspired by biological neurons. While the notion of neighbouring neurons is not well established in a fully connected network, our aim is to allow few neurons to be active and not only one, thus those few activations don’t have to be small to compensate for the penalty. $\sigma ^ { 2 }$ is a hyper parameter representing the scale at which neurons can affect each other. Note that this is somewhat more flexible than decorrelating neurons in fixed groups as used in Xiong et al. (2016). Our regularizer inhibits locally the active neurons leading to a sparse coding through local neural inhibition.
+
+# 3.3 NEURON IMPORTANCE FOR DISCOUNTING INHIBITION
+
+Our regularizer is to be applied for each task in the learning sequence. In the case of tasks with completely different input patterns, the active neurons of the previous tasks will not be activated given the new tasks input patterns. However, when the new tasks are of similar or shared patterns, neurons used for previous tasks will be active. In that case, our penalty would discourage other neurons from being active and encourage the new task to adapt the already active neurons instead. This would interfere with the previous tasks and could increase forgetting which is exactly what we want to overcome. To avoid such interference, we add a weight factor taking into account the importance of the neurons with respect to the previous tasks. To estimate the importance of the neurons, we use as a measure the sensitivity of the loss at the end of the training to their changes. This is approximated by the gradients of the loss w.r.t. the neurons outputs (before the activation function) evaluated at each data point. To get an importance value, we then accumulate the absolute value of the gradients over the given data points obtaining importance weight $\alpha _ { i }$ for neuron $n _ { i }$ :
+
+$$
+\alpha _ { i } = \frac { 1 } { M } \sum _ { m = 1 } ^ { M } \mid g _ { i } ( x _ { m } ) \mid , g _ { i } ( x _ { m } ) = \frac { \partial ( \mathcal { L } ( y _ { m } , f ( x _ { m } , \theta ^ { n } ) ) ) } { \partial n _ { i } ^ { m } }
+$$
+
+where $n _ { i } ^ { m }$ is the output of neuron $n _ { i }$ for a given input example $x _ { m }$ , and $\theta ^ { n }$ are the parameters after learning task $n$ . This is in line with the estimation of the parameters importance in Kirkpatrick et al. (2016) but considering the derivation variables to be the neurons outputs instead of the parameters. Instead of relying on the gradient of the loss, we can also use the gradient of the learned function, i.e. the output layer, as done in Aljundi et al. (2017) for estimating the parameters importance. During the early phases of this work, we experimented with both and observed a similar behaviour. For sake of consistency and computational efficiency we utilize the gradient of the function when using Aljundi et al. (2017) as LLL method and the gradient of the loss when experimenting with EWC (Kirkpatrick et al., 2016). Then, we can weight our regularizer as follows:
+
+$$
+R _ { \mathrm { S L N I D } } ( H _ { l } ) = \frac { 1 } { M } \sum _ { i , j } e ^ { - ( \alpha _ { i } + \alpha _ { j } ) } e ^ { - \frac { ( i - j ) ^ { 2 } } { 2 \sigma ^ { 2 } } } \sum _ { m } h _ { i } ^ { m } h _ { j } ^ { m } , \quad i \neq j
+$$
+
+which can be read as: if an important neuron for a previous task is active given an input pattern from the current task, it will not suppress the other neurons from being active neither be affected by other active neurons. For all other active neurons, local inhibition is deployed. The final objective for training is given in Eq. 2, setting $R _ { S S L } : = R _ { \mathbb { S } \mathrm { L N I D } }$ and $\lambda _ { \mathrm { S S L } } : = \lambda _ { \mathrm { S L N I D } }$ . We refer to our full method as Sparse coding through Local Neural Inhibition and Discounting (SLNID).
+
+# 4 EXPERIMENTS
+
+In this section we study the role of standard regularization techniques with a focus on sparsity and decorrelation of the representation in a sequential learning scenario. We first compare different activation functions and regularization techniques, including our proposed SLNID, on permuted MNIST (Sec. 4.1). Then, we compare the top competing techniques and our proposed method in the case of sequentially learning CIFAR-100 classes and Tiny Imagenet classes (Sec. 4.2). Our SLNID regularizer can be integrated in any importance weight-based lifelong learning approach such as (Kirkpatrick et al., 2016; Zenke et al., 2017; Aljundi et al., 2017). Here we focus on Memory Aware Synapses (Aljundi et al., 2017) (MAS), which is easy to integrate and experiment with and has shown superior performance (Aljundi et al., 2017). However, we also show results with Elastic weight consolidation (Kirkpatrick et al., 2016)(EWC) in Sec. 4.3. Further, we ablate the components of our regularizer, both in the standard setting (Sec. 4.4) as in a setting without hard task boundaries (Sec. 4.5). Finally, we show how our regularizer improves the state-of-the-art performance on a sequence of object recognition tasks (Sec. 4.6).
+
+![](images/a943710c123ca7497643734a2a7f38fc19498d05d01d613c87af1695bddd7fbd.jpg)  
+Figure 2: Comparison of different regularization techniques on 5 permuted MNIST sequence. Representation based regularizers are solid bars, bars with lines represent parameters regularizers, dotted bars represent activation functions. Average test accuracy over all tasks is given in the legend. Representation based regularizers achieve higher performance than other compared methods including parameters based regularizers. Our regularizer, SLNID, performs the best on the last two tasks indicating that more capacity is left to learn these tasks.
+
+# 4.1 AN IN-DEPTH COMPARISON OF REGULARIZERS AND ACTIVATION FUNCTIONS FOR SELFLESS SEQUENTIAL LEARNING
+
+We study possible regularization techniques that could lead to less interference between the different tasks in a sequential learning scenario either by enforcing sparsity or decorrelation. Additionally, we examine the use of activation functions that are inspired by lateral inhibition in biological neurons that could be advantageous in sequential learning. MAS Aljundi et al. (2017) is used in all cases as LLL method.
+
+# Representation Based methods:
+
+- L1-Rep: To promote representational sparsity, an $L _ { 1 }$ penalty on the activations is used. - Decov (Cogswell et al., 2015) aims at reducing overfitting by decorrelating neuron activations. To do so, it minimizes the Frobenius norm of the covariance matrix computed on the activations of the current batch after subtracting the diagonal to avoid penalizing independent neuron activations.
+
+# Activation functions:
+
+- Maxout network (Goodfellow et al., 2013b) utilizes the maxout activation function. For each group of neurons, based on a fixed window size, only the maximum activation is forwarded to the next layer. The activation function guarantees a minimum sparsity rate defined by the window size.
+
+- LWTA (Srivastava et al., 2013): similar idea to the Maxout network except that the non-maximum activations are set to zero while maintaining their connections. In contrast to Maxout, LWTA keeps the connections of the inactive neurons which can be occupied later once they are activated without changing the previously active neuron connections.
+
+- ReLU (Glorot et al., 2011) The rectifier activation function (ReLU) used as a baseline here and indicated in later experiments as No-Reg as it represents the standard setting of sequential learning on networks with ReLU. All the studied regularizers use ReLU as activation function.
+
+# Parameters based regularizers:
+
+- OrthReg (Rodríguez et al., 2016): Regularizing CNNs with locally constrained decorrelations. It aims at decorrelating the feature detectors by minimizing the cosine of the angle between the weight vectors resulting eventually in orthogonal weight vectors.
+
+- L2-WD: Weight decay with $L _ { 2 }$ norm (Krogh & Hertz, 1992) controls the complexity of the learned function by minimizing the magnitude of the weights.
+
+- L1-Param: $L _ { 1 }$ penalty on the parameters to encourage a solution with sparse parameters.
+
+Dropout is not considered as its role contradicts our goal. While dropout can improve each task performance and reduce overfitting, it acts as a model averaging technique. By randomly masking neurons, dropout forces the different neurons to work independently. As such it encourages a redundant representation. As shown by (Goodfellow et al., 2013a) the best network size for classifying MNIST digits when using dropout was about $5 0 \%$ more than without it. Dropout steers the learning of a task towards occupying a good portion of the network capacity, if not all of it, which contradicts the sequential learning needs.
+
+Experimental setup. We use the MNIST dataset (LeCun et al., 1998) as a first task in a sequence of 5 tasks, where we randomly permute all the input pixels differently for tasks 2 to 5. The goal is to classify MNIST digits from all the different permutations. The complete random permutation of the pixels in each task requires the neural network to instantiate a new neural representation for each pattern. A similar setup has been used by Kirkpatrick et al. (2016); Zenke et al. (2017); Goodfellow et al. (2013a) with different percentage of permutations or different number of tasks.
+
+As a base network, we employ a multi layer perceptron with two hidden layers and a Softmax loss.
+
+![](images/0a6264583240b617e1e040b41563598f810710fdf2d38c6e58badab807f261c6.jpg)  
+Figure 3: Comparison of different regularization techniques on a sequence of ten tasks from (a) Cifar split and (b) Tiny ImageNet split. The legend shows average test accuracy over all tasks. Simple L1-norm regularizer (L1-Rep) doesn’t help in such more complex tasks. Our regularizer SLNID achieves an improvement of $2 \%$ over Decov and $4 - 8 \%$ compared to No-Reg.
+
+We experiment with different number of neurons in the hidden layers $\{ 1 2 8 , 6 4 \}$ . For SLNID we evaluate the effect of $\lambda _ {  { \mathrm { S I N I D } } }$ on the performance and the obtained sparsity in Figure 4. In general, the best $\lambda _ {  { \mathrm { S I N I D } } }$ is the minimum value that maintains similar or better accuracy on the first task compared to the unregularized case, and we suggest to use this as a rule-of-thumb to set $\lambda _ {  { \mathrm { S I N I D } } }$ . For $\lambda _ { \Omega }$ , we have used a high $\lambda _ { \Omega }$ value that ensures the least forgetting which allows us to test the effect on the later tasks performance. Note that better average accuracies can be obtained with tuned $\lambda _ { \Omega }$ . Please refer to Appendix A for hyperparameters and other details.
+
+Results: Figure 2 presents the test accuracy on each task at the end of the sequence, achieved by the different regularizers and activation functions on the network with hidden layer of size 128. Results on a network with hidden layer size 64 are shown in the Appendix B. Clearly, in all the different tasks, the representational regularizers show a superior performance to the other studied techniques. For the regularizers applied to the parameters, $\mathbb { L } 2 - \mathbb { W } \mathbb { D }$ and L1-Param do not exhibit a clear trend and do not systematically show an improvement over the use of the different activation functions only. While OrthReg shows a consistently good performance, it is lower than what can be achieved by the representational regularizers. It is worth noting the L1-Rep yields superior performance over L1-Param. This observation is consistent across different sizes of the hidden layers (in Appendix B) and shows the advantage of encouraging sparsity in the activations compared to that in the parameters. Regarding the activation functions, Maxout and LWTA achieve a slightly higher performance than ReLU. We did not observe a significant difference between the two activation functions. However, the improvement over ReLU is only moderate and does not justify the use of a fixed window size and special architecture design. Our proposed regularizer SLNID achieves high if not the highest performance in all the tasks and succeeds in having a stable performance. This indicates the ability of SLNID to direct the learning process towards using minimum amount of neurons and hence more flexibility for upcoming tasks.
+
+Representation sparsity $\pmb { \& }$ important parameter sparsity. Here we want to examine the effect of our regularizer on the percentage of parameters that are utilized after each task and hence the capacity left for the later tasks. On the network with hidden layer size 128, we compute the percentage of parameters with $\dot { \Omega _ { k } } < 1 0 ^ { - 2 }$ , with $\Omega _ { k }$ , see Appendix A, the importance weight multiplier estimated and accumulated over tasks. Those parameters can be seen as unimportant and "free" for later tasks. Figure 4(top) shows the percentage of the unimportant (free) parameters in the first layer after each task for different $\lambda _ { \mathrm { S I N I D } }$ values along with the achieved average test accuracy at the end of the sequence. It is clear that the larger $\lambda _ {  { \mathrm { S I N I D } } }$ , i.e., the more neural inhibition, the smaller the percentage of important parameters. Apart from the highest $\lambda _ {  { \mathrm { S I N I D } } }$ where tasks couldn’t reach their top performance due to too strong inhibition, improvement over the $\mathtt { N O - R e g }$ is always observed. The optimal value for lambda seems to be the one that remains close to the optimal performance on the current task, while utilizing the minimum capacity feasible. Next, we compute the average activation per neuron, in the first layer, over all the examples and plot the corresponding histogram for SLNID, DeCov, L1-Rep, L1-Param and $\mathtt { N O - R e g }$ in Figure 4(bottom) at their setting that yielded the results shown in Figure 2. SLNID has a peak at zero indicating representation sparsity while the other methods values are spread along the line. This seems to hint at the effectiveness of our approach SLNID in learning a sparse yet powerful representation and in turn in a minimal interference between tasks.
+
+![](images/dae61b2dab28f2402f7e04446e1101438b885208128f4260d4474c7ed719ef34.jpg)  
+Figure 4: On the 5 permuted MNIST sequence, hidden layer ${ } = 1 2 8$ , Top: percentage of unused parameters in the 1st layer using different $\lambda _ {  { \mathrm { S I N I D } } }$ ; Bottom: histogram of neural activations on the first task.
+
+Table 1: SLNID ablation. Average test accuracy per task after training the last task in $\%$ . \* denotes that MultiTask Joint Training violates the LLL scenario as it has access to all tasks at once and thus can be seen as an upper bound.   
+
+<table><tr><td></td><td colspan="2">Permuted mnist</td><td colspan="2">Cifar</td></tr><tr><td>h-layer dim.</td><td>128</td><td>64</td><td>256</td><td>128</td></tr><tr><td>No-Reg</td><td>92.67</td><td>90.72</td><td>55.06</td><td>55.3</td></tr><tr><td>SNI</td><td>95.79</td><td>94.89</td><td>55.30</td><td>55.75</td></tr><tr><td>SNID</td><td>95.90</td><td>93.82</td><td>61.00</td><td>60.90</td></tr><tr><td>SLNI</td><td>95.95</td><td>94.87</td><td>56.06</td><td>55.79</td></tr><tr><td>SLNID</td><td>95.83</td><td>93.89</td><td>63.30</td><td>61.16</td></tr><tr><td>Multi-Task Joint Training*</td><td>97.30</td><td>96.80</td><td>70.99</td><td>71.95</td></tr></table>
+
+Table 2: 8 tasks object recognition sequence. Average test accuracy per task after training the last task in $\%$ .   
+
+<table><tr><td>Method</td><td>Avg-acc</td></tr><tr><td>Finetune LWF (Li &amp; Hoiem,2016)</td><td>32.67</td></tr><tr><td>EBLL (Triki et al., 2017)</td><td>49.49</td></tr><tr><td>IMM (Lee et al., 2017)</td><td>50.29</td></tr><tr><td></td><td>43.4</td></tr><tr><td>Path Integral (Zenke et al., 2017)</td><td>50.49</td></tr><tr><td>EWC (Kirkpatrick et al., 2016)</td><td>50.00</td></tr><tr><td>MAS (Aljundi et al.,2017)</td><td>52.69</td></tr><tr><td>SLNID-fc Pretrained (ours)</td><td>53.77</td></tr><tr><td>SLNID-fc randomly initialized (ours)</td><td>54.50</td></tr></table>
+
+# 4.2 10 TASK SEQUENCES ON CIFAR-100 AND TINY IMAGENET
+
+While the previous section focused on learning a sequence of tasks with completely different input patterns and same objective, we now study the case of learning different categories of one dataset. For this we split the CIFAR-100 and the Tiny ImageNet (Yao & Miller, 2015) dataset into ten tasks, respectively. We have 10 and 20 categories per task for CIFAR-100 and Tiny ImagNet, respectively. Further details about the experimental setup can be found in appendix A.
+
+We compare the top competing methods from the previous experiments, L1-Rep, DeCov and our SLNID, and No-Reg as a baseline, ReLU in previous experiment. Similarly, MAS Aljundi et al. (2017) is used in all cases as LLL method. Figures 3(a) and 3(b) show the performance on each of the ten tasks at the end of the sequence. For both datasets, we observe that our SLNID performs overall best. $\mathtt { I 1 - R e p }$ and DeCov continue to improve over the non regularized case No-Reg. These results confirm our proposal on the importance of sparsity and decorrelation in sequential learning.
+
+# 4.3 SLNID WITH EWC (KIRKPATRICK ET AL., 2016)
+
+We have shown that our proposed regularizer SLNID exhibits stable and superior performance on the different tested networks when using MAS as importance weight preservation method. To prove the effectiveness of our regularizer regardless of the used importance weight based method, we have tested SLNID on the 5 tasks permuted MNIST sequence in combination with Elastic Weight Consolidation (EWC,Kirkpatrick et al. (2016)) and obtained a boost in the average performance at the end of the learned sequence equal to $3 . 1 \%$ on the network with hidden layer size 128 and a boost of $2 . 8 \%$ with hidden layer size 64. Detailed accuracies are shown in Appendix B. It is worth noting that with both MAS and EWC our SLNID was able obtain better accuracy using a network with a 64-dimensional hidden size than when training without regularization $\mathtt { N O - R e g }$ on a network of double that size (128), indicating that SLNID allows to use neurons much more efficiently.
+
+# 4.4 ABLATION STUDY
+
+Our method can be seen as composed of three components: the neural inhibition, the locality relaxation and the neuron importance integration. To study how these components perform individually, Table 1 reports the average accuracy at the end of the Cifar 100 and permuted MNIST sequences for each variant, namely, SNID without neuron importance (SNI), SNID, SLNID without neuron importance (SLNI) in addition to our full SLNID regularizer. As we explained in Section 3, when tasks have completely different input patterns, the neurons that were activated on the previous task examples will not fire for new task samples and exclusion of important neurons is not mandatory. However, when sharing is present between the different tasks, a term to prevent SLNID from causing any interference is required. This is manifested in the reported results: for permuted MNIST, all the variants work nicely alone, as a result of the simplicity and the disjoint nature of this sequence. However, in the Cifar 100 sequence, the integration of the neuron importance in the SNID and SLNID regularizers exclude important neurons from the inhibition, resulting in a clearly better performance. The locality in SLNID improves the performance in the Cifar sequence, which suggests that a richer representation is needed and multiple active neurons should be tolerated.
+
+In the previous experiments, we considered the standard task based scenario as in (Li & Hoiem, 2016; Zenke et al., 2017; Aljundi et al., 2017; Serrà et al., 2018), where at each time step we receive a task along with its training data and a new classification layer is initiated for the new task, if needed. Here, we are interested in a more realistic scenario where the data distribution shifts gradually without hard task boundaries.
+
+To test this setting, we use the Cifar 100 dataset. Instead of considering a set of 10 disjoint tasks each composed of 10 classes, as in the previous experiment (Sec. 4.2), we now start by sampling with high probability $( 2 / 3 )$ from the first 10 classes and with low probability $( 1 / 3 )$ from the rest of the classes. We train the network (same architecture as in Sec. 4.2) for a few epochs and then change the sampling probabilities to be high $( 2 / 3 )$ for classes $1 1 - 2 0$ and low $( 1 / 3 )$ for the remaining classes. This process is repeated until sampling with high probability from the last 10 classes and low from the rest. We use one shared classification layer throughout and estimate the importance weights and the neurons importance after each training step (before changing the sampling probabilities). We consider 6 variants: our SLNID, the ablations SLNI and without regularizer $\mathtt { N O - R e g }$ , as in Section 4.4, as well each of these three trained without the MAS importance weight regularizer of Aljundi et al. (2017), denoted as $\mathtt { w } / \mathtt { o }$ MAS. Table 3 presents the accuracy averaged over the ten groups of ten classes, using each group model (i.e. the model trained when this group was sampled with high probability) in the top block and the average accuracy on each of the ten groups at the end of the training (middle and bottom block). We can deduce the following: 1) SLNID improves the performance considerably (by more than $4 \%$ ) even without importance weight regularizer. 2) In this scenario without hard task boundaries there is less forgetting than in the scenario with hard task boundaries studied in Section 4.2 for Cifar (difference between rows in top block to corresponding rows in middle block). As a result, the improvement obtained by deploying the importance weight regularizer is moderate: at $7 0 . 7 5 \%$ , SLNID $\mathtt { w } / \mathtt { o }$ MAS is already better than $\mathtt { N O - R e g }$ reaching $6 6 . 3 3 \%$ . 3) While SLNI without MAS improves the individual models performance $( 7 2 . 1 4 \%$ compared to $6 9 . 2 0 \%$ ), it fails to improve the overall performance at the end of the sequence $6 3 . 5 4 \%$ compared to $6 5 . 1 5 \%$ ), as important neurons are not excluded from the penalty and hence they are changed or inhibited leading to tasks interference and performance deterioration.
+
+Table 3: No tasks boundaries test case on Cifar 100. Top block, avg. acc on each group of classes using each group model. Bottom block, avg. acc. on each group at the end of the training.   
+
+<table><tr><td rowspan=1 colspan=1>Method</td><td rowspan=1 colspan=1>Avg.acc-tasks models</td></tr><tr><td rowspan=1 colspan=1>No-Reg w/o MASSLNI W/O MASSLNID W/O MASNo-RegSLNISLNID</td><td rowspan=1 colspan=1>69.20%72.14%73.03%66.88%71.32%72.33%</td></tr><tr><td rowspan=1 colspan=1>Method</td><td rowspan=1 colspan=1>Avg.acc-last model</td></tr><tr><td rowspan=1 colspan=1>No-Reg w/o MASSLNI W/O MASSLNID W/O MAS</td><td rowspan=1 colspan=1>65.15%63.54%70.75%</td></tr><tr><td rowspan=1 colspan=1>No-RegSLNISLNID</td><td rowspan=1 colspan=1>66.33%64.50%70.94%</td></tr></table>
+
+# 4.6 COMPARISON WITH THE STATE OF THE ART
+
+To compare our proposed approach with the different state-of-the-art sequential learning methods, we use a sequence of 8 different object recognition tasks, introduced in Aljundi et al. (2017). The sequence starts from AlexNet (Krizhevsky et al., 2012) pretrained on ImageNet (Russakovsky et al., 2015) as a base network, following the setting of Aljundi et al. (2017). More details are in Appendix A.4. We compare against the following: Learning without Forgetting (Li & Hoiem, 2016) (LwF), Incremental Moment Matching (Lee et al., 2017) (IMM), Path Integral (Zenke et al., 2017) and sequential finetuning (FineTuning), in addition to the case of MAS (Aljundi et al., 2017) alone, i.e. our No-Reg before. Compared methods were run with the exact same setup as in Aljundi et al. (2017). For our regularizer, we disable dropout, since dropout encourages redundant activations which contradicts our regularizer’s role. Also, since the network is pretrained, the locality introduced in SLNID may conflict with the already pretrained activations. For this reason, we also test SLNID with randomly initialized fully connected layers. Our regularizer is applied with MAS as a sequential learning method. Table 2 reports the average test accuracy at the end of the sequence achieved by each method. SLNID improves even when starting from a pretrained network and disabling dropout. Surprisingly, even with randomly initialized fully connected layers, SLNID improves $1 . 8 \%$ over the state of the art using a fully pretrained network.
+
+# 5 CONCLUSION
+
+In this paper we study the problem of sequential learning using a network with fixed capacity – a prerequisite for a scalable and computationally efficient solution. A key insight of our approach is that in the context of sequential learning (as opposed to other contexts where sparsity is imposed, such as network compression or avoiding overfitting), sparsity should be imposed at the level of the representation rather than at the level of the network parameters. Inspired by lateral inhibition in the mammalian brain, we impose sparsity by means of a new regularizer that decorrelates nearby active neurons. We integrate this in a model which learns selflessly a new task by leaving capacity for future tasks and at the same time avoids forgetting previous tasks by taking into account neurons importance.
+
+Acknowledgment: The first author’s PhD is funded by an FWO scholarship.
+
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+
+# APPENDIX
+
+# A DETAILS ON THE EXPERIMENTAL SETUP
+
+In all designed experiments, our regularizer is applied to the neurons of the fully connected layers. As a future work, we plan to integrate it in the convolutional layers.
+
+# A.1 PERMUTED MNIST
+
+The used network is composed of two fully connected layers. All tasks are trained for 10 epochs with a learning rate $1 0 ^ { - 2 }$ using SGD optimizer. ReLU is used as an activation function unless mentioned otherwise. Throughout the experiment, we used a scale $\sigma$ for the Gaussian function used for the local inhibition equal to $1 / 6$ of the hidden layer size. For all competing regularizers, we tested different hyper parameters from $\mathrm { i 0 ^ { - 2 } }$ to $\mathrm { i 0 ^ { - 9 } }$ and report the best one. For $\lambda _ { \Omega }$ , we have used a high $\lambda _ { \Omega }$ value that ensures the least forgetting. This allows us to examine the degradation in the performance on the later tasks compared to those learned previously as a result of lacking capacity. Note that better average accuracies can be obtained with tuned $\lambda _ { \Omega }$ .
+
+In section 4.1 we estimated the free capacity in the network with the percentage of $\Omega _ { k } < 1 0 ^ { - 2 }$ , with $\Omega _ { k }$ , the importance weight multiplier estimated and accumulated over tasks. We consider $\Omega _ { k } < 1 0 ^ { - 2 }$ of negligible importanc as in a network trained without a sparsity regularizer, $\Omega _ { i j } < 1 0 ^ { - 2 }$ covers the first 10 percentiles.
+
+# A.2 CIFAR-100
+
+As a base network, we use a network similar to the one used by Zenke et al. (2017) but without dropout. We evaluate two variants with hidden size $N = \{ 2 5 6 , 1 2 8 \}$ . Throughout the experiment, we again used a scale $\sigma$ for the Gaussian function equal to $1 / 6$ of the hidden layer size. We train the different tasks for 50 epochs with a learning rate of $1 0 ^ { - 2 }$ using SGD optimizer.
+
+# A.3 TINY IMAGENET
+
+We split the Tiny ImageNet dataset (Yao & Miller, 2015) into ten tasks, each containing twenty categories to be learned at once. As a base network, we use a variant of VGG (Simonyan & Zisserman, 2014). For architecture details, please refer to Table 4 below.
+
+Table 4: Architecture of the network used in the Tiny Imagenet experiment.   
+
+<table><tr><td>Layer</td><td>#filters/neurons</td></tr><tr><td>Convolution</td><td>64</td></tr><tr><td>Max Pooling</td><td>1</td></tr><tr><td>Convolution Max Pooling</td><td>128</td></tr><tr><td>Convolution</td><td>= 256</td></tr><tr><td>Max Pooling Convolution</td><td>-</td></tr><tr><td>Max Pooling</td><td>256 -</td></tr><tr><td>Convolution</td><td>512</td></tr><tr><td>Convolution</td><td>512</td></tr><tr><td>Fully connected</td><td>500</td></tr><tr><td>Fully connected</td><td>500</td></tr><tr><td>Fully connected</td><td>20</td></tr></table>
+
+Throughout the experiment, we again used a scale $\sigma$ for the Gaussian function equal to $1 / 6$ of the hidden layer size.
+
+# A.4 8 TASK OBJECT RECOGNITION SEQUENCE
+
+The 8 tasks sequence is composed of: 1. Oxford Flowers (Nilsback & Zisserman, 2008), 2. MIT Scenes (Quattoni & Torralba, 2009), 3. Caltech-UCSD Birds (Welinder et al., 2010), 4. Stanford Cars (Krause et al., 2013); 5. FGVC-Aircraft (Maji et al., 2013); 6. VOC Actions (Everingham et al.); 7. Letters (de Campos et al., 2009); and 8. SVHN (Netzer et al., 2011) datasets. We have rerun the different methods and obtain the same reported results as in Aljundi et al. (2017).
+
+# B EXTRA RESULTS
+
+# B.1 PERMUTED MNIST SEQUENCE
+
+In section 4.1, we have studied the performance of different regularizers and activation functions on 5 permuted Mnist tasks in a network with a hidden layer of size 128. Figure 5 shows the average accuracies achieved by each of the studied methods at the end of the learned sequence in a network with a hidden layer of size 64. Similar conclusions can be drawn. Maxout and LWTA perform similarly and improve slightly over ReLU. Regularizers applied to the representation are more powerful for sequential learning than regularizers applied directly to the parameters. Specifically, L1-Rep (orange) is consistently better than L1-Param (pink). Our SLNID is able of maintaining a good performance on all the tasks, achieving among the top average test accuracies. Admittedly, the performances of SLNID is very close to L1-Rep. The difference between these methods stands out more clearly for larger networks and more complex tasks.
+
+![](images/426ea32269169ea2b750be443a9e150e3399e3e4de3329a2a9c22532080af4fc.jpg)  
+Figure 5: Comparison of different regularization techniques on 5 permuted MNIST sequence of tasks, hidden siz $_ { = 6 4 }$ . Representation based regularizers are solid bars, bars with lines represent parameters regularizers, dotted bars represent activation functions. See Figure 2 for size 128.
+
+# B.2 SLNI WITH EWC
+
+To show that our approach is not limited to MAS (Aljundi et al., 2017), we have also experimented with EWC (Kirkpatrick et al., 2016) as another importance weight based method along with our regularize SLNID on the permuted Mnist sequence. Figure 6 shows the test accuracy of each task at the end of the 5 permuted Mnist sequence achieved by our SLNID combined with EWC and by No-Reg (here indicating EWC without regularization). It is clear that SLNID succeeds to improve the performance on all the learned tasks which validates the utility of our approach with different sequential learning methods.
+
+![](images/15353dabacd58a660d959f4d5f33614fda2ee3f66a2db60dc05cba778f6ba340.jpg)  
+Figure 6: (a) SLNID with EWC on 5 permuted Mnist sequence of tasks, hidden size ${ \mathrm { - } } 1 2 8$ , (b) hidden size $_ { = 6 4 }$
+
+# B.3 CIFAR 100 SEQUENCE
+
+In section 4.2 we have tested SLNID and other representation regularizers on the Cifar 100 sequence. In Figure 3(a) we compare their performance on a network with hidden layer size 256. Figure 7 repeats the same experiment for a network with hidden size 128. While DeCov and SLNID continue to improve over $\mathtt { N o - R e g }$ , L1-Rep seems to suffer in this case. Our interpretation is that L1-Rep here interferes with the previously learned tasks while penalizing activations and hence suffers from catastophic forgetting. In line with all the previous experiments SLNID achieves the best accuracies and manages here to improve over $6 \%$ compared to No-Reg.
+
+![](images/fd932cee5ace2998546abfcd322e62e314994e57d05fc4075ba9b69baba5ef5f.jpg)  
+Figure 7: Comparison of different regularization techniques on a sequence of ten tasks from Cifar split. Hidden size ${ = } 1 2 8$ . See Figure 3(a) for size 256.
+
+# B.4 SPATIAL LOCALITY TEST
+
+To avoid penalizing all the active neurons, our SLNID weights the correlation penalty between each two neurons based on their spatial distance using a Gaussian function. We want to visualize the effect of this spatial locality on the neurons activity. To achieve this, we have used the first 3 tasks of the Permuted Mnist sequence as a test case and visualized the neurons importance after each task. This is done using the network of hidden layer size 64. Figure 8, Figure 9 and Figure 10 show the neurons importance after each task. The left column is without locality, i.e. SLNID, and the right column is SLNID. Blue represents the first task, orange the second task and green the third task. When using SLNID, inhibition is applied in a local manner allowing more active neurons which could potentially improve the representation power. When learning the second task, new neurons become important regardless of their closeness to first task important neurons as those neurons are excluded from the inhibition. As such, new neurons are becoming active as new tasks are learned. For SLNID all neural correlation is penalized in the first task. And for later tasks, very few neurons are able to become active and important for the new task due to the strong global inhibition, where previous neurons that are excluded from the inhibition are easier to be re-used.
+
+![](images/d121a16b0b65d4110e291158fc5d80d4f8d1b4b3b13f90220de5cf29f8ad24c1.jpg)  
+Figure 8: First layer neuron importance after learning the first task (blue). Left: SNID, Right: SLNID. More active neurons are tolerated in SLNID.
+
+![](images/534c49cb2f6e67c941bc41d34b7785782b09e1c97da1c2f2202f42e2b61d9e5a.jpg)  
+Figure 9: First layer neuron importance after learning the second task (orange), superimposed on Figure 8. Left: SNID, Right: SLNID. SLNID allows new neurons, especially those that were close neighbours to previous important neurons, to become active and to be used for the new task. SNID penalizes all unimportant neurons equally. As a result, previous neurons are adapted for the new tasks and less new neurons are getting activated.
+
+![](images/3a8ed2ed7e84014a80d4130e98561e6c8d48831687301a1afe0f5451eae97906.jpg)  
+Figure 10: First layer neuron importance after learning the third task (green), superimposed on Figure 9. Left: SNID, Right: SLNID. SLNID allows previous neurons to be re-used for the third task. It avoids changing the previous important neurons by adding new neurons. For SNID, very few neurons are newly deployed. The new task is learned mostly by adapting previous important neurons, causing more interference.
+
+![](images/b9e18165821532a24d7e7ca918c946f828578386e81415ddd78f062fd5452737.jpg)  
+Figure 11: First layer neuron importance after learning the first task, sorted in descending order according to the first task neuron importance (blue). Left: SNID, Right: SLNID. More active neurons are tolerated in SLNID.
+
+![](images/5881d02fece910d1a8f8f1b183ef5dc576afe7c19526a3d3b56230976f3e9117.jpg)  
+Figure 12: First layer neuron importance after learning the second task sorted in descending order according to the first task neuron importance (orange), superimposed on top of figure 11. Left: SNID, Right: SLNID. SLNID allows new neurons to become active and be used for the new task. SNID penalizes all unimportant neurons equally and hence more neurons are re-used then initiated for the first time.
+
+![](images/b84013cb534a4daa9e19bdf1d57068a82731cf7b591edb3810098c4ad325266c.jpg)  
+Figure 13: First layer neuron importance after learning the third task sorted in descending order according to the first task neuron importance (green), superimposed on top of figure 12. Left: SNID, Right: SLNID. SLNID allows previous neurons to be re-used for the third task while activating new neurons to cope with the needs of the new task. For SNID, very few neurons are newly deployed while most previous important neurons for previous tasks are re-adapted to learn the new task.

parse/train/Bkxbrn0cYX/Bkxbrn0cYX_content_list.json

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+    {
+        "type": "text",
+        "text": "SELFLESS SEQUENTIAL LEARNING ",
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+        "type": "text",
+        "text": "Rahaf Aljundi   \nKU Leuven   \nESAT-PSI, Belgium   \nrahaf.aljundi@gmail.com   \nTinne Tuytelaars   \nKU Leuven   \nESAT-PSI, Belgium   \ntinne.tuytelaars@esat.kuleuven.be ",
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+    {
+        "type": "text",
+        "text": "ABSTRACT ",
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+        "type": "text",
+        "text": "Sequential learning, also called lifelong learning, studies the problem of learning tasks in a sequence with access restricted to only the data of the current task. In this paper we look at a scenario with fixed model capacity, and postulate that the learning process should not be selfish, i.e. it should account for future tasks to be added and thus leave enough capacity for them. To achieve Selfless Sequential Learning we study different regularization strategies and activation functions. We find that imposing sparsity at the level of the representation (i.e. neuron activations) is more beneficial for sequential learning than encouraging parameter sparsity. In particular, we propose a novel regularizer, that encourages representation sparsity by means of neural inhibition. It results in few active neurons which in turn leaves more free neurons to be utilized by upcoming tasks. As neural inhibition over an entire layer can be too drastic, especially for complex tasks requiring strong representations, our regularizer only inhibits other neurons in a local neighbourhood, inspired by lateral inhibition processes in the brain. We combine our novel regularizer with state-of-the-art lifelong learning methods that penalize changes to important previously learned parts of the network. We show that our new regularizer leads to increased sparsity which translates in consistent performance improvement on diverse datasets. ",
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+    {
+        "type": "text",
+        "text": "1 INTRODUCTION ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "Sequential learning, also referred to as continual, incremental, or lifelong learning (LLL), studies the problem of learning a sequence of tasks, one at a time, without access to the training data of previous or future tasks. When learning a new task, a key challenge in this context is how to avoid catastrophic interference with the tasks learned previously (French, 1999; Li & Hoiem, 2016). Some methods exploit an additional episodic memory to store a small amount of previous tasks data to regularize future task learning (e.g. Lopez-Paz et al. (2017)). Others store previous tasks models and at test time, select one model or merge the models (Rusu et al., 2016; Aljundi et al., 2016; Lee et al., 2017). In contrast, in this work we are interested in the challenging situation of learning a sequence of tasks without access to any previous or future task data and restricted to a fixed model capacity, as also studied in Kirkpatrick et al. (2016); Aljundi et al. (2017); Fernando et al. (2017); Mallya & Lazebnik (2017); Serrà et al. (2018). This scenario not only has many practical benefits, including privacy and scalability, but also resembles more closely how the mammalian brain learns tasks over time. ",
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+    {
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+        "text": "The mammalian brain is composed of billions of neurons. Yet at any given time, information is represented by only a few active neurons resulting in a sparsity of $9 0 { - } 9 5 \\%$ (Lennie, 2003). In neural biology, lateral inhibition describes the process where an activated neuron reduces the activity of its weaker neighbors. This creates a powerful decorrelated and compact representation with minimum interference between different input patterns in the brain (Yu et al., 2014). This is in stark contrast with artificial neural networks, which typically learn dense representations that are highly entangled (Bengio et al., 2009). Such an entangled representation is quite sensitive to changes in the input patterns, in that it responds differently to input patterns with only small variations. French (1999) suggests that an overlapped internal representation plays a crucial role in catastrophic forgetting and reducing this overlap would result in a reduced interference. Cogswell et al. (2015) show that when the amount of overfitting in a neural network is reduced, the representation correlation is also reduced. As such, learning a disentangled representation is more powerful and less vulnerable to catastrophic interference. However, if the learned disentangled representation at a given task is not sparse, only little capacity is left for the learning of new tasks. This would in turn result in either an underfitting to the new tasks or again a forgetting of previous tasks. In contrast, a sparse and decorrelated representation would lead to a powerful representation and at the same time enough free neurons that can be changed without interference with the neural activations learned for the previous tasks. ",
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+        "img_path": "images/a017368bfb65afbb42bcb6dc21d6c13cc867ef1742a3c8d5f95e153027446ce3.jpg",
+        "image_caption": [
+            "Figure 1: The difference between parameter sparsity (a) and representation sparsity (b) in a simple two tasks case. First layer indicates input patterns. Learning the first task utilizes parts indicated in red. Task 2 has different input patterns and uses parts shown in green. Orange indicates changed neurons activations as a result of the second task. In (a), when an example from the first task is encountered again, the activations of the first layer will not be affected by the changes, however, the second and later layer activations are changed. Such interference is largely reduced when imposing sparsity on the representation (b). "
+        ],
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+    {
+        "type": "text",
+        "text": "In general, sparsity in neural networks can be thought of either in terms of the network parameters or in terms of the representation (i.e., the activations). In this paper we postulate, and confirm experimentally, that a sparse and decorrelated representation is preferable over parameter sparsity in a sequential learning scenario. There are two arguments for this: first, a sparse representation is less sensitive to new and different patterns (such as data from new tasks) and second, the training procedure of the new tasks can use the free neurons leading to less interference with the previous tasks, hence reducing forgetting. In contrast, when the effective parameters are spread among different neurons, changing the ineffective ones would change the function of their corresponding neurons and hence interfere with previous tasks (see also Figure 1). Based on these observations, we propose a new regularizer that exhibits a behavior similar to the lateral inhibition in biological neurons. The main idea of our regularizer is to penalize neurons that are active at the same time. This leads to more sparsity and a decorrelated representation. However, complex tasks may actually require multiple active neurons in a layer at the same time to learn a strong representation. Therefore, our regularizer, Sparse coding through Local Neural Inhibition and Discounting (SLNID), only penalizes neurons locally. Furthermore, we don’t want inhibition to affect previously learned tasks, even if later tasks use neurons from earlier tasks. An important component of SLNID is thus to discount inhibition from/to neurons which have high neuron importance – a new concept that we introduce in analogy to parameter importance (Kirkpatrick et al., 2016; Zenke et al., 2017; Aljundi et al., 2017). When combined with a state-of-the-art important parameters preservation method (Aljundi et al., 2017; Kirkpatrick et al., 2016), our proposed regularizer leads to sparse and decorrelated representations which improves the lifelong learning performance. ",
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+    {
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+        "text": "Our contribution is threefold. First, we direct attention to Selfless Sequential Learning and study a diverse set of representation based regularizers, parameter based regularizers, as well as sparsity inducing activation functions to this end. These have not been studied extensively in the lifelong learning literature before. Second, we propose a novel regularizer, SLNID, which is inspired by lateral inhibition in the brain. Third, we show that our proposed regularizer consistently outperforms alternatives on three diverse datasets (Permuted MNIST, CIFAR, Tiny Imagenet) and we compare to and outperform state-of-the-art LLL approaches on an 8-task object classification challenge. SLNID can be applied to different regularization based LLL approaches, and we show experiments with MAS (Aljundi et al., 2017) and EWC (Kirkpatrick et al., 2016). ",
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+        "text": "In the following, we first discuss related approaches to LLL and different regularization criteria from a LLL perspective (Section 2). We proceed by introducing Selfless Sequential Learning and detailing our novel regularizer (Section 3). Section 4 describes our experimental evaluation, while Section 5 concludes the paper. ",
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+        "type": "text",
+        "text": "2 RELATED WORK ",
+        "text_level": 1,
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+        "text": "The goal in lifelong learning is to learn a sequence of tasks without catastrophic forgetting of previously learned ones (Thrun & Mitchell, 1995). One can identify different approaches to introducing lifelong learning in neural networks. Here, we focus on learning a sequence of tasks using a fixed model capacity, i.e. with a fixed architecture and fixed number of parameters. Under this setting, methods either follow a pseudo rehearsal approach, i.e. using the new task data to approximate the performance of the previous task (Li & Hoiem, 2016; Triki et al., 2017), or aim at identifying the important parameters used by the current set of tasks and penalizing changes to those parameters by new tasks (Kirkpatrick et al., 2016; Zenke et al., 2017; Aljundi et al., 2017; Chaudhry et al., 2018; Liu et al., 2018). To identify the important parameters for a given task, Elastic Weight Consolidation (Kirkpatrick et al., 2016) uses an approximation of the Fisher information matrix computed after training a given task. Liu et al. (2018) suggest a network reparameterization to obtain a better diagonal approximation of the Fisher Information matrix of the network parameters. Path Integral (Zenke et al., 2017) estimates the importance of the network parameters while learning a given task by accumulating the contribution of each parameter to the change in the loss. Chaudhry et al. (2018) suggest a KL-divergence based generalization of Elastic Weight Consolidation and Path Integral. Memory Aware Synapses (Aljundi et al., 2017) estimates the importance of the parameters in an online manner without supervision by measuring the sensitivity of the learned function to small perturbations on the parameters. This method is less sensitive to the data distribution shift, and a local version proposed by the authors resembles applying Hebb rule (Hebb, 2002) to consolidate the important parameters, making it more biologically plausible. ",
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+        "text": "A common drawback of all the above methods is that learning a task could utilize a good portion of the network capacity, leaving few \"free\" neurons to be adapted by the new task. This in turn leads to inferior performance on the newly learned tasks or forgetting the previously learned ones, as we will show in the experiments. Hence, we study the role of sparsity and representation decorrelation in sequential learning. This aspect has not received much attention in the literature yet. Very recently, (Serrà et al., 2018) proposed to overcome catastrophic forgetting through learned hard attention masks for each task with L1 regularization imposed on the accumulated hard attention masks. This comes closer to our approach although we study and propose a regularization scheme on the learned representation. ",
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+        "type": "text",
+        "text": "The concept of reducing the representation overlap has been suggested before in early attempts towards overcoming catastrophic forgetting in neural networks (French, 1999). This has led to several methods with the goal of orthogonalizing the activations (French, 1992; 1994; Kruschke, 1992; 1993; Sloman & Rumelhart, 1992). However, these approaches are mainly designed for specific architectures and activation functions, which makes it hard to integrate them in recent neural network structures. ",
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+        "type": "text",
+        "text": "The sparsification of neural networks has mostly been studied for compression. SVD decomposition can be applied to reduce the number of effective parameters (Xue et al., 2013). However, there is no guarantee that the training procedure converges to a low rank weight matrix. Other works iterate between pruning and retraining of a neural network as a post processing step (Liu et al., 2015; Sun et al., 2016; Aghasi et al., 2017; Louizos et al., 2017). While compressing a neural network by removing parameters leads to a sparser neural network, this does not necessarily lead to a sparser representation. Indeed, a weight vector can be highly sparse but spread among the different neurons. This reduces the effective size of a neural network, from a compression point of view, but it would not be beneficial for later tasks as most of the neurons are already occupied by the current set of tasks. In our experiments, we show the difference between using a sparse penalty on the representation versus applying it to the weights. ",
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+        "type": "text",
+        "text": "3 SELFLESS SEQUENTIAL LEARNING ",
+        "text_level": 1,
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+        "text": "One of the main challenges in single model sequential learning is to have capacity to learn new tasks and at the same time avoid catastrophic forgetting of previous tasks as a result of learning new tasks. In order to prevent catastrophic forgetting, importance weight based methods such as EWC (Kirkpatrick et al., 2016) or MAS (Aljundi et al., 2017) introduce an importance weight $\\Omega _ { k }$ for each parameter $\\theta _ { k }$ in the network. While these methods differ in how to estimate the important parameters, all of them penalize changes to important parameters when learning a new task $T _ { n }$ using $L _ { 2 }$ penalty: ",
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+        "type": "equation",
+        "img_path": "images/706098f4726f93046f994e9a504ebb5f6e5b3a270dfc7eff0d39a96bb48e5c06.jpg",
+        "text": "$$\nT _ { n } : \\quad \\operatorname* { m i n } _ { \\theta } { \\frac { 1 } { M } } \\sum _ { m = 1 } ^ { M } { \\mathcal { L } } ( y _ { m } , f ( x _ { m } , \\theta ^ { n } ) ) + \\lambda _ { \\Omega } \\sum _ { k } \\Omega _ { k } ( \\theta _ { k } ^ { n } - \\theta _ { k } ^ { n - 1 } ) ^ { 2 }\n$$",
+        "text_format": "latex",
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+        "type": "text",
+        "text": "where $\\theta ^ { n - 1 } = \\{ \\theta _ { k } ^ { n - 1 } \\}$ are the optimal parameters learned so far, i.e. before the current task. $\\{ x _ { m } \\}$ is the set of $M$ training inputs, with $\\{ f ( x _ { m } , \\theta ^ { n } ) \\}$ and $\\{ y _ { m } \\}$ the corresponding predicted and desired outputs, respectively. $\\lambda _ { \\Omega }$ is a trade-off parameter between the new task objective $\\mathcal { L }$ and the changes on the important parameters, i.e. the amount of forgetting. ",
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+        "text": "In this work we introduce an additional regularizer $R _ { \\mathrm { S S L } }$ which encourages sparsity in the activations $H _ { l } = \\{ h _ { i } ^ { m } \\}$ for each layer $l$ . ",
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+        "type": "equation",
+        "img_path": "images/0de6add6897df8e9f35d6532e0b31026ac434e5c3b2dfe89bca097015c84cc9e.jpg",
+        "text": "$$\nT _ { n } : \\quad \\operatorname* { m i n } _ { \\theta } { \\frac { 1 } { M } } \\sum _ { m = 1 } ^ { M } { \\mathcal { L } } ( y _ { m } , f ( x _ { m } , \\theta ^ { n } ) ) + \\lambda _ { \\Omega } \\sum _ { k } \\Omega _ { k } ( \\theta _ { k } ^ { n } - \\theta _ { k } ^ { n - 1 } ) ^ { 2 } + \\lambda _ { \\mathrm { S S L } } \\sum _ { l } R _ { \\mathrm { S S L } } ( H _ { l } )\n$$",
+        "text_format": "latex",
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+        "text": "$\\lambda _ { \\mathrm { S S L } }$ and $\\lambda _ { \\Omega }$ are trade-off parameters that control the contribution of each term. When training the first task $\\dot { n } = 1$ ), $\\Omega _ { k } = 0$ . ",
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+        "type": "text",
+        "text": "3.1 SPARSE CODING THROUGH NEURAL INHIBITION (SNI) ",
+        "text_level": 1,
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+        "text": "Now we describe how we obtain a sparse and decorrelated representation. In the literature sparsity has been proposed by Glorot et al. (2011) to be combined with the rectifier activation function (ReLU) to control unbounded activations and to increase sparsity. They minimize the $L _ { 1 }$ norm of the activations (since minimizing the $L _ { 0 }$ norm is an NP hard problem). However, $L _ { 1 }$ norm imposes an equal penalty on all the active neurons leading to small activation magnitude across the network. ",
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+        "text": "Learning a decorrelated representation has been explored before with the goal of reducing overfitting. This is usually done by minimizing the Frobenius norm of the covariance matrix corrected by the diagonal, as in Cogswell et al. (2015) or Xiong et al. (2016). Such a penalty results in a decorrelated representation but with activations that are mostly close to a non zero mean value. We merge the two objectives of sparse and decorrelated representation resulting in the following objective: ",
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+        "text": "$$\nR _ { \\mathrm { S N I } } ( H _ { l } ) = \\frac { 1 } { M } \\sum _ { i , j } \\sum _ { m } h _ { i } ^ { m } h _ { j } ^ { m } , \\quad i \\neq j\n$$",
+        "text_format": "latex",
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+        "type": "text",
+        "text": "where we consider a hidden layer $l$ with activations $H _ { l } = \\{ h _ { i } ^ { m } \\}$ for a set of inputs $X = \\{ x _ { m } \\}$ and $i , j \\in { 1 , . . , N }$ running over all $N$ neurons in the hidden layer. This formula differs from minimizing the Frobenius norm of the covariance matrix in two simple yet important aspects: ",
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+        "text": "(1) In the case of a ReLU activation function, used in most modern architectures, a neuron is active if its output is larger than zero, and zero otherwise. By assuming a close to zero mean of the activations, $\\mu _ { i } \\simeq 0 \\forall i \\in { 1 , . . , N }$ , we minimize the correlation between any two active neurons. ",
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+        "type": "text",
+        "text": "(2) By evaluating the derivative of the presented regularizer w.r.t. the activation, we get: ",
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+        "text": "$$\n\\frac { \\partial { R _ { \\mathrm { S N I } } ( H _ { l } ) } } { \\partial { h _ { i } ^ { m } } } = \\frac { 1 } { M } \\sum _ { j \\neq i } { h _ { j } ^ { m } }\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "i.e., each active neuron receives a penalty from every other active neuron that corresponds to that other neuron’s activation magnitude. In other words, if a neuron fires, with a high activation value, for a given example, it will suppress firing of other neurons for that same example. Hence, this results in a decorrelated sparse representation. ",
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+    {
+        "type": "text",
+        "text": "3.2 SPARSE CODING THROUGH LOCAL NEURAL INHIBITION (SLNI) ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "The loss imposed by the SNI objective will only be zero when there is at most one active neuron per example. This seems to be too harsh for complex tasks that need a richer representation. Thus, we suggest to relax the objective by imposing a spatial weighting to the correlation penalty. In other words, an active neuron penalizes mostly its close neighbours and this effect vanishes for neurons further away. Instead of uniformly penalizing all the correlated neurons, we weight the correlation penalty between two neurons with locations $i$ and $j$ using a Gaussian weighting. This gives ",
+        "bbox": [
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+        "type": "equation",
+        "img_path": "images/ece5b6e99d1295d97acd6a7acbb61e002ebcc0d4796ac8f5c8bc9c1219b0c05b.jpg",
+        "text": "$$\nR _ { \\mathrm { S L N I } } ( H _ { l } ) = \\frac { 1 } { M } \\sum _ { i , j } e ^ { - \\frac { ( i - j ) ^ { 2 } } { 2 \\sigma ^ { 2 } } } \\sum _ { m } h _ { i } ^ { m } h _ { j } ^ { m } , \\quad i \\neq j\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "As such, each active neuron inhibits its neighbours, introducing a locality in the network inspired by biological neurons. While the notion of neighbouring neurons is not well established in a fully connected network, our aim is to allow few neurons to be active and not only one, thus those few activations don’t have to be small to compensate for the penalty. $\\sigma ^ { 2 }$ is a hyper parameter representing the scale at which neurons can affect each other. Note that this is somewhat more flexible than decorrelating neurons in fixed groups as used in Xiong et al. (2016). Our regularizer inhibits locally the active neurons leading to a sparse coding through local neural inhibition. ",
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+        "page_idx": 4
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+    {
+        "type": "text",
+        "text": "3.3 NEURON IMPORTANCE FOR DISCOUNTING INHIBITION ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Our regularizer is to be applied for each task in the learning sequence. In the case of tasks with completely different input patterns, the active neurons of the previous tasks will not be activated given the new tasks input patterns. However, when the new tasks are of similar or shared patterns, neurons used for previous tasks will be active. In that case, our penalty would discourage other neurons from being active and encourage the new task to adapt the already active neurons instead. This would interfere with the previous tasks and could increase forgetting which is exactly what we want to overcome. To avoid such interference, we add a weight factor taking into account the importance of the neurons with respect to the previous tasks. To estimate the importance of the neurons, we use as a measure the sensitivity of the loss at the end of the training to their changes. This is approximated by the gradients of the loss w.r.t. the neurons outputs (before the activation function) evaluated at each data point. To get an importance value, we then accumulate the absolute value of the gradients over the given data points obtaining importance weight $\\alpha _ { i }$ for neuron $n _ { i }$ : ",
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+        "img_path": "images/5aa727ae6daededb5fcb351b37081b83c755c09ee2d179d119b663199c753be0.jpg",
+        "text": "$$\n\\alpha _ { i } = \\frac { 1 } { M } \\sum _ { m = 1 } ^ { M } \\mid g _ { i } ( x _ { m } ) \\mid , g _ { i } ( x _ { m } ) = \\frac { \\partial ( \\mathcal { L } ( y _ { m } , f ( x _ { m } , \\theta ^ { n } ) ) ) } { \\partial n _ { i } ^ { m } }\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "where $n _ { i } ^ { m }$ is the output of neuron $n _ { i }$ for a given input example $x _ { m }$ , and $\\theta ^ { n }$ are the parameters after learning task $n$ . This is in line with the estimation of the parameters importance in Kirkpatrick et al. (2016) but considering the derivation variables to be the neurons outputs instead of the parameters. Instead of relying on the gradient of the loss, we can also use the gradient of the learned function, i.e. the output layer, as done in Aljundi et al. (2017) for estimating the parameters importance. During the early phases of this work, we experimented with both and observed a similar behaviour. For sake of consistency and computational efficiency we utilize the gradient of the function when using Aljundi et al. (2017) as LLL method and the gradient of the loss when experimenting with EWC (Kirkpatrick et al., 2016). Then, we can weight our regularizer as follows: ",
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+        "text": "$$\nR _ { \\mathrm { S L N I D } } ( H _ { l } ) = \\frac { 1 } { M } \\sum _ { i , j } e ^ { - ( \\alpha _ { i } + \\alpha _ { j } ) } e ^ { - \\frac { ( i - j ) ^ { 2 } } { 2 \\sigma ^ { 2 } } } \\sum _ { m } h _ { i } ^ { m } h _ { j } ^ { m } , \\quad i \\neq j\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "which can be read as: if an important neuron for a previous task is active given an input pattern from the current task, it will not suppress the other neurons from being active neither be affected by other active neurons. For all other active neurons, local inhibition is deployed. The final objective for training is given in Eq. 2, setting $R _ { S S L } : = R _ { \\mathbb { S } \\mathrm { L N I D } }$ and $\\lambda _ { \\mathrm { S S L } } : = \\lambda _ { \\mathrm { S L N I D } }$ . We refer to our full method as Sparse coding through Local Neural Inhibition and Discounting (SLNID). ",
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+    {
+        "type": "text",
+        "text": "4 EXPERIMENTS ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "In this section we study the role of standard regularization techniques with a focus on sparsity and decorrelation of the representation in a sequential learning scenario. We first compare different activation functions and regularization techniques, including our proposed SLNID, on permuted MNIST (Sec. 4.1). Then, we compare the top competing techniques and our proposed method in the case of sequentially learning CIFAR-100 classes and Tiny Imagenet classes (Sec. 4.2). Our SLNID regularizer can be integrated in any importance weight-based lifelong learning approach such as (Kirkpatrick et al., 2016; Zenke et al., 2017; Aljundi et al., 2017). Here we focus on Memory Aware Synapses (Aljundi et al., 2017) (MAS), which is easy to integrate and experiment with and has shown superior performance (Aljundi et al., 2017). However, we also show results with Elastic weight consolidation (Kirkpatrick et al., 2016)(EWC) in Sec. 4.3. Further, we ablate the components of our regularizer, both in the standard setting (Sec. 4.4) as in a setting without hard task boundaries (Sec. 4.5). Finally, we show how our regularizer improves the state-of-the-art performance on a sequence of object recognition tasks (Sec. 4.6). ",
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+    {
+        "type": "image",
+        "img_path": "images/a943710c123ca7497643734a2a7f38fc19498d05d01d613c87af1695bddd7fbd.jpg",
+        "image_caption": [
+            "Figure 2: Comparison of different regularization techniques on 5 permuted MNIST sequence. Representation based regularizers are solid bars, bars with lines represent parameters regularizers, dotted bars represent activation functions. Average test accuracy over all tasks is given in the legend. Representation based regularizers achieve higher performance than other compared methods including parameters based regularizers. Our regularizer, SLNID, performs the best on the last two tasks indicating that more capacity is left to learn these tasks. "
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+    {
+        "type": "text",
+        "text": "4.1 AN IN-DEPTH COMPARISON OF REGULARIZERS AND ACTIVATION FUNCTIONS FOR SELFLESS SEQUENTIAL LEARNING ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "We study possible regularization techniques that could lead to less interference between the different tasks in a sequential learning scenario either by enforcing sparsity or decorrelation. Additionally, we examine the use of activation functions that are inspired by lateral inhibition in biological neurons that could be advantageous in sequential learning. MAS Aljundi et al. (2017) is used in all cases as LLL method. ",
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+    {
+        "type": "text",
+        "text": "Representation Based methods: ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "- L1-Rep: To promote representational sparsity, an $L _ { 1 }$ penalty on the activations is used. - Decov (Cogswell et al., 2015) aims at reducing overfitting by decorrelating neuron activations. To do so, it minimizes the Frobenius norm of the covariance matrix computed on the activations of the current batch after subtracting the diagonal to avoid penalizing independent neuron activations. ",
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+    {
+        "type": "text",
+        "text": "Activation functions: ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "- Maxout network (Goodfellow et al., 2013b) utilizes the maxout activation function. For each group of neurons, based on a fixed window size, only the maximum activation is forwarded to the next layer. The activation function guarantees a minimum sparsity rate defined by the window size. ",
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+        "text": "- LWTA (Srivastava et al., 2013): similar idea to the Maxout network except that the non-maximum activations are set to zero while maintaining their connections. In contrast to Maxout, LWTA keeps the connections of the inactive neurons which can be occupied later once they are activated without changing the previously active neuron connections. ",
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+        "type": "text",
+        "text": "- ReLU (Glorot et al., 2011) The rectifier activation function (ReLU) used as a baseline here and indicated in later experiments as No-Reg as it represents the standard setting of sequential learning on networks with ReLU. All the studied regularizers use ReLU as activation function. ",
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+        "type": "text",
+        "text": "Parameters based regularizers: ",
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+        "type": "text",
+        "text": "- OrthReg (Rodríguez et al., 2016): Regularizing CNNs with locally constrained decorrelations. It aims at decorrelating the feature detectors by minimizing the cosine of the angle between the weight vectors resulting eventually in orthogonal weight vectors. ",
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+        "type": "text",
+        "text": "- L2-WD: Weight decay with $L _ { 2 }$ norm (Krogh & Hertz, 1992) controls the complexity of the learned function by minimizing the magnitude of the weights. ",
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+        "type": "text",
+        "text": "- L1-Param: $L _ { 1 }$ penalty on the parameters to encourage a solution with sparse parameters. ",
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+        "type": "text",
+        "text": "Dropout is not considered as its role contradicts our goal. While dropout can improve each task performance and reduce overfitting, it acts as a model averaging technique. By randomly masking neurons, dropout forces the different neurons to work independently. As such it encourages a redundant representation. As shown by (Goodfellow et al., 2013a) the best network size for classifying MNIST digits when using dropout was about $5 0 \\%$ more than without it. Dropout steers the learning of a task towards occupying a good portion of the network capacity, if not all of it, which contradicts the sequential learning needs. ",
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+        "type": "text",
+        "text": "Experimental setup. We use the MNIST dataset (LeCun et al., 1998) as a first task in a sequence of 5 tasks, where we randomly permute all the input pixels differently for tasks 2 to 5. The goal is to classify MNIST digits from all the different permutations. The complete random permutation of the pixels in each task requires the neural network to instantiate a new neural representation for each pattern. A similar setup has been used by Kirkpatrick et al. (2016); Zenke et al. (2017); Goodfellow et al. (2013a) with different percentage of permutations or different number of tasks. ",
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+        "type": "text",
+        "text": "As a base network, we employ a multi layer perceptron with two hidden layers and a Softmax loss. ",
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+        "img_path": "images/0a6264583240b617e1e040b41563598f810710fdf2d38c6e58badab807f261c6.jpg",
+        "image_caption": [
+            "Figure 3: Comparison of different regularization techniques on a sequence of ten tasks from (a) Cifar split and (b) Tiny ImageNet split. The legend shows average test accuracy over all tasks. Simple L1-norm regularizer (L1-Rep) doesn’t help in such more complex tasks. Our regularizer SLNID achieves an improvement of $2 \\%$ over Decov and $4 - 8 \\%$ compared to No-Reg. "
+        ],
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+        "text": "We experiment with different number of neurons in the hidden layers $\\{ 1 2 8 , 6 4 \\}$ . For SLNID we evaluate the effect of $\\lambda _ {  { \\mathrm { S I N I D } } }$ on the performance and the obtained sparsity in Figure 4. In general, the best $\\lambda _ {  { \\mathrm { S I N I D } } }$ is the minimum value that maintains similar or better accuracy on the first task compared to the unregularized case, and we suggest to use this as a rule-of-thumb to set $\\lambda _ {  { \\mathrm { S I N I D } } }$ . For $\\lambda _ { \\Omega }$ , we have used a high $\\lambda _ { \\Omega }$ value that ensures the least forgetting which allows us to test the effect on the later tasks performance. Note that better average accuracies can be obtained with tuned $\\lambda _ { \\Omega }$ . Please refer to Appendix A for hyperparameters and other details. ",
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+        "text": "Results: Figure 2 presents the test accuracy on each task at the end of the sequence, achieved by the different regularizers and activation functions on the network with hidden layer of size 128. Results on a network with hidden layer size 64 are shown in the Appendix B. Clearly, in all the different tasks, the representational regularizers show a superior performance to the other studied techniques. For the regularizers applied to the parameters, $\\mathbb { L } 2 - \\mathbb { W } \\mathbb { D }$ and L1-Param do not exhibit a clear trend and do not systematically show an improvement over the use of the different activation functions only. While OrthReg shows a consistently good performance, it is lower than what can be achieved by the representational regularizers. It is worth noting the L1-Rep yields superior performance over L1-Param. This observation is consistent across different sizes of the hidden layers (in Appendix B) and shows the advantage of encouraging sparsity in the activations compared to that in the parameters. Regarding the activation functions, Maxout and LWTA achieve a slightly higher performance than ReLU. We did not observe a significant difference between the two activation functions. However, the improvement over ReLU is only moderate and does not justify the use of a fixed window size and special architecture design. Our proposed regularizer SLNID achieves high if not the highest performance in all the tasks and succeeds in having a stable performance. This indicates the ability of SLNID to direct the learning process towards using minimum amount of neurons and hence more flexibility for upcoming tasks. ",
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+        "type": "text",
+        "text": "Representation sparsity $\\pmb { \\& }$ important parameter sparsity. Here we want to examine the effect of our regularizer on the percentage of parameters that are utilized after each task and hence the capacity left for the later tasks. On the network with hidden layer size 128, we compute the percentage of parameters with $\\dot { \\Omega _ { k } } < 1 0 ^ { - 2 }$ , with $\\Omega _ { k }$ , see Appendix A, the importance weight multiplier estimated and accumulated over tasks. Those parameters can be seen as unimportant and \"free\" for later tasks. Figure 4(top) shows the percentage of the unimportant (free) parameters in the first layer after each task for different $\\lambda _ { \\mathrm { S I N I D } }$ values along with the achieved average test accuracy at the end of the sequence. It is clear that the larger $\\lambda _ {  { \\mathrm { S I N I D } } }$ , i.e., the more neural inhibition, the smaller the percentage of important parameters. Apart from the highest $\\lambda _ {  { \\mathrm { S I N I D } } }$ where tasks couldn’t reach their top performance due to too strong inhibition, improvement over the $\\mathtt { N O - R e g }$ is always observed. The optimal value for lambda seems to be the one that remains close to the optimal performance on the current task, while utilizing the minimum capacity feasible. Next, we compute the average activation per neuron, in the first layer, over all the examples and plot the corresponding histogram for SLNID, DeCov, L1-Rep, L1-Param and $\\mathtt { N O - R e g }$ in Figure 4(bottom) at their setting that yielded the results shown in Figure 2. SLNID has a peak at zero indicating representation sparsity while the other methods values are spread along the line. This seems to hint at the effectiveness of our approach SLNID in learning a sparse yet powerful representation and in turn in a minimal interference between tasks. ",
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+        "image_caption": [
+            "Figure 4: On the 5 permuted MNIST sequence, hidden layer ${ } = 1 2 8$ , Top: percentage of unused parameters in the 1st layer using different $\\lambda _ {  { \\mathrm { S I N I D } } }$ ; Bottom: histogram of neural activations on the first task. "
+        ],
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+    {
+        "type": "table",
+        "img_path": "images/434aa9e5f2ccd37579bfd4eeedb58d09a7b44958d02894b56f06f9f161b4e826.jpg",
+        "table_caption": [
+            "Table 1: SLNID ablation. Average test accuracy per task after training the last task in $\\%$ . \\* denotes that MultiTask Joint Training violates the LLL scenario as it has access to all tasks at once and thus can be seen as an upper bound. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td></td><td colspan=\"2\">Permuted mnist</td><td colspan=\"2\">Cifar</td></tr><tr><td>h-layer dim.</td><td>128</td><td>64</td><td>256</td><td>128</td></tr><tr><td>No-Reg</td><td>92.67</td><td>90.72</td><td>55.06</td><td>55.3</td></tr><tr><td>SNI</td><td>95.79</td><td>94.89</td><td>55.30</td><td>55.75</td></tr><tr><td>SNID</td><td>95.90</td><td>93.82</td><td>61.00</td><td>60.90</td></tr><tr><td>SLNI</td><td>95.95</td><td>94.87</td><td>56.06</td><td>55.79</td></tr><tr><td>SLNID</td><td>95.83</td><td>93.89</td><td>63.30</td><td>61.16</td></tr><tr><td>Multi-Task Joint Training*</td><td>97.30</td><td>96.80</td><td>70.99</td><td>71.95</td></tr></table>",
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+    {
+        "type": "table",
+        "img_path": "images/c27c4e99202c3ab2190e41e60ce2c1fce3b0ee10c9312674e7e4f8be844e2951.jpg",
+        "table_caption": [
+            "Table 2: 8 tasks object recognition sequence. Average test accuracy per task after training the last task in $\\%$ . "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Method</td><td>Avg-acc</td></tr><tr><td>Finetune LWF (Li &amp; Hoiem,2016)</td><td>32.67</td></tr><tr><td>EBLL (Triki et al., 2017)</td><td>49.49</td></tr><tr><td>IMM (Lee et al., 2017)</td><td>50.29</td></tr><tr><td></td><td>43.4</td></tr><tr><td>Path Integral (Zenke et al., 2017)</td><td>50.49</td></tr><tr><td>EWC (Kirkpatrick et al., 2016)</td><td>50.00</td></tr><tr><td>MAS (Aljundi et al.,2017)</td><td>52.69</td></tr><tr><td>SLNID-fc Pretrained (ours)</td><td>53.77</td></tr><tr><td>SLNID-fc randomly initialized (ours)</td><td>54.50</td></tr></table>",
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+    {
+        "type": "text",
+        "text": "4.2 10 TASK SEQUENCES ON CIFAR-100 AND TINY IMAGENET ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "While the previous section focused on learning a sequence of tasks with completely different input patterns and same objective, we now study the case of learning different categories of one dataset. For this we split the CIFAR-100 and the Tiny ImageNet (Yao & Miller, 2015) dataset into ten tasks, respectively. We have 10 and 20 categories per task for CIFAR-100 and Tiny ImagNet, respectively. Further details about the experimental setup can be found in appendix A. ",
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+    {
+        "type": "text",
+        "text": "We compare the top competing methods from the previous experiments, L1-Rep, DeCov and our SLNID, and No-Reg as a baseline, ReLU in previous experiment. Similarly, MAS Aljundi et al. (2017) is used in all cases as LLL method. Figures 3(a) and 3(b) show the performance on each of the ten tasks at the end of the sequence. For both datasets, we observe that our SLNID performs overall best. $\\mathtt { I 1 - R e p }$ and DeCov continue to improve over the non regularized case No-Reg. These results confirm our proposal on the importance of sparsity and decorrelation in sequential learning. ",
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+    {
+        "type": "text",
+        "text": "4.3 SLNID WITH EWC (KIRKPATRICK ET AL., 2016) ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "We have shown that our proposed regularizer SLNID exhibits stable and superior performance on the different tested networks when using MAS as importance weight preservation method. To prove the effectiveness of our regularizer regardless of the used importance weight based method, we have tested SLNID on the 5 tasks permuted MNIST sequence in combination with Elastic Weight Consolidation (EWC,Kirkpatrick et al. (2016)) and obtained a boost in the average performance at the end of the learned sequence equal to $3 . 1 \\%$ on the network with hidden layer size 128 and a boost of $2 . 8 \\%$ with hidden layer size 64. Detailed accuracies are shown in Appendix B. It is worth noting that with both MAS and EWC our SLNID was able obtain better accuracy using a network with a 64-dimensional hidden size than when training without regularization $\\mathtt { N O - R e g }$ on a network of double that size (128), indicating that SLNID allows to use neurons much more efficiently. ",
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+    {
+        "type": "text",
+        "text": "4.4 ABLATION STUDY ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "Our method can be seen as composed of three components: the neural inhibition, the locality relaxation and the neuron importance integration. To study how these components perform individually, Table 1 reports the average accuracy at the end of the Cifar 100 and permuted MNIST sequences for each variant, namely, SNID without neuron importance (SNI), SNID, SLNID without neuron importance (SLNI) in addition to our full SLNID regularizer. As we explained in Section 3, when tasks have completely different input patterns, the neurons that were activated on the previous task examples will not fire for new task samples and exclusion of important neurons is not mandatory. However, when sharing is present between the different tasks, a term to prevent SLNID from causing any interference is required. This is manifested in the reported results: for permuted MNIST, all the variants work nicely alone, as a result of the simplicity and the disjoint nature of this sequence. However, in the Cifar 100 sequence, the integration of the neuron importance in the SNID and SLNID regularizers exclude important neurons from the inhibition, resulting in a clearly better performance. The locality in SLNID improves the performance in the Cifar sequence, which suggests that a richer representation is needed and multiple active neurons should be tolerated. ",
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+    {
+        "type": "text",
+        "text": "In the previous experiments, we considered the standard task based scenario as in (Li & Hoiem, 2016; Zenke et al., 2017; Aljundi et al., 2017; Serrà et al., 2018), where at each time step we receive a task along with its training data and a new classification layer is initiated for the new task, if needed. Here, we are interested in a more realistic scenario where the data distribution shifts gradually without hard task boundaries. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "To test this setting, we use the Cifar 100 dataset. Instead of considering a set of 10 disjoint tasks each composed of 10 classes, as in the previous experiment (Sec. 4.2), we now start by sampling with high probability $( 2 / 3 )$ from the first 10 classes and with low probability $( 1 / 3 )$ from the rest of the classes. We train the network (same architecture as in Sec. 4.2) for a few epochs and then change the sampling probabilities to be high $( 2 / 3 )$ for classes $1 1 - 2 0$ and low $( 1 / 3 )$ for the remaining classes. This process is repeated until sampling with high probability from the last 10 classes and low from the rest. We use one shared classification layer throughout and estimate the importance weights and the neurons importance after each training step (before changing the sampling probabilities). We consider 6 variants: our SLNID, the ablations SLNI and without regularizer $\\mathtt { N O - R e g }$ , as in Section 4.4, as well each of these three trained without the MAS importance weight regularizer of Aljundi et al. (2017), denoted as $\\mathtt { w } / \\mathtt { o }$ MAS. Table 3 presents the accuracy averaged over the ten groups of ten classes, using each group model (i.e. the model trained when this group was sampled with high probability) in the top block and the average accuracy on each of the ten groups at the end of the training (middle and bottom block). We can deduce the following: 1) SLNID improves the performance considerably (by more than $4 \\%$ ) even without importance weight regularizer. 2) In this scenario without hard task boundaries there is less forgetting than in the scenario with hard task boundaries studied in Section 4.2 for Cifar (difference between rows in top block to corresponding rows in middle block). As a result, the improvement obtained by deploying the importance weight regularizer is moderate: at $7 0 . 7 5 \\%$ , SLNID $\\mathtt { w } / \\mathtt { o }$ MAS is already better than $\\mathtt { N O - R e g }$ reaching $6 6 . 3 3 \\%$ . 3) While SLNI without MAS improves the individual models performance $( 7 2 . 1 4 \\%$ compared to $6 9 . 2 0 \\%$ ), it fails to improve the overall performance at the end of the sequence $6 3 . 5 4 \\%$ compared to $6 5 . 1 5 \\%$ ), as important neurons are not excluded from the penalty and hence they are changed or inhibited leading to tasks interference and performance deterioration. ",
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+    },
+    {
+        "type": "table",
+        "img_path": "images/59c998eae0b1b157abeae1381b56a9eea92bac4e120f654039001b86e167aa3d.jpg",
+        "table_caption": [
+            "Table 3: No tasks boundaries test case on Cifar 100. Top block, avg. acc on each group of classes using each group model. Bottom block, avg. acc. on each group at the end of the training. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=1 colspan=1>Method</td><td rowspan=1 colspan=1>Avg.acc-tasks models</td></tr><tr><td rowspan=1 colspan=1>No-Reg w/o MASSLNI W/O MASSLNID W/O MASNo-RegSLNISLNID</td><td rowspan=1 colspan=1>69.20%72.14%73.03%66.88%71.32%72.33%</td></tr><tr><td rowspan=1 colspan=1>Method</td><td rowspan=1 colspan=1>Avg.acc-last model</td></tr><tr><td rowspan=1 colspan=1>No-Reg w/o MASSLNI W/O MASSLNID W/O MAS</td><td rowspan=1 colspan=1>65.15%63.54%70.75%</td></tr><tr><td rowspan=1 colspan=1>No-RegSLNISLNID</td><td rowspan=1 colspan=1>66.33%64.50%70.94%</td></tr></table>",
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+    {
+        "type": "text",
+        "text": "4.6 COMPARISON WITH THE STATE OF THE ART ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "To compare our proposed approach with the different state-of-the-art sequential learning methods, we use a sequence of 8 different object recognition tasks, introduced in Aljundi et al. (2017). The sequence starts from AlexNet (Krizhevsky et al., 2012) pretrained on ImageNet (Russakovsky et al., 2015) as a base network, following the setting of Aljundi et al. (2017). More details are in Appendix A.4. We compare against the following: Learning without Forgetting (Li & Hoiem, 2016) (LwF), Incremental Moment Matching (Lee et al., 2017) (IMM), Path Integral (Zenke et al., 2017) and sequential finetuning (FineTuning), in addition to the case of MAS (Aljundi et al., 2017) alone, i.e. our No-Reg before. Compared methods were run with the exact same setup as in Aljundi et al. (2017). For our regularizer, we disable dropout, since dropout encourages redundant activations which contradicts our regularizer’s role. Also, since the network is pretrained, the locality introduced in SLNID may conflict with the already pretrained activations. For this reason, we also test SLNID with randomly initialized fully connected layers. Our regularizer is applied with MAS as a sequential learning method. Table 2 reports the average test accuracy at the end of the sequence achieved by each method. SLNID improves even when starting from a pretrained network and disabling dropout. Surprisingly, even with randomly initialized fully connected layers, SLNID improves $1 . 8 \\%$ over the state of the art using a fully pretrained network. ",
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+    {
+        "type": "text",
+        "text": "5 CONCLUSION ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "In this paper we study the problem of sequential learning using a network with fixed capacity – a prerequisite for a scalable and computationally efficient solution. A key insight of our approach is that in the context of sequential learning (as opposed to other contexts where sparsity is imposed, such as network compression or avoiding overfitting), sparsity should be imposed at the level of the representation rather than at the level of the network parameters. Inspired by lateral inhibition in the mammalian brain, we impose sparsity by means of a new regularizer that decorrelates nearby active neurons. We integrate this in a model which learns selflessly a new task by leaving capacity for future tasks and at the same time avoids forgetting previous tasks by taking into account neurons importance. ",
+        "bbox": [
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+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Acknowledgment: The first author’s PhD is funded by an FWO scholarship. ",
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+    {
+        "type": "text",
+        "text": "REFERENCES ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            280,
+            285,
+            295
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
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+        "bbox": [
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+    {
+        "type": "text",
+        "text": "APPENDIX ",
+        "text_level": 1,
+        "bbox": [
+            176,
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+            263,
+            117
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "A DETAILS ON THE EXPERIMENTAL SETUP ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            135,
+            544,
+            150
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "In all designed experiments, our regularizer is applied to the neurons of the fully connected layers. As a future work, we plan to integrate it in the convolutional layers. ",
+        "bbox": [
+            171,
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+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "A.1 PERMUTED MNIST ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            205,
+            348,
+            219
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "The used network is composed of two fully connected layers. All tasks are trained for 10 epochs with a learning rate $1 0 ^ { - 2 }$ using SGD optimizer. ReLU is used as an activation function unless mentioned otherwise. Throughout the experiment, we used a scale $\\sigma$ for the Gaussian function used for the local inhibition equal to $1 / 6$ of the hidden layer size. For all competing regularizers, we tested different hyper parameters from $\\mathrm { i 0 ^ { - 2 } }$ to $\\mathrm { i 0 ^ { - 9 } }$ and report the best one. For $\\lambda _ { \\Omega }$ , we have used a high $\\lambda _ { \\Omega }$ value that ensures the least forgetting. This allows us to examine the degradation in the performance on the later tasks compared to those learned previously as a result of lacking capacity. Note that better average accuracies can be obtained with tuned $\\lambda _ { \\Omega }$ . ",
+        "bbox": [
+            174,
+            231,
+            825,
+            320
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "In section 4.1 we estimated the free capacity in the network with the percentage of $\\Omega _ { k } < 1 0 ^ { - 2 }$ , with $\\Omega _ { k }$ , the importance weight multiplier estimated and accumulated over tasks. We consider $\\Omega _ { k } < 1 0 ^ { - 2 }$ of negligible importanc as in a network trained without a sparsity regularizer, $\\Omega _ { i j } < 1 0 ^ { - 2 }$ covers the first 10 percentiles. ",
+        "bbox": [
+            174,
+            327,
+            823,
+            366
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "A.2 CIFAR-100 ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            381,
+            303,
+            395
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "As a base network, we use a network similar to the one used by Zenke et al. (2017) but without dropout. We evaluate two variants with hidden size $N = \\{ 2 5 6 , 1 2 8 \\}$ . Throughout the experiment, we again used a scale $\\sigma$ for the Gaussian function equal to $1 / 6$ of the hidden layer size. We train the different tasks for 50 epochs with a learning rate of $1 0 ^ { - 2 }$ using SGD optimizer. ",
+        "bbox": [
+            174,
+            406,
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+            457
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "A.3 TINY IMAGENET ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            473,
+            336,
+            487
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "We split the Tiny ImageNet dataset (Yao & Miller, 2015) into ten tasks, each containing twenty categories to be learned at once. As a base network, we use a variant of VGG (Simonyan & Zisserman, 2014). For architecture details, please refer to Table 4 below. ",
+        "bbox": [
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+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "table",
+        "img_path": "images/8e34d109126cacfe105971e2da22195df856e1e5a1338e27c2b70a900357b852.jpg",
+        "table_caption": [
+            "Table 4: Architecture of the network used in the Tiny Imagenet experiment. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Layer</td><td>#filters/neurons</td></tr><tr><td>Convolution</td><td>64</td></tr><tr><td>Max Pooling</td><td>1</td></tr><tr><td>Convolution Max Pooling</td><td>128</td></tr><tr><td>Convolution</td><td>= 256</td></tr><tr><td>Max Pooling Convolution</td><td>-</td></tr><tr><td>Max Pooling</td><td>256 -</td></tr><tr><td>Convolution</td><td>512</td></tr><tr><td>Convolution</td><td>512</td></tr><tr><td>Fully connected</td><td>500</td></tr><tr><td>Fully connected</td><td>500</td></tr><tr><td>Fully connected</td><td>20</td></tr></table>",
+        "bbox": [
+            370,
+            549,
+            624,
+            746
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "Throughout the experiment, we again used a scale $\\sigma$ for the Gaussian function equal to $1 / 6$ of the hidden layer size. ",
+        "bbox": [
+            173,
+            792,
+            825,
+            819
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "A.4 8 TASK OBJECT RECOGNITION SEQUENCE ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            834,
+            504,
+            848
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "The 8 tasks sequence is composed of: 1. Oxford Flowers (Nilsback & Zisserman, 2008), 2. MIT Scenes (Quattoni & Torralba, 2009), 3. Caltech-UCSD Birds (Welinder et al., 2010), 4. Stanford Cars (Krause et al., 2013); 5. FGVC-Aircraft (Maji et al., 2013); 6. VOC Actions (Everingham et al.); 7. Letters (de Campos et al., 2009); and 8. SVHN (Netzer et al., 2011) datasets. We have rerun the different methods and obtain the same reported results as in Aljundi et al. (2017). ",
+        "bbox": [
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+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "B EXTRA RESULTS ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            102,
+            349,
+            118
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "B.1 PERMUTED MNIST SEQUENCE ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            133,
+            428,
+            148
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "In section 4.1, we have studied the performance of different regularizers and activation functions on 5 permuted Mnist tasks in a network with a hidden layer of size 128. Figure 5 shows the average accuracies achieved by each of the studied methods at the end of the learned sequence in a network with a hidden layer of size 64. Similar conclusions can be drawn. Maxout and LWTA perform similarly and improve slightly over ReLU. Regularizers applied to the representation are more powerful for sequential learning than regularizers applied directly to the parameters. Specifically, L1-Rep (orange) is consistently better than L1-Param (pink). Our SLNID is able of maintaining a good performance on all the tasks, achieving among the top average test accuracies. Admittedly, the performances of SLNID is very close to L1-Rep. The difference between these methods stands out more clearly for larger networks and more complex tasks. ",
+        "bbox": [
+            173,
+            160,
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+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "image",
+        "img_path": "images/426ea32269169ea2b750be443a9e150e3399e3e4de3329a2a9c22532080af4fc.jpg",
+        "image_caption": [
+            "Figure 5: Comparison of different regularization techniques on 5 permuted MNIST sequence of tasks, hidden siz $_ { = 6 4 }$ . Representation based regularizers are solid bars, bars with lines represent parameters regularizers, dotted bars represent activation functions. See Figure 2 for size 128. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            191,
+            292,
+            805,
+            434
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "B.2 SLNI WITH EWC ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            515,
+            341,
+            530
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "To show that our approach is not limited to MAS (Aljundi et al., 2017), we have also experimented with EWC (Kirkpatrick et al., 2016) as another importance weight based method along with our regularize SLNID on the permuted Mnist sequence. Figure 6 shows the test accuracy of each task at the end of the 5 permuted Mnist sequence achieved by our SLNID combined with EWC and by No-Reg (here indicating EWC without regularization). It is clear that SLNID succeeds to improve the performance on all the learned tasks which validates the utility of our approach with different sequential learning methods. ",
+        "bbox": [
+            173,
+            541,
+            825,
+            618
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "image",
+        "img_path": "images/15353dabacd58a660d959f4d5f33614fda2ee3f66a2db60dc05cba778f6ba340.jpg",
+        "image_caption": [
+            "Figure 6: (a) SLNID with EWC on 5 permuted Mnist sequence of tasks, hidden size ${ \\mathrm { - } } 1 2 8$ , (b) hidden size $_ { = 6 4 }$ "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            176,
+            640,
+            818,
+            747
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "B.3 CIFAR 100 SEQUENCE ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            809,
+            370,
+            823
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "In section 4.2 we have tested SLNID and other representation regularizers on the Cifar 100 sequence. In Figure 3(a) we compare their performance on a network with hidden layer size 256. Figure 7 repeats the same experiment for a network with hidden size 128. While DeCov and SLNID continue to improve over $\\mathtt { N o - R e g }$ , L1-Rep seems to suffer in this case. Our interpretation is that L1-Rep here interferes with the previously learned tasks while penalizing activations and hence suffers from catastophic forgetting. In line with all the previous experiments SLNID achieves the best accuracies and manages here to improve over $6 \\%$ compared to No-Reg. ",
+        "bbox": [
+            173,
+            834,
+            825,
+            924
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "image",
+        "img_path": "images/fd932cee5ace2998546abfcd322e62e314994e57d05fc4075ba9b69baba5ef5f.jpg",
+        "image_caption": [
+            "Figure 7: Comparison of different regularization techniques on a sequence of ten tasks from Cifar split. Hidden size ${ = } 1 2 8$ . See Figure 3(a) for size 256. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            240,
+            102,
+            756,
+            242
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "B.4 SPATIAL LOCALITY TEST ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            308,
+            393,
+            320
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "To avoid penalizing all the active neurons, our SLNID weights the correlation penalty between each two neurons based on their spatial distance using a Gaussian function. We want to visualize the effect of this spatial locality on the neurons activity. To achieve this, we have used the first 3 tasks of the Permuted Mnist sequence as a test case and visualized the neurons importance after each task. This is done using the network of hidden layer size 64. Figure 8, Figure 9 and Figure 10 show the neurons importance after each task. The left column is without locality, i.e. SLNID, and the right column is SLNID. Blue represents the first task, orange the second task and green the third task. When using SLNID, inhibition is applied in a local manner allowing more active neurons which could potentially improve the representation power. When learning the second task, new neurons become important regardless of their closeness to first task important neurons as those neurons are excluded from the inhibition. As such, new neurons are becoming active as new tasks are learned. For SLNID all neural correlation is penalized in the first task. And for later tasks, very few neurons are able to become active and important for the new task due to the strong global inhibition, where previous neurons that are excluded from the inhibition are easier to be re-used. ",
+        "bbox": [
+            173,
+            332,
+            825,
+            496
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "image",
+        "img_path": "images/d121a16b0b65d4110e291158fc5d80d4f8d1b4b3b13f90220de5cf29f8ad24c1.jpg",
+        "image_caption": [
+            "Figure 8: First layer neuron importance after learning the first task (blue). Left: SNID, Right: SLNID. More active neurons are tolerated in SLNID. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            183,
+            123,
+            789,
+            272
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "image",
+        "img_path": "images/534c49cb2f6e67c941bc41d34b7785782b09e1c97da1c2f2202f42e2b61d9e5a.jpg",
+        "image_caption": [
+            "Figure 9: First layer neuron importance after learning the second task (orange), superimposed on Figure 8. Left: SNID, Right: SLNID. SLNID allows new neurons, especially those that were close neighbours to previous important neurons, to become active and to be used for the new task. SNID penalizes all unimportant neurons equally. As a result, previous neurons are adapted for the new tasks and less new neurons are getting activated. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            183,
+            354,
+            789,
+            503
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "image",
+        "img_path": "images/3a8ed2ed7e84014a80d4130e98561e6c8d48831687301a1afe0f5451eae97906.jpg",
+        "image_caption": [
+            "Figure 10: First layer neuron importance after learning the third task (green), superimposed on Figure 9. Left: SNID, Right: SLNID. SLNID allows previous neurons to be re-used for the third task. It avoids changing the previous important neurons by adding new neurons. For SNID, very few neurons are newly deployed. The new task is learned mostly by adapting previous important neurons, causing more interference. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            183,
+            612,
+            789,
+            761
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "image",
+        "img_path": "images/b9e18165821532a24d7e7ca918c946f828578386e81415ddd78f062fd5452737.jpg",
+        "image_caption": [
+            "Figure 11: First layer neuron importance after learning the first task, sorted in descending order according to the first task neuron importance (blue). Left: SNID, Right: SLNID. More active neurons are tolerated in SLNID. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            183,
+            123,
+            789,
+            272
+        ],
+        "page_idx": 16
+    },
+    {
+        "type": "image",
+        "img_path": "images/5881d02fece910d1a8f8f1b183ef5dc576afe7c19526a3d3b56230976f3e9117.jpg",
+        "image_caption": [
+            "Figure 12: First layer neuron importance after learning the second task sorted in descending order according to the first task neuron importance (orange), superimposed on top of figure 11. Left: SNID, Right: SLNID. SLNID allows new neurons to become active and be used for the new task. SNID penalizes all unimportant neurons equally and hence more neurons are re-used then initiated for the first time. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            183,
+            354,
+            789,
+            503
+        ],
+        "page_idx": 16
+    },
+    {
+        "type": "image",
+        "img_path": "images/b84013cb534a4daa9e19bdf1d57068a82731cf7b591edb3810098c4ad325266c.jpg",
+        "image_caption": [
+            "Figure 13: First layer neuron importance after learning the third task sorted in descending order according to the first task neuron importance (green), superimposed on top of figure 12. Left: SNID, Right: SLNID. SLNID allows previous neurons to be re-used for the third task while activating new neurons to cope with the needs of the new task. For SNID, very few neurons are newly deployed while most previous important neurons for previous tasks are re-adapted to learn the new task. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            183,
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+            753
+        ],
+        "page_idx": 16
+    }
+]

parse/train/H1q-TM-AW/H1q-TM-AW.md

@@ -0,0 +1,487 @@
+# A DIRT-T APPROACH TO UNSUPERVISED DOMAINADAPTATION
+
+Rui $\mathbf { S h u } ^ { \dagger }$ ∗, Hung H. Bui‡, Hirokazu Narui†, & Stefano Ermon†
+
+†Stanford University   
+‡DeepMind   
+†{ruishu,hirokaz2,ermon}@stanford.edu   
+‡{buih}@google.com
+
+# ABSTRACT
+
+Domain adaptation refers to the problem of leveraging labeled data in a source domain to learn an accurate model in a target domain where labels are scarce or unavailable. A recent approach for finding a common representation of the two domains is via domain adversarial training (Ganin & Lempitsky, 2015), which attempts to induce a feature extractor that matches the source and target feature distributions in some feature space. However, domain adversarial training faces two critical limitations: 1) if the feature extraction function has high-capacity, then feature distribution matching is a weak constraint, 2) in non-conservative domain adaptation (where no single classifier can perform well in both the source and target domains), training the model to do well on the source domain hurts performance on the target domain. In this paper, we address these issues through the lens of the cluster assumption, i.e., decision boundaries should not cross high-density data regions. We propose two novel and related models: 1) the Virtual Adversarial Domain Adaptation (VADA) model, which combines domain adversarial training with a penalty term that punishes violation of the cluster assumption; 2) the Decision-boundary Iterative Refinement Training with a Teacher (DIRT-T)1 model, which takes the VADA model as initialization and employs natural gradient steps to further minimize the cluster assumption violation. Extensive empirical results demonstrate that the combination of these two models significantly improve the state-of-the-art performance on the digit, traffic sign, and Wi-Fi recognition domain adaptation benchmarks.
+
+# 1 INTRODUCTION
+
+The development of deep neural networks has enabled impressive performance in a wide variety of machine learning tasks. However, these advancements often rely on the existence of a large amount of labeled training data. In many cases, direct access to vast quantities of labeled data for the task of interest (the target domain) is either costly or otherwise absent, but labels are readily available for related training sets (the source domain). A notable example of this scenario occurs when the source domain consists of richly-annotated synthetic or semi-synthetic data, but the target domain consists of unannotated real-world data (Sun & Saenko, 2014; Vazquez et al., 2014). However, the source data distribution is often dissimilar to the target data distribution, and the resulting significant covariate shift is detrimental to the performance of the source-trained model when applied to the target domain (Shimodaira, 2000).
+
+Solving the covariate shift problem of this nature is an instance of domain adaptation (Ben-David et al., 2010b). In this paper, we consider a challenging setting of domain adaptation where 1) we are provided with fully-labeled source samples and completely-unlabeled target samples, and 2) the existence of a classifier in the hypothesis space with low generalization error in both source and target domains is not guaranteed. Borrowing approximately the terminology from Ben-David et al. (2010b), we refer to this setting as unsupervised, non-conservative domain adaptation. We note that this is in contrast to conservative domain adaptation, where we assume our hypothesis space contains a classifier that performs well in both the source and target domains.
+
+To tackle unsupervised domain adaptation, Ganin & Lempitsky (2015) proposed to constrain the classifier to only rely on domain-invariant features. This is achieved by training the classifier to perform well on the source domain while minimizing the divergence between features extracted from the source versus target domains. To achieve divergence minimization, Ganin & Lempitsky (2015) employ domain adversarial training. We highlight two issues with this approach: 1) when the feature function has high-capacity and the source-target supports are disjoint, the domain-invariance constraint is potentially very weak (see Section 3), and 2) good generalization on the source domain hurts target performance in the non-conservative setting.
+
+Saito et al. (2017) addressed these issues by replacing domain adversarial training with asymmetric tri-training (ATT), which relies on the assumption that target samples that are labeled by a sourcetrained classifier with high confidence are correctly labeled by the source classifier. In this paper, we consider an orthogonal assumption: the cluster assumption (Chapelle & Zien, 2005), that the input distribution contains separated data clusters and that data samples in the same cluster share the same class label. This assumption introduces an additional bias where we seek decision boundaries that do not go through high-density regions. Based on this intuition, we propose two novel models: 1) the Virtual Adversarial Domain Adaptation (VADA) model which incorporates an additional virtual adversarial training (Miyato et al., 2017) and conditional entropy loss to push the decision boundaries away from the empirical data, and 2) the Decision-boundary Iterative Refinement Training with a Teacher (DIRT-T) model which uses natural gradients to further refine the output of the VADA model while focusing purely on the target domain. We demonstrate that
+
+1. In conservative domain adaptation, where the classifier is trained to perform well on the source domain, VADA can be used to further constrain the hypothesis space by penalizing violations of the cluster assumption, thereby improving domain adversarial training.   
+2. In non-conservative domain adaptation, where we account for the mismatch between the source and target optimal classifiers, DIRT-T allows us to transition from a joint (source and target) classifier (VADA) to a better target domain classifier. Interestingly, we demonstrate the advantage of natural gradients in DIRT-T refinement steps.
+
+We report results for domain adaptation in digits classification (MNIST-M, MNIST, SYN DIGITS, SVHN), traffic sign classification (SYN SIGNS, GTSRB), general object classification (STL-10, CIFAR-10), and Wi-Fi activity recognition (Yousefi et al., 2017). We show that, in nearly all experiments, VADA improves upon previous methods and that DIRT-T improves upon VADA, setting new state-of-the-art performances across a wide range of domain adaptation benchmarks. In adapting MNIST $ \mathrm { S V H N }$ , a very challenging task, we out-perform ATT by over $2 0 \%$ .
+
+# 2 RELATED WORK
+
+Given the extensive literature on domain adaptation, we highlight several works most relevant to our paper. Shimodaira (2000); Mansour et al. (2009) proposed to correct for covariate shift by re-weighting the source samples such that the discrepancy between the target distribution and reweighted source distribution is minimized. Such a procedure is problematic, however, if the source and target distributions do not contain sufficient overlap. Huang et al. (2007); Long et al. (2015); Ganin & Lempitsky (2015) proposed to instead project both distributions into some feature space and encourage distribution matching in the feature space. Ganin & Lempitsky (2015) in particular encouraged feature matching via domain adversarial training, which corresponds approximately to Jensen-Shannon divergence minimization (Goodfellow et al., 2014). To better perform nonconservative domain adaptation, Saito et al. (2017) proposed to modify tri-training (Zhou & Li, 2005) for domain adaptation, leveraging the assumption that highly-confident predictions are correct predictions (Zhu, 2005). Several of aforementioned methods are based on Ben-David et al. (2010a)’s theoretical analysis of domain adaptation, which states the following,
+
+Theorem 1 (Ben-David et al., 2010a) Let $\mathcal { H }$ be the hypothesis space and let $( X _ { s } , \epsilon _ { s } )$ and $( X _ { t } , \epsilon _ { t } )$ be the two domains and their corresponding generalization error functions. Then for any $h \in \mathcal H$ ,
+
+$$
+\epsilon _ { t } ( h ) \leq \frac { 1 } { 2 } d _ { { \mathcal { H } } \Delta { \mathcal { H } } } ( X _ { s } , X _ { t } ) + \epsilon _ { s } ( h ) + \operatorname* { m i n } _ { h ^ { \prime } \in { \mathcal { H } } } \epsilon _ { t } ( h ^ { \prime } ) + \epsilon _ { s } ( h ^ { \prime } ) ,
+$$
+
+where $d _ { \mathcal { H } \Delta \mathcal { H } }$ denotes the $\mathcal { H } \Delta \mathcal { H }$ -distance between the domains $X _ { s }$ and $X _ { t }$ ,
+
+$$
+d _ { { \mathcal H } \Delta { \mathcal H } } = 2 \operatorname* { s u p } _ { h , h ^ { \prime } \in { \mathcal H } } \left| \mathbb E _ { x \sim X _ { s } } \left[ h ( x ) \neq h ^ { \prime } ( x ) \right] - \mathbb E _ { x \sim X _ { t } } \left[ h ( x ) \neq h ^ { \prime } ( x ) \right] \right| .
+$$
+
+Intuitively, $d _ { \mathcal { H } \Delta \mathcal { H } }$ measures the extent to which small changes to the hypothesis in the source domain can lead to large changes in the target domain. It is evident that $d _ { \mathcal { H } \Delta \mathcal { H } }$ relates intimately to the complexity of the hypothesis space and the divergence between the source and target domains. For infinite-capacity models and domains with disjoint supports, $d _ { \mathcal { H } \Delta \mathcal { H } }$ is maximal.
+
+A critical component to our paper is the cluster assumption, which states that decision boundaries should not cross high-density regions (Chapelle & Zien, 2005). This assumption has been extensively studied and leveraged for semi-supervised learning, leading to proposals such as conditional entropy minimization (Grandvalet & Bengio, 2005) and pseudo-labeling (Lee, 2013). More recently, the cluster assumption has led to many successful deep semi-supervised learning algorithms such as semi-supervised generative adversarial networks (Dai et al., 2017), virtual adversarial training (Miyato et al., 2017), and self/temporal-ensembling (Laine & Aila, 2016; Tarvainen & Valpola, 2017). Given the success of the cluster assumption in semi-supervised learning, it is natural to consider its application to domain adaptation. Indeed, Ben-David & Urner (2014) formalized the cluster assumption through the lens of probabilistic Lipschitzness and proposed a nearest-neighbors model for domain adaptation. Our work extends this line of research by showing that the cluster assumption can be applied to deep neural networks to solve complex, high-dimensional domain adaptation problems. Independently of our work, French et al. (2017) demonstrated the application of selfensembling to domain adaptation. However, our work additionally considers the application of the cluster assumption to non-conservative domain adaptation.
+
+# 3 LIMITATION OF DOMAIN ADVERSARIAL TRAINING
+
+Before describing our model, we first highlight that domain adversarial training may not be sufficient for domain adaptation if the feature extraction function has high-capacity. Consider a classifier $h _ { \theta }$ , parameterized by $\theta$ , that maps inputs to the $( K - 1 )$ -simplex (denote as $\mathcal { C }$ ), where $K$ is the number of classes. Suppose the classifier $h = g \circ f$ can be decomposed as the composite of an embedding function $f _ { \theta } : \mathcal { X }  \mathcal { Z }$ and embedding classifier $g _ { \theta } : \mathcal { Z }  \mathcal { C }$ . For the source domain, let $\mathcal { D } _ { s }$ be the joint distribution over input $x$ and one-hot label $y$ and let $X _ { s }$ be the marginal input distribution. $( \mathcal { D } _ { t } , X _ { t } )$ are analogously defined for the target domain. Let $( \mathcal { L } _ { s } , \mathcal { L } _ { d } )$ be the loss functions
+
+$$
+\begin{array} { r l } & { \qquad \mathcal { L } _ { y } ( \theta ; \mathcal { D } _ { s } ) = \mathbb { E } _ { x , y \sim \mathcal { D } _ { s } } \left[ y ^ { \top } \ln h _ { \theta } ( x ) \right] } \\ & { \qquad \mathcal { L } _ { d } ( \theta ; \mathcal { D } _ { s } , \mathcal { D } _ { t } ) = \underset { D } { \operatorname* { s u p } } \mathbb { E } _ { x \sim \mathcal { D } _ { s } } \left[ \ln D ( f _ { \theta } ( x ) ) \right] + \mathbb { E } _ { x \sim \mathcal { D } _ { t } } \left[ \ln ( 1 - D ( f _ { \theta } ( x ) ) ) \right] , } \end{array}
+$$
+
+where the supremum ranges over discriminators $D : \mathcal { Z } \to ( 0 , 1 )$ . Then $\mathcal { L } _ { y }$ is the cross-entropy objective and $D$ is a domain discriminator. Domain adversarial training minimizes the objective
+
+$$
+\operatorname* { m i n } _ { \theta } . \mathcal { L } _ { y } ( \theta ; \mathcal { D } _ { s } ) + \lambda _ { d } \mathcal { L } _ { d } ( \theta ; \mathcal { D } _ { s } , \mathcal { D } _ { t } ) ,
+$$
+
+where $\lambda _ { d }$ is a weighting factor. Minimization of $\mathcal { L } _ { d }$ encourages the learning of a feature extractor $f$ for which the Jensen-Shannon divergence between $f ( X _ { s } )$ and $f ( X _ { t } )$ is small.2 Ganin & Lempitsky (2015) suggest that successful adaptation tends to occur when the source generalization error and feature divergence are both small.
+
+It is easy, however, to construct situations where this suggestion fails. In particular, if $f$ has infinitecapacity and the source-target supports are disjoint, then $f$ can employ arbitrary transformations to the target domain so as to match the source feature distribution (see Appendix $\mathrm { E }$ for formalization).
+
+We verify empirically that, for sufficiently deep layers, jointly achieving small source generalization error and feature divergence does not imply high accuracy on the target task (Table 5). Given the limitations of domain adversarial training, we wish to identify additional constraints that one can place on the model to achieve better, more reliable domain adaptation.
+
+![](images/0d85f8ff02d1a35a25681e0f1957f75fa2dddc26d594d675fa73e1e0f69e0d21.jpg)  
+4 CONSTRAINING VIA CONDITIONAL ENTROPY MINIMIZATION   
+Figure 1: VADA improves upon domain adversarial training by additionally penalizing violations of the cluster assumption.
+
+In this paper, we apply the cluster assumption to domain adaptation. The cluster assumption states that the input distribution $X$ contains clusters and that points in the same cluster come from the same class. This assumption has been extensively studied and applied successfully to a wide range of classification tasks (see Section 2). If the cluster assumption holds, the optimal decision boundaries should occur far away from data-dense regions in the space of $\mathcal { X }$ (Chapelle & Zien, 2005). Following Grandvalet & Bengio (2005), we achieve this behavior via minimization of the conditional entropy with respect to the target distribution,
+
+$$
+\begin{array} { r } { \mathcal { L } _ { c } ( \boldsymbol { \theta } ; \mathcal { D } _ { t } ) = - \mathbb { E } _ { \boldsymbol { x } \sim \mathcal { D } _ { t } } \left[ h _ { \boldsymbol { \theta } } ( \boldsymbol { x } ) ^ { \top } \ln h _ { \boldsymbol { \theta } } ( \boldsymbol { x } ) \right] . } \end{array}
+$$
+
+Intuitively, minimizing the conditional entropy forces the classifier to be confident on the unlabeled target data, thus driving the classifier’s decision boundaries away from the target data (Grandvalet & Bengio, 2005). In practice, the conditional entropy must be empirically estimated using the available data. However, Grandvalet & Bengio (2005) note that this approximation breaks down if the classifier $h$ is not locally-Lipschitz. Without the locally-Lipschitz constraint, the classifier is allowed to abruptly change its prediction in the vicinity of the training data points, which 1) results in a unreliable empirical estimate of conditional entropy and 2) allows placement of the classifier decision boundaries close to the training samples even when the empirical conditional entropy is minimized. To prevent this, we propose to explicitly incorporate the locally-Lipschitz constraint via virtual adversarial training (Miyato et al., 2017) and add to the objective function the additional term
+
+$$
+\mathcal { L } _ { v } ( \theta ; \mathcal { D } ) = \mathbb { E } _ { x \sim \mathcal { D } } \left[ \operatorname* { m a x } _ { \| r \| \leq \epsilon } \mathrm { D } _ { \mathrm { K L } } \big ( h _ { \theta } ( x ) \| h _ { \theta } ( x + r ) \big ) \right] ,
+$$
+
+which enforces classifier consistency within the norm-ball neighborhood of each sample $x$ . Note that virtual adversarial training can be applied with respect to either the target or source distributions. We can combine the conditional entropy minimization objective and domain adversarial training to yield
+
+$$
+\operatorname* { m i n } _ { \theta } . \mathcal { L } _ { y } ( \theta ; \mathcal { D } _ { s } ) + \lambda _ { d } \mathcal { L } _ { d } ( \theta ; \mathcal { D } _ { s } , \mathcal { D } _ { t } ) + \lambda _ { s } \mathcal { L } _ { v } ( \theta ; \mathcal { D } _ { s } ) + \lambda _ { t } \left[ \mathcal { L } _ { v } ( \theta ; \mathcal { D } _ { t } ) + \mathcal { L } _ { c } ( \theta ; \mathcal { D } _ { t } ) \right] ,
+$$
+
+a basic combination of domain adversarial training and semi-supervised training objectives. We refer to this as the Virtual Adversarial Domain Adaptation (VADA) model. Empirically, we observed that the hyperparameters $\left( \lambda _ { d } , \lambda _ { s } , \lambda _ { t } \right)$ are easy to choose and work well across multiple tasks (Appendix B).
+
+$\mathcal { H } \Delta \mathcal { H }$ -Distance Minimization. VADA aligns well with the theory of domain adaptation provided in Theorem 1. Let the loss,
+
+$$
+\begin{array} { r } { \mathcal L _ { t } ( \boldsymbol { \theta } ) = \mathcal L _ { v } ( \boldsymbol { \theta } ; \mathcal D _ { t } ) + \mathcal L _ { c } ( \boldsymbol { \theta } ; D _ { t } ) , } \end{array}
+$$
+
+denote the degree to which the target-side cluster assumption is violated. Modulating $\lambda _ { t }$ enables VADA to trade-off between hypotheses with low target-side cluster assumption violation and hypotheses with low source-side generalization error. Setting $\lambda _ { t } > 0$ allows rejection of hypotheses with high target-side cluster assumption violation. By rejecting such hypotheses from the hypothesis space $\mathcal { H }$ , VADA reduces $d _ { \mathcal { H } \Delta \mathcal { H } }$ and yields a tighter bound on the target generalization error. We verify empirically that VADA achieves significant improvements over existing models on multiple domain adaptation benchmarks (Table 1).
+
+# 5 DECISION-BOUNDARY ITERATIVE REFINEMENT TRAINING
+
+![](images/40ad8eacbf9405c2fd08272404c201ac0282afc878eaa8a382448c59712e7158.jpg)  
+Figure 2: DIRT-T uses VADA as initialization. After removing the source training signal, DIRTT minimizes cluster assumption violation in the target domain through a series of natural gradient steps.
+
+In non-conservative domain adaptation, we assume the following inequality,
+
+$$
+\operatorname* { m i n } _ { h \in \mathcal { H } } \epsilon _ { t } ( h ) < \epsilon _ { t } ( h ^ { a } ) \mathrm { ~ w h e r e ~ } h ^ { a } = \arg \operatorname* { m i n } _ { h \in \mathcal { H } } \epsilon _ { s } ( h ) + \epsilon _ { t } ( h ) ,
+$$
+
+where $( \epsilon _ { s } , \epsilon _ { t } )$ are generalization error functions for the source and target domains. This means that, for a given hypothesis class $\mathcal { H }$ , the optimal classifier in the source domain does not coincide with the optimal classifier in the target domain.
+
+We assume that the optimality gap in Eq. (10) results from violation of the cluster assumption. In other words, we suppose that any source-optimal classifier drawn from our hypothesis space necessarily violates the cluster assumption in the target domain. Insofar as VADA is trained on the source domain, we hypothesize that a better hypothesis is achievable by introducing a secondary training phase that solely minimizes the target-side cluster assumption violation.
+
+Under this assumption, the natural solution is to initialize with the VADA model and then further minimize the cluster assumption violation in the target domain. In particular, we first use VADA to learn an initial classifier $h _ { \theta _ { 0 } }$ . Next, we incrementally push the classifier’s decision boundaries away from data-dense regions by minimizing the target-side cluster assumption violation loss $\mathcal { L } _ { t }$ in Eq. (9). We denote this procedure Decision-boundary Iterative Refinement Training (DIRT).
+
+# 5.1 DECISION-BOUNDARY ITERATIVE REFINEMENT TRAINING WITH A TEACHER
+
+Stochastic gradient descent minimizes the loss $\scriptstyle { \mathcal { L } } _ { t }$ by selecting gradient steps $\Delta \theta$ according to the following objective,
+
+$$
+\begin{array} { r l } & { \underset { \Delta \theta } { \mathrm { m i n . } } ~ \mathcal { L } _ { t } ( \theta + \Delta \theta ) } \\ & { ~ \mathrm { s . t . } ~ \| \Delta \theta \| \leq \epsilon , } \end{array}
+$$
+
+which defines the neighborhood in the parameter space. This notion of neighborhood is sensitive to the parameterization of the model; depending on the parameterization, a seemingly small step $\Delta \theta$ may result in a vastly different classifier. This contradicts our intention of incrementally and locally pushing the decision boundaries to a local conditional entropy minimum, which requires that the decision boundaries of $h _ { \theta + \Delta \theta }$ stay close to that of $h _ { \theta }$ . It is therefore important to define a neighborhood that is parameterization-invariant. Following Pascanu $\&$ Bengio (2013), we instead select $\Delta \theta$ using the following objective,
+
+$$
+\begin{array} { r l } & { \underset { \Delta \theta } { \operatorname* { m i n } } \mathcal { L } _ { t } ( \theta + \Delta \theta ) } \\ & { \quad \mathrm { s . t . } \mathbb { E } _ { x \sim D _ { t } } \left[ \mathrm { D } _ { \mathrm { K L } } ( h _ { \theta } ( x ) \| h _ { \theta + \Delta \theta } ( x ) ) \right] \leq \epsilon . } \end{array}
+$$
+
+Each optimization step now solves for a gradient step $\Delta \theta$ that minimizes the conditional entropy, subject to the constraint that the Kullback-Leibler divergence between $h _ { \theta } ( x )$ and $h _ { \theta + \Delta \theta } ( x )$ is small for $x \sim \mathcal { X } _ { t }$ . The corresponding Lagrangian suggests that one can instead minimize a sequence of optimization problems
+
+$$
+\operatorname* { m i n . } _ { \theta _ { n } } \lambda _ { t } \mathcal { L } _ { t } ( \theta _ { n } ) + \beta _ { t } \mathbb { E } \left[ \mathrm { D } _ { \mathrm { K L } } \big ( h _ { \theta _ { n - 1 } } ( x ) \| h _ { \theta _ { n } } ( x ) \big ) \right] ,
+$$
+
+that approximates the application of a series of natural gradient steps.
+
+In practice, each of the optimization problems in Eq. (14) can be solved approximately via a finite number of stochastic gradient descent steps. We denote the number of steps taken to be the refinement interval $B$ . Similar to Tarvainen & Valpola (2017), we use the Adam Optimizer with Polyak averaging (Polyak & Juditsky, 1992). We interpret $h _ { \theta _ { n - 1 } }$ as a (sub-optimal) teacher for the student model $h _ { \theta _ { n } }$ , which is trained to stay close to the teacher model while seeking to reduce the cluster assumption violation. As a result, we denote this model as Decision-boundary Iterative Refinement Training with a Teacher (DIRT-T).
+
+Weakly-Supervised Learning. This sequence of optimization problems has a natural interpretation that exposes a connection to weakly-supervised learning. In each optimization problem, the teacher model $h _ { \theta _ { n - 1 } }$ pseudo-labels the target samples with noisy labels. Rather than naively training the student model $h _ { \theta _ { n } }$ on the noisy labels, the additional training signal $\mathcal { L } _ { t }$ allows the student model to place its decision boundaries further from the data. If the clustering assumption holds and the initial noisy labels are sufficiently similar to the true labels, conditional entropy minimization can improve the placement of the decision boundaries (Reed et al., 2014).
+
+Domain Adaptation. An alternative interpretation is that DIRT-T is the recursive extension of VADA, where the act of pseudo-labeling of the target distribution constructs a new “source” domain (i.e. target distribution $X _ { t }$ with pseudo-labels). The sequence of optimization problems can then be seen as a sequence of non-conservative domain adaptation problems in which $X _ { s } ~ = ~ X _ { t }$ but $p _ { s } ( y \mid x ) \neq p _ { t } \bar { ( } y \mid x )$ , where $p _ { s } ( y \mid x ) = h _ { \theta _ { n - 1 } } ( x )$ and $p _ { t } ( y \mid x )$ is the true conditional label distribution in the target domain. Since $d _ { \mathcal { H } \Delta \mathcal { H } }$ is strictly zero in this sequence of optimization problems, domain adversarial training is no longer necessary. Furthermore, if $\mathcal { L } _ { t }$ minimization does improve the student classifier, then the gap in Eq. (10) should get smaller each time the source domain is updated.
+
+# 6 EXPERIMENTS
+
+In principle, our method can be applied to any domain adaptation tasks so long as one can define a reasonable notion of neighborhood for virtual adversarial training (Miyato et al., 2016). For comparison against Saito et al. (2017) and French et al. (2017), we focus on visual domain adaptation and evaluate on MNIST, MNIST-M, Street View House Numbers (SVHN), Synthetic Digits (SYN DIGITS), Synthetic Traffic Signs (SYN SIGNS), the German Traffic Signs Recognition Benchmark (GTSRB), CIFAR-10, and STL-10. For non-visual domain adaptation, we evaluate on Wi-Fi activity recognition.
+
+# 6.1 IMPLEMENTATION DETAIL
+
+Architecture We use a small CNN for the digits, traffic sign, and Wi-Fi domain adaptation experiments, and a larger CNN for domain adaptation between CIFAR-10 and STL-10. Both architectures are available in Appendix A. For fair comparison, we additionally report the performance of source-only baseline models and demonstrate that the significant improvements are attributable to our proposed method.
+
+Replacing gradient reversal. In contrast to Ganin & Lempitsky (2015), which proposed to implement domain adversarial training via gradient reversal, we follow Goodfellow et al. (2014) and instead optimize via alternating updates to the discriminator and encoder (see Appendix C).
+
+Instance normalization. We explored the application of instance normalization as an image preprocessing step. This procedure makes the classifier invariant to channel-wide shifts and rescaling of pixel intensities. A discussion of instance normalization for domain adaptation is provided in Appendix D. We show in Figure 3 the effect of applying instance normalization to the input image.
+
+![](images/ca3ca66e9107950c36f6bddd2be1e33a5dfd7160e911c1f4b87611eadd799baf.jpg)  
+Figure 3: Effect of applying instance normalization to the input image. In clockwise direction: MNIST-M, GTSRB, SVHN, and CIFAR-10. In each quadrant, the top row is the original image, and the bottom row is the instance-normalized image.
+
+Hyperparameters. For each task, we tuned the four hyperparameters $\left( \lambda _ { d } , \lambda _ { s } , \lambda _ { t } , \beta \right)$ by randomly selecting 1000 labeled target samples from the training set and using that as our validation set. We observed that extensive hyperparameter-tuning is not necessary to achieve state-of-the-art performance. In all experiments with instance-normalized inputs, we restrict our hyperparameter search for each task to $\hat { \lambda } _ { d } = \{ 0 , 1 0 ^ { - 2 } \} , \lambda _ { s } = \{ 0 , 1 \} , \lambda _ { t } = \{ 1 \hat { 0 } ^ { - 2 } , 1 0 ^ { - 1 } \}$ . We fixed $\dot { \beta } = 1 0 ^ { - 2 }$ . Note that the decision to turn $( \lambda _ { d } , \lambda _ { s } )$ on or off that can often be determined a priori. A complete list of the hyperparameters is provided in Appendix B.
+
+# 6.2 MODEL EVALUATION
+
+<table><tr><td>Source Target</td><td>MNIST MNIST-M</td><td>SVHN MNIST</td><td>MNIST SVHN</td><td>DIGITS SVHN</td><td>SIGNS GTSRB</td><td>CIFAR STL</td><td>STL CIFAR</td></tr><tr><td>MMD (Long et al., 2015)</td><td>76.9</td><td>71.1</td><td>-</td><td>88.0</td><td>91.1</td><td>-</td><td>-</td></tr><tr><td>DANN (Ganin &amp; Lempitsky,2015)</td><td>81.5</td><td>71.1</td><td>35.7</td><td>90.3</td><td>88.7</td><td>1</td><td>-</td></tr><tr><td>DRCN (Ghifary et al.,2016)</td><td>-</td><td>82.0</td><td>40.1</td><td></td><td>=</td><td>66.4</td><td>58.7</td></tr><tr><td>DSN (Bousmalis et al.,2016b)</td><td>83.2</td><td>82.7</td><td>-</td><td>91.2</td><td>93.1</td><td>1</td><td>-</td></tr><tr><td>kNN-Ad (Sener et al., 2016)</td><td>86.7</td><td>78.8</td><td>40.3</td><td>-</td><td>1</td><td>=</td><td>-</td></tr><tr><td>PixelDA (Bousmalis etal., 2016a)</td><td>98.2 94.2</td><td>-</td><td>=</td><td>=</td><td>-</td><td>=</td><td>=</td></tr><tr><td>ATT (Saito et al., 2017) II-model (aug) (French et al.,2017)</td><td>-</td><td>86.2</td><td>52.8</td><td>92.9</td><td>96.2</td><td>-</td><td>1</td></tr><tr><td></td><td></td><td>92.0</td><td>71.4</td><td>94.2</td><td>98.4</td><td>76.3</td><td>64.2</td></tr><tr><td colspan="8">Without Instance-Normalized Input:</td></tr><tr><td>Source-Only</td><td>58.5</td><td>77.0</td><td>27.9</td><td>86.9</td><td>79.6</td><td>76.3</td><td>63.6</td></tr><tr><td>VADA</td><td>97.7</td><td>97.9</td><td>47.5</td><td>94.8</td><td>98.8</td><td>80.0</td><td>73.5</td></tr><tr><td>DIRT-T</td><td>98.9</td><td>99.4</td><td>54.5</td><td>96.1</td><td>99.5</td><td>=</td><td>75.3</td></tr><tr><td colspan="8">With Instance-Normalized Input:</td></tr><tr><td>Source-Only</td><td>59.9</td><td>82.4</td><td>40.9</td><td>88.6</td><td>86.2</td><td>77.0</td><td>62.6</td></tr><tr><td>VADA</td><td>95.7</td><td>94.5</td><td>73.3</td><td>94.9</td><td>99.2</td><td>78.3</td><td>71.4</td></tr><tr><td>DIRT-T</td><td>98.7</td><td>99.4</td><td>76.5</td><td>96.2</td><td>99.6</td><td>1</td><td>73.3</td></tr></table>
+
+Table 1: Test set accuracy on visual domain adaptation benchmarks. In all settings, both VADA and DIRT-T achieve state-of-the-art performance in all settings.
+
+$\mathbf { M N I S T } \to \mathbf { M N I S T - M }$ . We first evaluate the adaptation from MNIST to MNIST-M. MNIST-M is constructed by blending MNIST digits with random color patches from the BSDS500 dataset.
+
+$\mathbf { M N I S T }  \mathbf { S V H N }$ . The distribution shift is exacerbated when adapting between MNIST and SVHN. Whereas MNIST consists of black-and-white handwritten digits, SVHN consists of crops of colored, street house numbers. Because MNIST has a significantly lower intrinsic dimensionality that SVHN, the adaptation from M $\mathrm { I N I S T } \to \mathrm { S V H N }$ is especially challenging when the input is not pre-processed via instance normalization. When instance normalization is applied, we achieve a strong state-ofthe-art performance $7 6 . 5 \%$ and an equally impressive margin-of-improvement over source-only of $3 5 . 6 \%$ . Interestingly, by reducing the refinement interval $B$ and taking noisier natural gradient steps, we were occasionally able to achieve accuracies as high as $8 7 \%$ . However, due to the high-variance associated with this, we omit reporting this configuration in Table 1.
+
+SYN DIGITS ${ \bf \Gamma } \to { \bf S V H N }$ . The adaptation from SYN DIGITS $\mathrm { \Phi } \mathrm { \stackrel { . } { \to } } \mathrm { S V H N }$ reflect a common adaptation problem of transferring from synthetic images to real images. The SYN DIGITS dataset consist of 500000 images generated from Windows fonts by varying the text, positioning, orientation, background, stroke color, and the amount of blur.
+
+SYN SIGNS GTSRB. This setting provides an additional demonstration of adapting from synthetic images to real images. Unlike SYN DIGITS $ \mathrm { S V H N }$ , SYN SIGNS GTSRB contains 43 classes instead of 10.
+
+$\mathbf { S T L }  \mathbf { C I F A R }$ . Both STL-10 and CIFAR-10 are 10-class image datasets. These two datasets contain nine overlapping classes. Following the procedure in French et al. (2017), we removed the non-overlapping classes (“frog” and “monkey”) and reduce to a 9-class classification problem. We achieve state-of-the-art performance in both adaptation directions. In S $\mathrm { T L }  \mathrm { C }$ IFAR, we achieve a $1 1 . 7 \%$ margin-of-improvement and a performance accuracy of $7 3 . 3 \%$ . Note that because STL-10 contains a very small training set, it is difficult to estimate the conditional entropy, thus making DIRT-T unreliable for CIFAR $ \mathrm { S T L }$ .
+
+<table><tr><td>Source Target</td><td>Room A Room B</td></tr><tr><td colspan="2">With Instance-Normalized Input:</td></tr><tr><td>Source-Only</td><td>35.7</td></tr><tr><td>DANN</td><td>38.0</td></tr><tr><td>VADA</td><td>53.0</td></tr><tr><td>DIRT-T</td><td>53.0</td></tr></table>
+
+Table 2: Results of the domain adaptation experiments on Wi-Fi Activity Recognition Task
+
+Wi-Fi Activity Recognition. To evaluate the performance of our models on a non-visual domain adaptation task, we applied VADA and DIRT-T to the Wi-Fi Activity Recognition Dataset (Yousefi et al., 2017). The Wi-Fi Activity Recognition Dataset is a classification task that takes the WiFi Channel State Information (CSI) data stream as input $x$ to predict motion activity within an indoor area as output $y$ . Domain adaptation is necessary when the training and testing data are collected from different rooms, which we denote as Rooms A and B. Table 2 shows that VADA significantly improves classification accuracy compared to Source-Only and DANN by $1 7 . 3 \%$ and $1 \bar { 5 } \%$ respectively. However, DIRT-T does not lead to further improvements on this dataset. We perform experiments in Appendix F which suggests that VADA already achieves strong clustering in the target domain for this dataset, and therefore DIRT-T is not expected to yield further performance improvement.
+
+<table><tr><td>Source Target</td><td>MNIST MNIST-M</td><td>SVHN MNIST</td><td>MNIST SVHN</td><td>DIGITS SVHN</td><td>SIGNS GTSRB</td><td>CIFAR STL</td><td>STL CIFAR</td></tr><tr><td>ATT</td><td>37.1</td><td>16.1</td><td>17.9</td><td>9.0</td><td>20.5</td><td>-</td><td>-</td></tr><tr><td>II-model (aug)</td><td>-</td><td>3.7</td><td>18.1</td><td>10.6</td><td>1.0</td><td>4.5</td><td>7.4</td></tr><tr><td>DIRT-T</td><td>40.4</td><td>22.4</td><td>26.6</td><td>9.2</td><td>19.9</td><td>-</td><td>11.7</td></tr><tr><td>DIRT-T (W.I.N.I.)</td><td>38.8</td><td>17.0</td><td>35.6</td><td>7.6</td><td>13.4</td><td>-</td><td>10.7</td></tr></table>
+
+Table 3: Additional comparison of the margin of improvement computed by taking the reported performance of each model and subtracting the reported source-only performance in the respective papers. W.I.N.I. indicates “with instance-normalized input.”
+
+Overall. We achieve state-of-the-art results across all tasks. For a fairer comparison against ATT and the Π-model, Table 3 provides the improvement margin over the respective source-only performance reported in each paper. In four of the tasks (MNIST MNIST-M, $\mathrm { S V H N } \to \mathrm { M N I S T }$ , MNIST → SVHN, $\mathrm { S T L } \to \mathbf { C I F A R } )$ , we achieve substantial margin of improvement compared to previous models. In the remaining three tasks, our improvement margin over the source-only model is competitive against previous models. Our closest competitor is the Π-model. However, unlike the Π-model, we do not perform data augmentation.
+
+It is worth noting that DIRT-T consistently improves upon VADA. Since DIRT-T operates by incrementally pushing the decision boundaries away from the target domain data, it relies heavily on the cluster assumption. DIRT-T’s empirical success therefore demonstrates the effectiveness of leveraging the cluster assumption in unsupervised domain adaptation with deep neural networks.
+
+# 6.3 ANALYSIS OF VADA AND DIRT-T
+
+# 6.3.1 ROLE OF VIRTUAL ADVERSARIAL TRAINING
+
+To study the relative contribution of the virtual adversarial training in the VADA and DIRT-T objectives (Eq. (8) and Eq. (14) respectively), we perform an extensive ablation analysis in Table 4. The removal of the virtual adversarial training component is denoted by the “no-vat” subscript. Our results show that $\mathrm { \Delta V A D A _ { n o - V a t } }$ is sufficient for out-performing DANN in all but one task. The further ability for DIRT- $\scriptstyle \mathrm { { T } _ { n o - v a t } }$ to improve upon $\mathrm { \Delta V A D A _ { n o - V a t } }$ demonstrates the effectiveness of conditional entropy minimization. Ultimately, in six of the seven tasks, both virtual adversarial training and conditional entropy minimization are essential for achieving the best performance. The empirical importance of incorporating virtual adversarial training shows that the locally-Lipschitz constraint is beneficial for pushing the classifier decision boundaries away from data.
+
+<table><tr><td>Source Target</td><td>MNIST MNIST-M</td><td>SVHN MNIST</td><td>MNIST SVHN</td><td>DIGITS SVHN</td><td>SIGNS GTSRB</td><td>CIFAR STL</td><td>STL CIFAR</td></tr><tr><td colspan="8">With Instance-Normalized Input:</td></tr><tr><td>Source-Only</td><td>59.9</td><td>82.4</td><td>40.9</td><td>88.6</td><td>86.2</td><td>77.0</td><td>62.6</td></tr><tr><td>DANN (our implementation)</td><td>94.6</td><td>68.3</td><td>60.6</td><td>90.1</td><td>97.5</td><td>78.1</td><td>62.7</td></tr><tr><td>VADAno-vat</td><td>93.8</td><td>83.1</td><td>66.8</td><td>93.4</td><td>98.4</td><td>79.1</td><td>68.6</td></tr><tr><td>VADAno-vat→DIRT-Tno-vat</td><td>94.8</td><td>96.3</td><td>68.6</td><td>94.4</td><td>99.1</td><td>-</td><td>69.2</td></tr><tr><td>VADAno-vat →DIRT-T</td><td>98.3</td><td>99.4</td><td>69.8</td><td>95.3</td><td>99.6</td><td>-</td><td>71.0</td></tr><tr><td>VADA</td><td>95.7</td><td>94.5</td><td>73.3</td><td>94.9</td><td>99.2</td><td>78.3</td><td>71.4</td></tr><tr><td>VADA →DIRT-T</td><td>98.7</td><td>99.4</td><td>76.5</td><td>96.2</td><td>99.6</td><td>-</td><td>73.3</td></tr></table>
+
+Table 4: Test set accuracy in ablation experiments, starting from the DANN model. The “no-vat” subscript denote models where the virtual adversarial training component is removed.
+
+# 6.3.2 ROLE OF TEACHER MODEL IN DIRT-T
+
+![](images/d312b1ab2454d27054c9bdca71a76c4555b84afe884444c40817c1128bce2d64.jpg)  
+Figure 4: Comparing model behavior with and without the application of the KL-term. At iteration 0, we begin with the VADA initialization and apply the DIRT-T algorithm.
+
+When considering Eq. (14), it is natural to ask whether defining the neighborhood with respect to the classifier is truly necessary. In Figure 4, we demonstrate in $\mathrm { S V H N } \to \mathrm { M N I S T }$ and $\mathrm { S T L } \to \mathrm { C I F A R }$ that removal of the KL-term negatively impacts the model. Since the MNIST data manifold is low-dimensional and contains easily identifiable clusters, applying naive gradient descent (Eq. (12)) can also boost the test accuracy during initial training. However, without the KL constraint, the classifier can sometimes deviate significantly from the neighborhood of the previous classifier, and the resulting spikes in the KL-term correspond to sharp drops in target test accuracy. In $\mathrm { S T L } $ CIFAR, where the data manifold is much more complex and contains less obvious clusters, naive gradient descent causes immediate decline in the target test accuracy.
+
+# 6.3.3 VISUALIZATION OF REPRESENTATION
+
+![](images/3c8aec85d8234a9081f8a165eca04bcf0c233a289ae10e9a958859450fb84787.jpg)  
+Figure 5: T-SNE plot of the last hidden layer for MNIST (blue) $ \mathrm { S V H N }$ (red). We used the model without instance normalization to highlight the further improvement that DIRT-T provides.
+
+We further analyze the behavior of VADA and DIRT-T by showing T-SNE embeddings of the last hidden layer of the model trained to adapt from MNIST $ { \mathrm { S V H N } }$ . In Figure 5, source-only training shows strong clustering of the MNIST samples (blue) and performs poorly on SVHN (red). VADA offers significant improvement and exhibits signs of clustering on SVHN. DIRT-T begins with the VADA initialization and further enhances the clustering, resulting in the best performance on MNIST $ \mathrm { S V H N }$ .
+
+6.4 DOMAIN ADVERSARIAL TRAINING: LAYER ABLATION   
+
+<table><tr><td></td><td colspan="3">DANN</td><td colspan="3">VADA</td></tr><tr><td>Layer</td><td>JSD ≥</td><td>Source Accuracy</td><td>Target Accuracy</td><td>JSD ≥</td><td>Source Accuracy</td><td>Target Accuracy</td></tr><tr><td>L-0</td><td>0.001</td><td>78.0</td><td>24.7</td><td>0.001</td><td>24.9</td><td>18.4</td></tr><tr><td>L-1</td><td>0.002</td><td>98.6</td><td>35.0</td><td>0.007</td><td>12.0</td><td>11.6</td></tr><tr><td>L-2</td><td>0.353</td><td>16.4</td><td>10.3</td><td>0.383</td><td>11.5</td><td>9.9</td></tr><tr><td>L-3</td><td>0.036</td><td>94.8</td><td>33.8</td><td>0.034</td><td>67.8</td><td>37.1</td></tr><tr><td>L-4</td><td>0.012</td><td>97.0</td><td>40.0</td><td>0.020</td><td>96.8</td><td>61.5</td></tr><tr><td>L-5</td><td>0.235</td><td>99.3</td><td>57.9</td><td>0.244</td><td>99.4</td><td>73.3</td></tr><tr><td>L-6</td><td>0.486</td><td>99.2</td><td>60.3</td><td>0.509</td><td>99.3</td><td>70.4</td></tr><tr><td>L-7</td><td>0.644</td><td>99.0</td><td>52.5</td><td>0.608</td><td>99.1</td><td>70.5</td></tr></table>
+
+Table 5: Comparison of model behavior when domain adversarial training is applied to various layers. We denote the very last (simplex) layer of the neural network as $L$ and ablatively domain adversarial training to the last eight layers. A lower bound on the Jensen-Shannon Divergence is computed by training a logistic regression model to predict domain origin when given the layer embeddings.
+
+In Table 5, we applied domain adversarial training to various layers of a Domain Adversarial Neural Network (Ganin & Lempitsky, 2015) trained to adapt MNIST $ { \mathrm { S V H N } }$ . With the exception of layers $L - 2$ and $L - 0$ , which experienced training instability, the general observation is that as the layer gets deeper, the additional capacity of the corresponding embedding function allows better matching of the source and target distributions without hurting source generalization accuracy. This demonstrates that the combination of low divergence and high source accuracy does not imply better adaptation to the target domain. Interestingly, when the classifier is regularized to be locally-Lipschitz via VADA, the combination of low divergence and high source accuracy appears to correlate more strongly with better adaptation.
+
+# 7 CONCLUSION
+
+In this paper, we presented two novel models for domain adaptation inspired by the cluster assumption. Our first model, VADA, performs domain adversarial training with an added term that penalizes violations of the cluster assumption. Our second model, DIRT-T, is an extension of VADA that recursively refines the VADA classifier by untethering the model from the source training signal and applying approximate natural gradients to further minimize the cluster assumption violation. Our experiments demonstrate the effectiveness of the cluster assumption: VADA achieves strong performance across several domain adaptation benchmarks, and DIRT-T further improves VADA performance. Our proposed models open up several possibilities for future work. One possibility is to apply DIRT-T to weakly supervised learning; another is to improve the natural gradient approximation via K-FAC (Martens & Grosse, 2015) and PPO (Schulman et al., 2017). Given the strong performance of our models, we also recommend them for other downstream domain adaptation applications.
+
+# ACKNOWLEDGMENTS
+
+We gratefully acknowledge funding from Adobe, NSF (grants #1651565, #1522054, #1733686), Toyota Research Institute, Future of Life Institute, and Intel. We also thank Daniel Levy, Shengjia Zhao, and Jiaming Song for insightful discussions, and the anonymous reviewers for their helpful comments and suggestions.
+
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+
+# A ARCHITECTURES
+
+Table 6: Small and Large CNN architectures. Leaky ReLU parameter $a = 0 . 1$ . All convolutional and dense layers in the classifier are pre-activation batch-normalized. All images are resized to $3 2 \times$ $3 2 \times 3$ . Note the use of additive Gaussian noise: this addition was motivated by initial experiments in which we observed that domain adversarial training appears to contract the feature space.   
+
+<table><tr><td>Layer Index</td><td>Small CNN Large CNN</td></tr><tr><td>L-18 一</td><td>32× 32×3Image</td></tr><tr><td>L-17 1</td><td>Instance Normalization (optional)</td></tr><tr><td>L-16</td><td>3 × 3 conv. 64 IReLU 3 × 3 conv. 96 IReLU</td></tr><tr><td>L-15</td><td>3 × 3 conv. 64 IReLU 3 × 3 conv. 96 IReLU</td></tr><tr><td>L-14</td><td>3 × 3 conv. 64 IReLU 3 × 3 conv. 96 lReLU</td></tr><tr><td>L-13</td><td>2 x 2 max-pool, stride 2</td></tr><tr><td>L-12</td><td>dropout, p = 0.5</td></tr><tr><td>L-11</td><td>Gaussian noise,σ = 1</td></tr><tr><td>L-10</td><td>3 × 3 conv. 64 IReLU 3 × 3 conv. 192 IReLU</td></tr><tr><td>L-9</td><td>3 × 3 conv. 64 IReLU 3 × 3 conv. 192 IReLU</td></tr><tr><td>L-8</td><td>3 × 3 conv. 64 IReLU 3 × 3 conv. 192 IReLU</td></tr><tr><td>L-7</td><td>2 x 2 max-pool, stride 2</td></tr><tr><td>L-6</td><td>dropout, p = 0.5</td></tr><tr><td>L-5</td><td>Gaussian noise,σ = 1</td></tr><tr><td>L-4</td><td>3 × 3 conv. 64 IReLU 3 × 3 conv. 192 IReLU</td></tr><tr><td>L-3</td><td>3× 3 conv. .64IReLU 3 × 3 conv. 192 IReLU</td></tr><tr><td>L-2</td><td>3 × 3 conv. 64 lReLU 3 × 3 conv. 192 IReLU</td></tr><tr><td>L-1</td><td>global average pool</td></tr><tr><td></td><td></td></tr><tr><td>L-0</td><td>10 dense, softmax</td></tr></table>
+
+Table 7: Domain discriminator architecture.   
+
+<table><tr><td>Domain Discriminator</td></tr><tr><td>Layer L-5 Output</td></tr><tr><td>100 dense,ReLU</td></tr><tr><td>1 dense, sigmoid</td></tr></table>
+
+# B HYPERPARAMETERS
+
+We observed that extensive hyperparameter-tuning is not necessary to achieve state-of-the-art performance. To demonstrate this, we restrict our hyperparameter search for each task to $\lambda _ { d } ~ =$ $\{ 0 , 1 0 ^ { - 2 } \} , \lambda _ { s } = \{ 0 , 1 \} , \lambda _ { t } = \{ 1 0 ^ { - 2 } , 1 0 ^ { - 1 } \}$ , in all experiments with instance-normalized inputs. We fixed $\beta = 1 0 ^ { - 2 }$ . Note that the decision to turn $( \lambda _ { d } , \lambda _ { s } )$ on or off that can often be determined a priori based on prior belief regarding the extent to covariate shift. In the absence of such prior belief, a reliable choice is $( \lambda _ { d } = \mathsf { \breve { { 1 0 } } ^ { - 2 } } , \mathsf { \breve { { \lambda } } } _ { s } = 1 , \lambda _ { t } = 1 0 ^ { - 2 } , \beta = 1 0 ^ { - 2 } )$ .
+
+<table><tr><td>Task</td><td>Instance-Normalized</td><td>入d</td><td>入s</td><td>\t</td><td>β</td></tr><tr><td>MNIST→MNIST-M</td><td>Yes,No</td><td>10-2</td><td>0</td><td>10-2</td><td>10-2</td></tr><tr><td>SVHN→MNIST</td><td>Yes,No</td><td>10-2</td><td>0</td><td>10-2</td><td>10-2</td></tr><tr><td>MNIST→SVHN</td><td>Yes</td><td>10-2</td><td>1</td><td>10-2</td><td>10-2</td></tr><tr><td>MNIST→SVHN</td><td>No</td><td>10-2</td><td>1</td><td>10-2</td><td>10-3</td></tr><tr><td>DIGITS→SVHN</td><td>Yes,No</td><td>10- -2</td><td>1</td><td>10-1</td><td>10-2</td></tr><tr><td>SIGNS →GTSRB</td><td>Yes,No</td><td>10-2</td><td>1</td><td>10-2</td><td>10-2</td></tr><tr><td>CIFAR→STL</td><td>Yes,No</td><td>0</td><td>1</td><td>10-1</td><td>10-2</td></tr><tr><td>STL→CIFAR</td><td>Yes,No</td><td>0</td><td>0</td><td>10-1</td><td>10-2</td></tr><tr><td>Room A→B</td><td>Yes</td><td>0</td><td>0</td><td>10-2</td><td>10-2</td></tr></table>
+
+Table 8: Hyperparameters for each task, both with and without instance-normalized input. The only exception is $\mathrm { M N I S T } \to \mathrm { S V H N }$ without instance-normalized input. In this specific case, $d _ { \mathcal { H } \Delta \mathcal { H } }$ is sufficiently large that conditional entropy minimization quickly finds a degenerate solution in the target domain. To counter this, we remove conditional entropy minimization (but keep the target-side virtual adversarial training) only during VADA. We apply target-side conditional entropy minimization and virtual adversarial training during DIRT-T. To compensate, we use a lower $\beta$ during the DIRT-T phase to allow for larger natural gradient steps.
+
+When the target domain is MNIST/MNIST-M, the task is sufficiently simple that we only allocate $B = 5 0 0$ iterations to each optimization problem in Eq. (14). In all other cases, we set the refinement interval $B = 5 0 0 0$ . We apply Adam Optimizer (learning rate $= 0 . 0 0 1 , \beta _ { 1 } = 0 . 5 , \beta _ { 2 } = 0 . 9 9 9 )$ with Polyak averaging (more accurately, we apply an exponential moving average with momentum $= \ 0 . 9 9 8$ to the parameter trajectory). VADA was trained for 80000 iterations and DIRT-T takes VADA as initialization and was trained for $\{ 2 0 0 0 0 , 4 0 0 0 0 , 6 0 0 0 0 , 8 0 0 0 0 \}$ iterations, with number of iterations chosen as hyperparameter.
+
+# C REPLACING GRADIENT REVERSAL
+
+We note from Goodfellow et al. (2014) that the gradient of $\nabla _ { \theta } \ln ( 1 - D ( f _ { \theta } ( x ) ) )$ is tends to have smaller norm than $- \nabla _ { \theta } \ln D ( f _ { \theta } ( x ) )$ during initial training since the latter rescales the gradient by $1 / D ( f _ { \theta } ( x ) )$ . Following this observation, we replace the gradient reversal procedure with alternating minimization of
+
+$$
+\begin{array} { r l } & { \underset { D } { \mathop { \operatorname* { m i n } } } - \mathbb { E } _ { x \sim \mathcal { D } _ { s } } \left[ \ln D ( f _ { \theta } ( x ) ) \right] - \mathbb { E } _ { x \sim \mathcal { D } _ { t } } \left[ \ln 1 - D ( f _ { \theta } ( x ) ) \right] } \\ & { \underset { \theta } { \mathop { \operatorname* { m i n } } } - \mathbb { E } _ { x \sim \mathcal { D } _ { t } } \left[ \ln D ( f _ { \theta } ( x ) ) \right] - \mathbb { E } _ { x \sim \mathcal { D } _ { s } } \left[ \ln 1 - D ( f _ { \theta } ( x ) ) \right] . } \end{array}
+$$
+
+The choice of using gradient reversal versus alternating minimization reflects a difference in choice of approximating the mini-max using saturating versus non-saturating optimization (Fedus et al., 2017). In some of our initial experiments, we observed the replacement of gradient reversal with alternating minimization stabilizes domain adversarial training. However, we encourage practitioners to try either optimization strategy when applying VADA.
+
+# D INSTANCE NORMALIZATION FOR DOMAIN ADAPTATION
+
+Theorem 1 suggests that we should identify ways of constraining the hypothesis space without hurting the global optimal classifier for the joint task. We propose to further constrain our model by
+
+introducing instance normalization as an image pre-processing step for the input data. Instance normalization was proposed for style transfer Ulyanov et al. (2016) and applies the operation
+
+$$
+\ell ( x ^ { ( i ) } ) = \frac { x ^ { ( i ) } - \mu ( x ^ { ( i ) } ) } { \sigma ( x ^ { ( i ) } ) } ,
+$$
+
+where $\boldsymbol { x } ^ { ( i ) } \in \mathbb { R } ^ { H \times W \times C }$ denotes the $i ^ { \mathrm { { t h } } }$ sample with $( H , W , C )$ corresponding to the height, width, ∈ and channel dimensions, and where $\mu , \sigma : \dot { \mathbb R } ^ { H \times W \times C }  \mathbb R ^ { C }$ are functions that compute the mean and standard deviation across the spatial dimensions. A notable property of instance normalization is that it is invariant to channel-wide scaling and shifting of the input elements. Formally, consider scaling and shift variables $\gamma , \beta \in \mathbb { R } ^ { C }$ . If $\gamma \succ 0$ and $\sigma ( x ^ { ( i ) } ) \succ 0$ , then
+
+$$
+\ell ( x ^ { ( i ) } ) = \ell ( \gamma x ^ { ( i ) } + \beta ) .
+$$
+
+For visual data the application of instance normalization to the input layer makes the classifier invariant to channel-wide shifts and scaling of the pixel intensities. For most visual tasks, sensitivity to channel-wide pixel intensity changes is not critical to the success of the classifier. As such, instance normalization of the input may help reduce $d _ { \mathcal { H } \Delta \mathcal { H } }$ without hurting the globally optimal classifier. Interestingly, Figure 3 shows that input instance normalization is not equivalent to gray-scaling, since color is partially preserved. To test the effect of instance normalization, we report results both with and without the use of instance-normalized inputs.
+
+# E LIMITATION OF DOMAIN ADVERSARIAL TRAINING
+
+We denote the source and target distributions respectively as $p _ { s } ( x , y )$ and $p _ { t } ( x , y )$ . Let the source covariate distribution $p _ { s } ( x )$ define the random variable $X _ { s }$ that have support $\mathrm { s u p p } ( X _ { s } ) = \mathcal { X } _ { s }$ and let $( X _ { t } , \mathcal { X } _ { t } )$ be analogously defined for the target domain. Both $\mathcal { X } _ { s }$ and $\mathcal { X } _ { t }$ are subsets of $\mathbb { R } ^ { n }$ . Let $p _ { s } ( y )$ and $p _ { t } ( y )$ define probabilities over the support $\mathcal { V } = \{ 1 , \ldots , K \}$ . We consider any embedding function $f : \mathbb { R } ^ { n }  \mathbb { R } ^ { m }$ , where $\mathbb { R } ^ { m }$ is the embedding space, and any embedding classifier $g : \mathbb { R } ^ { m } $ $\mathcal { C }$ , where $\mathcal { C }$ is the $( K - 1 )$ -simplex. We denote a classifier $h = g \circ f$ has the composite of an embedding function with an embedding classifier.
+
+For simplicity, we restrict our analysis to the simple case where $K = 2$ , i.e. where $\mathcal { V } = \{ 0 , 1 \}$ .   
+Furthermore, we assume that for any $\delta \in [ 0 , 1 ]$ , there exists a subset $\Omega \subseteq \mathbb { R } ^ { n }$ where $p _ { s } ( x \in \Omega ) = \delta$ .   
+We impose a similar condition on $p _ { t } ( x )$ .
+
+For a joint distribution $p ( x , y )$ , we denote the generalization error of a classifier as
+
+$$
+\epsilon _ { p } ( h ) = \mathbb { E } _ { p ( x , y ) } \left| y - h ( x ) \right| .
+$$
+
+Note that for a given classifier $h : \mathbb { R } ^ { n }  [ 0 , 1 ]$ , the corresponding hard classifier is $k ( x ) \ =$ $\mathbb { 1 } \{ h ( x ) > 0 . 5 \}$ . We further define the set $\Omega \subseteq \mathbb { R } ^ { n }$ such that
+
+$$
+\Omega = \{ x \in \mathbb { R } ^ { n } \mid k ( x ) = 1 \} \iff k ( x ) = \mathbb { 1 } \{ x \in \Omega \} .
+$$
+
+In a slight abuse of notation, we define the generalization error $\epsilon ( \Omega )$ with respect to $\Omega$ as
+
+$$
+\epsilon _ { p } ( \Omega ) = \mathbb { E } _ { p ( x , y ) } \mathbb { 1 } \{ x \in \Omega \} = \epsilon ( k ) .
+$$
+
+An optimal $\Omega _ { p } ^ { * }$ is a partitioning of $\mathbb { R } ^ { n }$
+
+$$
+\epsilon _ { p } ( \Omega _ { p } ^ { * } ) = \operatorname* { m i n } _ { \Omega \subseteq \mathbb { R } ^ { n } } \epsilon _ { p } ( \Omega )
+$$
+
+such that generalization error under the distribution $p ( x , y )$ is minimized.
+
+# E.1 GOOD TARGET-DOMAIN ACCURACY IS NOT GUARANTEED
+
+Domain adversarial training seeks to find a single classifier $h$ used for both the source $p _ { s }$ and target $p _ { t }$ distributions. To do so, domain adversarial training sets up the objective
+
+$$
+\begin{array} { r l } & { \underset { f \in \mathcal { F } , g \in \mathcal { G } } { \operatorname* { m i n } } \ \epsilon _ { p _ { s } } ( g \circ f ) } \\ & { \qquad \mathrm { s . t . } \ g ( X _ { s } ) = g ( X _ { t } ) , } \end{array}
+$$
+
+where $\mathcal { F }$ and $\mathcal { G }$ are the hypothesis spaces for the embedding function and embedding classifier. Intuitively, domain adversarial training operates under the hypothesis that good source generalization error in conjunction with source-target feature matching implies good target generalization error. We shall see, however, that if $\boldsymbol { \mathcal { X } } _ { s } \cap \boldsymbol { \mathcal { X } } _ { t } = \boldsymbol { \mathcal { O } }$ and $\mathcal { F }$ is sufficiently complex, this implication does not necessarily hold.
+
+Let $\mathcal { F }$ contain all functions mapping $\mathbb { R } ^ { n } \to \mathbb { R } ^ { m }$ , i.e. $\mathcal { F }$ has infinite capacity. Suppose $\mathcal { G }$ contains the function $g ( z ) = 1 \{ z = 1 _ { m } \}$ and $\mathcal { X } _ { s } \cap \mathcal { X } _ { t } = \mathcal { O }$ . We consider the set
+
+$$
+\mathcal { H } ^ { \ast } = \left\{ g \circ f | \exists g \in \mathcal { G } , f \in \mathcal { F } \mathrm { ~ s . t . ~ } \epsilon _ { p _ { s } } ( g \circ f ) \leq \epsilon _ { p _ { s } } ( \Omega _ { p _ { s } } ^ { \ast } ) , f ( X _ { s } ) = f ( X _ { t } ) \right\} .
+$$
+
+Such a set of classifiers satisfies the feature-matching constraint while achieving source generalization error no worse than the optimal source-domain hard classifier. It suffices to show that $\mathcal { H } ^ { \ast }$ includes hypotheses that perform poorly in the target domain.
+
+We first show $\mathcal { H } ^ { \ast }$ is not an empty set by constructing an element of this set. Choose a partitioning $\Omega$ where
+
+$$
+p _ { t } ( x \in \Omega ) = p _ { s } ( x \in \Omega _ { p _ { s } } ^ { * } ) .
+$$
+
+Consider the embedding function
+
+$$
+f _ { \Omega } ( x ) = { \left\{ \begin{array} { l l } { \mathbf { 1 } _ { m } } & { { \mathrm { i f ~ } } ( x \in { \mathcal { X } } _ { s } \cap \Omega _ { p _ { s } } ^ { * } ) \lor ( x \in { \mathcal { X } } _ { t } \cap \Omega ) } \\ { \mathbf { 0 } _ { m } } & { { \mathrm { o t h e r w i s e } } . } \end{array} \right. }
+$$
+
+Let $g ( z ) = 1 \{ z = 1 _ { m } \}$ . It follows that the composite classifier $h _ { \Omega } = g \circ f _ { \Omega }$ is an element of $\mathcal { H } ^ { \ast }$ .
+
+Next, we show that a classifier $h \in \mathcal { H } ^ { * }$ does not necessarily achieve good target generalization error. Consider the partitioning $\hat { \Omega }$ which solves the following optimization problem
+
+$$
+\begin{array} { r l } & { \underset { \Omega \subseteq \mathbb { R } ^ { n } } { \mathrm { m a x } } ~ \epsilon _ { p _ { t } } ( \Omega ) } \\ & { \mathrm { s . t . } ~ p _ { t } ( x \in \Omega ) = p _ { s } ( x \in \Omega _ { p _ { s } } ^ { * } ) . } \end{array}
+$$
+
+Such a partitioning $\hat { \Omega }$ is the worst-case partitioning subject to the probability mass constraint. It follows that worse case $h ^ { \prime } \in \mathcal { H } ^ { * }$ has generalization error
+
+$$
+\epsilon _ { p _ { t } } ( h ^ { \prime } ) = \operatorname* { m a x } _ { h \in \mathcal { H } ^ { * } } \epsilon _ { p _ { t } } ( h ) \geq \epsilon _ { p _ { t } } ( h _ { \hat { \Omega } } ) .
+$$
+
+To provide intuition that $\epsilon _ { p _ { t } } ( h ^ { \prime } )$ is potentially very large, consider hypothetical source and target domains where $\boldsymbol { \mathcal { X } } _ { s } \cap \boldsymbol { \mathcal { X } } _ { t } = \boldsymbol { \varpi }$ and $\bar { p _ { t } } ( x \in \Omega _ { p _ { t } } ^ { * } ) = \bar { p } _ { s } ( x \overbar { \in \Omega } _ { p _ { s } } ^ { * } ) = 0 . 5$ . The worst-case partitioning subject to the probability mass constraint is simply $\hat { \Omega } = \mathbb { R } ^ { n } \backslash \Omega _ { p _ { t } } ^ { * }$ (which flips the labels) and consequently, $\mathcal { H } ^ { \ast }$ contains solutions
+
+$$
+\operatorname* { m a x } _ { h \in \mathcal { H } ^ { \ast } } \epsilon _ { p _ { t } } ( h ) \geq 1 - \epsilon _ { p _ { t } } ( \Omega _ { p _ { t } } ^ { \ast } )
+$$
+
+no better than the worst-case partitioning of the target domain.
+
+# E.2 CONNECTION TO THEOREM 1
+
+Let $\mathcal { F }$ contain all functions mapping $\mathbb { R } ^ { n } \to \mathbb { R } ^ { m }$ , i.e. $\mathcal { F }$ has infinite capacity. Suppose $\mathcal { G }$ contains the function $g ( z ) = \mathbb { 1 } \{ z = \mathbf { 1 } _ { m } \}$ and $\mathcal { X } _ { s } \cap \mathcal { X } _ { t } = \mathcal { O }$ . We consider the sets
+
+$$
+\begin{array} { r l } & { \mathcal { H } = \left\{ g \circ f \ | \ g \in { \mathcal { G } } , f \in { \mathcal { F } } \right\} } \\ & { \bar { \mathcal { H } } = \left\{ g \circ f \ | \ \exists g \in { \mathcal { G } } , f \in { \mathcal { F } } \mathrm { ~ s . t . ~ } f ( X _ { s } ) = f ( X _ { t } ) \right\} . } \end{array}
+$$
+
+A justification for domain adversarial training is that the $\bar { \mathcal { H } } \Delta \bar { \mathcal { H } }$ -divergence term is smaller than the $\mathcal { H } \Delta \mathcal { H }$ -divergence, thus yielding a tighter upper bound for Theorem 1. However, we shall see that the $\bar { \mathcal { H } } \Delta \bar { \mathcal { H } }$ -divergence term is in fact maximal.
+
+Choose partitionings $\Omega _ { s } , \Omega _ { t } \subseteq \mathbb { R } ^ { n }$ such that
+
+$$
+p _ { s } ( x \in \Omega _ { s } ) = p _ { t } ( x \in \Omega _ { t } ) = 0 . 5 .
+$$
+
+Define the embedding functions
+
+$$
+\begin{array} { r l } { f ( x ) = \left\{ \mathbf { 1 } _ { m } \right. } & { \mathrm { i f ~ } ( x \in \mathcal { X } _ { s } \cap \Omega _ { s } ) \vee ( x \in \mathcal { X } _ { t } \cap \Omega _ { t } ) } \\ { \mathbf { 0 } _ { m } } & { \mathrm { o t h e r w i s e } . } \end{array}
+$$
+
+Let ${ { g } ^ { \prime } } ( z ) = g ( z ) = \mathbb { 1 } \left\{ z = \mathbf { 1 } _ { m } \right\}$ . It follows that the composite classifiers $h = g \circ f$ and $h ^ { \prime } = g ^ { \prime } \circ f ^ { \prime }$ are elements of $\bar { \mathcal { H } }$ .
+
+From the definition of $d _ { \mathcal { H } \Delta \mathcal { H } }$ , we see that
+
+$$
+\begin{array} { r l } & { d _ { \bar { \mathcal { H } } \Delta \bar { \mathcal { H } } } \geq 2 \left| \mathbb { E } _ { x \sim X _ { s } } \left[ h ( x ) \neq h ^ { \prime } ( x ) \right] - \mathbb { E } _ { x \sim X _ { t } } \left[ h ( x ) \neq h ^ { \prime } ( x ) \right] \right| } \\ & { \qquad = 2 \cdot \left| 0 - 1 \right| = 2 . } \end{array}
+$$
+
+The $\bar { \mathcal { H } } \Delta \bar { \mathcal { H } }$ -divergence thus achieves the maximum value of 2.
+
+# E.3 IMPLICATIONS
+
+Our analysis assumes infinite capacity embedding functions and the ability to solve optimization problems exactly. The empirical success of domain adversarial training suggests that the use of finite-capacity convolutional neural networks combined with stochastic gradient-based optimization provides the necessary regularization for domain adversarial training to work. The theoretical characterization of domain adversarial training in the case finite-capacity convolutional neural networks and gradient-based learning remains a challenging but important open research problem.
+
+# F NON-VISUAL DOMAIN ADAPTATION TASK
+
+To evaluate the performance of our models on a non-visual domain adaptation task, we applied VADA and DIRT-T to the Wi-Fi Activity Recognition Dataset (Yousefi et al., 2017). The Wi-Fi Activity Recognition Dataset is a classification task that takes the Wi-Fi Channel State Information (CSI) data stream as input $x$ to predict motion activity within an indoor area as output $y$ . The dataset collected the CSI data stream samples associated with seven activities, denoted as “bed”, “fall”, “walk”, “pick up”, “run”, “sit down”, and “stand up”.
+
+However, the joint distribution over the CSI data stream and motion activity changes depending on the room in which the data was collected. Since the data was collected for multiple rooms, we selected two rooms (denoted here as Room A and Room B) and constructed the unsupervised domain adaptation task by using Room A as the source domain and Room B as the target domain. We compare the performance of DANN, VADA, and DIRT-T on the Wi-Fi domain adaptation task in Table 2, using the hyperparameters $( \lambda _ { d } = 0 , \lambda _ { s } = 0 , \lambda _ { t } = 1 0 ^ { - 2 } , \beta = 1 0 ^ { - 2 } )$ .
+
+Table 2 shows that VADA significantly improves classification accuracy compared to Source-Only and DANN. However, DIRT-T does not lead to further improvements on this dataset. We believe this is attributable to VADA successfully pushing the decision boundary away from data-dense regions in the target domain. As a result, further application of DIRT-T would not lead to better decision boundaries. To validate this hypothesis, we visualize the t-SNE embeddings for VADA and DIRT-T in Figure 6 and show that VADA is already capable of yielding strong clustering in the target domain. To verify that the decision boundary indeed did not change significantly, we additionally provide the confusion matrix between the VADA and DIRT-T predictions in the target domain (Fig. 7).
+
+![](images/78c92b10e5e6c3f8c513377fea99f29619a11867564123ab0bd5b657b6819b60.jpg)  
+Figure 6: T-SNE plot of the last hidden layer for Room A (blue) Room B (red)
+
+![](images/aa4937a0ea5ebc3ef71a233b39fdb63e36a50a67c34398625889953d9916bc85.jpg)  
+Figure 7: Confusion matrix between VADA and and DIRT-T prediction labels.

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+# LEARNING A SAT SOLVER FROM SINGLE-BIT SUPER-VISION
+
+Daniel Selsam, Matthew Lamm, Benedikt Bunz, Percy Liang, David L. Dill ¨
+
+Department of Computer Science   
+Stanford University   
+Stanford, CA 94305   
+{dselsam,mlamm,buenz,pliang,dill}@cs.stanford.edu
+
+Leonardo de Moura Microsoft Research Redmond, WA 98052 leonardo@microsoft.com
+
+# ABSTRACT
+
+We present NeuroSAT, a message passing neural network that learns to solve SAT problems after only being trained as a classifier to predict satisfiability. Although it is not competitive with state-of-the-art SAT solvers, NeuroSAT can solve problems that are substantially larger and more difficult than it ever saw during training by simply running for more iterations. Moreover, NeuroSAT generalizes to novel distributions; after training only on random SAT problems, at test time it can solve SAT problems encoding graph coloring, clique detection, dominating set, and vertex cover problems, all on a range of distributions over small random graphs.
+
+# 1 INTRODUCTION
+
+The propositional satisfiability problem (SAT) is one of the most fundamental problems of computer science. Cook (1971) showed that the problem is NP-complete, which means that searching for any kind of efficiently-checkable certificate in any context can be reduced to finding a satisfying assignment of a propositional formula. In practice, search problems arising from a wide range of domains such as hardware and software verification, test pattern generation, planning, scheduling, and combinatorics are all routinely solved by constructing an appropriate SAT problem and then calling a SAT solver (Gomes et al., 2008). Modern SAT solvers based on backtracking search are extremely well-engineered and have been able to solve problems of practical interest with millions of variables (Biere et al., 2009).
+
+We consider the question: can a neural network learn to solve SAT problems? To answer, we develop a novel message passing neural network (MPNN) (Scarselli et al., 2009; Li et al., 2015; Gilmer et al., 2017), NeuroSAT, and train it as a classifier to predict satisfiability on a dataset of random SAT problems. We provide NeuroSAT with only a single bit of supervision for each SAT problem that indicates whether or not the problem is satisfiable. When making a prediction about a new SAT problem, we find that NeuroSAT guesses unsatisfiable with low confidence until it finds a solution, at which point it converges and guesses satisfiable with very high confidence. The solution itself can almost always be automatically decoded from the network’s activations, making NeuroSAT an end-to-end SAT solver. See Figure 1 for an illustration of the train and test regimes.
+
+Although it is not competitive with state-of-the-art SAT solvers, NeuroSAT can solve SAT problems that are substantially larger and more difficult than it ever saw during training by simply performing more iterations of message passing. Despite only running for a few dozen iterations during training, at test time NeuroSAT continues to find solutions to harder problems after hundreds and even thousands of iterations. The learning process has yielded not a traditional classifier but rather a procedure that can be run indefinitely to search for solutions to problems of varying difficulty.
+
+![](images/1d225b957fb03a6856771f4a7ff69ce2bb215e56e9ecf9dadd0703c9dec0e7b8.jpg)  
+Figure 1: We train NeuroSAT to predict whether SAT problems are satisfiable, providing only a single bit of supervision for each problem. At test time, when NeuroSAT predicts satisfiable, we can almost always extract a satisfying assignment from the network’s activations. The problems at test time can also be substantially larger, more difficult, and even from entirely different domains than the problems seen during training.
+
+Moreover, NeuroSAT generalizes to entirely new domains. Since NeuroSAT operates on SAT problems and since SAT is NP-complete, NeuroSAT can be queried on SAT problems encoding any kind of search problem for which certificates can be checked in polynomial time. Although we train it using only problems from a single random problem generator, at test time it can solve SAT problems encoding graph coloring, clique detection, dominating set, and vertex cover problems, all on a range of distributions over small random graphs.
+
+The same neural network architecture can also be used to help construct proofs for unsatisfiable problems. When we train it on a different dataset in which every unsatisfiable problem contains a small contradiction (call this trained model NeuroUNSAT), it learns to detect these contradictions instead of searching for satisfying assignments. Just as we can extract solutions from NeuroSAT’s activations, we can extract the variables involved in the contradiction from NeuroUNSAT’s activations. When the number of variables involved in the contradiction is small relative to the total number of variables, knowing which variables are involved in the contradiction can enable constructing a resolution proof more efficiently.
+
+# 2 PROBLEM SETUP
+
+Background. A formula of propositional logic is a boolean expression built using the constants true (1) and false (0), variables, negations, conjunctions, and disjunctions. A formula is satisfiable provided there exists an assignment of boolean values to its variables such that the formula evaluates to 1. For example, the formula $( x _ { 1 } \lor x _ { 2 } \lor x _ { 3 } ) \land \lnot ( x _ { 1 } \land x _ { 2 } \land x _ { 3 } )$ is satisfiable because it will evaluate to 1 under every assignment that does not map $x _ { 1 }$ , $x _ { 2 }$ and $x _ { 3 }$ to the same value. For every formula, there exists an equisatisfiable formula in conjunctive normal form (CNF), expressed as a conjunction of disjunctions of (possibly negated) variables.1 Each conjunct of a formula in CNF is called a clause, and each (possibly negated) variable within a clause is called a literal. The formula above is equivalent to the CNF formula $\left( x _ { 1 } \lor x _ { 2 } \lor x _ { 3 } \right) \land \left( \lnot x _ { 1 } \lor \lnot x _ { 2 } \lor \lnot x _ { 3 } \right)$ , which we can represent more concisely as $\{ 1 | 2 | 3 , { \overline { { 1 } } } | { \overline { { 2 } } } | { \overline { { 3 } } } \}$ . A formula in CNF has a satisfying assignment if and only if it has an assignment such that every clause has at least one literal mapped to 1. A SAT problem is a formula in CNF, where the goal is to determine if the formula is satisfiable, and if so, to produce a satisfying assignment of truth values to variables. We use $n$ to denote the number of of variables in a SAT problem, and $m$ to denote the number of clauses.
+
+Classification task. For a SAT problem $P$ , we define $\phi ( P )$ to be true if and only if $P$ is satisfiable. Our first goal is to learn a classifier that approximates $\phi$ . Given a distribution $\Psi$ over SAT problems, we can construct datasets $\mathcal { D } _ { \mathrm { t r a i n } }$ and $\mathcal { D } _ { \mathrm { t e s t } }$ with examples of the form $( P , \phi ( P ) )$ by sampling problems $P \sim \Psi$ and computing $\phi ( P )$ using an existing SAT solver. At test time, we get only the problem $P$ and the goal is to predict $\phi ( P )$ , i.e. to determine if $P$ is satisfiable. Ultimately we care about the solving task, which also includes finding solutions to satisfiable problems.
+
+# 3 MODEL
+
+A SAT problem has a simple syntactic structure and therefore could be encoded into a vector space using standard methods such as an RNN. However, the semantics of propositional logic induce rich invariances that such a syntactic method would ignore, such as permutation invariance and negation invariance. Specifically, the satisfiability of a formula is not affected by permuting the variables (e.g. swapping $x _ { 1 }$ and $x _ { 2 }$ throughout the formula), by permuting the clauses (e.g. swapping the first clause with the second clause), or by permuting the literals within a clause (e.g. replacing the clause $1 | \overline { { 2 } }$ with $\overline { { 2 } } | 1$ . The satisfiability of a formula is also not affected by negating every literal corresponding to a given variable (e.g. negating all occurrences of $x _ { 1 }$ in $\{ 1 | \overline { { 2 } } , \overline { { 1 } } | \overline { { 3 } } \}$ to yield $\{ { \overline { { 1 } } } | { \overline { { 2 } } } , 1 | { \overline { { 3 } } } \} )$ .
+
+We now describe our neural network architecture, NeuroSAT, that enforces both permutation invariance and negation invariance. We encode a SAT problem as an undirected graph with one node for every literal, one node for every clause, an edge between every literal and every clause it appears in, and a different type of edge between each pair of complementary literals (e.g. between $x _ { i }$ and $\overline { { x _ { i } } }$ ). NeuroSAT iteratively refines a vector space embedding for each node by passing “messages” back and forth along the edges of the graph as described in Gilmer et al. (2017). At every time step, we have an embedding for every literal and every clause. An iteration consists of two stages. First, each clause receives messages from its neighboring literals and updates its embedding accordingly. Next, each literal receives messages from its neighboring clauses as well as from its complement, then updates its embedding accordingly. Figure 2 provides a high-level illustration of the architecture.
+
+![](images/f0aae2ef71377da7802b98440976e9a312f6b983b1f98b853dd17e23a76f7d0b.jpg)  
+Figure 2: High-level illustration of NeuroSAT operating on the graph representation of $\{ 1 | 2 , { \overline { { 1 } } } | { \overline { { 2 } } } \}$ . On the top of both figures are nodes for each of the four literals, and on the bottom are nodes for each of the two clauses. At every time step $t$ , we have an embedding for every literal and every clause. An iteration consists of two stages. First, each clause receives messages from its neighboring literals and updates it embedding accordingly (Figure 2a). Next, each literal receives messages from its neighboring clause as well as from its complement, and updates its embedding accordingly (Figure 2b).
+
+More formally, our model is parameterized by two vectors $\mathbf { L } _ { \mathrm { i n i t } }$ , $\mathbf { C } _ { \mathrm { i n i t } } )$ , three multilayer perceptrons $( \mathbf { L } _ { \mathrm { m s g } } , \mathbf { C } _ { \mathrm { m s g } } , \mathbf { L } _ { \mathrm { v o t e } } )$ and two layer-norm LSTMs (Ba et al., 2016; Hochreiter & Schmidhuber, 1997) $( { \bf L } _ { \bf u } , { \bf C } _ { \bf u } )$ . At every time step $t$ , we have a matrix $L ^ { ( t ) } \in \mathbb { R } ^ { 2 n \times d }$ whose ith row contains the embedding for the literal $\ell _ { i }$ and a matrix $C ^ { ( t ) } \in \mathbb { R } ^ { m \times d }$ whose $j$ th row contains the embedding for the clause $c _ { j }$ , which we initialize by tiling $\mathbf { L } _ { \mathrm { i n i t } }$ and $\mathbf { C } _ { \mathrm { i n i t } }$ respectively. We also have hidden states $L _ { h } ^ { ( t ) } \in \mathbb { R } ^ { 2 n \times d }$ and artite $C _ { h } ^ { ( t ) } \in \mathbb { R } ^ { m \times d }$ for atrix $\mathbf { L } _ { \mathrm { u } }$ and fined $\mathbf { C } _ { \mathbf { u } }$ alized to zero matrices. Letand let Flip be the operator $M$ $M ( i , j ) = \mathrm { ~ \bar { 1 } ~ } \{ \ell _ { i } \in c _ { j } \}$   
+that takes a matrix $L$ and swaps each row of $L$ with the row corresponding to the literal’s negation. A single iteration consists of applying the following two updates:
+
+$$
+\begin{array} { r l } & { ( C ^ { ( t + 1 ) } , C _ { h } ^ { ( t + 1 ) } ) \gets \mathbf { C } _ { \mathbf { u } } ( [ C _ { h } ^ { ( t ) } , M ^ { \top } \mathbf { L } _ { \mathrm { m s g } } ( L ^ { ( t ) } ) ] ) } \\ & { ( L ^ { ( t + 1 ) } , L _ { h } ^ { ( t + 1 ) } ) \gets \mathbf { L } _ { \mathbf { u } } ( [ L _ { h } ^ { ( t ) } , \mathrm { F l i p } ( L ^ { ( t ) } ) , M \mathbf { C } _ { \mathrm { m s g } } ( C ^ { ( t + 1 ) } ) ] ) } \end{array}
+$$
+
+After $T$ iterations, we compute $L _ { * } ^ { ( T ) }  \mathbf { L } _ { \mathrm { v o t e } } ( L ^ { ( T ) } ) \in \mathbb { R } ^ { 2 n }$ , which contains a single scalar for each literal (the literal’s vote), and then we compute the average of the literal votes $y ^ { ( T ) } \gets \mathrm { m e a n } ( L _ { * } ^ { ( T ) } ) \in$ $\mathbb { R }$ . We train the network to minimize the sigmoid cross-entropy loss between the logit $y ^ { ( T ) }$ and the true label $\phi ( P )$ .
+
+Our architecture enforces permutation invariance by operating on nodes and edges according to the topology of the graph without any additional ordering over nodes or edges. Likewise, it enforces negation invariance by treating all literals the same no matter whether they originated as a positive or negative occurrence of a variable.
+
+![](images/eddc765e938654a29717017c9c02dd342e57d38a5f900cee4678095d179f8f86.jpg)  
+Iteration   
+Figure 3: The sequence of literal votes $L _ { * } ^ { ( 1 ) }$ 1) to L(24)∗ as NeuroSAT runs on a satisfiable problem from SR(20). For clarity, we reshape each $L _ { * } ^ { ( t ) }$ to be an $\mathbb { R } ^ { n \times 2 }$ matrix so that each literal is paired with its complement; specifically, the ith row contains the scalar votes for $x _ { i }$ and $\overline { { x _ { i } } }$ . Here white represents zero, blue negative and red positive. For several iterations, almost every literal is voting unsat with low confidence (light blue). Then a few scattered literals start voting sat for the next few iterations, but not enough to affect the mean vote. Suddenly there is a phase transition and all the literals (and hence the network as a whole) start to vote sat with very high confidence (dark red). After the phase transition, the vote for each literal converges and the network stops evolving.
+
+We stress that none of the learned parameters depend on the size of the SAT problem and that a single model can be trained and tested on problems of arbitrary and varying sizes. At both train and test time, the input to the model is simply any bipartite adjacency matrix $M$ over any number of literals and clauses. The learned parameters only determine how each individual literal and clause behaves in terms of its neighbors in the graph. Variation in problem size is handled by the aggregation operators: we sum the outgoing messages of each of a node’s neighbors to form the incoming message, and we take the mean of the literal votes at the end of message passing to form the logit $y ^ { ( T ) }$ .
+
+# 4 TRAINING DATA
+
+We want our neural network to be able to classify (and ultimately solve) SAT problems from a variety of domains that it never trained on. One can easily construct distributions over SAT problems for which it would be possible to predict satisfiability with perfect accuracy based only on crude statistics; however, a neural network trained on such a distribution would be unlikely to generalize to problems from other domains. To force our network to learn something substantive, we create a distribution $\mathbf { S R } ( n )$ over pairs of random SAT problems on $n$ variables with the following property: one element of the pair is satisfiable, the other is unsatisfiable, and the two differ by negating only a single literal occurrence in a single clause. To generate a random clause on $n$ variables, $\mathbf { S R } ( n )$ first samples a small integer $k$ (with mean 5) 2 then samples $k$ variables uniformly at random without replacement, and finally negates each one with independent probability $50 \%$ . It continues to generate clauses $c _ { i }$ in this fashion, adding them to the SAT problem, and then querying a traditional SAT solver (we used Minisat Sorensson & Een (2005)), until adding the clause $c _ { m }$ finally makes the problem unsatisfiable. Since $\{ c _ { 1 } , \ldots , c _ { m - 1 } \}$ had a satisfying assignment, negating a single literal in $c _ { m }$ must yield a satisfiable problem $\{ c _ { 1 } , \ldots , c _ { m - 1 } , c _ { m } ^ { \prime } \}$ . The pair $( \{ c _ { 1 } , \hdots , c _ { m - 1 } , c _ { m } \} , \{ c _ { 1 } , \hdots , c _ { m - 1 } , c _ { m } ^ { \prime } \} )$ are a sample from $\mathbf { S R } ( n )$ .
+
+# 5 PREDICTING SATISFIABILITY
+
+Although our ultimate goal is to solve SAT problems arising from a variety of domains, we begin by training NeuroSAT as a classifier to predict satisfiability on SR(40). Problems in $\mathbf { S R } ( 4 0 )$ are small enough to be solved efficiently by modern SAT solvers—a fact we rely on to generate the problems—but the classification problem is highly non-trivial from a machine learning perspective. Each problem has 40 variables and over 200 clauses on average, and the positive and negative examples differ by negating only a single literal occurrence out of a thousand. We were unable to train an LSTM on a many-hot encoding of clauses (specialized to problems with 40 variables) to predict with $5 5 0 \%$ accuracy on its training set. Even the canonical SAT solver MiniSAT (Sorensson & Een, 2005) needs to backjump3 almost ten times on average, and needs to perform over a hundred primitive logical inferences (i.e. unit propagations) to solve each problem.
+
+We instantiated the NeuroSAT architecture described in $\ S 3$ with $d = 1 2 8$ dimensions for the literal embeddings, the clause embeddings, and all the hidden units; 3 hidden layers and a linear output layer for each of the MLPs $\mathbf { L } _ { \mathrm { m s g } }$ , $\mathbf { C } _ { \mathrm { m s g } }$ , and $\scriptstyle \mathbf { L } _ { \mathrm { v o t e } }$ ; and rectified linear units for all non-linearities. We regularized by the $\ell _ { 2 }$ norm of the parameters scaled by $1 0 ^ { - 1 0 }$ , and performed $T = 2 6$ iterations of message passing on every problem. We trained our model using the ADAM optimizer (Kingma & Ba, 2014) with a learning rate of $2 \times 1 0 ^ { - 5 }$ , clipping the gradients by global norm with clipping ratio 0.65 (Pascanu et al., 2012). We batched multiple problems together, with each batch containing up to 12,000 nodes (i.e. literals plus clauses). To accelerate the learning, we sampled the number of variables $n$ uniformly from between 10 and 40 during training (i.e. we trained on $\mathbf { S R ( U ( 1 0 , 4 0 ) ) } )$ , though we only evaluate on SR(40). We trained on millions of problems.
+
+After training, NeuroSAT is able to classify the test set correctly with $8 5 \%$ accuracy. In the next section, we examine how NeuroSAT manages to do so and show how we can decode solutions to satisfiable problems from its activations. Note: for the entire rest of the paper, NeuroSAT refers to the specific trained model that has only been trained on $\mathbf { S R } ( \mathbf { U } ( 1 0 , 4 0 ) )$ .
+
+# 6 DECODING SATISFYING ASSIGNMENTS
+
+Let us try to understand what NeuroSAT (trained on $\mathbf { S R ( U ( 1 0 , 4 0 ) ) } )$ is computing as it runs on new problems at test time. For a given run, we can compute and visualize the $2 n$ -dimensional vector of literal votes $L _ { * } ^ { ( t ) } \gets \mathbf { L } _ { \mathrm { v o t e } } ( L ^ { ( t ) } )$ at every iteration $t$ . Figure 3 illustrates the sequence of literal votes $L _ { * } ^ { ( 1 ) }$ ) to L(24)∗ a s NeuroSAT runs on a satisfiable problem from SR(20). For clarity, we reshape each $L _ { * } ^ { ( t ) }$ to be an $\mathbb { R } ^ { n \times 2 }$ matrix so that each literal is paired with its complement; specifically, the ith row contains the scalar votes for $x _ { i }$ and $\overline { { x _ { i } } }$ . Here white represents zero, blue negative and red positive. For several iterations, almost every literal is voting unsat with low confidence (light blue). Then a few scattered literals start voting sat for the next few iterations, but not enough to affect the mean vote. Suddenly, there is a phase transition and all the literals (and hence the network as a whole) start to vote sat with very high confidence (dark red). After the phase transition, the vote for each literal converges and the network stops evolving.
+
+NeuroSAT seems to exhibit qualitatively similar behavior on every satisfiable problem that it predicts correctly. The problems for which NeuroSAT guesses unsat are similar except without the phase change: it continues to guess unsat with low-confidence for as many iterations as NeuroSAT runs for. NeuroSAT never becomes highly confident that a problem is unsat, and it almost never guesses sat on an unsat problem. These results suggest that NeuroSAT searches for a certificate of satisfiability, and that it only guesses sat once it has found one.
+
+Let us look more carefully at the literal votes $L _ { * } ^ { ( 2 4 ) }$ from Figure 3 after convergence. Note that most of the variables have one literal vote distinctly darker than the other. Moreover, the dark votes are all approximately equal to each other, and the light votes are all approximately equal to each other as well. Thus the votes seem to encode one bit for each variable. It turns out that these bits encode a satisfying assignment in this case, but they do not do so reliably in general. Recall from $\ S 3$ that NeuroSAT projects the higher dimensional literal embeddings $L ^ { ( \bar { T } ) } \in \bar { \mathbb { R } } ^ { 2 n \times d }$ to the literal votes $L _ { * } ^ { ( T ) }$ using the MLP $\scriptstyle \mathbf { L } _ { \mathrm { v o t e } }$ . Figure 4 illustrates the two-dimensional PCA embeddings for $L ^ { ( 1 2 ) }$ to $L ^ { ( 2 6 ) }$ (skipping every other time step) as NeuroSAT runs on a satisfiable problem from SR(40). Blue and red dots indicate literals that are set to 0 and 1 in the satisfying assignment that it eventually finds, respectively. The blue and red dots cannot be linearly separated until the phase transition at the end, at which point they form two distinct clusters according to the satisfying assignment. We observe a similar clustering almost every time the network guesses sat. Thus the literal votes $L _ { * } ^ { ( T ) }$ only ever encode the satisfying assignment by chance, when the projection $\scriptstyle \mathbf { L } _ { \mathrm { v o t e } }$ happens to preserve this clustering.
+
+<table><tr><td>Trained on: Trained with: Tested on: Tested with: Overall test accuracy: Accuracy on unsat problems: Accuracy on sat problems: Percent of sat problems solved:</td><td>SR(U(10,40)) 26 iterations SR(40) 26 iterations 85% 96%</td></tr></table>
+
+Our analysis suggests a more reliable way to decode solutions from NeuroSAT’s internal activations: 2-cluster $L ^ { ( T ) }$ to get cluster centers $\Delta _ { 1 }$ and $\Delta _ { 2 }$ , partition the variables according to the predicate $\lVert x _ { i } - \Delta _ { 1 } \rVert ^ { 2 } + \lVert \overline { { x _ { i } } } - \Delta _ { 2 } \rVert ^ { 2 } < \lVert x _ { i } - \Delta _ { 2 } \rVert ^ { 2 } + \lVert \overline { { x _ { i } } } - \bar { \Delta _ { 1 } } \rVert ^ { 2 }$ , and then try both candidate assignments that result from mapping the partitions to truth values. This decoding procedure (using $k$ -means to find the two cluster centers) successfully decodes a satisfying assignment for over $7 0 \%$ of the satisfiable problems in the $\mathbf { S R } ( 4 0 )$ test set. Table 1 summarizes the results when training on $\mathbf { S R } ( \mathbf { U } ( 1 0 , 4 0 ) )$ ) and testing on SR(40).
+
+![](images/b2b93ec940f8c8d4be4ceb084ea9fda5f802889227023d9d651d2520db825f72.jpg)  
+Table 1: NeuroSAT’s performance at test time on $\mathbf { S R } ( 4 0 )$ after training on $\mathbf { S R } ( \mathbf { U } ( 1 0 , 4 0 ) )$ . It almost never guesses sat on unsatisfiable problems. On satisfiable problems, it correctly guesses sat $73 \%$ of the time, and we can decode a satisfying assignment for $70 \%$ of the satisfiable problems by clustering the literal embeddings $L ^ { ( T ) }$ as described in $\ S 6$ .   
+Figure 4: PCA projections for the high-dimensional literal embeddings $L ^ { ( 1 2 ) }$ to $L ^ { ( 2 6 ) }$ (skipping every other time step) as NeuroSAT runs on a satisfiable problem from SR(40). Blue and red dots indicate literals that are set to 0 and 1 in the satisfying assignment that it eventually finds, respectively. We see that the blue and red dots are mixed up and cannot be linearly separated until the phase transition at the end, at which point they form two distinct clusters according to the satisfying assignment.
+
+Recall that at training time, NeuroSAT is only given a single bit of supervision for each SAT problem. Moreover, the positive and negative examples in the dataset differ only by the placement of a single edge. NeuroSAT has learned to search for satisfying assignments solely to explain that single bit of supervision.
+
+# 7 EXTRAPOLATING TO OTHER PROBLEM DISTRIBUTIONS
+
+# 7.1 BIGGER PROBLEMS
+
+Even though we only train NeuroSAT on $\mathbf { S R } ( \mathbf { U } ( 1 0 , 4 0 ) )$ , it is able to solve SAT problems sampled from $\mathbf { S R } ( n )$ for $n$ much larger than 40 by simply running for more iterations of message passing. Figure 5 shows NeuroSAT’s success rate on $\mathbf { S R } ( n )$ for a range of $n$ as a function of the number of iterations $T$ . For $n = 2 0 0$ , there are $2 ^ { 1 6 0 }$ times more possible assignments to the variables than any problem it saw during training, and yet it can solve $2 5 \%$ of the satisfiable problems in SR(200) by running for four times more iterations than it performed during training. On the other hand, when restricted to the number of iterations it was trained with, it solves under $10 \%$ of them. Thus we see that its ability to solve bigger and harder problems depends on the fact that the dynamical system it has learned encodes generic procedural knowledge that can operate effectively over a wide range of time frames.
+
+![](images/dbe3a32da539b3b746da55e6820e1c1309ef869418443eaf7cb15324a6c0c755.jpg)  
+Figure 5: NeuroSAT’s success rate on $\mathbf { S R } ( n )$ for a range of $n$ as a function of the number of iterations $T$ . Even though we only train NeuroSAT on $\mathbf { S R } ( 4 0 )$ and below, it is able to solve SAT problems sampled from $\mathbf { S R } ( n )$ for $n$ much larger than 40 by simply running for more iterations.
+
+![](images/2057d6fd092a8ad0636c64dc52a0e726556d1b2ea9e61829c6e767ae383eebb6.jpg)  
+Figure 6: Example graph from the Forest-Fire distribution. The graph has a coloring for $k \geq 5$ , a clique for $k \leq 3$ , a dominating set for $k \geq 3$ , and a vertex cover for $k \geq 6$ . However, these properties are not perceptually obvious and require deliberate computation to determine.
+
+# 7.2 DIFFERENT PROBLEMS
+
+Every problem in NP can be reduced to SAT in polynomial time, and SAT problems arising from different domains may have radically different structural and statistical properties. Even though NeuroSAT has learned to search for satisfying assignments on problems from $\mathbf { S R } ( n )$ , we may still find that the dynamical system it has learned only works properly on problems similar to those it was trained on.
+
+To assess NeuroSAT’s ability to extrapolate to different classes of problems, we generated problems in several other domains and then encoded them all into SAT problems (using standard encodings). In particular, we started by generating one hundred graphs from each of six different random graph distributions (Barabasi, Erdos-Renyi, Forest-Fire, Random-¨ $k$ -Regular, Random-Static-Power-Law, and Random-Geometric).4 We found parameters for the random graph generators such that each graph has ten nodes and seventeen edges on average. For each graph in each collection, we generated graph coloring problems $( 3 \leq k \leq 5 )$ ), dominating-set problems $( 2 \leq k \leq 4 )$ ), clique-detection problems $( 3 ~ \leq ~ k ~ \leq ~ 5 )$ , and vertex cover problems $( 4 ~ \leq ~ k ~ \leq ~ 6 )$ .5 We chose the range of $k$ for each problem to include the threshold for most of the graphs while avoiding trivial problems such as 2-clique. As before, we used Minisat Sorensson & Een (2005) to determine satisfiability. Figure 6 shows an example graph from the distribution. Note that the trained network does not know anything a priori about these tasks; the generated SAT problems need to encode not only the graphs themselves but also formal descriptions of the tasks to be solved.
+
+Out of the 7,200 generated problems, we kept only the 4,888 satisfiable problems. On average these problems contained over two and a half times as many clauses as the problems in SR(40). We ran NeuroSAT for 512 iterations on each of them and found that we could successfully decode solutions for $8 5 \%$ of them. In contrast, Survey Propagation (SP) (Braunstein et al., 2005), the canonical (learning-free) message passing algorithm for satisfiability, does not on its own converge to a satisfying assignment on any of these problems.6 This suggests that NeuroSAT has not simply found a way to approximate SP, but rather has synthesized a qualitatively different algorithm.
+
+# 8 FINDING UNSAT CORES
+
+NeuroSAT (trained on $\mathbf { S R ( U ( 1 0 , 4 0 ) ) } \rangle$ can find satisfying assignments but is not helpful in constructing proofs of unsatisfiability. When it runs on an unsatisfiable problem, it keeps searching for a satisfying assignment indefinitely and non-systematically. However, when we train the same architecture on a dataset in which each unsatisfiable problem has a small subset of clauses that are already unsatisfiable (called an unsat core), it learns to detect these unsat cores instead of searching for satisfying assignments. The literals involved in the unsat core can be decoded from its internal activations. When the number of literals involved in the unsat core is small relative to the total number of literals, knowing the literals involved in the unsat core can enable constructing a resolution proof more efficiently.
+
+We generated a new distribution $\mathbf { S R C } ( n , u )$ that is similar to $\mathbf { S R } ( n )$ except that every unsatisfiable problem contains a small unsat core. Here $n$ is the number of variables as before, and $u$ is an unsat core over $x _ { 1 } , \ldots , x _ { k }$ $( k < n )$ that can be made into a satisfiable set of clauses $u ^ { \prime }$ by negating a single literal. We sample a pair from $\mathbf { S R C } ( n , u )$ as follows. First, we initialize a problem with $u ^ { \prime }$ , and then we sample clauses (over $x _ { 1 }$ to $x _ { n }$ ) just as we did for $\mathbf { S R } ( n )$ until the problem becomes unsatisfiable. We can now negate a literal in the final clause to get a satisfiable problem $p _ { s }$ , and then we can swap $u ^ { \prime }$ for $u$ in $p _ { s }$ to get $p _ { u }$ , which is unsatisfiable since it contains the unsat core $u$ . We created train and test datasets from $\mathbf { S R C } ( 4 0 , u )$ with $u$ sampled at random for each problem from a collection of three unsat cores ranging from three clauses to nine clauses: the unsat core $R$ from Knuth (2015), and the two unsat cores resulting from encoding the pigeonhole principles $\mathbf { P P } ( 2 , 1 )$ and $\mathbf { P P } ( 3 , 2 )$ . 7 We trained our architecture on this dataset, and we refer to the trained model as NeuroUNSAT.
+
+![](images/2066957dcb07a7a9e9535cf1b143befffa21bfff5771e38c35e096823d40388f.jpg)  
+(b) NeuroUNSAT running on an unsatisfiable problem.
+
+(a) NeuroUNSAT running on a satisfiable problem.
+
+![](images/eb4246e96b30294841909d61559d94429bb6ad22b8aa7eb85c820bd8894b2939.jpg)  
+Figure 7: The sequence of literal votes $L _ { * } ^ { ( t ) }$ as NeuroUNSAT runs on a pair of problems from $\mathbf { S } \bar { \mathbf { R C } } ( 3 0 , \mathbf { P P } ( 3 , \bar { 2 } ) )$ . In both cases, the literals in the first six rows are involved in the unsat core. In 7a, NeuroUNSAT inspects the modified core $u ^ { \prime }$ of the satisfiable problem but concludes that it does not match the pattern. In 7b, NeuroUNSAT finds the unsat core $u$ and votes unsat with high confidence (dark blue).
+
+NeuroUNSAT is able to predict satisfiability on the test set with $100 \%$ accuracy. Upon inspection, it seems to do so by learning to recognize the unsat cores. Figure 7 shows NeuroUNSAT running on a pair of problems from $\bar { \mathbf { S } } \mathbf { R } \mathbf { C } ( 3 0 , \mathbf { \bar { P } } \mathbf { P } ( 3 , 2 ) )$ . In both cases, the literals in the first six rows are involved in the unsat core. In Figure 7a, NeuroUNSAT inspects the modified core $u ^ { \prime }$ of the satisfiable problem but concludes that it does not match the pattern exactly. In Figure 7b, NeuroUNSAT finds the unsat core $u$ and votes unsat with high confidence (dark blue). As in $\ S 6$ , the literals involved in the unsat core can sometimes be decoded from the literal votes $L _ { * } ^ { ( T ) }$ , but it is more reliable to 2- cluster the higher-dimensional literal embeddings $L ^ { ( T ) }$ . On the test set, the small number of literals involved in the unsat core end up in their own cluster $98 \%$ of the time.
+
+Note that we do not expect NeuroUNSAT to generalize to arbitary unsat cores: as far as we know it is simply memorizing a collection of specific subgraphs, and there is no evidence it has learned a generic procedure to prove unsat.
+
+# 9 RELATED WORK
+
+There have been many attempts over the years to apply statistical learning to various aspects of the SAT problem: restart strategies (Haim & Walsh, 2009), branching heuristics (Liang et al., 2016;
+
+Grozea & Popescu, 2014; Flint & Blaschko, 2012), parameter tuning (Singh et al., 2009), and solver selection (Xu et al., 2008). None of these approaches use neural networks, and instead make use of both generic graph features and features extracted from the runs of SAT solvers. Moreover, these approaches are designed to assist existing solvers and do not aim to solve SAT problems on their own.
+
+From the machine learning perspective, the closest work to ours is Palm et al. (2017), which showed that an MPNN can be trained to predict the unique solutions of Sudoku puzzles. We believe that their network’s success is an instance of the phenomenon we study in this paper, namely that MPNNs can synthesize local search algorithms for constraint satisfaction problems. Evans et al. (2018) present a neural network architecture that can learn to predict whether one propositional formula entails another by randomly sampling and evaluating candidate assignments. Unlike NeuroSAT, their network does not perform heuristic search and can only work on simple problems for which random guessing is tractable. There have also been several recent papers showing that various neural network architectures can learn good heuristics for NP-hard combinatorial optimization problems (Vinyals et al., 2015; Bello et al., 2016; Dai et al., 2017); however, finding low-cost solutions to optimization problems requires less precise reasoning than finding satisfying assignments.
+
+# 10 DISCUSSION
+
+Our main motivation has been scientific: to better understand the extent to which neural networks are capable of precise, logical reasoning. Our work has definitively established that neural networks can learn to perform discrete search on their own without the help of hard-coded search procedures, even after only end-to-end training with minimal supervision. We found this result surprising and think it constitutes an important contribution to the community’s evolving understanding of the capabilities and limitations of neural networks.
+
+Although not our primary concern, we also hope that our findings eventually lead to improvements in practical SAT solving. As we stressed early on, as an end-to-end SAT solver the trained NeuroSAT system discussed in this paper is still vastly less reliable than the state-of-the-art. We concede that we see no obvious path to beating existing SAT solvers. One approach might be to continue to train NeuroSAT as an end-to-end solver on increasingly difficult problems. A second approach might be to use a system like NeuroSAT to help guide decisions within a more traditional SAT solver, though it is not clear that NeuroSAT provides any useful information before it finds a satisfying assignment. However, as we discussed in $\ S 8$ , when we trained our architecture on different data it learned an entirely different procedure. In a separate experiment omitted for space reasons, we also trained our architecture to predict whether there is a satisfying assignment involving each individual literal in the problem and found that it was able to predict these bits with high accuracy as well. Unlike NeuroSAT, it made both type I and type II errors, had no discernable phase transition, and could make reasonable predictions within only a few rounds. We believe that architectures descended from NeuroSAT will be able to learn very different mechanisms and heuristics depending on the data they are trained on and the details of their objective functions. We are cautiously optimistic that a descendant of NeuroSAT will one day lead to improvements to the state-of-the-art.
+
+# ACKNOWLEDGEMENTS
+
+We thank Steve Mussmann, Alexander Ratner, Nathaniel Thomas, Vatsal Sharan and Cristina White for providing valuable feedback on early drafts. We also thank William Hamilton, Geoffrey Irving and Arun Chaganty for helpful discussions. This work was supported by Future of Life Institute grant 2017-158712.
+
+# REFERENCES
+
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+
+Irwan Bello, Hieu Pham, Quoc V Le, Mohammad Norouzi, and Samy Bengio. Neural combinatorial optimization with reinforcement learning. arXiv preprint arXiv:1611.09940, 2016.
+
+Armin Biere, Marijn Heule, Hans van Maaren, and Toby Walsh. Conflict-driven clause learning sat solvers. Handbook of Satisfiability, Frontiers in Artificial Intelligence and Applications, pp. 131–153, 2009.
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+Alfredo Braunstein, Marc Mezard, and Riccardo Zecchina. Survey propagation: An algorithm for ´ satisfiability. Random Structures & Algorithms, 27(2):201–226, 2005.
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+Stephen A Cook. The complexity of theorem-proving procedures. In Proceedings of the third annual ACM symposium on Theory of computing, pp. 151–158. ACM, 1971.
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+Hanjun Dai, Elias B Khalil, Yuyu Zhang, Bistra Dilkina, and Le Song. Learning combinatorial optimization algorithms over graphs. arXiv preprint arXiv:1704.01665, 2017.
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+Richard Evans, David Saxton, David Amos, Pushmeet Kohli, and Edward Grefenstette. Can neural networks understand logical entailment? arXiv preprint arXiv:1802.08535, 2018.
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+Alex Flint and Matthew Blaschko. Perceptron learning of sat. In Advances in Neural Information Processing Systems, pp. 2771–2779, 2012.
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+Justin Gilmer, Samuel S Schoenholz, Patrick F Riley, Oriol Vinyals, and George E Dahl. Neural message passing for quantum chemistry. arXiv preprint arXiv:1704.01212, 2017.
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+Carla P Gomes, Henry Kautz, Ashish Sabharwal, and Bart Selman. Satisfiability solvers. Foundations of Artificial Intelligence, 3:89–134, 2008.
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+Cristian Grozea and Marius Popescu. Can machine learning learn a decision oracle for np problems? a test on sat. Fundamenta Informaticae, 131(3-4):441–450, 2014.
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+Shai Haim and Toby Walsh. Restart strategy selection using machine learning techniques. CoRR, abs/0907.5032, 2009.
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+S. Hochreiter and J. Schmidhuber. Long short-term memory. Neural Computation, 9(8):1735–1780, 1997.
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+Diederik Kingma and Jimmy Ba. Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980, 2014.
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+Donald E Knuth. The art of computer programming, volume 4, fascicle 6: Satisfiability, 2015.
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+Harry R Lewis. Computers and intractability. a guide to the theory of np-completeness, 1983.
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+Yujia Li, Daniel Tarlow, Marc Brockschmidt, and Richard Zemel. Gated graph sequence neural networks. arXiv preprint arXiv:1511.05493, 2015.
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+Jia Hui Liang, Vijay Ganesh, Pascal Poupart, and Krzysztof Czarnecki. Learning rate based branching heuristic for sat solvers. In International Conference on Theory and Applications of Satisfiability Testing, pp. 123–140. Springer, 2016.
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+Mark Newman. Networks: an introduction. Oxford university press, 2010.
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+Rasmus Berg Palm, Ulrich Paquet, and Ole Winther. Recurrent relational networks for complex relational reasoning. arXiv preprint arXiv:1711.08028, 2017.
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+Razvan Pascanu, Tomas Mikolov, and Yoshua Bengio. Understanding the exploding gradient problem. CoRR, abs/1211.5063, 2012.
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+Franco Scarselli, Marco Gori, Ah Chung Tsoi, Markus Hagenbuchner, and Gabriele Monfardini. The graph neural network model. IEEE Transactions on Neural Networks, 20(1):61–80, 2009.
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+Rishabh Singh, Joseph P Near, Vijay Ganesh, and Martin Rinard. Avatarsat: An auto-tuning boolean sat solver. 2009.
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+Niklas Sorensson and Niklas Een. Minisat v1. 13-a sat solver with conflict-clause minimization. SAT, 2005(53):1–2, 2005.
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+G Tseitin. On the complexity ofderivation in propositional calculus. Studies in Constrained Mathematics and Mathematical Logic, 1968.   
+O. Vinyals, M. Fortunato, and N. Jaitly. Pointer networks. In Advances in Neural Information Processing Systems (NIPS), pp. 2674–2682, 2015.   
+L. Xu, F. Hutter, H. H. Hoos, and K. Leyton-Brown. SATzilla: portfolio-based algorithm selection for SAT. Journal of Artificial Intelligence Research (JAIR), 32:565–606, 2008.

parse/train/HJMC_iA5tm/HJMC_iA5tm_content_list.json

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+    {
+        "type": "text",
+        "text": "LEARNING A SAT SOLVER FROM SINGLE-BIT SUPER-VISION",
+        "text_level": 1,
+        "bbox": [
+            174,
+            98,
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+            145
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Daniel Selsam, Matthew Lamm, Benedikt Bunz, Percy Liang, David L. Dill ¨ ",
+        "bbox": [
+            184,
+            170,
+            704,
+            185
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Department of Computer Science   \nStanford University   \nStanford, CA 94305   \n{dselsam,mlamm,buenz,pliang,dill}@cs.stanford.edu ",
+        "bbox": [
+            184,
+            186,
+            663,
+            239
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Leonardo de Moura Microsoft Research Redmond, WA 98052 leonardo@microsoft.com ",
+        "bbox": [
+            184,
+            261,
+            401,
+            316
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "ABSTRACT ",
+        "text_level": 1,
+        "bbox": [
+            454,
+            354,
+            544,
+            369
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "We present NeuroSAT, a message passing neural network that learns to solve SAT problems after only being trained as a classifier to predict satisfiability. Although it is not competitive with state-of-the-art SAT solvers, NeuroSAT can solve problems that are substantially larger and more difficult than it ever saw during training by simply running for more iterations. Moreover, NeuroSAT generalizes to novel distributions; after training only on random SAT problems, at test time it can solve SAT problems encoding graph coloring, clique detection, dominating set, and vertex cover problems, all on a range of distributions over small random graphs. ",
+        "bbox": [
+            232,
+            390,
+            764,
+            501
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "1 INTRODUCTION ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            540,
+            336,
+            556
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "The propositional satisfiability problem (SAT) is one of the most fundamental problems of computer science. Cook (1971) showed that the problem is NP-complete, which means that searching for any kind of efficiently-checkable certificate in any context can be reduced to finding a satisfying assignment of a propositional formula. In practice, search problems arising from a wide range of domains such as hardware and software verification, test pattern generation, planning, scheduling, and combinatorics are all routinely solved by constructing an appropriate SAT problem and then calling a SAT solver (Gomes et al., 2008). Modern SAT solvers based on backtracking search are extremely well-engineered and have been able to solve problems of practical interest with millions of variables (Biere et al., 2009). ",
+        "bbox": [
+            174,
+            575,
+            825,
+            700
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "We consider the question: can a neural network learn to solve SAT problems? To answer, we develop a novel message passing neural network (MPNN) (Scarselli et al., 2009; Li et al., 2015; Gilmer et al., 2017), NeuroSAT, and train it as a classifier to predict satisfiability on a dataset of random SAT problems. We provide NeuroSAT with only a single bit of supervision for each SAT problem that indicates whether or not the problem is satisfiable. When making a prediction about a new SAT problem, we find that NeuroSAT guesses unsatisfiable with low confidence until it finds a solution, at which point it converges and guesses satisfiable with very high confidence. The solution itself can almost always be automatically decoded from the network’s activations, making NeuroSAT an end-to-end SAT solver. See Figure 1 for an illustration of the train and test regimes. ",
+        "bbox": [
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+            833
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Although it is not competitive with state-of-the-art SAT solvers, NeuroSAT can solve SAT problems that are substantially larger and more difficult than it ever saw during training by simply performing more iterations of message passing. Despite only running for a few dozen iterations during training, at test time NeuroSAT continues to find solutions to harder problems after hundreds and even thousands of iterations. The learning process has yielded not a traditional classifier but rather a procedure that can be run indefinitely to search for solutions to problems of varying difficulty. ",
+        "bbox": [
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+            840,
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+            924
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+        "page_idx": 0
+    },
+    {
+        "type": "image",
+        "img_path": "images/1d225b957fb03a6856771f4a7ff69ce2bb215e56e9ecf9dadd0703c9dec0e7b8.jpg",
+        "image_caption": [
+            "Figure 1: We train NeuroSAT to predict whether SAT problems are satisfiable, providing only a single bit of supervision for each problem. At test time, when NeuroSAT predicts satisfiable, we can almost always extract a satisfying assignment from the network’s activations. The problems at test time can also be substantially larger, more difficult, and even from entirely different domains than the problems seen during training. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            220,
+            99,
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Moreover, NeuroSAT generalizes to entirely new domains. Since NeuroSAT operates on SAT problems and since SAT is NP-complete, NeuroSAT can be queried on SAT problems encoding any kind of search problem for which certificates can be checked in polynomial time. Although we train it using only problems from a single random problem generator, at test time it can solve SAT problems encoding graph coloring, clique detection, dominating set, and vertex cover problems, all on a range of distributions over small random graphs. ",
+        "bbox": [
+            174,
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "The same neural network architecture can also be used to help construct proofs for unsatisfiable problems. When we train it on a different dataset in which every unsatisfiable problem contains a small contradiction (call this trained model NeuroUNSAT), it learns to detect these contradictions instead of searching for satisfying assignments. Just as we can extract solutions from NeuroSAT’s activations, we can extract the variables involved in the contradiction from NeuroUNSAT’s activations. When the number of variables involved in the contradiction is small relative to the total number of variables, knowing which variables are involved in the contradiction can enable constructing a resolution proof more efficiently. ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "2 PROBLEM SETUP ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            551,
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+        ],
+        "page_idx": 1
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+    {
+        "type": "text",
+        "text": "Background. A formula of propositional logic is a boolean expression built using the constants true (1) and false (0), variables, negations, conjunctions, and disjunctions. A formula is satisfiable provided there exists an assignment of boolean values to its variables such that the formula evaluates to 1. For example, the formula $( x _ { 1 } \\lor x _ { 2 } \\lor x _ { 3 } ) \\land \\lnot ( x _ { 1 } \\land x _ { 2 } \\land x _ { 3 } )$ is satisfiable because it will evaluate to 1 under every assignment that does not map $x _ { 1 }$ , $x _ { 2 }$ and $x _ { 3 }$ to the same value. For every formula, there exists an equisatisfiable formula in conjunctive normal form (CNF), expressed as a conjunction of disjunctions of (possibly negated) variables.1 Each conjunct of a formula in CNF is called a clause, and each (possibly negated) variable within a clause is called a literal. The formula above is equivalent to the CNF formula $\\left( x _ { 1 } \\lor x _ { 2 } \\lor x _ { 3 } \\right) \\land \\left( \\lnot x _ { 1 } \\lor \\lnot x _ { 2 } \\lor \\lnot x _ { 3 } \\right)$ , which we can represent more concisely as $\\{ 1 | 2 | 3 , { \\overline { { 1 } } } | { \\overline { { 2 } } } | { \\overline { { 3 } } } \\}$ . A formula in CNF has a satisfying assignment if and only if it has an assignment such that every clause has at least one literal mapped to 1. A SAT problem is a formula in CNF, where the goal is to determine if the formula is satisfiable, and if so, to produce a satisfying assignment of truth values to variables. We use $n$ to denote the number of of variables in a SAT problem, and $m$ to denote the number of clauses. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Classification task. For a SAT problem $P$ , we define $\\phi ( P )$ to be true if and only if $P$ is satisfiable. Our first goal is to learn a classifier that approximates $\\phi$ . Given a distribution $\\Psi$ over SAT problems, we can construct datasets $\\mathcal { D } _ { \\mathrm { t r a i n } }$ and $\\mathcal { D } _ { \\mathrm { t e s t } }$ with examples of the form $( P , \\phi ( P ) )$ by sampling problems $P \\sim \\Psi$ and computing $\\phi ( P )$ using an existing SAT solver. At test time, we get only the problem $P$ and the goal is to predict $\\phi ( P )$ , i.e. to determine if $P$ is satisfiable. Ultimately we care about the solving task, which also includes finding solutions to satisfiable problems. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "3 MODEL ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "A SAT problem has a simple syntactic structure and therefore could be encoded into a vector space using standard methods such as an RNN. However, the semantics of propositional logic induce rich invariances that such a syntactic method would ignore, such as permutation invariance and negation invariance. Specifically, the satisfiability of a formula is not affected by permuting the variables (e.g. swapping $x _ { 1 }$ and $x _ { 2 }$ throughout the formula), by permuting the clauses (e.g. swapping the first clause with the second clause), or by permuting the literals within a clause (e.g. replacing the clause $1 | \\overline { { 2 } }$ with $\\overline { { 2 } } | 1$ . The satisfiability of a formula is also not affected by negating every literal corresponding to a given variable (e.g. negating all occurrences of $x _ { 1 }$ in $\\{ 1 | \\overline { { 2 } } , \\overline { { 1 } } | \\overline { { 3 } } \\}$ to yield $\\{ { \\overline { { 1 } } } | { \\overline { { 2 } } } , 1 | { \\overline { { 3 } } } \\} )$ . ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "We now describe our neural network architecture, NeuroSAT, that enforces both permutation invariance and negation invariance. We encode a SAT problem as an undirected graph with one node for every literal, one node for every clause, an edge between every literal and every clause it appears in, and a different type of edge between each pair of complementary literals (e.g. between $x _ { i }$ and $\\overline { { x _ { i } } }$ ). NeuroSAT iteratively refines a vector space embedding for each node by passing “messages” back and forth along the edges of the graph as described in Gilmer et al. (2017). At every time step, we have an embedding for every literal and every clause. An iteration consists of two stages. First, each clause receives messages from its neighboring literals and updates its embedding accordingly. Next, each literal receives messages from its neighboring clauses as well as from its complement, then updates its embedding accordingly. Figure 2 provides a high-level illustration of the architecture. ",
+        "bbox": [
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+    },
+    {
+        "type": "image",
+        "img_path": "images/f0aae2ef71377da7802b98440976e9a312f6b983b1f98b853dd17e23a76f7d0b.jpg",
+        "image_caption": [
+            "Figure 2: High-level illustration of NeuroSAT operating on the graph representation of $\\{ 1 | 2 , { \\overline { { 1 } } } | { \\overline { { 2 } } } \\}$ . On the top of both figures are nodes for each of the four literals, and on the bottom are nodes for each of the two clauses. At every time step $t$ , we have an embedding for every literal and every clause. An iteration consists of two stages. First, each clause receives messages from its neighboring literals and updates it embedding accordingly (Figure 2a). Next, each literal receives messages from its neighboring clause as well as from its complement, and updates its embedding accordingly (Figure 2b). "
+        ],
+        "image_footnote": [],
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+    },
+    {
+        "type": "text",
+        "text": "More formally, our model is parameterized by two vectors $\\mathbf { L } _ { \\mathrm { i n i t } }$ , $\\mathbf { C } _ { \\mathrm { i n i t } } )$ , three multilayer perceptrons $( \\mathbf { L } _ { \\mathrm { m s g } } , \\mathbf { C } _ { \\mathrm { m s g } } , \\mathbf { L } _ { \\mathrm { v o t e } } )$ and two layer-norm LSTMs (Ba et al., 2016; Hochreiter & Schmidhuber, 1997) $( { \\bf L } _ { \\bf u } , { \\bf C } _ { \\bf u } )$ . At every time step $t$ , we have a matrix $L ^ { ( t ) } \\in \\mathbb { R } ^ { 2 n \\times d }$ whose ith row contains the embedding for the literal $\\ell _ { i }$ and a matrix $C ^ { ( t ) } \\in \\mathbb { R } ^ { m \\times d }$ whose $j$ th row contains the embedding for the clause $c _ { j }$ , which we initialize by tiling $\\mathbf { L } _ { \\mathrm { i n i t } }$ and $\\mathbf { C } _ { \\mathrm { i n i t } }$ respectively. We also have hidden states $L _ { h } ^ { ( t ) } \\in \\mathbb { R } ^ { 2 n \\times d }$ and artite $C _ { h } ^ { ( t ) } \\in \\mathbb { R } ^ { m \\times d }$ for atrix $\\mathbf { L } _ { \\mathrm { u } }$ and fined $\\mathbf { C } _ { \\mathbf { u } }$ alized to zero matrices. Letand let Flip be the operator $M$ $M ( i , j ) = \\mathrm { ~ \\bar { 1 } ~ } \\{ \\ell _ { i } \\in c _ { j } \\}$   \nthat takes a matrix $L$ and swaps each row of $L$ with the row corresponding to the literal’s negation. A single iteration consists of applying the following two updates: ",
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+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/8fecf9fe01fdde458c770ab0c826da48485cfe6f33b7fb5c706c74658825ae23.jpg",
+        "text": "$$\n\\begin{array} { r l } & { ( C ^ { ( t + 1 ) } , C _ { h } ^ { ( t + 1 ) } ) \\gets \\mathbf { C } _ { \\mathbf { u } } ( [ C _ { h } ^ { ( t ) } , M ^ { \\top } \\mathbf { L } _ { \\mathrm { m s g } } ( L ^ { ( t ) } ) ] ) } \\\\ & { ( L ^ { ( t + 1 ) } , L _ { h } ^ { ( t + 1 ) } ) \\gets \\mathbf { L } _ { \\mathbf { u } } ( [ L _ { h } ^ { ( t ) } , \\mathrm { F l i p } ( L ^ { ( t ) } ) , M \\mathbf { C } _ { \\mathrm { m s g } } ( C ^ { ( t + 1 ) } ) ] ) } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            299,
+            744,
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+            791
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "After $T$ iterations, we compute $L _ { * } ^ { ( T ) }  \\mathbf { L } _ { \\mathrm { v o t e } } ( L ^ { ( T ) } ) \\in \\mathbb { R } ^ { 2 n }$ , which contains a single scalar for each literal (the literal’s vote), and then we compute the average of the literal votes $y ^ { ( T ) } \\gets \\mathrm { m e a n } ( L _ { * } ^ { ( T ) } ) \\in$ $\\mathbb { R }$ . We train the network to minimize the sigmoid cross-entropy loss between the logit $y ^ { ( T ) }$ and the true label $\\phi ( P )$ . ",
+        "bbox": [
+            174,
+            796,
+            825,
+            862
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Our architecture enforces permutation invariance by operating on nodes and edges according to the topology of the graph without any additional ordering over nodes or edges. Likewise, it enforces negation invariance by treating all literals the same no matter whether they originated as a positive or negative occurrence of a variable. ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "image",
+        "img_path": "images/eddc765e938654a29717017c9c02dd342e57d38a5f900cee4678095d179f8f86.jpg",
+        "image_caption": [
+            "Iteration ",
+            "Figure 3: The sequence of literal votes $L _ { * } ^ { ( 1 ) }$ 1) to L(24)∗ as NeuroSAT runs on a satisfiable problem from SR(20). For clarity, we reshape each $L _ { * } ^ { ( t ) }$ to be an $\\mathbb { R } ^ { n \\times 2 }$ matrix so that each literal is paired with its complement; specifically, the ith row contains the scalar votes for $x _ { i }$ and $\\overline { { x _ { i } } }$ . Here white represents zero, blue negative and red positive. For several iterations, almost every literal is voting unsat with low confidence (light blue). Then a few scattered literals start voting sat for the next few iterations, but not enough to affect the mean vote. Suddenly there is a phase transition and all the literals (and hence the network as a whole) start to vote sat with very high confidence (dark red). After the phase transition, the vote for each literal converges and the network stops evolving. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            174,
+            97,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "We stress that none of the learned parameters depend on the size of the SAT problem and that a single model can be trained and tested on problems of arbitrary and varying sizes. At both train and test time, the input to the model is simply any bipartite adjacency matrix $M$ over any number of literals and clauses. The learned parameters only determine how each individual literal and clause behaves in terms of its neighbors in the graph. Variation in problem size is handled by the aggregation operators: we sum the outgoing messages of each of a node’s neighbors to form the incoming message, and we take the mean of the literal votes at the end of message passing to form the logit $y ^ { ( T ) }$ . ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "4 TRAINING DATA ",
+        "text_level": 1,
+        "bbox": [
+            176,
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+    },
+    {
+        "type": "text",
+        "text": "We want our neural network to be able to classify (and ultimately solve) SAT problems from a variety of domains that it never trained on. One can easily construct distributions over SAT problems for which it would be possible to predict satisfiability with perfect accuracy based only on crude statistics; however, a neural network trained on such a distribution would be unlikely to generalize to problems from other domains. To force our network to learn something substantive, we create a distribution $\\mathbf { S R } ( n )$ over pairs of random SAT problems on $n$ variables with the following property: one element of the pair is satisfiable, the other is unsatisfiable, and the two differ by negating only a single literal occurrence in a single clause. To generate a random clause on $n$ variables, $\\mathbf { S R } ( n )$ first samples a small integer $k$ (with mean 5) 2 then samples $k$ variables uniformly at random without replacement, and finally negates each one with independent probability $50 \\%$ . It continues to generate clauses $c _ { i }$ in this fashion, adding them to the SAT problem, and then querying a traditional SAT solver (we used Minisat Sorensson & Een (2005)), until adding the clause $c _ { m }$ finally makes the problem unsatisfiable. Since $\\{ c _ { 1 } , \\ldots , c _ { m - 1 } \\}$ had a satisfying assignment, negating a single literal in $c _ { m }$ must yield a satisfiable problem $\\{ c _ { 1 } , \\ldots , c _ { m - 1 } , c _ { m } ^ { \\prime } \\}$ . The pair $( \\{ c _ { 1 } , \\hdots , c _ { m - 1 } , c _ { m } \\} , \\{ c _ { 1 } , \\hdots , c _ { m - 1 } , c _ { m } ^ { \\prime } \\} )$ are a sample from $\\mathbf { S R } ( n )$ . ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "5 PREDICTING SATISFIABILITY ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Although our ultimate goal is to solve SAT problems arising from a variety of domains, we begin by training NeuroSAT as a classifier to predict satisfiability on SR(40). Problems in $\\mathbf { S R } ( 4 0 )$ are small enough to be solved efficiently by modern SAT solvers—a fact we rely on to generate the problems—but the classification problem is highly non-trivial from a machine learning perspective. Each problem has 40 variables and over 200 clauses on average, and the positive and negative examples differ by negating only a single literal occurrence out of a thousand. We were unable to train an LSTM on a many-hot encoding of clauses (specialized to problems with 40 variables) to predict with $5 5 0 \\%$ accuracy on its training set. Even the canonical SAT solver MiniSAT (Sorensson & Een, 2005) needs to backjump3 almost ten times on average, and needs to perform over a hundred primitive logical inferences (i.e. unit propagations) to solve each problem. ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "We instantiated the NeuroSAT architecture described in $\\ S 3$ with $d = 1 2 8$ dimensions for the literal embeddings, the clause embeddings, and all the hidden units; 3 hidden layers and a linear output layer for each of the MLPs $\\mathbf { L } _ { \\mathrm { m s g } }$ , $\\mathbf { C } _ { \\mathrm { m s g } }$ , and $\\scriptstyle \\mathbf { L } _ { \\mathrm { v o t e } }$ ; and rectified linear units for all non-linearities. We regularized by the $\\ell _ { 2 }$ norm of the parameters scaled by $1 0 ^ { - 1 0 }$ , and performed $T = 2 6$ iterations of message passing on every problem. We trained our model using the ADAM optimizer (Kingma & Ba, 2014) with a learning rate of $2 \\times 1 0 ^ { - 5 }$ , clipping the gradients by global norm with clipping ratio 0.65 (Pascanu et al., 2012). We batched multiple problems together, with each batch containing up to 12,000 nodes (i.e. literals plus clauses). To accelerate the learning, we sampled the number of variables $n$ uniformly from between 10 and 40 during training (i.e. we trained on $\\mathbf { S R ( U ( 1 0 , 4 0 ) ) } )$ , though we only evaluate on SR(40). We trained on millions of problems. ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "After training, NeuroSAT is able to classify the test set correctly with $8 5 \\%$ accuracy. In the next section, we examine how NeuroSAT manages to do so and show how we can decode solutions to satisfiable problems from its activations. Note: for the entire rest of the paper, NeuroSAT refers to the specific trained model that has only been trained on $\\mathbf { S R } ( \\mathbf { U } ( 1 0 , 4 0 ) )$ . ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "6 DECODING SATISFYING ASSIGNMENTS ",
+        "text_level": 1,
+        "bbox": [
+            174,
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Let us try to understand what NeuroSAT (trained on $\\mathbf { S R ( U ( 1 0 , 4 0 ) ) } )$ is computing as it runs on new problems at test time. For a given run, we can compute and visualize the $2 n$ -dimensional vector of literal votes $L _ { * } ^ { ( t ) } \\gets \\mathbf { L } _ { \\mathrm { v o t e } } ( L ^ { ( t ) } )$ at every iteration $t$ . Figure 3 illustrates the sequence of literal votes $L _ { * } ^ { ( 1 ) }$ ) to L(24)∗ a s NeuroSAT runs on a satisfiable problem from SR(20). For clarity, we reshape each $L _ { * } ^ { ( t ) }$ to be an $\\mathbb { R } ^ { n \\times 2 }$ matrix so that each literal is paired with its complement; specifically, the ith row contains the scalar votes for $x _ { i }$ and $\\overline { { x _ { i } } }$ . Here white represents zero, blue negative and red positive. For several iterations, almost every literal is voting unsat with low confidence (light blue). Then a few scattered literals start voting sat for the next few iterations, but not enough to affect the mean vote. Suddenly, there is a phase transition and all the literals (and hence the network as a whole) start to vote sat with very high confidence (dark red). After the phase transition, the vote for each literal converges and the network stops evolving. ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "NeuroSAT seems to exhibit qualitatively similar behavior on every satisfiable problem that it predicts correctly. The problems for which NeuroSAT guesses unsat are similar except without the phase change: it continues to guess unsat with low-confidence for as many iterations as NeuroSAT runs for. NeuroSAT never becomes highly confident that a problem is unsat, and it almost never guesses sat on an unsat problem. These results suggest that NeuroSAT searches for a certificate of satisfiability, and that it only guesses sat once it has found one. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Let us look more carefully at the literal votes $L _ { * } ^ { ( 2 4 ) }$ from Figure 3 after convergence. Note that most of the variables have one literal vote distinctly darker than the other. Moreover, the dark votes are all approximately equal to each other, and the light votes are all approximately equal to each other as well. Thus the votes seem to encode one bit for each variable. It turns out that these bits encode a satisfying assignment in this case, but they do not do so reliably in general. Recall from $\\ S 3$ that NeuroSAT projects the higher dimensional literal embeddings $L ^ { ( \\bar { T } ) } \\in \\bar { \\mathbb { R } } ^ { 2 n \\times d }$ to the literal votes $L _ { * } ^ { ( T ) }$ using the MLP $\\scriptstyle \\mathbf { L } _ { \\mathrm { v o t e } }$ . Figure 4 illustrates the two-dimensional PCA embeddings for $L ^ { ( 1 2 ) }$ to $L ^ { ( 2 6 ) }$ (skipping every other time step) as NeuroSAT runs on a satisfiable problem from SR(40). Blue and red dots indicate literals that are set to 0 and 1 in the satisfying assignment that it eventually finds, respectively. The blue and red dots cannot be linearly separated until the phase transition at the end, at which point they form two distinct clusters according to the satisfying assignment. We observe a similar clustering almost every time the network guesses sat. Thus the literal votes $L _ { * } ^ { ( T ) }$ only ever encode the satisfying assignment by chance, when the projection $\\scriptstyle \\mathbf { L } _ { \\mathrm { v o t e } }$ happens to preserve this clustering. ",
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+    },
+    {
+        "type": "table",
+        "img_path": "images/12b9e073509ac9586437bdefcc495afd0bc0bd4e0ae56f807795d375dd89475f.jpg",
+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Trained on: Trained with: Tested on: Tested with: Overall test accuracy: Accuracy on unsat problems: Accuracy on sat problems: Percent of sat problems solved:</td><td>SR(U(10,40)) 26 iterations SR(40) 26 iterations 85% 96%</td></tr></table>",
+        "bbox": [
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+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "",
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+    {
+        "type": "text",
+        "text": "Our analysis suggests a more reliable way to decode solutions from NeuroSAT’s internal activations: 2-cluster $L ^ { ( T ) }$ to get cluster centers $\\Delta _ { 1 }$ and $\\Delta _ { 2 }$ , partition the variables according to the predicate $\\lVert x _ { i } - \\Delta _ { 1 } \\rVert ^ { 2 } + \\lVert \\overline { { x _ { i } } } - \\Delta _ { 2 } \\rVert ^ { 2 } < \\lVert x _ { i } - \\Delta _ { 2 } \\rVert ^ { 2 } + \\lVert \\overline { { x _ { i } } } - \\bar { \\Delta _ { 1 } } \\rVert ^ { 2 }$ , and then try both candidate assignments that result from mapping the partitions to truth values. This decoding procedure (using $k$ -means to find the two cluster centers) successfully decodes a satisfying assignment for over $7 0 \\%$ of the satisfiable problems in the $\\mathbf { S R } ( 4 0 )$ test set. Table 1 summarizes the results when training on $\\mathbf { S R } ( \\mathbf { U } ( 1 0 , 4 0 ) )$ ) and testing on SR(40). ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "image",
+        "img_path": "images/b2b93ec940f8c8d4be4ceb084ea9fda5f802889227023d9d651d2520db825f72.jpg",
+        "image_caption": [
+            "Table 1: NeuroSAT’s performance at test time on $\\mathbf { S R } ( 4 0 )$ after training on $\\mathbf { S R } ( \\mathbf { U } ( 1 0 , 4 0 ) )$ . It almost never guesses sat on unsatisfiable problems. On satisfiable problems, it correctly guesses sat $73 \\%$ of the time, and we can decode a satisfying assignment for $70 \\%$ of the satisfiable problems by clustering the literal embeddings $L ^ { ( T ) }$ as described in $\\ S 6$ . ",
+            "Figure 4: PCA projections for the high-dimensional literal embeddings $L ^ { ( 1 2 ) }$ to $L ^ { ( 2 6 ) }$ (skipping every other time step) as NeuroSAT runs on a satisfiable problem from SR(40). Blue and red dots indicate literals that are set to 0 and 1 in the satisfying assignment that it eventually finds, respectively. We see that the blue and red dots are mixed up and cannot be linearly separated until the phase transition at the end, at which point they form two distinct clusters according to the satisfying assignment. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Recall that at training time, NeuroSAT is only given a single bit of supervision for each SAT problem. Moreover, the positive and negative examples in the dataset differ only by the placement of a single edge. NeuroSAT has learned to search for satisfying assignments solely to explain that single bit of supervision. ",
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+    {
+        "type": "text",
+        "text": "7 EXTRAPOLATING TO OTHER PROBLEM DISTRIBUTIONS ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "7.1 BIGGER PROBLEMS ",
+        "text_level": 1,
+        "bbox": [
+            176,
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+    {
+        "type": "text",
+        "text": "Even though we only train NeuroSAT on $\\mathbf { S R } ( \\mathbf { U } ( 1 0 , 4 0 ) )$ , it is able to solve SAT problems sampled from $\\mathbf { S R } ( n )$ for $n$ much larger than 40 by simply running for more iterations of message passing. Figure 5 shows NeuroSAT’s success rate on $\\mathbf { S R } ( n )$ for a range of $n$ as a function of the number of iterations $T$ . For $n = 2 0 0$ , there are $2 ^ { 1 6 0 }$ times more possible assignments to the variables than any problem it saw during training, and yet it can solve $2 5 \\%$ of the satisfiable problems in SR(200) by running for four times more iterations than it performed during training. On the other hand, when restricted to the number of iterations it was trained with, it solves under $10 \\%$ of them. Thus we see that its ability to solve bigger and harder problems depends on the fact that the dynamical system it has learned encodes generic procedural knowledge that can operate effectively over a wide range of time frames. ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "image",
+        "img_path": "images/dbe3a32da539b3b746da55e6820e1c1309ef869418443eaf7cb15324a6c0c755.jpg",
+        "image_caption": [
+            "Figure 5: NeuroSAT’s success rate on $\\mathbf { S R } ( n )$ for a range of $n$ as a function of the number of iterations $T$ . Even though we only train NeuroSAT on $\\mathbf { S R } ( 4 0 )$ and below, it is able to solve SAT problems sampled from $\\mathbf { S R } ( n )$ for $n$ much larger than 40 by simply running for more iterations. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            196,
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "image",
+        "img_path": "images/2057d6fd092a8ad0636c64dc52a0e726556d1b2ea9e61829c6e767ae383eebb6.jpg",
+        "image_caption": [
+            "Figure 6: Example graph from the Forest-Fire distribution. The graph has a coloring for $k \\geq 5$ , a clique for $k \\leq 3$ , a dominating set for $k \\geq 3$ , and a vertex cover for $k \\geq 6$ . However, these properties are not perceptually obvious and require deliberate computation to determine. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "7.2 DIFFERENT PROBLEMS ",
+        "text_level": 1,
+        "bbox": [
+            176,
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Every problem in NP can be reduced to SAT in polynomial time, and SAT problems arising from different domains may have radically different structural and statistical properties. Even though NeuroSAT has learned to search for satisfying assignments on problems from $\\mathbf { S R } ( n )$ , we may still find that the dynamical system it has learned only works properly on problems similar to those it was trained on. ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "To assess NeuroSAT’s ability to extrapolate to different classes of problems, we generated problems in several other domains and then encoded them all into SAT problems (using standard encodings). In particular, we started by generating one hundred graphs from each of six different random graph distributions (Barabasi, Erdos-Renyi, Forest-Fire, Random-¨ $k$ -Regular, Random-Static-Power-Law, and Random-Geometric).4 We found parameters for the random graph generators such that each graph has ten nodes and seventeen edges on average. For each graph in each collection, we generated graph coloring problems $( 3 \\leq k \\leq 5 )$ ), dominating-set problems $( 2 \\leq k \\leq 4 )$ ), clique-detection problems $( 3 ~ \\leq ~ k ~ \\leq ~ 5 )$ , and vertex cover problems $( 4 ~ \\leq ~ k ~ \\leq ~ 6 )$ .5 We chose the range of $k$ for each problem to include the threshold for most of the graphs while avoiding trivial problems such as 2-clique. As before, we used Minisat Sorensson & Een (2005) to determine satisfiability. Figure 6 shows an example graph from the distribution. Note that the trained network does not know anything a priori about these tasks; the generated SAT problems need to encode not only the graphs themselves but also formal descriptions of the tasks to be solved. ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Out of the 7,200 generated problems, we kept only the 4,888 satisfiable problems. On average these problems contained over two and a half times as many clauses as the problems in SR(40). We ran NeuroSAT for 512 iterations on each of them and found that we could successfully decode solutions for $8 5 \\%$ of them. In contrast, Survey Propagation (SP) (Braunstein et al., 2005), the canonical (learning-free) message passing algorithm for satisfiability, does not on its own converge to a satisfying assignment on any of these problems.6 This suggests that NeuroSAT has not simply found a way to approximate SP, but rather has synthesized a qualitatively different algorithm. ",
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "8 FINDING UNSAT CORES ",
+        "text_level": 1,
+        "bbox": [
+            176,
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "NeuroSAT (trained on $\\mathbf { S R ( U ( 1 0 , 4 0 ) ) } \\rangle$ can find satisfying assignments but is not helpful in constructing proofs of unsatisfiability. When it runs on an unsatisfiable problem, it keeps searching for a satisfying assignment indefinitely and non-systematically. However, when we train the same architecture on a dataset in which each unsatisfiable problem has a small subset of clauses that are already unsatisfiable (called an unsat core), it learns to detect these unsat cores instead of searching for satisfying assignments. The literals involved in the unsat core can be decoded from its internal activations. When the number of literals involved in the unsat core is small relative to the total number of literals, knowing the literals involved in the unsat core can enable constructing a resolution proof more efficiently. ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "We generated a new distribution $\\mathbf { S R C } ( n , u )$ that is similar to $\\mathbf { S R } ( n )$ except that every unsatisfiable problem contains a small unsat core. Here $n$ is the number of variables as before, and $u$ is an unsat core over $x _ { 1 } , \\ldots , x _ { k }$ $( k < n )$ that can be made into a satisfiable set of clauses $u ^ { \\prime }$ by negating a single literal. We sample a pair from $\\mathbf { S R C } ( n , u )$ as follows. First, we initialize a problem with $u ^ { \\prime }$ , and then we sample clauses (over $x _ { 1 }$ to $x _ { n }$ ) just as we did for $\\mathbf { S R } ( n )$ until the problem becomes unsatisfiable. We can now negate a literal in the final clause to get a satisfiable problem $p _ { s }$ , and then we can swap $u ^ { \\prime }$ for $u$ in $p _ { s }$ to get $p _ { u }$ , which is unsatisfiable since it contains the unsat core $u$ . We created train and test datasets from $\\mathbf { S R C } ( 4 0 , u )$ with $u$ sampled at random for each problem from a collection of three unsat cores ranging from three clauses to nine clauses: the unsat core $R$ from Knuth (2015), and the two unsat cores resulting from encoding the pigeonhole principles $\\mathbf { P P } ( 2 , 1 )$ and $\\mathbf { P P } ( 3 , 2 )$ . 7 We trained our architecture on this dataset, and we refer to the trained model as NeuroUNSAT. ",
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+        "page_idx": 7
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+        "img_path": "images/2066957dcb07a7a9e9535cf1b143befffa21bfff5771e38c35e096823d40388f.jpg",
+        "image_caption": [
+            "(b) NeuroUNSAT running on an unsatisfiable problem. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "(a) NeuroUNSAT running on a satisfiable problem. ",
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+    {
+        "type": "image",
+        "img_path": "images/eb4246e96b30294841909d61559d94429bb6ad22b8aa7eb85c820bd8894b2939.jpg",
+        "image_caption": [
+            "Figure 7: The sequence of literal votes $L _ { * } ^ { ( t ) }$ as NeuroUNSAT runs on a pair of problems from $\\mathbf { S } \\bar { \\mathbf { R C } } ( 3 0 , \\mathbf { P P } ( 3 , \\bar { 2 } ) )$ . In both cases, the literals in the first six rows are involved in the unsat core. In 7a, NeuroUNSAT inspects the modified core $u ^ { \\prime }$ of the satisfiable problem but concludes that it does not match the pattern. In 7b, NeuroUNSAT finds the unsat core $u$ and votes unsat with high confidence (dark blue). "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            522,
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "NeuroUNSAT is able to predict satisfiability on the test set with $100 \\%$ accuracy. Upon inspection, it seems to do so by learning to recognize the unsat cores. Figure 7 shows NeuroUNSAT running on a pair of problems from $\\bar { \\mathbf { S } } \\mathbf { R } \\mathbf { C } ( 3 0 , \\mathbf { \\bar { P } } \\mathbf { P } ( 3 , 2 ) )$ . In both cases, the literals in the first six rows are involved in the unsat core. In Figure 7a, NeuroUNSAT inspects the modified core $u ^ { \\prime }$ of the satisfiable problem but concludes that it does not match the pattern exactly. In Figure 7b, NeuroUNSAT finds the unsat core $u$ and votes unsat with high confidence (dark blue). As in $\\ S 6$ , the literals involved in the unsat core can sometimes be decoded from the literal votes $L _ { * } ^ { ( T ) }$ , but it is more reliable to 2- cluster the higher-dimensional literal embeddings $L ^ { ( T ) }$ . On the test set, the small number of literals involved in the unsat core end up in their own cluster $98 \\%$ of the time. ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Note that we do not expect NeuroUNSAT to generalize to arbitary unsat cores: as far as we know it is simply memorizing a collection of specific subgraphs, and there is no evidence it has learned a generic procedure to prove unsat. ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "9 RELATED WORK ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            840,
+            341,
+            856
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "There have been many attempts over the years to apply statistical learning to various aspects of the SAT problem: restart strategies (Haim & Walsh, 2009), branching heuristics (Liang et al., 2016; ",
+        "bbox": [
+            176,
+            872,
+            825,
+            900
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Grozea & Popescu, 2014; Flint & Blaschko, 2012), parameter tuning (Singh et al., 2009), and solver selection (Xu et al., 2008). None of these approaches use neural networks, and instead make use of both generic graph features and features extracted from the runs of SAT solvers. Moreover, these approaches are designed to assist existing solvers and do not aim to solve SAT problems on their own. ",
+        "bbox": [
+            174,
+            103,
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+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "From the machine learning perspective, the closest work to ours is Palm et al. (2017), which showed that an MPNN can be trained to predict the unique solutions of Sudoku puzzles. We believe that their network’s success is an instance of the phenomenon we study in this paper, namely that MPNNs can synthesize local search algorithms for constraint satisfaction problems. Evans et al. (2018) present a neural network architecture that can learn to predict whether one propositional formula entails another by randomly sampling and evaluating candidate assignments. Unlike NeuroSAT, their network does not perform heuristic search and can only work on simple problems for which random guessing is tractable. There have also been several recent papers showing that various neural network architectures can learn good heuristics for NP-hard combinatorial optimization problems (Vinyals et al., 2015; Bello et al., 2016; Dai et al., 2017); however, finding low-cost solutions to optimization problems requires less precise reasoning than finding satisfying assignments. ",
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "10 DISCUSSION ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            356,
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+            372
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Our main motivation has been scientific: to better understand the extent to which neural networks are capable of precise, logical reasoning. Our work has definitively established that neural networks can learn to perform discrete search on their own without the help of hard-coded search procedures, even after only end-to-end training with minimal supervision. We found this result surprising and think it constitutes an important contribution to the community’s evolving understanding of the capabilities and limitations of neural networks. ",
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Although not our primary concern, we also hope that our findings eventually lead to improvements in practical SAT solving. As we stressed early on, as an end-to-end SAT solver the trained NeuroSAT system discussed in this paper is still vastly less reliable than the state-of-the-art. We concede that we see no obvious path to beating existing SAT solvers. One approach might be to continue to train NeuroSAT as an end-to-end solver on increasingly difficult problems. A second approach might be to use a system like NeuroSAT to help guide decisions within a more traditional SAT solver, though it is not clear that NeuroSAT provides any useful information before it finds a satisfying assignment. However, as we discussed in $\\ S 8$ , when we trained our architecture on different data it learned an entirely different procedure. In a separate experiment omitted for space reasons, we also trained our architecture to predict whether there is a satisfying assignment involving each individual literal in the problem and found that it was able to predict these bits with high accuracy as well. Unlike NeuroSAT, it made both type I and type II errors, had no discernable phase transition, and could make reasonable predictions within only a few rounds. We believe that architectures descended from NeuroSAT will be able to learn very different mechanisms and heuristics depending on the data they are trained on and the details of their objective functions. We are cautiously optimistic that a descendant of NeuroSAT will one day lead to improvements to the state-of-the-art. ",
+        "bbox": [
+            174,
+            481,
+            825,
+            702
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "ACKNOWLEDGEMENTS ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            723,
+            334,
+            734
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "We thank Steve Mussmann, Alexander Ratner, Nathaniel Thomas, Vatsal Sharan and Cristina White for providing valuable feedback on early drafts. We also thank William Hamilton, Geoffrey Irving and Arun Chaganty for helpful discussions. This work was supported by Future of Life Institute grant 2017-158712. ",
+        "bbox": [
+            174,
+            747,
+            825,
+            803
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
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parse/train/O358nrve1W/O358nrve1W.md

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+# NEURALLY GUIDED GENETIC PROGRAMMING FOR TURING COMPLETE PROGRAMMING BY EXAMPLE
+
+Anonymous authors Paper under double-blind review
+
+# ABSTRACT
+
+The ability to synthesise source code from input/output examples allows nonexperts to generate programs, and experts to abstract away a wide range of simple programming tasks. Current research in this area has explored neural synthesis, SMT solvers, and genetic programming; each of these approaches is limited, however, often using highly specialised target languages for synthesis. In this paper we present a novel hybrid approach using neural networks to guide genetic programming (GP), which allows us to successfully synthesise code from just ten I/O examples in a generalised Turing complete target language, up to and including a sorting algorithm. We show that GP by itself is able to synthesise a set of simple programs, and show which hints (suggested lines of code for inclusion) are of most utility to GP in solving harder problems. Using a form of unstructured curriculum learning, we then demonstrate that neural networks can be used to determine when to make use of these high-utility hints for specific I/O problems and thus enable complex functions to be successfully synthesised. We apply our approach to two different problem sets: common array-to-array programs (including sorting), and a canvas drawing problem set inspired by So & Oh (2018).
+
+# 1 INTRODUCTION
+
+The ability to synthesise source code from examples, in which a source code implementation of a function is created based on one or more demonstrations of input-output mapping, is a fundamental question in machine learning. We specifically study this question in the form of scenarios where large corpora of existing code are not available (e.g., human-written programs in open-source repositories). The immediate applications of this would allow non-programmers to generate programs, or experts to abstract away trivial coding tasks. In addition, from a machine learning perspective, it allows complex functions to be generated in a symbolic and human-readable form – which can be subjected to a wide range of static analysis tools to model generality or correctness.
+
+To date this challenge has been studied using neural-network-driven synthesis, genetic programming, and SMT solvers. However, at present these approaches are significantly constrained in the complexity of the target language in which code is synthesised. Neural synthesis for example, such as the DeepCoder architecture (Balog et al., 2017; Zohar & Wolf, 2018), shows success on simple domain-specific languages, but the search space of more realistic Turing-complete languages is vast by comparison and is unlikely to be representable in a neural network on current or near-future hardware. Genetic programming, meanwhile, is limited by our ability to specify a fitness function which can successfully navigate to a solution for a particular problem – in the highly irregular and often flat fitness landscape of program space (Kinnear, 1994; Renzullo et al., 2018). SMT solvers by comparison lack the analytical power to handle loops without human guidance, constraining their applicability (Srivastava et al., 2010a; So & Oh, 2018; Srivastava et al., 2010b).
+
+In this paper we examine code synthesis from examples for a Turing-complete language which can be cross-compiled into C/Java. We use just 10 input/output examples to describe each problem for which we need to synthesise a matching function (i.e., providing unsorted and sorted arrays of integers to describe sorting); because our target language yields a total search space size of $\mathrm { \dot { 5 } * 1 0 ^ { 1 1 9 } }$ possible program permutations, scalability of the synthesis technique is crucial. In this context, we use a novel combination of genetic programming (GP) and neural networks (NNs); GP is used to navigate within program space from a given starting point using a general-purpose fitness function, while NN methods are used to provide prediction of high-level features which help guide the GP to higher-probability success areas in which to search. In essence this technique allows the NN to model only highly abstract features of program space, allowing it to scale to vast program search spaces, while the GP can then incrementally traverse program space from an NN-derived starting point to a correct solution. We bootstrap this process by using an unstructured form of curriculum learning, in which successfully-found functions are used as seed programs to generate synthetic corpora on which to train new neural networks, leading to further high-utility source code feature predictions for new problems. Initially this curriculum learning is based on programs that can be successfully found using GP alone with our generic fitness function, which then allows us to synthesise more complex programs using NN inference.
+
+Our key results demonstrate that GP is able to solve simple synthesis problems unaided, and that synthetic corpora generated from these problems allow popular neural network architectures to identify high-utility search hints on more complex problems to guide the genetic programming search. In one of our problem sets, this has the effect of allowing the framework to successfully synthesise 7 of the 10 programs that had never been found by GP alone, and among other programs moves success rates from $38 \%$ to $55 \%$ . All of our source code is provided online (pending de-anonymisation).
+
+# 2 RELATED WORK
+
+Code synthesis from I/O examples has been studied using three major approaches: deductive solvers; neural networks (NNs) with search; and genetic programming (GP). For code synthesis in a Turingcomplete language, deductive solvers have yet been shown to operate well with loop-based flow control operators (although frameworks which manually define any non-linear program flow can yield good performance (So & Oh, 2018)). Neural synthesis, by comparison, is limited by how much of program space for a general-purpose target language can be captured in a NN model, while GP is limited by the difficulty of deriving a fitness function to navigate to a solution (Renzullo et al., 2018). In the remainder of this section we focus on NN and GP approaches in more detail.
+
+Neural synthesis Neural synthesis works by training a neural network on a sub-sample of the entirety of program space for the target language (e.g., sampling at a uniform interval or at random). When presented with a new problem as an I/O example, the neural network will then be asked to predict which lines of code (or particular operators) are likely to be present in the solution based on similar I/O transforms observed in the training set from the above sub-sample. The system will then perform an exhaustive search of program space to fill in the remaining (non-predicted) features. Notable examples here include DeepCoder and RobustFill, among others (e.g., Balog et al. (2017); Zohar & Wolf (2018); Devlin et al. (2017); Chen et al. (2019); Singh & Kohli (2017))
+
+The key limitation to this approach is that it must be able to train on a detailed enough sub-sample of program space, and store this sample inside a neural network, to make meaningful predictions on program features for unseen problems. While this works for highly simplified languages (DeepCoder, for example, has no loop operators (Balog et al., 2017)), the search space size of a Turingcomplete language is astronomical by comparison. If we consider that the capacity of a feed-forward ReLu neural network to differentiate between classes (in our case programs), termed its VapnikChervonenkis (VC) dimension, at best grows as a linear function of $w * L * l o g ( w )$ where $w$ is the total number of weights and $L$ the number of layers (Bartlett et al., 2019), it is unlikely that a neural network on current hardware would be able to represent a useful sub-sample of possible program permutations yielded by the search space of a general-purpose language.
+
+Genetic programming GP relies on iterative travel through program space from a starting point (often an empty program) to the solution, guided by a fitness function (Vanneschi & Poli, 2012; Taleby Ahvanooey et al., 2019). The field has a long history (Forsyth, 1981) but still shows results (Miranda et al., 2019) that are competitive with neural networks (Ain et al., 2020), and an ability to tackle complex problems mixing diverse datatypes (Pantridge & Spector, 2020).
+
+Unlike neural synthesis, a GP approach does not need to encode the entirety of program space in a model, and so can in theory work in a scalable fashion on high dimensional search spaces as long as a fitness function is provided which can guide the search incrementally towards a solution. The key problem with GP for code synthesis is that large areas of program space are difficult to navigate, exhibiting large plateaus of neutrality (no behavioural change despite significant code change) and highly irregular responses to code change (jagged fitness landscape) (Renzullo et al., 2018; Kinnear, 1994). A fitness function may also fail to capture higher level properties, for example “all outputs are even” or “all values from the input are repeated in the output”, which may otherwise be identified by a neural network.
+
+Combining neural network prediction with genetic programming Given the limitations of both NNs and GP in themselves, we hypothesise that the combination of the two techniques may provide the best features of both while mitigating their respective limitations in the context of code synthesis in a Turing-complete language with a very large program search space.
+
+While a NN cannot feasibly model all of program space, and so cannot be expected to predict each line to be synthesised, it does have the potential to predicting a small number of higher-level features which only require a very limited internal model of program space. This can be combined with a GP search process which fixes these lines in place to constrain the search area, and can use this partially-constructed program in combination with one or more generic fitness functions to guide the search to a successful result. We further find that we can use successfully-found programs to generate synthetic training sets for a NN focused on that area of program space, which then leads to further high-value predictions of likely features for more complex programs.
+
+# 3 METHODOLOGY
+
+Our system accepts up to ten I/O examples from a user and synthesises the source code of a function which converts those inputs to their corresponding outputs. We use 10 examples as a number which a user may be willing to input while representing lower effort than writing the function manually. If our system fails to synthesise source code for a given problem, the user can specify a simpler but related I/O problem; the successful synthesis of this simpler problem can then lead to subsequent success on the more complex problem. As a target programming language for synthesis, we use a Turing-complete language (previously used by the authors of (Wild & Porter, 2019)) which can be cross-compiled directly into C/Java/Python. The language features primitive loop operators, variable declarations, and conditional branch operators; a full listing of its operators is given in appendix A.1.
+
+We use a combination of GP guided by NN prediction to approach this problem, and show successful synthesis of a wide range of programs including sorting. In this paper we study three specific parts of the above overall framework: (i) the success rate of GP by itself of finding programs in our problem sets; (ii) the effect of different hints (suggested lines of code) on the success rate of GP; (iii) the ability of a NN to predict high-utility hints, after being trained on a synthetic corpus derived from high-success program finds from the GP alone.
+
+In the remainder of this section we describe our problem sets, genetic programming and NN implementations in more detail, then present the set of experiments that we conduct.
+
+# 3.1 SPECIFICATION OF PROBLEM
+
+We use two different problem sets which represent human-useful programs of the kind that may be input into our system. Our first problem set is array-to-array programs such as extract even numbers, append arrays, or sort. Our second problem set is provided with an image to draw on a canvas and must synthesise the program which draws that image (this problem set is taken from So & Oh (2018)).
+
+Each problem within each problem set is presented to the system as a set of 10 IO examples. These are generated for each problem by feeding in 10 inputs and corresponding outputs of the form the problem requires (either a randomly generated input array and integer, or a canvas size). These 10 inputs are randomly generated from a fixed seed, to reduce internal variability between tests, allowing more accurate evaluation of the changes made by alterations to parameters or by guidance to the GP. This is designed to represent an unbiased input set.
+
+# 3.2 GENETIC PROGRAMMING
+
+Our GP algorithm creates a population of 2,000 programs, each one the result of crossover and mutation from among the 10 highest-ranked parents of the previous generation (or, for the first generation, mutations of the empty program). Each program in a population is then ranked using a general-purpose fitness function which was experimentally found to be good at locating programs from various search space starting points. We integrate novelty search (Lehman & Stanley, 2010; Doncieux et al., 2019) as part of our fitness function to avoid falling into the same local minima repeatedly, which was found to further boost search success. Once all programs have been ranked, a new generation of 2,000 members is then created, repeatedly, until the target program is found or the system reaches a maximum number of generations (3,000) and reports failure.
+
+Our fitness function, in detail, is formulated as error ∗ novelty penalty. The value for error is calculated as the number of elements in the output of a candidate function which do not match those expected for the corresponding input; this value is normalised into the range $[ 0 . 0 , - 1 . 0 ]$ by dividing it by the corresponding error which would have been produced by an empty output for that input.
+
+The value for novelty penalty is calculated by first extracting the highest-fitness member of a population, and removing all lines of code which do not contribute to its behaviour. This program is stored as a repulsor which indicates how well-explored a given region of program space is. When calculating the fitness of a new program, each repulsor adds a multiplicative value depending on the distance $D$ of that program from the repulsor. $D$ itself is calculated by examining each line in the first program and calculating how far away (in lines of code) the same line is in a second program. Each repulsor adds $m a x ( 0 , 1 - D / 1 5 )$ to novelty penalty, such that multiple repulsors can exist in the same area of program space, leading to stronger avoidance of those areas.
+
+For each new generation during the GP search, parents are chosen using tournament selection with tournament size of 10 (Miller & Goldberg, 1995). Crossover occurs by taking the first half of the first parent, and appending the second half of the second parent (syntactically flawed offspring are accepted into the population but receive minimal fitness when evaluated). Following crossover, we apply a single mutation to a program with a probability of 0.35. Following each mutation a random boolean value is selected, and if true another mutation is applied, to a maximum of 8 mutations.
+
+Each mutation takes one of the following forms, selected uniformly at random:
+
+1. Insert: this inserts a random line from all possible lines available in the language, and will delete last lines of program if program is already at maximum length. It automatically adds an ENDBLOCK operator if a flow-control operator was inserted.   
+2. Delete: sets a random line to NO-OP.   
+3. Mutate: changes a random operator/parameter on a line, ensuring the line remains valid.   
+4. Swap: exchanges position of any two lines of the program.
+
+In all of the above, the parameter values relating to $D$ and mutation probabilities were chosen as values found to experimentally work well.
+
+# 3.3 NEURAL NETWORK DESIGN
+
+While no programs have yet been successfully synthesised by our system, we initially rely on GP as above to locate simpler programs. Once at least one program has been located it is added to our set of successfully found programs $S _ { F }$ . We then augment GP with a neural network which provides hints of one or more likely lines of code for a given unsolved problem, which correspond to likely areas of program space in which the GP will search (rather than the GP starting its search from an empty program). Our NN is trained on a synthesised training set of programs; the programs in this training set are generated at random but must exhibit some of the program features present in programs in $S _ { F }$ . The intuition here is that the neural network will thus learn to recognise which program features are likely to be present for unseen I/O examples, based on program features known to be useful in other programs requested by users, which are then useful to guide the GP search.
+
+Each synthesised training corpus based on $S _ { F }$ has 20,000 programs for training and 2,000 for testing. The 20,000 programs are divided into 10,000 which do have a particular program fragment of interest, and 10,000 which do not have that fragment, with the NN trained to determine whether or not a given I/O example is likely to have that fragment in the corresponding implementation program. Each program in each set of 10,000 is assured to be distinct in functionality from every other program in the set, tested on its behaviour with respect to a fixed $1 0 ~ \mathrm { { J / O } }$ examples. Each new program in a training set is generated by selecting uniformly at random two programs already accepted, applying a crossover, then applying between 1 and 8 mutations (as described above in the GP section) while assuring that the fragment of interest still exists.
+
+We use a range of NN architectures to study which ones work best in each problem set. Specifically we use a feed-forward NN in both problem sets, as a shared baseline; in our array-to-array problem set we then also use an LSTM-based recurrent network, and in our canvas problem set we use a CNN. The details of all network architectures are described in appendix A.3.
+
+# 3.4 EXPERIMENT SETUP
+
+We use three different experiments to study each element of our approach. The details of each experiment are given below, while the next section presents the results.
+
+Experiment 1: Exhaustive Fragment Evaluation In our first experiment we examine the baseline performance of GP alone on our two problem sets, then study the effect of each possible source code hint that can be given for each problem (in terms of its effect on success rate).
+
+For the latter we examine the set of 1 or 2-line code fragments which can be cleanly isolated (with no dependencies) in the ground truth solution to each problem. We then run a GP pass with the code fragment as a forced requirement, such that any program produced by the GP which does not include them automatically receives a penalty fitness of -10,000. Each experiment in this series is repeated 30 times to account for the inherent stochasticity in the GP process.
+
+These results demonstrate the kinds of problem that can be solved using GP alone, and the extent to which a GP algorithm can have its probability of finding a solution increased by constraining its sampled programs to contain certain lines of code – which help identify the highest utility hints that a NN can seek to find.
+
+Experiment 2: Fragment Recognisability by Neural Network Having established that code fragments can be used to improve GP find rates (including from zero to non-zero find rates), our second experiment studies the use of synthetic NN training corpora based around these fragments – and the extent to which NNs can successfully predict these features in unseen problems to assist GP.
+
+For this experiment we need to assume that some programs have already been found by the GP alone, and useful fragments identified, from which to automatically synthesise NN training sets as described above.
+
+For our array-to-array problem set, we select all programs which have find rates of $90 \%$ and above using the GP alone from experiment 1. For our canvas problem set, we select the single highest findrate program using the GP alone for each ‘class’ of problem identified by So & Oh (2018) (such as ‘triangles’). These programs represent those which are most likely to have been found first without the aid of the NN. From within the source code of these programs we select a set of individual fragments from which to generate our synthetic NN training corpora. These fragments are chosen using high-utility fragments identified from experiment 1, and augmented with further fragments of interest that were manually chosen to gain a wider coverage of fragment predictability by a NN.
+
+Altogether, these experiments indicate how well NNs can predict the presence of different program features based on our synthetic training sets – a mechanism by which the solution to easy (high-findrate) problems can be used to find solutions to hard (low-find-rate) problems.
+
+Experiment 3: Success of chosen fragments In the third experiment we test the success of our GP algorithm when guided by NN-predicted source code fragments using the trained NNs from experiment 2, using all problems in each problem set. To do this we apply a simple selection process, choosing uniformly at random from among all fragments which were estimated by an NN to have a probability of $> = 0 . 5$ presence in a program, and provide one such fragment to the GP process. This shows how a full neurally-guided GP process would perform in our end-to-end system.
+
+Table 1: Find rates for guided Genetic Programming Algorithm with forced inclusion of code fragments from ground truth. Baseline is unguided GP. Maximum is single best performing fragments. Best operators are those used in the highest-scoring fragment (first if tied) $\mathrm { n } { = } 3 0$ per fragment, percentage success)   
+
+<table><tr><td>Problem</td><td>Baseline</td><td>Maximum</td><td>Best Operators</td></tr><tr><td>Append</td><td>0%</td><td>27%</td><td>Var=Literal1;Addition</td></tr><tr><td>Cumulative Abs Sum Keep Evens</td><td>0% 0%</td><td>3% 7%</td><td>Loop; Read Var=Literal 2; Make Array</td></tr><tr><td>Retain First Half Reverse</td><td>0% 58%</td><td>13% 80%</td><td>Var=Literal 2; Divide Var=Literal 1; Make Array</td></tr><tr><td>Shift Right</td><td>0%</td><td>13%</td><td>Var=Literal 1; Loop</td></tr><tr><td>Shift Right Lossy Sort (Bubblesort)</td><td>84% 0%</td><td>80% 0%</td><td>Var=Literal 1 (None)</td></tr><tr><td></td><td></td><td></td><td></td></tr><tr><td></td><td></td><td></td><td></td></tr><tr><td>Parallelogram</td><td>7%</td><td>30%</td><td>Var=Literal2; Divide</td></tr><tr><td>Mirrored Hollow Parallelogram</td><td>13%</td><td>60%</td><td>Var=Literal2;Divide</td></tr><tr><td>Hollow Right Triangle</td><td></td><td>90%</td><td></td></tr><tr><td></td><td>87%</td><td></td><td>Var=Literal1; Subtract</td></tr><tr><td>Hollow Mirrored Right Triangle</td><td>63%</td><td>93%</td><td>Var=Literal1; Subtract</td></tr><tr><td></td><td></td><td></td><td></td></tr><tr><td>Inverted Isosceles Triangle Trapezoid</td><td>46% 7%</td><td>23% 10%</td><td>Var=Literal 2 Var=Literal 1</td></tr></table>
+
+# 4 EVALUATION
+
+Our evaluation was conducted on Tensorflow 1.14, Python 3.6.9, Java OpenJDK 11.0.6. Our source code will be made available in camera-ready version
+
+4.1 EXPERIMENT 1: GP BASELINE AND ITERATIVE REQUIREMENT SEARCH
+
+In this experiment we first determine the baseline find rate of the GP with no assistance, and also select a number of problems to study by supplying certain subsets of the lines as guidance to the GP. This is done by fully running $3 0 \mathrm { G P }$ search repeats with each valid (as described above) 2-line fragment from the ground truth. From the first corpus we selected 8 problems, from the second we select 6. From the first corpus we select mostly low-find-rate problems, to study which form of fragments would be useful to provide to achieve success in the GP, while in the second corpus we select a more representative sample, to study how constraint-forcing behaves in general. We select two problems of the same class from the 2nd corpus (right triangles), to ensure that similar fragments provide similar results in similar circumstances (to give confidence in generality of these results). The baseline success, best find rate increase, and best fragment’s operators for selected problems are presented in Table 4.1.
+
+Here we clearly see that forcing the inclusion of even the simplest code elements (one or two lines of the ground truth) into the GP’s population allows the GP to find previously unsolvable problems. We can also see that fragments containing arithmetic operators (especially literal assignment) appear to have a stronger influence on success, possibly as they are harder to find, as their effects are far more subtle and complex than, say presence of a loop operator. These should therefore be studied as high-utility candidates for deployment into GP processes we wish to guide in the future.
+
+We also note that some examples show a decrease in success rate (e.g. “Shift Right Lossy”). While no fragment reduced find rates to zero, we speculate that in some cases the provision of a fragment places the GP into an area of program space from which it is harder to reach the solution using our general-purpose fitness function (for example, meaning that this point in program space has larger regions of neutral landscape around it which are harder to traverse over).
+
+A full breakdown of all baseline GP find rates is presented in Appendix A.4, with fragments and their successes presented in Appendix A.5.
+
+<table><tr><td>Fragment</td><td>FFNN</td><td>FFNN Test</td><td>RNN</td><td>RNN Test</td></tr><tr><td></td><td></td><td></td><td></td><td></td></tr><tr><td>Add</td><td>58%</td><td>62 %</td><td>58%</td><td>62 %</td></tr><tr><td>+1 Offset Loop</td><td>74 %</td><td>67 %</td><td>71 %</td><td>66 %</td></tr><tr><td>Length -1 Loop</td><td>76 %</td><td>66 %</td><td>77 %</td><td>65 %</td></tr><tr><td>Literal (2)</td><td>76%</td><td>63%</td><td>71 %</td><td>60 %</td></tr><tr><td>Loop</td><td>97 %</td><td>78%</td><td>98 %</td><td>79 %</td></tr><tr><td>Loop+Conditional</td><td>72 %</td><td>62%</td><td>70 %</td><td>59 %</td></tr><tr><td>Loop+Read</td><td>65 %</td><td>61%</td><td>62 %</td><td>58%</td></tr><tr><td>Nonstandard Array</td><td>81%</td><td>68%</td><td>80%</td><td>66 %</td></tr><tr><td>Read</td><td>76%</td><td>63%</td><td>60 %</td><td>58%</td></tr><tr><td>Subtract</td><td>61 %</td><td>56%</td><td>62 %</td><td>56 %</td></tr><tr><td>Average</td><td>74 %</td><td>65 %</td><td>71 %</td><td>63 %</td></tr><tr><td>Fragment</td><td>FFNN</td><td>FFNN Test</td><td>CNN HU</td><td>CNN Test</td></tr><tr><td></td><td></td><td></td><td></td><td></td></tr><tr><td>Add</td><td>58 %</td><td>67 %</td><td>78 %</td><td></td></tr><tr><td>Conditional</td><td>61 %</td><td>57 %</td><td></td><td>86 %</td></tr><tr><td>Half</td><td>61 %</td><td></td><td>64 %</td><td>85 %</td></tr><tr><td>Half Loop</td><td></td><td>62 %</td><td>87 %</td><td>87 %</td></tr><tr><td></td><td>65%</td><td>60 %</td><td>82%</td><td>87 %</td></tr><tr><td>Half Loop Depend</td><td>70 %</td><td>59 %</td><td>82 %</td><td>90 %</td></tr><tr><td>Loop Conditional</td><td>64 %</td><td>58 %</td><td>57 %</td><td>85 %</td></tr><tr><td>Loop Draw</td><td>65 %</td><td>58%</td><td>70 %</td><td>85%</td></tr><tr><td>Loop Loop</td><td>73 %</td><td>57 %</td><td>83 %</td><td>75%</td></tr><tr><td>Loop Loop Subtract</td><td>60 %</td><td>55%</td><td>64 %</td><td>77 %</td></tr><tr><td>Draw Draw Average</td><td>52 % 63 %</td><td>52 % 58 %</td><td>74 % 74 %</td><td>71 % 83 %</td></tr></table>
+
+Table 2: Percentage accuracy of two neural networks on two corpora, with regards to ability to recognise presence of a code fragment within the source code of the function whose IO mapping they are receiving as feature inputs. First NN architecture is a common FFNN implementation. Accuracy is only that of the NNs which show success on known-ground-truth seed fragments (as described in methodology, others discarded from results). Testing accuracy is accuracy on the 2,000 synthetic testing programs. Full description of fragments in Appendix Tables 24 and 25 $\mathrm { ( n = } 3 0$ )
+
+# 4.2 EXPERIMENT 2: FRAGMENT RECOGNISABILITY
+
+In this experiment we start with selected high-success seed programs from our first experiment. For array-to-array problems, these are all programs with a $90 \%$ find rate or better via GP alone, while for our canvas drawing problem set, these are the single highest-find-rate program in each category such as ‘triangles’. We then take 10 source code fragments present in these seed programs which exhibit a positive effect of find rates, and use these fragments to generate completely synthetic training corpora for neural networks trained to predict whether or not an I/O example will include a particular fragment in its source code solution.
+
+Table 4.2 shows how effective our different trained neural networks architectures are at then correctly predicting the presence of these fragments in the I/O examples from our two problem sets (which are not part of the synthetic training sets).
+
+This data show significant variance of prediction success (from $58 \%$ up to $9 7 \%$ ) but overall shows that our neural networks do exhibit the ability to accurately predict that a particular source code fragment will exist in the solution to a given I/O problem – even though these neural networks are trained on entirely synthetic data.
+
+Examining the different neural network architectures, our baseline feed-forward NN shows success in both problem sets and so is able to act as a viable generalist model. Our recurrent network, used in the array-to-array corpus, shows prediction success that is generally lower than the feed-forward NN. Our CNN meanwhile, used in our canvas drawing corpus, shows some significant gains in prediction success compared to the feed-forward NN, though also shows some lower results (for example on predicting the presence of a loop with a conditional). This suggests that a mixture-ofexperts approach may be desirable here, but also that a general feed-forward model is viable.
+
+In practice we would use these trained networks alongside a threshold of prediction when determining which fragments to recommend for inclusion in a GP search; we present this in the following section.
+
+# 4.3 EXPERIMENT 3: CHOSEN FRAGMENT DEPLOYMENT
+
+In this experiment, the fragments from the above NN experiments were employed to guide the GP process, based on the average estimates by the trained neural networks. This demonstrates the efficacy of our end-to-end system in taking successfully-found solutions to easy problems and employing their characteristics, via trained NN predictors, to find solutions to more difficult problems.
+
+We report the success rates here using two different approaches to selecting fragments for the GP. We do this either using a uniform random choice of fragments which had an average presence probability estimate of $> = 0 . 5$ (termed Uniform); or using a fragment which is predicted for the problem being solved but has the lowest prediction rate across all other problems (which ideally therefore may be the most information bearing fragment). For our array-to-array problem set we use both approaches, while for our canvas problem set we see less clarity in lowest prediction rates across all problems and so only use the uniform random style.
+
+<table><tr><td>Corpus And Approach</td><td>Find Rate (vs Baseline)</td><td>Gained</td><td>Lost</td></tr><tr><td></td><td></td><td></td><td></td></tr><tr><td>1st (1D Arrays), Uniform</td><td>46 % (38%)</td><td>5</td><td>1</td></tr><tr><td>1st (1D Arrays),Rarest Preferred</td><td>55 % (38%)</td><td>7</td><td>0</td></tr><tr><td>2nd (2D Array), Uniform</td><td>39 % (36%)</td><td>4</td><td>1</td></tr></table>
+
+Table 3: Success of the GP when guided by the fragments selected by the neural network. ‘Gained’ are problems which have $a > 0 \%$ find rate which have a baseline of $0 \%$ , ‘Lost’ are those which previously had $> 0 \%$ but now have $0 \%$ . $\mathrm { ( n { = } } 3 0$ )
+
+As can be seen in Table 4.3, guiding the GP using this process produces notable improvements to the overall program synthesis success rates. Average find rates were boosted, and crucially a number of problems became solvable which were not before, with only a single problem in each problem set failing to be solved. This set of newly-solved problems here includes instance in which the bubblesort algorithm, considered by the authors to be the hardest problem in the set, was successfully found.
+
+This improvement was strongest on the approach which selected the estimated-rarest fragment for use, clearly indicating that the fragments are not equivalent in their usefulness. The ideal way in which to extract the most useful fragments for the GP to use, based on those available from NN predictions, therefore remains a topic of future work.
+
+The full set of guided GP results is available in detail in Appendix A.7.
+
+# 5 CONCLUSION
+
+In this paper we have presented a novel combination of genetic programming and neural network prediction to synthesise code from just $1 0 \mathrm { { I / O } }$ examples.
+
+Our framework demonstrates the potential to create a system which searches for solutions to problems within a corpus, finds a subset, extracts code fragments from the successes, then trains a neural network to recognise the presence of these fragments and determines which unsolved problems would benefit from NN-guidance – thus boosting GP find rates on a subsequent pass. We demonstrate that this process can render previously unfindable problems findable, and boost overall find rates, including the successful synthesis of bubble sort.
+
+Our approach is scalable to the search space of Turing-complete languages, and has been demonstrated to work successfully in at least two distinct domains using a common target language which can be cross-compiled to C/Java.
+
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+
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+
+# A APPENDIX
+
+# A.1 OPERATORS OF THE LANGUAGE USED
+
+Tables A.1 and A.1 provide lists of all the operators used for the two corpora used in the language employed in the experiments. Language variants are improved in order to allow effective processing of the two corpora’s specific problem domains.
+
+# Operator
+
+Assign Variable To Array   
+Assign Variable From Array   
+Make Array   
+Variable To Literal   
+Add   
+Subtract   
+Multiply   
+Divide   
+Modulo   
+Assign Var from Var   
+Loop   
+Conditional (var $> 0$ )   
+Conditional (var1 $= =$ var2)   
+Conditional (var1 $>$ var2)
+
+Table 4: Operators available for GP, when using the 1st (array-to-array) corpus
+
+# Operator
+
+Assign Variable To Array   
+Assign Variable From Array   
+Make Array   
+Variable To Literal   
+Add   
+Subtract   
+Multiply   
+Divide   
+Modulo   
+Assign Var from Var   
+Loop   
+Conditional (var $> 0$ )   
+Conditional $\operatorname { v a r } 1 = = \operatorname { v a r } 2$ )   
+Create 2D Array   
+Get 2D Array Size   
+Var to XY Point from 2D Array   
+Set 2D Array to 0 at XY Point   
+Set 2D Array to 1 at XY Point
+
+Table 5: Operators available for GP, when using the 2nd (2D pattern) corpus
+
+A.2 PROBLEMS IN CORPORA
+
+Tables 6 and 7 provide a list of all human-provided problems used to test the system across the experiments. These are defined by a source-code implementation in both the custom language used in this paper, as well as in Java. An example of the behaviour of the array-to-array problems (First corpus), has been given, using a fixed sample input, to illustrate the behaviour.
+
+<table><tr><td rowspan=1 colspan=6>Problem                               Example</td></tr><tr><td rowspan=1 colspan=5>CumulativeAbsoluteSum  [4, -2,1,0,3,-5] - &gt; [4,6,7,7,10,15]</td><td></td></tr><tr><td rowspan=1 colspan=5>CumulativeSum            [4,-2,1,0,3,-5] - &gt; [4,2,3,3,6,1]</td><td></td></tr><tr><td rowspan=1 colspan=5>DivergentSequence        [4,-2,1,0,3,-5] − &gt;[0,0,1,-1,2,-2]</td><td></td></tr><tr><td rowspan=1 colspan=5>FirstElementOnly          [4,-2,1,0,3,-5] - &gt;[4]</td><td></td></tr><tr><td rowspan=1 colspan=5>Identity                     [4,-2,1,0,3,-5] − &gt; [4,-2,1,0,3,-5]</td><td></td></tr><tr><td rowspan=1 colspan=1>IndexParity</td><td rowspan=1 colspan=4>[4,-2,1,0,3,-5]- ></td><td></td></tr><tr><td rowspan=1 colspan=1>IterativeDifference</td><td rowspan=1 colspan=4>[4,-2,1,0,3,-5] - &gt;[4,-6,3,-1,3,-8]</td><td></td></tr><tr><td rowspan=1 colspan=1>KeepEvens</td><td rowspan=1 colspan=4>[4,-2,1,0,3,-5]- ></td><td></td></tr><tr><td rowspan=1 colspan=1>KeepNegatives</td><td rowspan=1 colspan=4>[4,-2,1,0,3,-5] - &gt; [0, -2,0,0, 0, -5]</td><td></td></tr><tr><td rowspan=1 colspan=1>KeepOdds</td><td rowspan=1 colspan=4>[4,-2,1,0,3,-5]- ></td><td></td></tr><tr><td rowspan=1 colspan=1>KeepPositives</td><td rowspan=1 colspan=4>[4,-2,1,0,3,-5] - &gt;[4,0,1,0,3,0]]</td><td></td></tr><tr><td rowspan=1 colspan=1>Negative</td><td rowspan=1 colspan=4>[4,-2,1,0,3,-5]-></td><td></td></tr><tr><td rowspan=1 colspan=1>Pop</td><td rowspan=1 colspan=1>[4, -2,1,0</td><td rowspan=1 colspan=3>3,-5]-></td><td></td></tr><tr><td rowspan=1 colspan=1>RemoveFirstElement</td><td rowspan=1 colspan=1>[4,-2,1,0</td><td rowspan=1 colspan=1>3,</td><td rowspan=1 colspan=2>-5]-></td><td rowspan=1 colspan=1>[-2,1,0,3,-5]</td></tr><tr><td rowspan=1 colspan=1>RetainFirstHalf</td><td rowspan=1 colspan=1>[4,-2,1,0</td><td rowspan=1 colspan=3>3,-5]-></td><td rowspan=1 colspan=1>[4，-2,1]</td></tr><tr><td rowspan=1 colspan=1>Reverse</td><td rowspan=1 colspan=1>[4,-2.1,0</td><td rowspan=1 colspan=3>3,-5]-></td><td></td></tr><tr><td rowspan=1 colspan=1>ShiftLeft</td><td rowspan=1 colspan=1>[4,-2,1,0</td><td rowspan=1 colspan=1>3,</td><td rowspan=1 colspan=2>-5-></td><td rowspan=1 colspan=1>[-2,1,0,3,-5]</td></tr><tr><td rowspan=1 colspan=1>ShiftLeftZeroPadded</td><td rowspan=1 colspan=1>[4,-2,1,0</td><td rowspan=1 colspan=1>3,</td><td rowspan=1 colspan=2>-5]-></td><td></td></tr><tr><td rowspan=1 colspan=1>ShiftRight</td><td rowspan=1 colspan=1>[4,-2,1,0</td><td rowspan=1 colspan=1>3,</td><td rowspan=1 colspan=2>-5]-></td><td></td></tr><tr><td rowspan=1 colspan=1>ShiftRightLossy</td><td rowspan=1 colspan=1>[4,-2,1,0</td><td rowspan=1 colspan=1>3,</td><td rowspan=1 colspan=2>-5]-></td><td></td></tr><tr><td rowspan=1 colspan=1>ShuffleZerosToBack</td><td rowspan=1 colspan=1>[4,-2,1,0</td><td rowspan=1 colspan=1>3,</td><td rowspan=1 colspan=2>-5]-></td><td></td></tr><tr><td rowspan=1 colspan=1>Signum</td><td rowspan=1 colspan=2>[4,-2,1,0,3,</td><td rowspan=1 colspan=2>-5]-></td><td></td></tr><tr><td rowspan=1 colspan=1>Sort</td><td rowspan=1 colspan=2>[4,-2,1,0,3,</td><td rowspan=1 colspan=2>-5]-></td><td></td></tr><tr><td rowspan=1 colspan=1>SquareValues</td><td rowspan=1 colspan=2>[4, -2,1,0, 3,</td><td rowspan=1 colspan=2>-5]-></td><td rowspan=2 colspan=1>[16,4,1,0,9,25]</td></tr><tr><td rowspan=1 colspan=1>ToIterator</td><td rowspan=1 colspan=1>[4,-2,1,0</td><td rowspan=1 colspan=1>3,</td><td rowspan=1 colspan=2>-5]-></td></tr><tr><td rowspan=1 colspan=1>Add</td><td rowspan=1 colspan=1>[4,-2,1,0</td><td rowspan=1 colspan=1>3,</td><td rowspan=1 colspan=2>-5]，4</td><td rowspan=2 colspan=1>[4,-2,1,0,3,-5],4- &gt;[4,-2,1,0,3,-5,4]</td></tr><tr><td rowspan=1 colspan=1>Append</td><td rowspan=1 colspan=1>[4,-2,1,0</td><td rowspan=1 colspan=1>3,</td><td rowspan=1 colspan=2>-5]，4-</td></tr><tr><td rowspan=1 colspan=1>ClipToMax</td><td rowspan=1 colspan=1>[4,-2,1,0</td><td rowspan=1 colspan=1>3,</td><td rowspan=1 colspan=2>-5]，4-</td><td rowspan=1 colspan=1>>[4,-2,1,0,3,-5]</td></tr><tr><td rowspan=1 colspan=1>ClipToMin</td><td rowspan=1 colspan=4>[4,-2,1,0,3,-5],4-</td><td rowspan=3 colspan=1>[4,-2,1,0,3,-5],4- &gt;[4,2,9,12,19,15]</td></tr><tr><td rowspan=1 colspan=1>ConstantAddition</td><td rowspan=1 colspan=4>[4,-2,1,0,3,-5]，4</td></tr><tr><td rowspan=1 colspan=1>FillArray</td><td rowspan=1 colspan=4>[4,-2,1,0,3,-5]，4-</td></tr><tr><td rowspan=1 colspan=1>GreaterThan</td><td rowspan=1 colspan=3>[4,-2,1,0,3,-5</td><td rowspan=1 colspan=1>，41</td><td rowspan=1 colspan=1>>[-1,-1,-1,-1,-1,-1]</td></tr><tr><td rowspan=1 colspan=1>IterateFromStart</td><td rowspan=1 colspan=3>[4,-2,1,0,3,-5</td><td rowspan=1 colspan=1>，41</td><td rowspan=1 colspan=1>V</td></tr><tr><td rowspan=1 colspan=1>LessThan</td><td rowspan=1 colspan=3>[4,-2,1,0,3,-5</td><td rowspan=1 colspan=1>，41</td><td rowspan=1 colspan=1>V</td></tr><tr><td rowspan=1 colspan=1>MultiplesOf</td><td rowspan=1 colspan=2>[4,-2,1,0,3,</td><td rowspan=1 colspan=1>-5]</td><td rowspan=1 colspan=1>,41V</td><td></td></tr><tr><td rowspan=1 colspan=1>Multiply</td><td rowspan=1 colspan=2>[4,-2,1,0,3,</td><td rowspan=1 colspan=1>-5]</td><td rowspan=1 colspan=1>,41V</td><td rowspan=1 colspan=1>[16,-8,4,0,12,-20]</td></tr><tr><td rowspan=1 colspan=1>Subtract</td><td rowspan=1 colspan=4>[4,-2,1,0,3,-5],4- &gt;[0,-6,-3,-4,-1,-9]</td><td rowspan=1 colspan=1>[0,-6,-3,-4, -1, -9]</td></tr></table>
+
+Table 6: The first corpus of problems, taking either a single array, or an array and an integer. Example provided of the behaviour of each problem, given a standard example input.
+
+# Problem
+
+Square   
+HollowSquare   
+Parallelogram   
+HollowParallelogram   
+MirroredParallelogram   
+MirroredHollowParallelogram   
+RightTriangle   
+HollowRightTriangle   
+MirroredRightTriangle   
+HollowMirroredRightTriangle   
+InvertedRightTriangle   
+HollowInvertedRightTriangle   
+InvertedMirroredRightTriangle   
+InvertedHollowMirroredRightTriangle   
+IsoceleseTriangle   
+HollowIsoceleseTriangle   
+InvertedIsoceleseTriangle   
+HollowInvertedIsoceleseTriangle   
+RectangleWithEmptyTrapezoid   
+InvertedRectangle   
+obtuseTriangle   
+hollowObtuseTriangle   
+mirroredObtuseTriangle   
+mirroredHollowObtuseTriangle   
+invertedObtuseTriangle   
+hollowInvertedObtuseTriangle   
+invertedMirroredObtuseTriangle   
+hollowMirroredInvertedObtuseTriangle   
+VShape   
+Trapezoid
+
+Table 7: The second corpus, a set of 2D image generation tasks, drawing simple geometric shapes.
+
+# A.3 NEURAL NETWORK ARCHITECTURES
+
+Feed Forward Neural Network The FFNN is a 5 layer structure with 128 nodes per layer, seLu activation. Each layer is connected to all layers below (dense block). Each layer other than the last has a dropout component, with a training dropout rate of $p K e e p = 0 . 7 5$ . The output is a single sigmoidal unit, loss function is mean squared error. Batch size of 32, maximum steps of 512, early stopping after 12 non-progress epochs on validation loss.
+
+Recurrent Neural Network The RNN encodes both the input and output arrays in two architecturally symmetric branches for every example (so 20 branches). Each branch begins by encoding the values in 2 dense layers with 11 nodes, reLu activation, with dropout with $p K e e p = 0 . 7 5$ after each. After this encoding layer, a layer of 8 LSTM nodes was connected. These LSTM layers represented the end of the two branches. All LSTM outputs are concatenated, along with the input parameters, into a single representation of the IO examples. These are then processed by a set of 2 dense layers with 64 nodes, reLu activation. The output is a single sigmoidal unit, loss function is mean squared error. Batch size of 32, maximum steps of 512, early stopping after 12 non-progress epochs on validation loss.
+
+Convolutional Neural Network Each of the 10 examples of problem output are split into their own branch. Each branch contains a 2D input with width/height equal to the maximum input size, 32. We feed this into a 2D convolutional layer, with stride of 2 and kernel size of 3, reLu activation. We then feed this into a max pooling layer of 2 by 2. We then feed through a second convolutional layer of identical configuration to the first, and a second max pooling layer, again identical. Each branch then terminates in a single dense layer, 64 nodes, reLu activation. All branches are then concatenated into a single dense layer, 64 nodes, reLu activation. The output is a single sigmoidal unit, loss function is mean squared error. Batch size of 32, maximum steps of 512, early stopping after 12 non-progress epochs on validation loss.
+
+# A.4 PROBLEM FIND RATES AND DESCRIPTIONS
+
+The tables89 are the find rates for the genetic programming algorithm, without any constraints. Note the high degree of variability between problems, with a number from both corpora achieving either a $100 \%$ success rate of a $0 \%$ .
+
+One remark is that the “Square” problem, which simply requires the entire canvas to be covered (therefore requiring two loops and a write), did not achieve $100 \%$ success. This is believed to be due to the genetic algorithm starting with a flawed partial solution, and being unable to move away from this detrimental start due to the nature of the fitness landscape (trapped in local maximum).
+
+# A.5 FULL BREAKDOWN OF FRAGMENTS EVALUATED IN EXPERIMENT 1
+
+Tables 10 to 23 describe each fragment evaluated by the exhaustive fragment testing process. Each fragment is at most two lines, and has no variables which depend on being set in lines outside the fragment (therefore the fragment stands alone in terms of functionality). The source code of the ground-truth implementation is given, firstly as simply the operator used on that line, and secondly in a C-like fashion (excluding braces). This C-like fashion is a programmatically generated translation of the source code of the custom language implementation, provided for ease of readability (due to the difficult-to-parse structure of the custom language). We then refer to the lines in this source code by line number. Fragments cannot contain end-of-block operators (used to indicate the end point of blocks started by the flow-control operators loop and conditional), nor can they contain the initial definition of the 2D canvas.
+
+# A.6 FRAGMENTS EVALUATED FOR NN RECOGNISABILITY IN EXPERIMENT 2
+
+The tables2425 describe the fragments (some of which contain requirements about variable dependencies) used in experiment 2.
+
+# A.7 FULL RESULTS OF NN-SELECTED GUIDANCE FOR GP FROM EXPERIMENT 3
+
+Tables 26,27,28 shows the success rate of the GP, if provided with hints by the neural network sets. Two sets of experiments are done on the array to array corpus, one on the canvas corpus. Most problems showed a success increase, including a number from both corpora which increased in success chance from $0 \%$ to a non-zero value. Two problems were made unfindable by the lesseffective uniform fragment selection process, Iterative Difference and Trapezoid. As the baseline find rate was low, this does not represent a major drop in success, and may potentially simply be due to insufficient samples to determine the true success probability. We do not however reject the possibility that our approach has a negative effect on find-rates for certain problems. It was seen in Experiment 1 that some fragments, known to be present in the ground-truth implementation, decreased find-rates. It is possible that the neural networks correctly identified fragment presence, but that these degraded the GP’s performance. It is, of course, also possible that the NN incorrectly estimated that a fragment was present when it was not, and that this erroneous hint harmed the GP.
+
+Table 8: Find rates for a Genetic Algorithm as implemented above on the problems of the first corpus, a set of functions which take an input array of integers only (functions above dividing line) or an input array of integers and an integer. Both forms return a single array. $\scriptstyle \mathrm { n = } 3 0$   
+
+<table><tr><td>Abs</td><td>8%</td></tr><tr><td>ArrayLength</td><td>100 %</td></tr><tr><td>ArrayToZero</td><td>100 %</td></tr><tr><td>CumulativeAbsoluteSum</td><td>0%</td></tr><tr><td>CumulativeSum</td><td>4%</td></tr><tr><td>DivergentSequence</td><td>58%</td></tr><tr><td>FirstElementOnly</td><td>28 %</td></tr><tr><td>Identity</td><td>100 %</td></tr><tr><td>IndexParity</td><td>100 %</td></tr><tr><td>IterativeDifference</td><td>4%</td></tr><tr><td>KeepEvens</td><td>0%</td></tr><tr><td>KeepNegatives</td><td>0%</td></tr><tr><td>KeepOdds</td><td>0%</td></tr><tr><td>KeepPositives</td><td>60 %</td></tr><tr><td>Negative</td><td>68 %</td></tr><tr><td>Pop</td><td>30%</td></tr><tr><td>RemoveFirstElement</td><td>10 %</td></tr><tr><td>RetainFirstHalf</td><td>0%</td></tr><tr><td>Reverse</td><td>58%</td></tr><tr><td>ShiftLeft</td><td>10 %</td></tr><tr><td>ShiftLeftZeroPadded</td><td>40%</td></tr><tr><td>ShiftRight</td><td>0%</td></tr><tr><td>ShiftRightLossy</td><td>84 %</td></tr><tr><td>ShuffleZerosToBack</td><td>100 %</td></tr><tr><td>Signum</td><td>0%</td></tr><tr><td>Sort</td><td>0%</td></tr><tr><td>SquareValues ToIterator</td><td>70 %</td></tr><tr><td></td><td>100 %</td></tr><tr><td>Add</td><td>24 %</td></tr><tr><td>Append</td><td>0%</td></tr><tr><td>ClipToMax</td><td>18 %</td></tr><tr><td>ClipToMin</td><td>4%</td></tr><tr><td>ConstantAddition</td><td>0%</td></tr><tr><td>FillArray</td><td>100 %</td></tr><tr><td>GreaterThan</td><td>10 %</td></tr><tr><td>IterateFromStart</td><td>98 %</td></tr><tr><td>LessThan</td><td>8%</td></tr><tr><td>MultiplesOf</td><td>90 %</td></tr><tr><td>Multiply</td><td>20 %</td></tr><tr><td>Subtract</td><td></td></tr><tr><td></td><td>18 %</td></tr></table>
+
+Square 97 % Hollow Square 100 % Parallelogram 0 % Hollow Parallelogram 0 % Mirrored Parallelogram 7 % Mirrored Hollow Parallelogram 13 % Right Triangle 97 % Hollow Right Triangle 87 % Mirrored Right Triangle 60 % Hollow Mirrored Right Triangle 63 % Inverted Right Triangle 60 % Hollow Inverted Right Triangle 83 % Inverted Mirrored Right Triangle 100 % Inverted Hollow Mirrored Right Triangle 100 % Isocelese Triangle 0 % Hollow Isocelese Triangle 13 % Inverted Isocelese Triangle 47 % Hollow Inverted Isocelese Triangle $50 \%$ Rectangle With Empty Trapezoid 3 % Inverted Rectangle With Empty Trapezoid 3 % Obtuse Triangle $3 \%$ Hollow Obtuse Triangle $27 \%$ Mirrored Obtuse Triangle $0 \%$ Mirrored Hollow Obtuse Triangle $0 \%$ Inverted Obtuse Triangle $0 \%$ Hollow Inverted Obtuse Triangle $10 \%$ Inverted Mirrored Obtuse Triangle $0 \%$ Hollow Mirrored Inverted Obtuse Triangle $3 \%$ V Shape $47 \%$ Trapezoid 7 %
+
+Table 10: Fragments assessed from program “Append”. Program’s code listed, in C-like format, with operators listed ahead of each line for each of readability. Fragments then described, in reference to lines used followed by success rate using fragment as GP guidance $\mathrm { ( n = } 3 0$ )   
+
+<table><tr><td>1 2 Add 3 4 5 Read 6 7 8</td><td>Literal Make Array Loop Write Endloop Write</td><td>variables[6] = 1; variables[7]= variables[O] + variables[6]; arrays[1]= new int[vars[7]] for(variables[2]=O;variables[2]&lt;variables[O];variables[2]++) variables[5]=arrays[O][variables[2]]; arrays[1][variables[2]]=variables[5];</td></tr><tr><td>3%</td><td colspan="2">Fragment Success Rate</td></tr><tr><td>1</td><td colspan="2"></td></tr><tr><td>1,2</td><td colspan="2">27%</td></tr><tr><td>1,4</td><td colspan="2">10%</td></tr><tr><td>4</td><td colspan="2">0%</td></tr><tr><td>4,5</td><td colspan="2">0%</td></tr><tr><td>4,6</td><td colspan="2">0%</td></tr><tr><td>4,8</td><td colspan="2">0%</td></tr></table>
+
+# Fragment Success Rate
+
+Table 11: Fragments assessed from program “Cumulative Absolute Sum”. Program’s code listed, in C-like format, with operators listed ahead of each line for each of readability. Fragments then described, in reference to lines used followed by success rate using fragment as GP guidance $\mathrm { ( n = } 3 0$ )   
+
+<table><tr><td>1</td><td>Make Array</td><td>arrays[1] =new int[vars[O]]</td></tr><tr><td>2</td><td>Loop</td><td>for(variables[2]=O;variables[2]&lt;variables[O];variables[2]++)</td></tr><tr><td>3</td><td>Literal</td><td>variables[5] = -1;</td></tr><tr><td>4</td><td>Read</td><td>variables[3]=arrays[O][variables[2]];</td></tr><tr><td>5</td><td>Condition</td><td>if (variables[3]&gt;0)</td></tr><tr><td>6</td><td>Else</td><td>else</td></tr><tr><td>7</td><td>Multiply</td><td>variables[3] = variables[3] * variables[5];</td></tr><tr><td>8</td><td>Endloop</td><td></td></tr><tr><td>9 10</td><td>Add</td><td>variables[4]= variables[4]+ variables[3];</td></tr><tr><td></td><td>Write</td><td>arrays[1][variables[2]]=variables[4];</td></tr><tr><td>11</td><td>Endloop</td><td></td></tr></table>
+
+Table 12: Fragments assessed from program “Keep Evens”. Program’s code listed, in C-like format, with operators listed ahead of each line for each of readability. Fragments then described, in reference to lines used followed by success rate using fragment as GP guidance $\mathrm { ( n = } 3 0$ )   
+
+<table><tr><td>Line</td><td>Operator</td><td>As Code</td></tr><tr><td>1 ２3</td><td>Literal Make Array Loop</td><td>variables[4] = 2; arrays[1] = new int[vars[O]] for (variables[2]=O;variables[2]&lt;variables[O];variables[2]++)</td></tr><tr><td>5 Modulo 6 Condition 7 Write 8 Endloop</td><td>Read Endloop</td><td>variables[3]=arrays[O][variables[2]]; variables[5]=variables[3] % variables[4]; if (variables[5]==variables[6]) arrays[1][variables[2]]= variables[3];</td></tr><tr><td>9</td><td colspan="2"></td></tr><tr><td>Fragment</td><td>Success Rate</td><td></td></tr><tr><td>1 1,2 1,3 1,5</td><td>0% 6% 3% 3% 3%</td><td></td></tr></table>
+
+Table 13: Fragments assessed from program “Retain First Half”. Program’s code listed, in C-like format, with operators listed ahead of each line for each of readability. Fragments then described, in reference to lines used followed by success rate using fragment as GP guidance $\mathrm { ( n = } 3 0$ )   
+
+<table><tr><td>Line</td><td>Operator</td><td>As Code</td></tr><tr><td>Literal</td><td rowspan="3">variables[6] = 2;</td><td></td></tr><tr><td>1</td><td>Divide variables[3]= variables[O]/variables[6];</td></tr><tr><td></td><td>arrays[1]=new int[vars[3]]</td></tr><tr><td>２３４</td><td>Make Array Loop</td><td rowspan="3">for (variables[2]=O;variables[2]&lt;variables[3];variables[2]++) variables[5]=arrays[O][variables[2]];</td></tr><tr><td>5</td><td>Read</td></tr><tr><td>6</td><td>Write</td></tr><tr><td>7</td><td>Endloop</td><td>arrays[1][variables[2]]=variables[5];</td></tr><tr><td colspan="3">Fragment Success Rate</td></tr><tr><td>1 0%</td><td colspan="3"></td></tr><tr><td>1,2</td><td colspan="3"></td></tr><tr><td></td><td colspan="3">13%</td></tr></table>
+
+Table 14: Fragments assessed from program “Reverse”. Program’s code listed, in C-like format, with operators listed ahead of each line for each of readability. Fragments then described, in reference to lines used followed by success rate using fragment as GP guidance $\mathrm { ( n = } 3 0$ )   
+
+<table><tr><td>Line</td><td colspan="2">Operator As Code</td></tr><tr><td>1</td><td>Literal</td><td>variables[7] = 2;</td></tr><tr><td>2</td><td>Make Array</td><td>arrays[1] = new int[vars[O]]</td></tr><tr><td>3</td><td>Loop</td><td>for (variables[2]=O;variables[2]ivariables[O];variables[2]++)</td></tr><tr><td>4</td><td>Subtract</td><td>variables[6] = variables[O] - variables[2];</td></tr><tr><td>5</td><td>Subtract</td><td>variables[6]= variables[6] - variables[7];</td></tr><tr><td>6</td><td>Read</td><td>variables[5]=arrays[O][variables[6]];</td></tr><tr><td>7</td><td>Write</td><td>arrays[1][variables[2]]=variables[5];</td></tr><tr><td>8</td><td>Endloop</td><td></td></tr><tr><td>Fragment Success Rate</td><td colspan="2">63%</td></tr><tr><td>1</td><td colspan="2"></td></tr><tr><td></td><td colspan="2">80%</td></tr><tr><td>1,2</td><td colspan="2">77%</td></tr><tr><td>1,3</td><td colspan="2">73%</td></tr><tr><td>2</td><td colspan="2"></td></tr><tr><td>2,3</td><td colspan="2">60%</td></tr><tr><td>3</td><td colspan="2">63%</td></tr><tr><td>3,4</td><td colspan="2">80%</td></tr><tr><td>3,7</td><td colspan="2">77%</td></tr></table>
+
+Table 15: Fragments assessed from program “Shift Right”. Program’s code listed, in C-like format, with operators listed ahead of each line for each of readability. Fragments then described, in reference to lines used followed by success rate using fragment as GP guidance $\mathrm { ( n = } 3 0$ )   
+
+<table><tr><td>1 ２3 4 5 6 7 8</td><td>Literal Add Make Array Loop Add Read Write Endloop</td><td>variables[6] = 1; variables[8]= variables[O] + variables[6]; arrays[1]= new int[vars[8]] for (variables[2]=O;variables[2]&lt;variables[O];variables[2]++) variables[7]= variables[2] + variables[6]; variables[5]= arrays[O][variables[2]]; arrays[1][variables[7]]= variables[5];</td></tr><tr><td colspan="2">1 3%</td><td></td></tr><tr><td>1,2 13%</td><td colspan="2"></td></tr><tr><td>1,4</td><td colspan="2">20%</td></tr><tr><td></td><td colspan="2">0%</td></tr><tr><td>4</td><td colspan="2">0%</td></tr><tr><td>4,6</td><td colspan="2"></td></tr></table>
+
+Table 16: Fragments assessed from program “Shift Right Lossy”. Program’s code listed, in C-like format, with operators listed ahead of each line for each of readability. Fragments then described, in reference to lines used followed by success rate using fragment as GP guidance $\mathrm { ( n = } 3 0$ )   
+
+<table><tr><td colspan="3">variables[6]= 2; 2 Add variables[8]= variables[O] + variables[6]; 3 Make Array arrays[1]= new int[vars[O]] 4 Subtract variables[9]= variables[O]-variables[6]; 5 Loop for (variables[2]=O;variables[2]ivariables[9];variables[2]++) 6 Add variables[7]= variables[2]+ variables[6]; 7 Read variables[5] =arrays[O][variables[2]]; 8 Write arrays[1][variables[7]]= variables[5]; 9 Endloop</td></tr><tr><td colspan="3">Fragment Success Rate</td></tr><tr><td>1 80%</td><td colspan="3"></td></tr><tr><td>1,2</td><td colspan="3">73%</td></tr><tr><td>1,3</td><td colspan="3">63%</td></tr><tr><td>1,4</td><td colspan="3">63%</td></tr><tr><td>3</td><td colspan="3">67%</td></tr><tr><td></td><td colspan="3"></td></tr></table>
+
+# Fragment Success Rate
+
+Table 17: Fragments assessed from program “Sort”. Program’s code listed, in C-like format, with operators listed ahead of each line for each of readability. Fragments then described, in reference to lines used followed by success rate using fragment as GP guidance $\mathrm { ( n = } 3 0$ )   
+
+<table><tr><td>1</td><td>Literal</td><td>variables[5] =1;</td></tr><tr><td>2</td><td>Subtract</td><td>variables[1]= variables[O]-variables[5];</td></tr><tr><td>3</td><td>Loop</td><td>for (variables[2]=O;variables[2]&lt;variables[O];variables[2]++)</td></tr><tr><td>4</td><td>Loop</td><td>for (variables[3]=O;variables[3]&lt;variables[1];variables[3]++)</td></tr><tr><td>5</td><td>Add</td><td>variables[6] = variables[3]+ variables[5];</td></tr><tr><td>6</td><td>Read</td><td>variables[4] =arrays[O][variables[3]];</td></tr><tr><td>7</td><td>Read</td><td>variables[7]=arrays[O][variables[6]];</td></tr><tr><td>8</td><td>Subtract</td><td>variables[8] = variables[4] - variables[7];</td></tr><tr><td>9</td><td>Condition</td><td>if (variables[8]&gt;0)</td></tr><tr><td>10</td><td>Write</td><td>arrays[O][variables[6]]= variables[4];</td></tr><tr><td>11</td><td>Write</td><td>arrays[O][variables[3]]= variables[7];</td></tr><tr><td>12</td><td>Endloop</td><td></td></tr><tr><td>13</td><td>Endloop</td><td></td></tr><tr><td>14</td><td>Endloop</td><td></td></tr><tr><td>15</td><td>Make Array</td><td>arrays[1] =new int[vars[O]]</td></tr><tr><td>16</td><td>Loop</td><td>for (variables[2]=O;variables[2]&lt;variables[O];variables[2]++)</td></tr><tr><td>17</td><td>Read Write</td><td>variables[5]=arrays[O][variables[2]];</td></tr><tr><td>18</td><td></td><td>arrays[1][variables[2]]=variables[5];</td></tr><tr><td>19</td><td>Endloop</td><td></td></tr></table>
+
+Table 18: Fragments assessed from program “Mirrored Parallelogram”. Program’s code listed, in C-like format, with operators listed ahead of each line for each of readability. Fragments then described, in reference to lines used followed by success rate using fragment as GP guidance $\mathrm { ( n = } 3 0$ )   
+
+<table><tr><td>Line</td><td>Operator</td><td>As Code</td></tr><tr><td>1</td><td>Make 2D Array</td><td>new 2DArray(size=variables[O]);</td></tr><tr><td>2</td><td>Literal</td><td>variables[6] = 2;</td></tr><tr><td>3</td><td>Divide</td><td>variables[4]= variables[O]/variables[6];</td></tr><tr><td>4</td><td>Loop</td><td>for (variables[2]=O;variables[2]&lt;variables[4];variables[2]++)</td></tr><tr><td>5</td><td>Loop</td><td>for (variables[3]=O;variables[3]&lt;variables[4];variables[3]++)</td></tr><tr><td>6</td><td>Add</td><td>variables[7]= variables[2] +variables[3];</td></tr><tr><td>7</td><td>Write to 2D</td><td>array[variables[7][variables[3]]=1;</td></tr><tr><td>8 9</td><td>Endloop</td><td></td></tr><tr><td></td><td>Endloop</td><td></td></tr><tr><td></td><td></td><td></td></tr><tr><td>Fragment</td><td>Success Rate</td><td></td></tr><tr><td>2</td><td>23%</td><td></td></tr><tr><td>2,3</td><td></td><td></td></tr><tr><td></td><td>30%</td><td></td></tr></table>
+
+Table 19: Fragments assessed from program “Mirrored Hollow Parallelogram”. Program’s code listed, in C-like format, with operators listed ahead of each line for each of readability. Fragments then described, in reference to lines used followed by success rate using fragment as GP guidance $\scriptstyle ( \mathrm { n } = 3 0 )$ )   
+
+<table><tr><td>1 Make 2D Array</td><td></td><td>new 2DArray(size=variables[O]);</td></tr><tr><td>2</td><td>Literal</td><td>variables[6] = 2;</td></tr><tr><td>3</td><td>Divide</td><td>variables[4]= variables[O] /variables[6];</td></tr><tr><td>4</td><td>Loop</td><td>for(variables[2]=O;variables[2]&lt;variables[4];variables[2]++)</td></tr><tr><td>5</td><td>Add</td><td>variables[5]= variables[2] + variables[4];</td></tr><tr><td>6</td><td>Write to 2D</td><td>array[variables[5][variables[10]]=1;</td></tr><tr><td>7</td><td>Write to 2D</td><td>array[variables[2][variables[4]]=1;</td></tr><tr><td>8</td><td>Subtract</td><td>variables[6]= variables[4] -variables[2];</td></tr><tr><td>9</td><td>Write to 2D</td><td>array[variables[2][variables[6]]=1;</td></tr><tr><td>10</td><td>Write to 2D</td><td>array[variables[5][variables[6]]=1;</td></tr><tr><td>11 12</td><td>Endloop Literal</td><td></td></tr><tr><td>13</td><td>Subtract</td><td>variables[8] = 1;</td></tr><tr><td></td><td>Write to 2D</td><td>variables[7]= variables[O] -variables[8];</td></tr><tr><td>14</td><td></td><td>array[variables[7][variables[10]]=1;</td></tr><tr><td>Fragment</td><td>Success Rate</td><td></td></tr><tr><td>2</td><td>13%</td><td></td></tr><tr><td>2,3</td><td>60%</td><td></td></tr><tr><td>2,12</td><td>10%</td><td></td></tr><tr><td>12</td><td>13%</td><td></td></tr><tr><td>12,13</td><td>40%</td><td></td></tr><tr><td></td><td></td><td></td></tr></table>
+
+Table 20: Fragments assessed from program “Hollow Right Triangle”. Program’s code listed, in C-like format, with operators listed ahead of each line for each of readability. Fragments then described, in reference to lines used followed by success rate using fragment as GP guidance $\mathrm { ( n = } 3 0$ )   
+
+<table><tr><td>Line</td><td>Operator</td><td>As Code</td></tr><tr><td>1</td><td>Make 2D Array</td><td>new 2DArray(size=variables[O]);</td></tr><tr><td>2</td><td>Literal</td><td>variables[1] = 1;</td></tr><tr><td>3</td><td>Subtract</td><td>variables[4] = variables[O] - variables[1];</td></tr><tr><td>4</td><td>Loop</td><td>for(variables[2]=O;variables[2]&lt;variables[O];variables[2]++)</td></tr><tr><td>5</td><td>Write to 2D</td><td>array[variables[2][variables[4]]=1;</td></tr><tr><td>6</td><td>Write to 2D</td><td>array[variables[5][variables[2]]=1;</td></tr><tr><td>7</td><td>Write to 2D</td><td>array[variables[2][variables[2]]=1;</td></tr><tr><td>8</td><td>Endloop</td><td></td></tr><tr><td></td><td></td><td></td></tr><tr><td>Fragment</td><td>Success Rate</td><td></td></tr><tr><td>2 2,3</td><td>80% 90%</td><td></td></tr><tr><td></td><td>80%</td><td></td></tr><tr><td>2,4</td><td></td><td></td></tr><tr><td>4</td><td>90%</td><td></td></tr><tr><td>4,6</td><td>63%</td><td></td></tr><tr><td>4,7</td><td>87%</td><td></td></tr><tr><td></td><td></td><td></td></tr></table>
+
+Table 21: Fragments assessed from program “Hollow Mirrored Right Triangle”. Program’s code listed, in C-like format, with operators listed ahead of each line for each of readability. Fragments then described, in reference to lines used followed by success rate using fragment as GP guidance $\scriptstyle ( \mathrm { n } = 3 0 )$ )   
+
+<table><tr><td>Line</td><td>Operator</td><td>AsCode</td></tr><tr><td colspan="3"></td></tr><tr><td>1</td><td>Make 2D Array</td><td>new 2DArray(size=variables[O]);</td></tr><tr><td>2</td><td>Literal</td><td>variables[3]=1;</td></tr><tr><td>3</td><td>Subtract</td><td>variables[4]= variables[O] -variables[3];</td></tr><tr><td>4</td><td>Loop</td><td>for (variables[2]=O;variables[2]&lt;variables[O];variables[2]++)</td></tr><tr><td>5</td><td>Write to 2D</td><td>array[variables[2][variables[4]]=1;</td></tr><tr><td>6</td><td>Write to 2D</td><td>array[variables[4][variables[2]]=1;</td></tr><tr><td>7</td><td>Subtract</td><td>variables[5]= variables[O]- variables[2];</td></tr><tr><td>8</td><td>Subtract</td><td>variables[5]= variables[5]- variables[3];</td></tr><tr><td>9</td><td>Write to 2D</td><td>array[variables[2][variables[5]]=1;</td></tr><tr><td>10</td><td>Endloop</td><td></td></tr><tr><td colspan="3"></td></tr><tr><td>Fragment</td><td colspan="2">Success Rate</td></tr><tr><td>2</td><td colspan="2">67%</td></tr><tr><td>2,3</td><td colspan="2">93%</td></tr><tr><td>2,4</td><td colspan="2">80%</td></tr><tr><td></td><td colspan="2"></td></tr><tr><td>4</td><td colspan="2">67%</td></tr><tr><td>4,7</td><td colspan="2">80%</td></tr><tr><td>Line</td><td>Operator</td><td>As Code</td></tr><tr><td colspan="3"></td></tr><tr><td>1</td><td>Make 2D Array</td><td>new 2DArray(size=variables[0]);</td></tr><tr><td>2</td><td>Literal</td><td>variables[4] = 2;</td></tr><tr><td>3</td><td>Loop</td><td>for(variables[2]=O;variables[2]&lt;variables[O];variables[2]++)</td></tr><tr><td>4</td><td>Multiply</td><td>variables[6]= variables[2] * variables[4];</td></tr><tr><td>5</td><td>Subtract</td><td>variables[5]= variables[O] - variables[6];</td></tr><tr><td>6</td><td>Loop</td><td>for(variables[3]=O;variables[3]&lt;variables[5];variables[3]++)</td></tr><tr><td>7 8</td><td>Add</td><td>variables[7]= variables[3] + variables[2];</td></tr><tr><td>9</td><td>Write to 2D</td><td>array[variables[7][variables[2]]=1;</td></tr><tr><td>10</td><td>Endloop</td><td></td></tr><tr><td></td><td>Endloop</td><td></td></tr><tr><td colspan="3"></td></tr><tr><td>Fragment</td><td colspan="2">Success Rate</td></tr><tr><td>2</td><td colspan="2">23%</td></tr><tr><td>2,3</td><td colspan="2">20%</td></tr><tr><td>3</td><td colspan="2">20%</td></tr><tr><td></td><td colspan="2"></td></tr></table>
+
+Table 22: Fragments assessed from program “Inverted Isoceles Triangle”. Program’s code listed, in C-like format, with operators listed ahead of each line for each of readability. Fragments then described, in reference to lines used followed by success rate using fragment as GP guidance $\mathrm { ( n = } 3 0$ )
+
+Table 23: Fragments assessed from program “Trapezoid”. Program’s code listed, in C-like format, with operators listed ahead of each line for each of readability. Fragments then described, in reference to lines used followed by success rate using fragment as GP guidance $\mathrm { ( n = } 3 0$ )   
+
+<table><tr><td>1</td><td>Make 2D Array</td><td>new 2DArray(size=variables[O]);</td></tr><tr><td>２34</td><td>Literal</td><td>variables[7]=1;</td></tr><tr><td></td><td>Literal</td><td>variables[4] = 2;</td></tr><tr><td></td><td>Divide</td><td>variables[5]= variables[O] /variables[4];</td></tr><tr><td>5</td><td>Loop</td><td>for (variables[2]=O;variables[2]&lt;variables[O];variables[2]++)</td></tr><tr><td>6</td><td>Loop</td><td>for (variables[3]=O;variables[3]&lt;variables[5];variables[3]++)</td></tr><tr><td>7</td><td>Subtract</td><td>variables[8]= variables[O] - variables[5];</td></tr><tr><td>8</td><td>Divide</td><td>variables[8] = variables[8]/variables[4];</td></tr><tr><td>9</td><td>Subtract</td><td>variables[8]= variables[8]- variables[3];</td></tr><tr><td>10</td><td>Add</td><td>variables[9]= variables[8] +variables[7];</td></tr><tr><td>11</td><td>Condition</td><td>if (variables[9]&gt;0)</td></tr><tr><td>12</td><td>Subtract</td><td>variables[9] = variables[2] - variables[8];</td></tr><tr><td>13</td><td>Condition</td><td>if (variables[9]&gt;0)</td></tr><tr><td>14</td><td>Subtract</td><td>variables[9]= variables[O] - variables[8];</td></tr><tr><td>15</td><td>Subtract</td><td>variables[9]= variables[9]- variables[2];</td></tr><tr><td>16</td><td>Condition</td><td>if (variables[9]&gt;0)</td></tr><tr><td>17</td><td>Write to 2D</td><td>array[variables[2][variables[3]]=1;</td></tr><tr><td>18</td><td>Endloop</td><td></td></tr><tr><td>19</td><td>Endloop</td><td></td></tr><tr><td>20</td><td>Endloop</td><td></td></tr><tr><td>21</td><td>Endloop</td><td></td></tr><tr><td>22</td><td>Endloop</td><td></td></tr><tr><td colspan="2">Fragment Success Rate</td><td></td></tr><tr><td>2</td><td colspan="2">10%</td></tr><tr><td>2,3</td><td colspan="2">10%</td></tr><tr><td></td><td colspan="2">10%</td></tr><tr><td>2,5</td><td colspan="2"></td></tr><tr><td>3</td><td colspan="2">3%</td></tr><tr><td>3,4</td><td colspan="2">0%</td></tr><tr><td></td><td colspan="2">10%</td></tr><tr><td>3,5 5</td><td colspan="2">3%</td></tr><tr><td></td><td colspan="3"></td></tr></table>
+
+Table 24: Fragments used in experiment 2, corpus 1, to determine whether the NN can recognise their presence in a program’s source code based on its behaviour. If multiple lines are required they are required to exist in order, but not necessarily consecutively. Variable numbering is not reflective of source-code implementation and for descriptive purposes only.   
+
+<table><tr><td>Add</td><td>Simple addition. Requires the program to at some point contain an addition operation</td></tr><tr><td>+1 Offset Loop</td><td>Three Line Fragment The first sets Var1 to1 The second is a loop operator The third requires an addition operator such that</td></tr><tr><td>Length -1 Loop</td><td>Var2=loop_iterator+Varl Three Line Fragment The first sets Var1 to 1 The second requires an addition operator such that Var2= input_array_size+Varl</td></tr><tr><td>Literal (2)</td><td>The third is a loop operator bounded to Var2 Requires a variable to be set to 2</td></tr><tr><td>Loop</td><td>Requires the program to contain a loop</td></tr><tr><td>Loop Conditional</td><td>Two line fragment. The first line requires a loop The second line requires a conditional (var &gt; O)</td></tr><tr><td>Loop Read</td><td>Two line fragment. The first line requires a loop</td></tr><tr><td>Nonstandard Array</td><td>The second line requires a read operation such that the indexread is the loop&#x27;s iterator One line fragment. Requires the output array to be created,</td></tr><tr><td></td><td>witha sizeVarl such that Varl is not the variable defaulting to input array size</td></tr><tr><td>Read</td><td></td></tr><tr><td>Add</td><td>Requires the program to read from the input array Simple subtraction. Requires the program to at some point contain</td></tr></table>
+
+Table 25: Fragments used in experiment 2, corpus 2, to determine whether the NN can recognise their presence in a program’s source code based on its behaviour. If multiple lines are required they are required to exist in order, but not necessarily consecutively. Variable numbering is not reflective of source-code implementation and for descriptive purposes only.   
+
+<table><tr><td>Add</td><td>Simple addition. Requires the program to at some point contain an addition operation</td></tr><tr><td>Conditional</td><td>Greater than O operator. The program must contain an operator which executes a non-empty code block if a variable is greater than zero</td></tr><tr><td>Half</td><td>Two line fragment. The first sets a variable Var1 to the literal 2, The second assigns a variable to desired_output_size/Var1</td></tr><tr><td>Half Loop</td><td>Three line fragment. The first sets a variable Var1 to the literal 2, The second assigns a variable Var2 to desired_output_size/Var1 The third defines a loop operator which runs from O to Var2</td></tr><tr><td>Half Loop Depends</td><td>Four line fragment. The first sets a variable Varl to the literal 2, The second assigns a variable Var2 to desired_output_size/Var1 The third defines a loop operator which runs from O to Var2 The fourth defines an operation setting a value on the 2D canvas, with the requirement that the X position of the point be logically</td></tr><tr><td>Loop Conditional</td><td>dependent on the loop iterator Two line fragment. The first requires a loop operator The second requires a conditional (var &gt; O) operator</td></tr><tr><td>Loop Draw</td><td>Two line fragment. The first requires a loop operator The second requires a 2D array write operation in which</td></tr><tr><td>Loop Loop</td><td>the X position drawn to depends logically on the loop&#x27;s iterator Two line fragment. The program must have two loops (not necessarily nested)</td></tr><tr><td>Loop Loop Subtract</td><td>Three line fragment. The program must have two loops (not necessarily nested)</td></tr><tr><td>Draw Draw</td><td>It must then have a subtract operator Two line fragment. The program must have two draw-to-2D-array operators</td></tr></table>
+
+Table 26: Success rates for problems of the 1st corpus, using the rarest-first guidance strategy, with the aggregate estimates from the feed-forward architecture network $\mathrm { ( n = } 3 0$ )   
+
+<table><tr><td>Problem</td><td>Success Rate</td><td>Baseline</td></tr><tr><td></td><td></td><td></td></tr><tr><td>Abs</td><td>73%</td><td>7%</td></tr><tr><td>ArrayLength</td><td>100%</td><td>100%</td></tr><tr><td>ArrayToZero</td><td>100%</td><td>100%</td></tr><tr><td>CumulativeAbsoluteSum</td><td>0%</td><td>0%</td></tr><tr><td>CumulativeSum</td><td>33%</td><td>3%</td></tr><tr><td>DivergentSequence</td><td>63%</td><td>57%</td></tr><tr><td>FirstElementOnly</td><td>100%</td><td>27%</td></tr><tr><td>Identity</td><td>100%</td><td>100%</td></tr><tr><td>IndexParity</td><td>100%</td><td>100%</td></tr><tr><td>IterativeDifference</td><td>3%</td><td>3%</td></tr><tr><td>KeepEvens</td><td>0%</td><td>0%</td></tr><tr><td>KeepNegatives</td><td>27%</td><td>0%</td></tr><tr><td>KeepOdds</td><td>10%</td><td>0%</td></tr><tr><td>KeepPositives</td><td>47%</td><td>60%</td></tr><tr><td>Negative</td><td>67%</td><td>67%</td></tr><tr><td>Pop</td><td>100%</td><td>30%</td></tr><tr><td>RemoveFirstElement</td><td>83%</td><td>10%</td></tr><tr><td>RetainFirstHalf</td><td>0%</td><td>0%</td></tr><tr><td>Reverse</td><td>77%</td><td>57%</td></tr><tr><td>ShiftLeft</td><td>80%</td><td>10%</td></tr><tr><td>ShiftLeftZeroPadded</td><td>83%</td><td>40%</td></tr><tr><td>ShiftRight</td><td>17%</td><td>0%</td></tr><tr><td>ShiftRightLossy</td><td>77%</td><td>83%</td></tr><tr><td>ShuffleZerosToBack</td><td>80%</td><td>100%</td></tr><tr><td>Signum</td><td>10%</td><td>0%</td></tr><tr><td>Sort</td><td>3%</td><td>0%</td></tr><tr><td>SquareValues</td><td>77%</td><td>70%</td></tr><tr><td>ToIterator</td><td>100%</td><td>100%</td></tr><tr><td>Add</td><td>37%</td><td>23%</td></tr><tr><td>Append</td><td>47%</td><td>0%</td></tr><tr><td>ClipToMax</td><td>43%</td><td>17%</td></tr><tr><td>ClipToMin</td><td>70%</td><td>3%</td></tr><tr><td>ConstantAddition</td><td>7%</td><td>0%</td></tr><tr><td>FillArray</td><td>100%</td><td>100%</td></tr><tr><td>GreaterThan</td><td>23%</td><td>10%</td></tr><tr><td>IterateFromStart</td><td>100%</td><td>97%</td></tr><tr><td>LessThan</td><td>17%</td><td>7%</td></tr><tr><td>MultiplesOf</td><td>87%</td><td>90%</td></tr><tr><td>Multiply</td><td>30%</td><td>20%</td></tr><tr><td>Subtract</td><td>43%</td><td>17%</td></tr></table>
+
+Table 27: Success rates for problems of the 1st corpus, using the uniform guidance strategy, with the aggregate estimates from the feed-forward architecture network $\mathrm { ( n = } 3 0$ )   
+
+<table><tr><td>Problem</td><td>Success Rate</td><td>Baseline</td></tr><tr><td>Abs</td><td>27%</td><td>7%</td></tr><tr><td>ArrayLength</td><td>100%</td><td>100%</td></tr><tr><td></td><td></td><td>100%</td></tr><tr><td>ArrayToZero</td><td>100%</td><td></td></tr><tr><td>CumulativeAbsoluteSum</td><td>0%</td><td>0%</td></tr><tr><td>CumulativeSum</td><td>20%</td><td>3%</td></tr><tr><td>DivergentSequence</td><td>83%</td><td>57%</td></tr><tr><td>FirstElementOnly Identity</td><td>83%</td><td>27%</td></tr><tr><td>IndexParity</td><td>100%</td><td>100%</td></tr><tr><td>IterativeDifference</td><td>100%</td><td>100%</td></tr><tr><td>KeepEvens</td><td>0%</td><td>3%</td></tr><tr><td>KeepNegatives</td><td>0%</td><td>0%</td></tr><tr><td>KeepOdds</td><td>53%</td><td>0%</td></tr><tr><td>KeepPositives</td><td>10%</td><td>0% 60%</td></tr><tr><td>Negative</td><td>40%</td><td>67%</td></tr><tr><td>Pop</td><td>63%</td><td>30%</td></tr><tr><td>RemoveFirstElement</td><td>73%</td><td>10%</td></tr><tr><td>RetainFirstHalf</td><td>57%</td><td>0%</td></tr><tr><td>Reverse</td><td>0%</td><td>57%</td></tr><tr><td>ShiftLeft</td><td>53%</td><td></td></tr><tr><td>ShiftLeftZeroPadded</td><td>43%</td><td>10%</td></tr><tr><td>ShiftRight</td><td>63%</td><td>40%</td></tr><tr><td>ShiftRightLossy</td><td>0%</td><td>0%</td></tr><tr><td>ShuffleZerosToBack</td><td>53%</td><td>83%</td></tr><tr><td>Signum</td><td>77%</td><td>100%</td></tr><tr><td>Sort</td><td>10%</td><td>0%</td></tr><tr><td></td><td>0%</td><td>0%</td></tr><tr><td>SquareValues</td><td>77%</td><td>70%</td></tr><tr><td>ToIterator Add</td><td>100%</td><td>100%</td></tr><tr><td></td><td>23%</td><td>23%</td></tr><tr><td>Append</td><td>7%</td><td>0%</td></tr><tr><td>ClipToMax ClipToMin</td><td>33%</td><td>17%</td></tr><tr><td></td><td>20%</td><td>3%</td></tr><tr><td>ConstantAddition</td><td>3%</td><td>0%</td></tr><tr><td>FillArray</td><td>100%</td><td>100%</td></tr><tr><td>GreaterThan</td><td>13%</td><td>10%</td></tr><tr><td>IterateFromStart</td><td>97%</td><td>97%</td></tr><tr><td>LessThan</td><td>30%</td><td>7%</td></tr><tr><td>MultiplesOf</td><td>93%</td><td>90%</td></tr><tr><td>Multiply</td><td>30%</td><td>20%</td></tr><tr><td>Subtract</td><td>17%</td><td>17%</td></tr></table>
+
+Table 28: Success rates for problems of the 2nd corpus, using the uniform guidance strategy, with the aggregate estimates from the convolutional architecture network $\mathrm { ( n = } 3 0$ )   
+
+<table><tr><td>Square</td><td>90%</td><td>97%</td></tr><tr><td>HollowSquare</td><td>100%</td><td>100% 0%</td></tr><tr><td>Parallelogram</td><td>23%</td><td>0%</td></tr><tr><td>HollowParallelogram</td><td>7%</td><td>7% 13%</td></tr><tr><td>MirroredParallelogram MirroredHollowParallelogram</td><td>43%</td><td></td></tr><tr><td>RightTriangle</td><td>17% 93%</td><td>97%</td></tr><tr><td>HollowRightTriangle</td><td>97%</td><td>87%</td></tr><tr><td>MirroredRightTriangle</td><td>50%</td><td>60%</td></tr><tr><td>HollowMirroredRightTriangle</td><td>60%</td><td>63%</td></tr><tr><td>InvertedRightTriangle</td><td>67%</td><td>60%</td></tr><tr><td>HollowInvertedRightTriangle</td><td>57%</td><td>83%</td></tr><tr><td>InvertedMirroredRightTriangle</td><td>97%</td><td>100%</td></tr><tr><td>InvertedHollowMirroredRightTriangle</td><td>100%</td><td>100%</td></tr><tr><td>IsoceleseTriangle</td><td>10%</td><td>0%</td></tr><tr><td>HollowIsoceleseTriangle</td><td>43%</td><td>13%</td></tr><tr><td>InvertedIsoceleseTriangle</td><td>37%</td><td>47%</td></tr><tr><td>HollowInvertedIsoceleseTriangle</td><td>33%</td><td>50%</td></tr><tr><td>RectangleWithEmptyTrapezoid</td><td>3%</td><td>3%</td></tr><tr><td>InvertedRectangle</td><td>10%</td><td>3%</td></tr><tr><td>obtuseTriangle</td><td>23%</td><td>3%</td></tr><tr><td>hollowObtuseTriangle</td><td>53%</td><td>27%</td></tr><tr><td>mirroredObtuseTriangle</td><td>0%</td><td>0%</td></tr><tr><td>mirroredHollowObtuseTriangle</td><td>0%</td><td>0%</td></tr><tr><td>invertedObtuseTriangle</td><td>0%</td><td>0%</td></tr><tr><td>hollowInvertedObtuseTriangle</td><td>17%</td><td>10%</td></tr><tr><td>invertedMirroredObtuseTriangle</td><td>7%</td><td>0%</td></tr><tr><td>hollowMirroredInvertedObtuseTriangle</td><td>7%</td><td>3%</td></tr><tr><td>VShape</td><td>40%</td><td>47%</td></tr><tr><td>Trapezoid</td><td>0%</td><td>7%</td></tr></table>

parse/train/a5KvtsZ14ev/a5KvtsZ14ev.md

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+# SLAPS: SELF-SUPERVISION IMPROVES STRUCTURE LEARNING FOR GRAPH NEURAL NETWORKS
+
+Anonymous authors Paper under double-blind review
+
+# ABSTRACT
+
+Graph neural networks (GNNs) work well when the graph structure is provided. However, this structure may not always be available in real-world applications. One solution to this problem is to infer the latent structure and then apply a GNN to the inferred graph. Unfortunately, the space of possible graph structures grows super-exponentially with the number of nodes and so the available node labels may be insufficient for learning both the structure and the GNN parameters. In this work, we propose the Simultaneous Learning of Adjacency and GNN Parameters with Self-supervision, or SLAPS, a method that provides more supervision for inferring a graph structure. This approach consists of training a denoising autoencoder GNN in parallel with the task-specific GNN. The autoencoder is trained to reconstruct the initial node features given noisy node features as well as a structure provided by a learnable graph generator. We explore the design space of SLAPS by comparing different graph generation and symmetrization approaches. A comprehensive experimental study demonstrates that SLAPS scales to large graphs with hundreds of thousands of nodes and outperforms several models that have been proposed to learn a task-specific graph structure on established benchmarks.
+
+# 1 INTRODUCTION
+
+Graph representation learning has grown rapidly and found applications in domains where data points define a graph (Chami et al., 2020; Kazemi et al., 2020). Graph neural networks (GNNs) (Scarselli et al., 2008) have been a key component to the success of the research in this area. Following the success of graph convolutional networks (GCNs) (Kipf & Welling, 2017) on semi-supervised node classification, several other GNN variants have been proposed for different prediction tasks on graphs (Hamilton et al., 2017; Velickovi ˇ c et al., 2018; Gilmer et al., 2017; Battaglia et al., 2018) and ´ the power of these models has been studied theoretically (Xu et al., 2019; Sato, 2020).
+
+GNNs take as input a set of node features and an adjacency matrix corresponding to the graph structure, and, for each node, output an embedding that captures not only the initial features of the node but also the features and embeddings of its neighbors. The performance of GNNs highly depends on the quality of the input graph structure and deteriorates substantially when the graph structure is noisy (see Zugner et al., 2018; Dai et al., 2018; Fox & Rajamanickam, 2019). The need ¨ for both node features and a clean graph structure impedes the applicability of GNNs to domains where one has access to a set of nodes and their features but not to their underlying graph structure, or only has access to a noisy structure. Examples of such domains include brain signal classification (Jang et al., 2019), computer-aided diagnosis (Cosmo et al., 2020), analysis of computer programs (Johnson et al., 2020), and particle reconstruction (Qasim et al., 2019).
+
+In this paper, we address this limitation by developing a model that learns both the GNN parameters as well as an adjacency matrix simultaneously. Since the number of possible graph structures grows super-exponentially with the number of nodes (Stanley, 1973) and obtaining node labels is typically costly, the number of available labels may not be enough for learning both the GNN parameters and an adjacency matrix–especially for semi-supervised node classification. Our main contribution is to supplement the classification task with a self-supervised task that helps learn a high-quality adjacency matrix. Our self-supervision approach masks some input features (or adds noise to them) and trains a separate GNN aiming at updating the adjacency matrix in such a way that it can recover the masked (or noisy) features. Introducing this self-supervision adds the inductive bias that a graph structure suitable for predicting the node features is also suitable for predicting the node labels.
+
+We experiment with several classification datasets. For datasets with a graph structure, we only feed the node features to our model. The model operates on the node features and an adjacency that is learned simultaneously from data. We compare our model with different classes of methods: some which do not use the graph structure for predicting labels, some which use a fixed $\mathbf { k }$ -Nearest Neighbors (kNN) graph built based on a chosen similarity metric, and some which initialize the graph with kNN but then revise it throughout the training. We show that our model consistently outperforms these methods. We also show that the self-supervised task is key to the high performance of our model. As an additional contribution, we provide an implementation for simultaneous structure and parameter learning that scales to graphs with hundreds of thousands of nodes.
+
+# 2 RELATED WORK
+
+Existing methods that relate to this work can be grouped into the following categories.
+
+Similarity Graph: One approach for inferring a graph structure is to select a similarity metric and set the edge weight between two nodes to be their similarity (Roweis & Saul, 2000; Tenenbaum et al., 2000). To obtain a sparse structure, one may create a kNN similarity graph, only connect pairs of nodes whose similarity surpasses some predefined threshold, or do sampling. As an example, Gidaris & Komodakis (2019) create a (fixed) kNN graph using the cosine similarity of the node features. Wang et al. (2019b) extend this idea by creating a fresh graph in each layer of the GNN based on the node embedding similarities in that layer as opposed to fixing a graph solely based on the initial features. Instead of choosing a single similarity metric, Halcrow et al. (2020) fuse several (potentially weak) measures of similarity. The quality of the predictions of these methods depends heavily on the choice of the similarity metric(s) and the value of $k$ for the kNN graph, or the threshold on similarity. Furthermore, designing an appropriate similarity metric may not be straightforward in some applications.
+
+Fully-connected Graph: Another approach is to assume a fully-connected graph and employ GNN variants such as graph attention networks (Velickovi ˇ c et al., 2018; Zhang et al., 2018) or the trans- ´ former (Vaswani et al., 2017) which infer the graph structure via an attention mechanism, or infer the graph structure using additional information. This approach has been used in computer vision (e.g., Suhail & Sigal, 2019), natural language processing (e.g., Zhu et al., 2019), and few-shot learning (e.g., Garcia & Bruna, 2017), where there are not many nodes. The complexity of this approach, however, grows rapidly making it applicable only to small-sized graphs with a few thousand nodes and not scalable to the datasets we use in our experiments.
+
+Learnable Graph: Instead of computing a similarity graph on the initial features, one may use a graph generator with learnable parameters. Li et al. (2018b) create a fully-connected graph based on a biliear similarity function with learnable parameters. A common approach is to learn to project the nodes to a latent space where node similarities correspond to edge weights. Wu et al. (2018) project the nodes to a latent space by learning weights for each of the input features. Cosmo et al. (2020) and Qasim et al. (2019) use a multi-layer perceptron for projection. Yu et al. (2020) use a GNN that projects the nodes into a latent space using the initial node features as well as an initial graph structure, aiming at providing a revised graph structure to the task-specific GNN. Franceschi et al. (2019) propose a model named LDS with a bi-level optimization setup for simultaneously learning the GNN parameters and a full adjacency matrix. Yang et al. (2019) update the input adjacency matrix based on the inductive bias that nodes belonging to the same class should be connected to each other and nodes belonging to different classes should be disconnected. Chen et al. (2020) propose an iterative approach that iterates over projecting the nodes to a latent space and constructing an adjacency matrix from the latent representations multiple times. In our experiments, we compare with several approaches from this category.
+
+Leveraging Domain Knowledge: In applications where specific domain knowledge is available, one may leverage this to guide the model toward learning specific structures. For example, Johnson et al. (2020) leverage abstract syntax trees and regular languages in learning graph structures of Python programs that aid reasoning for downstream tasks. Jin et al. (2020b) train GNNs that are robust to adversarial attack by learning a cleaned version of the input poisoned adjacency matrix using the domain knowledge that clean adjacency matrices are often sparse and low-rank and exhibit feature smoothness along connected nodes.
+
+Proposed Method: Our model falls within the learnable graph category in which we use graph generators with learnable parameters to infer the adjacency matrix. We supplement the training with a self-supervised objective to increase the amount of supervision in learning a graph structure. The self-supervised objective is generic and can be combined with many of the models described above. It is in the same vein as the auxiliary tasks in the context of multi-task learning used in computer vision, natural language processing, and reinforcement learning (see, e.g., Jaderberg et al., 2016; Liebel & Korner, 2018; Alonso & Plank, 2016). The self-supervised task is inspired by the ¨ successful training procedures of several recent language models such as BERT (Devlin et al., 2018) and RoBERTa (Liu et al., 2019b). Similar self-supervision techniques have also been employed for GNNs (Hu et al., 2020b;c) (Jin et al., 2020a; You et al., 2020; Zhu et al., 2020). While we employ similar self-supervision techniques, our work differs from this line of work as we use selfsupervision for learning a graph structure whereas the above methods use it to learn better (and, in some cases, transferable) GNN parameters. Specifically, we adopt the multi-task learning framework of You et al. (2020) with two differences: 1- we do not used shared parameters for the task-specific and self-supervised GNNs, and 2- instead of using a fixed adjacency matrix provided as input, we allow both GNNs to provide gradients for a generator that learns to generate a graph structure suitable for the downstream task.
+
+# 3 BACKGROUND AND NOTATION
+
+We use lowercase letters to denote scalars, bold lowercase letters to denote vectors and bold uppercase letters to denote matrices. $\pmb { I }$ represents an identity matrix. For a vector $\textbf {  { v } }$ , we represent its $i ^ { \mathrm { { t h } } }$ element as ${ \mathbf { } } v _ { i }$ . For a matrix $M$ , we represent the $i ^ { \mathrm { t h } }$ row as $M _ { i }$ and the element at the $i ^ { \mathrm { { \bar { t h } } } }$ row and $j ^ { \mathrm { t h } }$ column as $M _ { i j }$ . For an attributed graph, we use $n , m$ and $f$ to represent the number of nodes, edges, and features respectively, and denote the graph as ${ \mathcal { G } } = \{ \gamma , A , X \}$ where $\mathcal { V } = \{ v _ { 1 } , \ldots , v _ { n } \}$ is a set of nodes, $A \in \mathbb { R } ^ { n \times n }$ is a (sparse) adjacency matrix with $\boldsymbol { A } _ { i j }$ indicating the weight of the edge from $v _ { i }$ to $v _ { j }$ $A _ { i j } = 0$ implies no edge), and $\ b { X } \in \mathbb { R } ^ { n \times f }$ is a matrix whose rows correspond to node features or attributes. A degree matrix $_ D$ for a graph $\mathcal { G }$ is a diagonal matrix where $\begin{array} { r } { D _ { i i } = \sum _ { j } A _ { i j } } \end{array}$ .
+
+Graph convolutional networks (GCNs) are a powerful variant of GNNs. For a graph ${ \mathcal { G } } = \{ \gamma , A , X \}$ with a degree matrix $_ D$ , layer $l$ of the GCN architecture can be defined as $\pmb { H } ^ { ( l ) } = \sigma ( \hat { A } \pmb { H } ^ { ( l - 1 ) } \pmb { W } ^ { ( l ) } )$ where $\hat { A }$ represents a normalized adjacency matrix, $\pmb { H } ^ { ( l - 1 ) } \in \mathbb { R } ^ { n \times d _ { l - 1 } }$ represents the node representations in layer ${ l - I }$ with $H ^ { ( 0 ) } = X$ , $\dot { \mathbf W } ^ { ( l ) } \in \mathbb R ^ { d _ { l - 1 } \times d _ { l } }$ is a weight matrix, $\sigma$ is an activation function such as ReLU, and $\pmb { H } ^ { ( l ) } \in \mathbb { R } ^ { n \times d _ { l } }$ is the updated node embeddings. For undirected graphs where the adjacency is symmetric, $\hat { A } = D ^ { - \frac { 1 } { 2 } } ( A \overset { \cdot } { + } I ) D ^ { - \frac { 1 } { 2 } }$ corresponds to a row-and-column normalized adjacency with self-loops, and for directed graphs where the adjacency is not necessarily symmetric, $\pmb { \hat { A } } = \pmb { D } ^ { - 1 } ( \pmb { A } + \pmb { I } )$ corresponds to a row normalized adjacency matrix with self-loops.
+
+# 4 SLAPS: SIMULTANEOUS LEARNING OF ADJACENCY MATRIX AND GNN PARAMETERS WITH SELF-SUPERVISION
+
+We break our model into four components: 1) generator, 2) adjacency processor, 3) classifier, and 4) self-supervision. The generator takes the node features as input and generates a (perhaps sparse, non-normalized, and non-symmetric) matrix $\tilde { \pmb { A } } \in \mathbb { R } ^ { n \times n }$ . $\tilde { A }$ is then fed into the adjacency processor which outputs $\ b { A } \in \mathbb { R } ^ { n \times n }$ corresponding to a normalized, and in some cases symmetric version of $\tilde { A }$ . The classifier is a GNN that receives $\pmb { A }$ as well as the node features as input and classifies the nodes into a set of predefined classes. The self-supervision component is a GNN that receives noisy features and the generated adjacency as input and aims at denoising the features. Figure 1 illustrates the different components and in what follows, we describe each component in more detail.
+
+![](images/0827eb618483d5ff57f7e7964a33d063294b766380ce088904e40e94caffe3c0.jpg)  
+Figure 1: Overview of SLAPS. At the top, a generator receives the node features and produces a non-symmetric, non-normalized adjacency having (potentially) both positive and negative values (Section 4.1). The adjacency processor makes the values positive, symmetrizes and normalizes the adjacency (Section 4.2). The resulting adjacency and the node features go into $\mathsf { G N N } _ { \mathsf { C } }$ which predicts the node classes (Section 4.3). At the bottom, some noise is added to the node features. The resulting noisy features and the generated adjacency go into $\mathsf { G N N } _ { \mathsf { D A E } }$ which then denoises the features (Section 4.4).
+
+# 4.1 GENERATOR
+
+The generator is a function $\mathsf { G } : \mathbb { R } ^ { n \times f } \to \mathbb { R } ^ { n \times n }$ with parameters $\theta _ { \mathsf { G } }$ which takes the node features $\boldsymbol { X }$ as input and produces $\tilde { \pmb { A } } \in \mathbb { R } ^ { n \times n }$ as output. We consider the following two generators.
+
+Full Parameterization (FP): In this case, $\theta _ { \mathsf { G } }$ is a single matrix in $\mathbb { R } ^ { n \times n }$ and the generator function is defined as $\tilde { A } = \mathsf { G } _ { F P } ( X ; \theta _ { \mathsf { G } } ) = \theta _ { \mathsf { G } }$ . That is, the generator ignores the input node features and directly optimizes the adjacency matrix. The disadvantages of this generator include adding $n ^ { 2 }$ parameters to the model, which limits scalability and makes the model susceptible to overfitting, and not being applicable to inductive settings where during test time predictions are to be made for nodes unseen during training. This generator is similar to the one proposed by Franceschi et al. (2019) except that they treat each element of $\tilde { A }$ as the parameter of a Bernoulli distribution and sample graph structures from these Bernoulli distributions.
+
+MLP-kNN: In this case, $\theta _ { \mathsf { G } }$ corresponds to the weights of a multi-layer perceptron (MLP) and $\tilde { A } = \mathsf { G } _ { \mathsf { M L P } } ( X ; \theta _ { \mathsf { G } } ) = \mathsf { k N N } ( \mathsf { M L P } ( X ) )$ , where MLP : $\mathbb { R } ^ { n \times f } \to \mathbb { R } ^ { n \times f ^ { \prime } }$ is an MLP that produces a matrix with updated node representations $X ^ { \prime }$ $; \mathsf { k N N } : \mathbb { R } ^ { n \times f ^ { \prime } } \to \mathbb { R } ^ { n \times n }$ produces a sparse matrix. Let $M \in \mathbb { R } ^ { n \times n }$ with $M _ { i j } = 1$ if $v _ { j }$ is among the top $k$ similar nodes to $v _ { i }$ and 0 otherwise, and let $S \in \mathbb { R } ^ { n \times n }$ such that $S _ { i j } = \mathsf { S i m } ( X _ { i } ^ { \prime } , X _ { j } ^ { \prime } )$ for some differentiable similarity function $\mathsf { S i m }$ (we used cosine in our experiments). Then $\tilde { A } = \mathsf { k N N } ( X ^ { \prime } ) = M \odot S$ where $\odot$ represents the Hadamard (element-wise) product. Since $\boldsymbol { s }$ is computed based on $X ^ { \prime }$ , the gradients flow to the elements in $X ^ { \prime }$ (and consequently to the weights of the MLP) through $\boldsymbol { s }$ . In the backward phase of our model, we compute the gradients only with respect to those elements in $\boldsymbol { s }$ whose corresponding value in $_ M$ is 1 (i.e. those elements $S _ { i j }$ such that $M _ { i j } = 1$ ); the gradient with respect to the other elements is 0. With this formulation, in the forward phase of the network, one can first compute the matrix $^ { M }$ using an off-the-shelf k-nearest neighbors algorithm and then compute the similarities in $S$ only for pairs of nodes where $M _ { i j } = 1$ . Unlike FP, this generator can be used for the inductive setting.
+
+Smart initialization: In our experiments, we found the initialization of the generator parameters (i.e. $\theta _ { \mathsf { G } }$ ) to be important. Let $A ^ { k N N }$ represent an adjacency matrix created by applying a kNN function on the initial node features. One smart initialization for $\theta _ { \mathsf { G } }$ is to initialize them in a way that the generator generates $A ^ { k N N }$ before training starts (i.e. $\ddot { A } = A ^ { k N N }$ before training starts). Such an initialization can be trivially done for the FP generator by initializing $\theta _ { \mathsf { G } }$ to $A ^ { k N N }$ , but may not be straightforward for the MLP-kNN generator. To enable initializing the parameters of the MLPkNN generator in a way that it generates $A ^ { k N N }$ before training starts, we consider two variants of this generator. In one, hereafter referred to simply as MLP, we keep the input dimension the same throughout the layers. In the other, hereafter referred to as MLP-D, we consider MLPs with diagonal weight matrices (i.e., except the main diagonal, all other parameters in the weight matrices are zero). For both variants, we initialize the weight matrices in $\theta _ { \mathsf { G } }$ with the identity matrix to ensure that the output of the MLP is initially the same as its input and the kNN graph created on these outputs is equivalent to $A ^ { k N N }$ . MLP-D can be thought of as assigning different weights to different features and then computing node similarities. Note that, alternatively, one may use other MLP variants but pre-train the weights to output $A ^ { k N N }$ before the main training starts.
+
+# 4.2 ADJACENCY PROCESSOR
+
+The output $\tilde { A }$ of the generator may have both positive and negative values, may be non-symmetric and non-normalized. To ensure all values of the adjacency are positive and make the adjacency symmetric and normalized, we apply the following function to $\tilde { A }$ :
+
+$$
+\pmb { A } = \pmb { D } ^ { - \frac 1 2 } \bigg ( \frac { \mathsf { P } ( \tilde { \pmb { A } } ) + \mathsf { P } ( \tilde { \pmb { A } } ) ^ { T } } { 2 } \bigg ) \pmb { D } ^ { - \frac 1 2 }
+$$
+
+Here $\mathsf { P }$ is a function with a non-negative range. In our experiments, when using an MLP generator, we apply the ReLU function to the elements of $\tilde { A }$ . When using the fully-parameterized (FP) generator, applying ReLU results in a gradient flow problem as any edge whose corresponding value in $\tilde { A }$ becomes less than or equal to zero stops receiving gradient updates. For this reason, for FP we apply the ELU function to the elements of $\tilde { A }$ and then add a value of 1. The sub-expression $\frac { \mathsf { P } ( \tilde { A } ) + \mathsf { P } ( \bar { \tilde { A } } ) ^ { \tilde { T } } } { 2 }$ makes the resulting matrix $\mathsf { P } ( \tilde { A } )$ symmetric. To understand the reason for taking the mean of $\mathsf { P } ( \tilde { A } )$ and $\mathsf { P } ( \tilde { \mathbf { A } } ) ^ { T }$ , assume $\tilde { A }$ is generated by $G _ { \mathsf { M L P } }$ . If $v _ { j }$ is among the $k$ most similar nodes to $v _ { i }$ and vice versa, then the strength of the connection between $v _ { i }$ and $v _ { j }$ will remain the same. However, if, say, $v _ { j }$ is among the $k$ most similar nodes to $v _ { i }$ but $v _ { i }$ is not among the top $\mathbf { k }$ for $v _ { j }$ , then taking the average of the similarities reduces the strength of the connection between $v _ { i }$ and $v _ { j }$ . Finally, once we have a symmetric adjacency with non-negative values, we compute the degree matrix $_ { D }$ for $\frac { \mathsf { P } ( \tilde { A } ) + \mathsf { P } ( \tilde { A } ) ^ { T } } { 2 }$ and normalize $\frac { \mathsf { P } ( \tilde { A } ) + \mathsf { P } ( \tilde { A } ) ^ { T } } { 2 }$ by multiplying it left and right with $D ^ { - { \frac { 1 } { 2 } } }$ .
+
+# 4.3 CLASSIFIER
+
+The classifier is a function $\mathsf { G N N } _ { \mathsf { C } } : \mathbb { R } ^ { n \times f } \times \mathbb { R } ^ { n \times n } \to \mathbb { R } ^ { n \times | \mathcal { C } | }$ with parameters $\theta _ { \mathsf { G N N } _ { \mathsf { C } } }$ . It takes the node features $\boldsymbol { X }$ and the generated adjacency $\pmb { A }$ as input and provides for each node the logits for each class. $\mathcal { C }$ corresponds to the classes and $| { \mathcal { C } } |$ corresponds to the number of classes. We use a twolayer GCN for which $\theta _ { \mathsf { G N N } _ { \mathsf { C } } } = \{ W ^ { ( 1 ) } , W ^ { ( 2 ) } \}$ and define our classifier as $\mathsf { G N N } _ { \mathsf { C } } ( A , X ; \theta _ { \mathsf { G N N } _ { \mathsf { C } } } ) =$ $A { \mathsf { R e L U } } ( A X W ^ { ( 1 ) } ) W ^ { ( 2 ) }$ but other GNN variants can be used as well (recall that $\pmb { A }$ is normalized). The training loss $\mathcal { L } _ { C }$ for the classification task is computed by taking the softmax of the logits to produce a probability distribution for each node and then computing the cross-entropy loss.
+
+# 4.4 ADDING SELF-SUPERVISION
+
+As explained in Section 1, in many domains, the number of labeled nodes may be insufficient for learning both the structure and the GNN parameters from data. To increase the amount of supervision for learning the structure, we propose a self-supervised approach based on denoising autoencoders (Vincent et al., 2008). Let $\mathsf { \bar { G } N N } _ { \mathsf { D A E } } : \mathbb { R } ^ { n \times f } \times \mathbb { R } ^ { n \times n } \ \overset { \texttt { c . } } {  } \mathbb { R } ^ { n \times f }$ be a GNN that takes node features as well as a normalized adjacency produced by a generator as input and provides updated node features with the same dimension as input. We train $\mathsf { G N N } _ { \mathsf { D A E } }$ such that it receives a noisy version $\tilde { X }$ of the features $\boldsymbol { X }$ as input and produces the denoised features $\boldsymbol { X }$ as output. Let idx represent the indices corresponding to the elements of $\boldsymbol { X }$ to which we have added noise, and $X _ { i d x }$ represent the values at these indices. The aim of the training procedure is to minimize:
+
+$$
+\mathcal { L } _ { D A E } = \mathsf { L } ( X _ { i d x } , \mathsf { G N N } _ { \mathsf { D A E } } ( \tilde { X } , A ; \theta _ { \mathsf { G N N } _ { \mathsf { D A E } } } ) _ { i d x } )
+$$
+
+where $\pmb { A }$ is the generated adjacency matrix and $\mathsf { L }$ is a loss function. To add noise to the input features for datasets where features consist of binary vectors, in each iteration, $i d x$ consists of $r$ percent of the indices of $\boldsymbol { X }$ whose values are ones and $r \eta$ percent of the indices whose values are zeros, both selected uniformly at random in each epoch. Both $r$ and $\eta$ (corresponding to the negative ratio) are hyperparameters. In this case, we add noise by setting the ones in the selected mask to zeros and L is the binary cross-entropy loss. For datasets where the input features are continuous numbers, idx consists of $r$ percent of the indices of $\boldsymbol { X }$ selected uniformly at random in each epoch. We add noise by either replacing the values at idx with zeros or by adding independent Gaussian noises to each of the features. In this case, L is the mean-squared error loss.
+
+To understand the cruciality of the proposed self-supervision, let us consider a scenario for training the model in Figure 1 (or a model with a similar architecture) but without the self-supervised task. As training proceeds, assume that two unlabeled nodes $v _ { i }$ and $v _ { j }$ are not directly connected to any labeled nodes. Then, since a two-layer GCN makes predictions for the nodes based on their two-hop neighbors, the edge between $v _ { i }$ and $v _ { j }$ receives no supervision1. Figure 2 provides an example of such a scenario.
+
+As a quantitative example, in the original structures and train/validation/test splits of Cora and Citeseer, $8 0 . 4 \%$ and $8 9 . 9 \%$ of the nodes, and consequently $6 4 . 6 \%$ and $8 0 . 8 \%$ of pairs of nodes, are not connected to any labeled/train nodes. If no supervision is provided for some edges in the graph, after training the existence and weights of these edges may end up being set randomly (or the same as their initialization value), which may be problematic during testing. With the self-supervised task, however, although these edges may not receive supervision from the main task (i.e. from ${ \mathsf { G C N } } _ { \mathsf { C } } ,$ ), the supervision provided by the self-supervised task (i.e. from $G C N _ { \mathsf { D A E } } ^ { - }$ ) helps learn an appropriate weight for them.
+
+![](images/4923c24ed8b9e3535be51258c2a80a91cdbacee7a54088e137697d7c2e2de50d.jpg)  
+Figure 2: The dashed edge receives no supervision when training a two-layer GCN as it is not in the two-hop neighborhood of any labeled node.
+
+# 4.5 SLAPS
+
+Our final model, dubbed SLAPS, is trained to minimize $\mathcal { L } = \mathcal { L } _ { C } + \lambda \mathcal { L } _ { D A E }$ where $\mathcal { L } _ { C }$ is the classification loss, $\mathcal { L } _ { D A E }$ is the denoising autoencoder loss (see Equation 2), and $\lambda$ is a hyperparameter controlling the relative importance of the two losses.
+
+To verify the merit of the $\mathsf { G N N } _ { \mathsf { D A E } }$ for learning an adjacency matrix in isolation, we also consider a variant of SLAPS named $S L A P S _ { 2 s }$ that is trained in two stages. We first train the $\mathsf { G N N } _ { \mathsf { D A E } }$ model by minimizing the loss function described in Equation 2. Note that the loss function in Equation 2 depends on the parameters $\theta _ { \mathsf { G } }$ of the generator and the parameters $\theta _ { \mathsf { G N N } _ { \mathsf { D A E } } }$ of the denoising autoencoder. After every $t$ epochs of training, we fix the adjacency matrix, train a classifier with the fixed adjacency matrix, and measure classification accuracy on the validation set. We select the epoch that produces the adjacency providing the best validation accuracy for the classifier. Note that in $S L \bar { A } P S _ { 2 s }$ , the adjacency matrix is trained only based on $\mathsf { G N N } _ { \mathsf { D A E } }$ .
+
+# 5 EXPERIMENTS
+
+Baselines: We compare our proposal to several baselines with different properties. The first baseline is a multi-layer perceptron (MLP) which does not take the graph structure into account. We also compare against MLP-GAM\* (Stretcu et al., 2019) which learns a fully-connected graph structure and uses this structure to supplement the loss function of the MLP toward predicting similar labels for neighboring nodes. Similar to Franceschi et al. (2019), we also consider a baseline named kNNGCN where we create a kNN graph based on the node features and feed this graph to a GCN. The graph structure remains fixed in this approach. We also compare with baselines that learn the graph structure from data including LDS (Franceschi et al., 2019), GRCN (Yu et al., 2020), DGCNN (Wang et al., 2019b), and IDGL (Chen et al., 2020). We feed a kNN graph to the models requiring an initial graph structure.
+
+Datasets: We use three established benchmarks in the GNN literature namely Cora, Citeseer, and Pubmed (Sen et al., 2008) as well as a newly released dataset for node classification named ogbnarxiv (Hu et al., 2020a) that is orders of magnitude larger than the other three datasets and is more challenging due to the more realistic split of the data into train, validation, and test sets. For all the datasets, we only feed the node features to the models and not the graph structure. Following Franceschi et al. (2019), we also experiment with several classification (non-graph) datasets available in scikit-learn (Pedregosa et al., 2011) including Wine, Cancer, Digits, and 20News. Dataset statistics can be found in the Appendix. For Cora and Citeseer, the LDS model uses the train data for learning the parameters of their classification GCN, half of the validation for learning the parameters of the adjacency matrix (in their bi-level optimization setup, these are considered as hyperparameters), and the other half of the validation set for early stopping and tuning the other hyperparameters. Besides experimenting with the original setups of these two datasets, we also consider a setup that is closer (although not identical) to that of LDS: we use the train set and half of the validation set for training and the other half of validation for early stopping and hyperparameter tuning. We name the modified versions Cora390 and Citeseer370 respectively where the number proceeding the dataset name corresponds to the number of labels used for training. We also follow a similar procedure for the scikit-learn datasets.
+
+Table 1: Results of SLAPS and the baselines on established node classification benchmarks. $\dagger$ indicates results have been taken from Franceschi et al. (2019). $^ \ddag$ indicates results have been taken from Stretcu et al. (2019). Bold and underlined values indicate best and second-best mean performances respectively. OOM indicates out of memory.   
+
+<table><tr><td>Model</td><td>Generator</td><td>Cora</td><td>Citeseer</td><td>Cora390</td><td>Citeseer370</td><td>Pubmed</td><td>ogbn-arxiv</td></tr><tr><td>MLP</td><td rowspan="6"></td><td>56.1 ± 1.6†</td><td>56.7 ± 1.7†</td><td>65.8±0.4</td><td>67.1±0.5</td><td>71.4±0.0</td><td>54.7± 0.1</td></tr><tr><td>MLP-GAM*</td><td>70.7</td><td>70.3t</td><td></td><td></td><td>71.9‡</td><td>一</td></tr><tr><td>kNN-GCN</td><td>66.5 ± 0.4†</td><td>68.3 ± 1.3†</td><td>72.5 ±0.5</td><td>71.8 ±0.8</td><td>70.4± 0.4</td><td>49.1 ± 0.3</td></tr><tr><td>LDS</td><td></td><td></td><td>71.5 ± 0.8†</td><td>71.5 ± 1.1†</td><td>OOM</td><td>OOM</td></tr><tr><td>GRCN</td><td>67.4 ± 0.3</td><td>67.3 ± 0.8</td><td>71.3± 0.9</td><td>70.9 ±0.7</td><td>67.3± 0.3</td><td>OOM</td></tr><tr><td>DGCNN IDGL</td><td>56.5 ± 1.2</td><td>55.1 ± 1.4</td><td>67.3 ± 0.7</td><td>66.6±0.8</td><td>70.1 ± 1.3</td><td>OOM</td></tr><tr><td>SLAPS</td><td>FP</td><td>70.9 ± 0.6</td><td>68.2±0.6 70.7±0.4</td><td>73.4 ± 0.5</td><td>72.7± 0.4</td><td>72.3 ± 0.4</td><td>OOM</td></tr><tr><td>SLAPS</td><td>MLP</td><td>72.4± 0.4</td><td>70.5 ± 1.1</td><td>76.6± 0.4</td><td>73.1± 0.6</td><td>0OM</td><td>OOM</td></tr><tr><td></td><td></td><td>72.8± 0.8</td><td></td><td>75.3 ± 1.0</td><td>73.0 ± 0.9</td><td>74.4 ± 0.6</td><td>56.6 ±0.1</td></tr><tr><td>SLAPS</td><td>MLP-D</td><td>73.4 ± 0.3</td><td>72.6±0.6</td><td>75.1 ± 0.5</td><td>73.9 ±0.4</td><td>73.1 ± 0.7</td><td>52.9 ±0.1</td></tr></table>
+
+# 5.1 COMPARATIVE RESULTS
+
+The results of SLAPS and the baselines on the node classification benchmarks are reported in Table 1. Considering only the baselines first, we see that kNN-GCN significantly outperforms MLP on Cora and Citeseer but underperforms on Pubmed and ogbn-arxiv. This shows the importance of the similarity metric and the graph structure that is fed into GCN as a low-quality structure can harm model performance. LDS outperforms MLP but the fully parameterized adjacency matrix of LDS results in memory issues for Pubmed and ogbn-arxiv. As for GRCN, it was shown in the original paper that GRCN can revise a good initial adjacency matrix and provide a substantial boost in performance. However, as evidenced by the results, if the initial graph structure is somewhat poor, GRCN’s performance becomes on-par with kNN-GCN. IDGL is the best performing baseline but the iterative nature of it makes it slow to train and test.
+
+SLAPS consistently outperforms the baselines on all datasets, in some cases by large margins. Among the generators, the winner is dataset-dependent with MLP-D mostly outperforming MLP on datasets with many features and MLP outperforming on datasets with small numbers of features. Using the software that was publicly released by the authors, all baselines that learn a graph structure fail on ogbn-arxiv and our implementation is the first that generalizes to such large graphs.
+
+Table 2 reports the results for the scikit-learn datasets and compares with LDS and IDGL. On three out of four datasets, SLAPS outperforms the other two baselines. Among the datasets on which we can train SLAPS with the FP generator, 20news has the largest number of nodes. On this dataset, we observed that an FP generator suffers from overfitting and produces weaker results compared to other generators due to its large number of parameters.
+
+Table 2: Results on classification datasets. $\dagger$ indicates results have been taken from Franceschi et al. (2019). Bold and underlined values indicate best and second-best mean performances respectively.   
+
+<table><tr><td>Model</td><td>Generator</td><td>Wine</td><td>Cancer</td><td>Digits</td><td>20news</td></tr><tr><td>LDS IDGL</td><td></td><td>97.3 ± 0.4†</td><td>94.4± 1.9†</td><td>92.5± 0.7†</td><td>46.4±1.6†</td></tr><tr><td>SLAPS</td><td>FP</td><td>97.0 ± 0.7 96.6±0.4</td><td>94.2 ± 2.3 94.6± 0.3</td><td>92.5 ± 1.3</td><td>48.5 ± 0.6</td></tr><tr><td>SLAPS</td><td>MLP</td><td>96.3 ± 1.0</td><td>96.0 ± 0.8</td><td>94.4± 0.7 92.4± 0.6</td><td>44.4± 0.8 50.4± 0.7</td></tr><tr><td>SLAPS</td><td>MLP-D</td><td>96.5 ± 0.8</td><td></td><td></td><td></td></tr><tr><td></td><td></td><td></td><td>96.6 ± 0.2</td><td>93.2 ± 0.6</td><td>49.8 ± 0.9</td></tr></table>
+
+# 5.2 FURTHER ANALYSIS
+
+$\mathbf { S L A P S } _ { 2 s }$ : To provide more insight into the value provided by the self-supervision task on the learned adjacency, we conduct experiments with $\mathrm { S L A P S } _ { 2 s }$ . Recall from Section 4.5 that in $\mathrm { S L A P S } _ { 2 s }$ , the adjacency is learned only based on the self-supervision task and the node labels are only used for early stopping, hyperparameter tuning, and training $\mathsf { G C N } _ { \mathsf { C } }$ . Figure 3(a) shows the performance of SLAPS and $\mathrm { S L A P S } _ { 2 s }$ on Cora and compares them with kNN-GCN. Although $\mathrm { S L A P S } _ { 2 s }$ does not use the node labels in learning an adjacency matrix, it outperforms kNN-GCN $8 . 4 \%$ improvement when using an FP generator). With an FP generator, $\mathrm { S L A P S } _ { 2 s }$ even achieves competitive performance with SLAPS; this is mainly because FP does not leverage the supervision provided by $\mathsf { G C N } _ { \mathsf { C } }$ toward learning generalizable patterns that can be used for nodes other than those in the training set. These results corroborate the effectiveness of the self-supervision task for learning an adjacency matrix. Besides, the results show that learning the adjacency using both self-supervision and the task-specific node labels results in higher predictive accuracy.
+
+The value of λ: Figure 3(b) shows the performance of SLAPS2 on Cora and Citeseer with different values of $\lambda$ . When $\lambda = 0$ , corresponding to removing self-supervision, the model performance is somewhat poor. As soon as $\lambda$ becomes positive, both models see a large boost in performance showing that self-supervision is crucial to the high performance of SLAPS. Increasing $\lambda$ further provides larger boosts until it becomes so large that the self-supervision loss dominates the classification loss and the performance deteriorates. Note that with $\lambda = 0$ , SLAPS with the MLP generator becomes a variant of the model proposed by Cosmo et al. (2020), but with a different similarity function.
+
+Importance of $\boldsymbol { k }$ in kNN: Figure 3(c) shows the performance of SLAPS on Cora for three graph generators as a function of $k$ in kNN. For all three cases, the value of $k$ plays a major role in model performance. The FP generator is the least sensitive because in FP, $k$ only affects the initialization of the adjacency matrix but then the model can change the number of neighbors of each node. For MLP and MLP-D, however, the number of neighbors of each node remains close to $k$ (but not necessarily equal as the adjacency processor can add or remove some edges) and the two generators become more sensitive to $k$ . For larger values of $k$ , the extra flexibility of the MLP generator enables removing some of the unwanted edges through the function P or reducing the weights of the unwanted edges resulting in MLP being less sensitive to large values of $k$ compared to MLP-D.
+
+Symmetrization: To symmetrize the adjacency, in Equation 1 we took the average of $\mathsf { P } ( \tilde { A } )$ and $\mathsf { P } ( \tilde { \mathbf { A } } ) ^ { T }$ . Here we also consider two other choices: 1) $\mathsf { m a x } ( \mathsf { P } ( \tilde { \boldsymbol { A } } ) , \mathsf { P } ( \tilde { \boldsymbol { A } } ) ^ { T } )$ , and 2) not symmetrizing the adjacency (i.e. using $\mathsf { P } ( \tilde { A } ) )$ . Figure 3(d) compares these three choices on Cora and Citeseer with an MLP generator (other generators produced similar results). On both datasets, symmetrizing the adjacency provides a performance boost. Compared to mean symmetrization, max symmetrization performs slightly worse. This may be because max symmetrization does not distinguish between the case where both $v _ { i }$ and $v _ { j }$ are among the $k$ most similar nodes of each other and the case where only one of them is among the $k$ most similar nodes of the other.
+
+Analysing the learned adjacency: Many graph-based semi-supervised classification models are based on the cluster assumption according to which nearby nodes are more likely to share the same label (Chapelle & Zien, 2005). To verify the quality of the adjacency matrix learned using SLAPS, for every pair of nodes in the test set, we compute the odds of the two nodes sharing the same label as a function of the normalized weight of the edge connecting them. Figure 3(e) represents the odds for different weight intervals. For both Cora and Citeseer, nodes connected with higher edge weights are more likely to share the same label compared to nodes with lower or zero edge weights. As a specific example, when $A _ { i j } ~ \geq ~ 0 . 1$ , $v _ { i }$ and $v _ { j }$ are almost 2.5 times more likely to share the same label on Cora and almost 2.0 times more likely on Citeseer. Note that SLAPS may connect nodes based on a different criterion than the one used in the original datasets and so the learned adjacencies do not necessarily resemble the original structures.
+
+![](images/8c71bf8b447f2bec3821f9a1380d5933093b4c3ff3510bd92988cee66b2f9096.jpg)  
+Figure 3: The performance of SLAPS (a) compared to $\mathrm { S L A P S } _ { 2 s }$ on Cora with different generators, (b) with MLP graph generator on Cora and Citeseer as a function of $\lambda$ , (c) with different graph generators on Cora as a function of $k$ in kNN, and (d) on Cora and Citeseer with different adjacency symmetrizations. (e) The odds of two nodes in the test set sharing the same label as a function of the edge weights learned by SLAPS.
+
+# 6 CONCLUSION
+
+In this paper, we proposed a model for learning the parameters of a graph neural network and the graph structure of the nodes simultaneously. We showed the effectiveness of our model using a comprehensive set of experiments and analyses. In the future, we would like to try more sophisticated graph generation models (e.g., GraphRNN (You et al., 2018) and GNF (Liu et al., 2019a)) and extend our approach to applications with a temporal aspect where node features are observed over time but their connections are not provided as input.
+
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+
+# A APPENDIX
+
+Implementation Detail: We implemented our model in PyTorch (Paszke et al., 2017), used deep graph library (DGL) (Wang et al., 2019a) for the sparse operations, and used Adam (Kingma & Ba, 2014) as the optimizer. We performed early stopping and hyperparameter tuning based on the accuracy on the validation set for all datasets except Wine and Cancer. For these two datasets, validation accuracy reached 100 percent with many hyperparameter settings, making it difficult to select the best set of hyperparameters so instead, we used the validation cross-entropy loss. We fixed the maximum number of epochs to 2000. We use two-layer GCNs for both $G N N _ { \mathrm { { C } } }$ and $\mathsf { G N N } _ { \mathsf { D A E } }$ as well as for baselines and two-layer MLPs throughout the paper (for experiments on ogbn-arxiv, although the original paper uses models with three layers and with batch normalization after each layer, to be consistent with our other experiments we used two layers and removed the normalization). We used two learning rates, one for $\mathsf { G C N } _ { \mathsf { C } }$ and one for the other parameters of the models. We tuned the two learning rates from the set $\lbrace 0 . 0 1 , 0 . 0 0 1 \rbrace$ . We added dropout layers with dropout probabilities of 0.5 after the first layer of the GNNs. We also added dropout to the adjacency matrix and tuned the values from the set $\{ 0 . 2 5 , 0 . 5 \}$ . We set the hidden dimension of $\mathsf { G N N } _ { \mathsf { C } }$ to 32 for all datasets except for ogbn-arxiv for which we set it to 256. We used cosine similarity for building the kNN graphs and tuned the value of $k$ from the set $\{ 1 5 , 2 0 , 3 0 \}$ . We tuned $\lambda$ ( $\lambda$ controls the relative importance of the two losses) from the set $\{ 0 . 1 , 1 , 1 0 , 1 0 0 \}$ . The code of our experiments will be available upon acceptance of the paper.
+
+For GRCN (Yu et al., 2020), DGCNN (Wang et al., 2019b), and IDGL (Chen et al., 2020), we used the code released by the authors and tuned the hyperparameters as suggested in the original papers. The results of LDS (Franceschi et al., 2019) are directly taken from the original paper. All the results for our model and the baselines are averaged over 10 runs. We report the mean and standard deviation.
+
+Dataset statistics: The statistics of the datasets used in the experiments can be found in Table 3.
+
+Table 3: Dataset statistics.   
+
+<table><tr><td>Dataset</td><td>Nodes</td><td>Edges</td><td>Classes</td><td>Features</td><td>Label rate</td></tr><tr><td>Cora</td><td>2,708</td><td>5,429</td><td>7</td><td>1,433</td><td>0.052</td></tr><tr><td>Citeseer</td><td>3,327</td><td>4,732</td><td>6</td><td>3,703</td><td>0.036</td></tr><tr><td>Wine</td><td>178</td><td>0</td><td>3</td><td>13</td><td>0.112</td></tr><tr><td>Cancer</td><td>569</td><td>0</td><td>2</td><td>30</td><td>0.035</td></tr><tr><td>Digits</td><td>1,797</td><td>0</td><td>10</td><td>64</td><td>0.056</td></tr><tr><td>20news</td><td>9,607</td><td>0</td><td>10</td><td>236</td><td>0.021</td></tr><tr><td>Pubmed</td><td>19,717</td><td>44,338</td><td>3</td><td>500</td><td>0.003</td></tr><tr><td>ogbn-arxiv</td><td>169,343</td><td>1,166,243</td><td>40</td><td>128</td><td>0.537</td></tr></table>

parse/train/a5KvtsZ14ev/a5KvtsZ14ev_content_list.json

@@ -0,0 +1,1293 @@
+[
+    {
+        "type": "text",
+        "text": "SLAPS: SELF-SUPERVISION IMPROVES STRUCTURE LEARNING FOR GRAPH NEURAL NETWORKS ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Anonymous authors Paper under double-blind review ",
+        "bbox": [
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "ABSTRACT ",
+        "text_level": 1,
+        "bbox": [
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+            234,
+            544,
+            251
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Graph neural networks (GNNs) work well when the graph structure is provided. However, this structure may not always be available in real-world applications. One solution to this problem is to infer the latent structure and then apply a GNN to the inferred graph. Unfortunately, the space of possible graph structures grows super-exponentially with the number of nodes and so the available node labels may be insufficient for learning both the structure and the GNN parameters. In this work, we propose the Simultaneous Learning of Adjacency and GNN Parameters with Self-supervision, or SLAPS, a method that provides more supervision for inferring a graph structure. This approach consists of training a denoising autoencoder GNN in parallel with the task-specific GNN. The autoencoder is trained to reconstruct the initial node features given noisy node features as well as a structure provided by a learnable graph generator. We explore the design space of SLAPS by comparing different graph generation and symmetrization approaches. A comprehensive experimental study demonstrates that SLAPS scales to large graphs with hundreds of thousands of nodes and outperforms several models that have been proposed to learn a task-specific graph structure on established benchmarks. ",
+        "bbox": [
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "1 INTRODUCTION ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            527,
+            336,
+            542
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Graph representation learning has grown rapidly and found applications in domains where data points define a graph (Chami et al., 2020; Kazemi et al., 2020). Graph neural networks (GNNs) (Scarselli et al., 2008) have been a key component to the success of the research in this area. Following the success of graph convolutional networks (GCNs) (Kipf & Welling, 2017) on semi-supervised node classification, several other GNN variants have been proposed for different prediction tasks on graphs (Hamilton et al., 2017; Velickovi ˇ c et al., 2018; Gilmer et al., 2017; Battaglia et al., 2018) and ´ the power of these models has been studied theoretically (Xu et al., 2019; Sato, 2020). ",
+        "bbox": [
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "GNNs take as input a set of node features and an adjacency matrix corresponding to the graph structure, and, for each node, output an embedding that captures not only the initial features of the node but also the features and embeddings of its neighbors. The performance of GNNs highly depends on the quality of the input graph structure and deteriorates substantially when the graph structure is noisy (see Zugner et al., 2018; Dai et al., 2018; Fox & Rajamanickam, 2019). The need ¨ for both node features and a clean graph structure impedes the applicability of GNNs to domains where one has access to a set of nodes and their features but not to their underlying graph structure, or only has access to a noisy structure. Examples of such domains include brain signal classification (Jang et al., 2019), computer-aided diagnosis (Cosmo et al., 2020), analysis of computer programs (Johnson et al., 2020), and particle reconstruction (Qasim et al., 2019). ",
+        "bbox": [
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "In this paper, we address this limitation by developing a model that learns both the GNN parameters as well as an adjacency matrix simultaneously. Since the number of possible graph structures grows super-exponentially with the number of nodes (Stanley, 1973) and obtaining node labels is typically costly, the number of available labels may not be enough for learning both the GNN parameters and an adjacency matrix–especially for semi-supervised node classification. Our main contribution is to supplement the classification task with a self-supervised task that helps learn a high-quality adjacency matrix. Our self-supervision approach masks some input features (or adds noise to them) and trains a separate GNN aiming at updating the adjacency matrix in such a way that it can recover the masked (or noisy) features. Introducing this self-supervision adds the inductive bias that a graph structure suitable for predicting the node features is also suitable for predicting the node labels. ",
+        "bbox": [
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+        "page_idx": 0
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+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "We experiment with several classification datasets. For datasets with a graph structure, we only feed the node features to our model. The model operates on the node features and an adjacency that is learned simultaneously from data. We compare our model with different classes of methods: some which do not use the graph structure for predicting labels, some which use a fixed $\\mathbf { k }$ -Nearest Neighbors (kNN) graph built based on a chosen similarity metric, and some which initialize the graph with kNN but then revise it throughout the training. We show that our model consistently outperforms these methods. We also show that the self-supervised task is key to the high performance of our model. As an additional contribution, we provide an implementation for simultaneous structure and parameter learning that scales to graphs with hundreds of thousands of nodes. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "2 RELATED WORK ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            290,
+            344,
+            306
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Existing methods that relate to this work can be grouped into the following categories. ",
+        "bbox": [
+            176,
+            325,
+            736,
+            340
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Similarity Graph: One approach for inferring a graph structure is to select a similarity metric and set the edge weight between two nodes to be their similarity (Roweis & Saul, 2000; Tenenbaum et al., 2000). To obtain a sparse structure, one may create a kNN similarity graph, only connect pairs of nodes whose similarity surpasses some predefined threshold, or do sampling. As an example, Gidaris & Komodakis (2019) create a (fixed) kNN graph using the cosine similarity of the node features. Wang et al. (2019b) extend this idea by creating a fresh graph in each layer of the GNN based on the node embedding similarities in that layer as opposed to fixing a graph solely based on the initial features. Instead of choosing a single similarity metric, Halcrow et al. (2020) fuse several (potentially weak) measures of similarity. The quality of the predictions of these methods depends heavily on the choice of the similarity metric(s) and the value of $k$ for the kNN graph, or the threshold on similarity. Furthermore, designing an appropriate similarity metric may not be straightforward in some applications. ",
+        "bbox": [
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Fully-connected Graph: Another approach is to assume a fully-connected graph and employ GNN variants such as graph attention networks (Velickovi ˇ c et al., 2018; Zhang et al., 2018) or the trans- ´ former (Vaswani et al., 2017) which infer the graph structure via an attention mechanism, or infer the graph structure using additional information. This approach has been used in computer vision (e.g., Suhail & Sigal, 2019), natural language processing (e.g., Zhu et al., 2019), and few-shot learning (e.g., Garcia & Bruna, 2017), where there are not many nodes. The complexity of this approach, however, grows rapidly making it applicable only to small-sized graphs with a few thousand nodes and not scalable to the datasets we use in our experiments. ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Learnable Graph: Instead of computing a similarity graph on the initial features, one may use a graph generator with learnable parameters. Li et al. (2018b) create a fully-connected graph based on a biliear similarity function with learnable parameters. A common approach is to learn to project the nodes to a latent space where node similarities correspond to edge weights. Wu et al. (2018) project the nodes to a latent space by learning weights for each of the input features. Cosmo et al. (2020) and Qasim et al. (2019) use a multi-layer perceptron for projection. Yu et al. (2020) use a GNN that projects the nodes into a latent space using the initial node features as well as an initial graph structure, aiming at providing a revised graph structure to the task-specific GNN. Franceschi et al. (2019) propose a model named LDS with a bi-level optimization setup for simultaneously learning the GNN parameters and a full adjacency matrix. Yang et al. (2019) update the input adjacency matrix based on the inductive bias that nodes belonging to the same class should be connected to each other and nodes belonging to different classes should be disconnected. Chen et al. (2020) propose an iterative approach that iterates over projecting the nodes to a latent space and constructing an adjacency matrix from the latent representations multiple times. In our experiments, we compare with several approaches from this category. ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Leveraging Domain Knowledge: In applications where specific domain knowledge is available, one may leverage this to guide the model toward learning specific structures. For example, Johnson et al. (2020) leverage abstract syntax trees and regular languages in learning graph structures of Python programs that aid reasoning for downstream tasks. Jin et al. (2020b) train GNNs that are robust to adversarial attack by learning a cleaned version of the input poisoned adjacency matrix using the domain knowledge that clean adjacency matrices are often sparse and low-rank and exhibit feature smoothness along connected nodes. ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+        "page_idx": 2
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+    {
+        "type": "text",
+        "text": "Proposed Method: Our model falls within the learnable graph category in which we use graph generators with learnable parameters to infer the adjacency matrix. We supplement the training with a self-supervised objective to increase the amount of supervision in learning a graph structure. The self-supervised objective is generic and can be combined with many of the models described above. It is in the same vein as the auxiliary tasks in the context of multi-task learning used in computer vision, natural language processing, and reinforcement learning (see, e.g., Jaderberg et al., 2016; Liebel & Korner, 2018; Alonso & Plank, 2016). The self-supervised task is inspired by the ¨ successful training procedures of several recent language models such as BERT (Devlin et al., 2018) and RoBERTa (Liu et al., 2019b). Similar self-supervision techniques have also been employed for GNNs (Hu et al., 2020b;c) (Jin et al., 2020a; You et al., 2020; Zhu et al., 2020). While we employ similar self-supervision techniques, our work differs from this line of work as we use selfsupervision for learning a graph structure whereas the above methods use it to learn better (and, in some cases, transferable) GNN parameters. Specifically, we adopt the multi-task learning framework of You et al. (2020) with two differences: 1- we do not used shared parameters for the task-specific and self-supervised GNNs, and 2- instead of using a fixed adjacency matrix provided as input, we allow both GNNs to provide gradients for a generator that learns to generate a graph structure suitable for the downstream task. ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "3 BACKGROUND AND NOTATION",
+        "text_level": 1,
+        "bbox": [
+            176,
+            416,
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "We use lowercase letters to denote scalars, bold lowercase letters to denote vectors and bold uppercase letters to denote matrices. $\\pmb { I }$ represents an identity matrix. For a vector $\\textbf {  { v } }$ , we represent its $i ^ { \\mathrm { { t h } } }$ element as ${ \\mathbf { } } v _ { i }$ . For a matrix $M$ , we represent the $i ^ { \\mathrm { t h } }$ row as $M _ { i }$ and the element at the $i ^ { \\mathrm { { \\bar { t h } } } }$ row and $j ^ { \\mathrm { t h } }$ column as $M _ { i j }$ . For an attributed graph, we use $n , m$ and $f$ to represent the number of nodes, edges, and features respectively, and denote the graph as ${ \\mathcal { G } } = \\{ \\gamma , A , X \\}$ where $\\mathcal { V } = \\{ v _ { 1 } , \\ldots , v _ { n } \\}$ is a set of nodes, $A \\in \\mathbb { R } ^ { n \\times n }$ is a (sparse) adjacency matrix with $\\boldsymbol { A } _ { i j }$ indicating the weight of the edge from $v _ { i }$ to $v _ { j }$ $A _ { i j } = 0$ implies no edge), and $\\ b { X } \\in \\mathbb { R } ^ { n \\times f }$ is a matrix whose rows correspond to node features or attributes. A degree matrix $_ D$ for a graph $\\mathcal { G }$ is a diagonal matrix where $\\begin{array} { r } { D _ { i i } = \\sum _ { j } A _ { i j } } \\end{array}$ . ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Graph convolutional networks (GCNs) are a powerful variant of GNNs. For a graph ${ \\mathcal { G } } = \\{ \\gamma , A , X \\}$ with a degree matrix $_ D$ , layer $l$ of the GCN architecture can be defined as $\\pmb { H } ^ { ( l ) } = \\sigma ( \\hat { A } \\pmb { H } ^ { ( l - 1 ) } \\pmb { W } ^ { ( l ) } )$ where $\\hat { A }$ represents a normalized adjacency matrix, $\\pmb { H } ^ { ( l - 1 ) } \\in \\mathbb { R } ^ { n \\times d _ { l - 1 } }$ represents the node representations in layer ${ l - I }$ with $H ^ { ( 0 ) } = X$ , $\\dot { \\mathbf W } ^ { ( l ) } \\in \\mathbb R ^ { d _ { l - 1 } \\times d _ { l } }$ is a weight matrix, $\\sigma$ is an activation function such as ReLU, and $\\pmb { H } ^ { ( l ) } \\in \\mathbb { R } ^ { n \\times d _ { l } }$ is the updated node embeddings. For undirected graphs where the adjacency is symmetric, $\\hat { A } = D ^ { - \\frac { 1 } { 2 } } ( A \\overset { \\cdot } { + } I ) D ^ { - \\frac { 1 } { 2 } }$ corresponds to a row-and-column normalized adjacency with self-loops, and for directed graphs where the adjacency is not necessarily symmetric, $\\pmb { \\hat { A } } = \\pmb { D } ^ { - 1 } ( \\pmb { A } + \\pmb { I } )$ corresponds to a row normalized adjacency matrix with self-loops. ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "4 SLAPS: SIMULTANEOUS LEARNING OF ADJACENCY MATRIX AND GNN PARAMETERS WITH SELF-SUPERVISION ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "We break our model into four components: 1) generator, 2) adjacency processor, 3) classifier, and 4) self-supervision. The generator takes the node features as input and generates a (perhaps sparse, non-normalized, and non-symmetric) matrix $\\tilde { \\pmb { A } } \\in \\mathbb { R } ^ { n \\times n }$ . $\\tilde { A }$ is then fed into the adjacency processor which outputs $\\ b { A } \\in \\mathbb { R } ^ { n \\times n }$ corresponding to a normalized, and in some cases symmetric version of $\\tilde { A }$ . The classifier is a GNN that receives $\\pmb { A }$ as well as the node features as input and classifies the nodes into a set of predefined classes. The self-supervision component is a GNN that receives noisy features and the generated adjacency as input and aims at denoising the features. Figure 1 illustrates the different components and in what follows, we describe each component in more detail. ",
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+        "type": "image",
+        "img_path": "images/0827eb618483d5ff57f7e7964a33d063294b766380ce088904e40e94caffe3c0.jpg",
+        "image_caption": [
+            "Figure 1: Overview of SLAPS. At the top, a generator receives the node features and produces a non-symmetric, non-normalized adjacency having (potentially) both positive and negative values (Section 4.1). The adjacency processor makes the values positive, symmetrizes and normalizes the adjacency (Section 4.2). The resulting adjacency and the node features go into $\\mathsf { G N N } _ { \\mathsf { C } }$ which predicts the node classes (Section 4.3). At the bottom, some noise is added to the node features. The resulting noisy features and the generated adjacency go into $\\mathsf { G N N } _ { \\mathsf { D A E } }$ which then denoises the features (Section 4.4). "
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+    {
+        "type": "text",
+        "text": "4.1 GENERATOR ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "The generator is a function $\\mathsf { G } : \\mathbb { R } ^ { n \\times f } \\to \\mathbb { R } ^ { n \\times n }$ with parameters $\\theta _ { \\mathsf { G } }$ which takes the node features $\\boldsymbol { X }$ as input and produces $\\tilde { \\pmb { A } } \\in \\mathbb { R } ^ { n \\times n }$ as output. We consider the following two generators. ",
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+    {
+        "type": "text",
+        "text": "Full Parameterization (FP): In this case, $\\theta _ { \\mathsf { G } }$ is a single matrix in $\\mathbb { R } ^ { n \\times n }$ and the generator function is defined as $\\tilde { A } = \\mathsf { G } _ { F P } ( X ; \\theta _ { \\mathsf { G } } ) = \\theta _ { \\mathsf { G } }$ . That is, the generator ignores the input node features and directly optimizes the adjacency matrix. The disadvantages of this generator include adding $n ^ { 2 }$ parameters to the model, which limits scalability and makes the model susceptible to overfitting, and not being applicable to inductive settings where during test time predictions are to be made for nodes unseen during training. This generator is similar to the one proposed by Franceschi et al. (2019) except that they treat each element of $\\tilde { A }$ as the parameter of a Bernoulli distribution and sample graph structures from these Bernoulli distributions. ",
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+    {
+        "type": "text",
+        "text": "MLP-kNN: In this case, $\\theta _ { \\mathsf { G } }$ corresponds to the weights of a multi-layer perceptron (MLP) and $\\tilde { A } = \\mathsf { G } _ { \\mathsf { M L P } } ( X ; \\theta _ { \\mathsf { G } } ) = \\mathsf { k N N } ( \\mathsf { M L P } ( X ) )$ , where MLP : $\\mathbb { R } ^ { n \\times f } \\to \\mathbb { R } ^ { n \\times f ^ { \\prime } }$ is an MLP that produces a matrix with updated node representations $X ^ { \\prime }$ $; \\mathsf { k N N } : \\mathbb { R } ^ { n \\times f ^ { \\prime } } \\to \\mathbb { R } ^ { n \\times n }$ produces a sparse matrix. Let $M \\in \\mathbb { R } ^ { n \\times n }$ with $M _ { i j } = 1$ if $v _ { j }$ is among the top $k$ similar nodes to $v _ { i }$ and 0 otherwise, and let $S \\in \\mathbb { R } ^ { n \\times n }$ such that $S _ { i j } = \\mathsf { S i m } ( X _ { i } ^ { \\prime } , X _ { j } ^ { \\prime } )$ for some differentiable similarity function $\\mathsf { S i m }$ (we used cosine in our experiments). Then $\\tilde { A } = \\mathsf { k N N } ( X ^ { \\prime } ) = M \\odot S$ where $\\odot$ represents the Hadamard (element-wise) product. Since $\\boldsymbol { s }$ is computed based on $X ^ { \\prime }$ , the gradients flow to the elements in $X ^ { \\prime }$ (and consequently to the weights of the MLP) through $\\boldsymbol { s }$ . In the backward phase of our model, we compute the gradients only with respect to those elements in $\\boldsymbol { s }$ whose corresponding value in $_ M$ is 1 (i.e. those elements $S _ { i j }$ such that $M _ { i j } = 1$ ); the gradient with respect to the other elements is 0. With this formulation, in the forward phase of the network, one can first compute the matrix $^ { M }$ using an off-the-shelf k-nearest neighbors algorithm and then compute the similarities in $S$ only for pairs of nodes where $M _ { i j } = 1$ . Unlike FP, this generator can be used for the inductive setting. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Smart initialization: In our experiments, we found the initialization of the generator parameters (i.e. $\\theta _ { \\mathsf { G } }$ ) to be important. Let $A ^ { k N N }$ represent an adjacency matrix created by applying a kNN function on the initial node features. One smart initialization for $\\theta _ { \\mathsf { G } }$ is to initialize them in a way that the generator generates $A ^ { k N N }$ before training starts (i.e. $\\ddot { A } = A ^ { k N N }$ before training starts). Such an initialization can be trivially done for the FP generator by initializing $\\theta _ { \\mathsf { G } }$ to $A ^ { k N N }$ , but may not be straightforward for the MLP-kNN generator. To enable initializing the parameters of the MLPkNN generator in a way that it generates $A ^ { k N N }$ before training starts, we consider two variants of this generator. In one, hereafter referred to simply as MLP, we keep the input dimension the same throughout the layers. In the other, hereafter referred to as MLP-D, we consider MLPs with diagonal weight matrices (i.e., except the main diagonal, all other parameters in the weight matrices are zero). For both variants, we initialize the weight matrices in $\\theta _ { \\mathsf { G } }$ with the identity matrix to ensure that the output of the MLP is initially the same as its input and the kNN graph created on these outputs is equivalent to $A ^ { k N N }$ . MLP-D can be thought of as assigning different weights to different features and then computing node similarities. Note that, alternatively, one may use other MLP variants but pre-train the weights to output $A ^ { k N N }$ before the main training starts. ",
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+        "text": "4.2 ADJACENCY PROCESSOR ",
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+        "text": "The output $\\tilde { A }$ of the generator may have both positive and negative values, may be non-symmetric and non-normalized. To ensure all values of the adjacency are positive and make the adjacency symmetric and normalized, we apply the following function to $\\tilde { A }$ : ",
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+        "img_path": "images/f72616c747fe27811f3ef5deb9324a4668fe8fd5ee5f75e3f18ae3bde52ae4e4.jpg",
+        "text": "$$\n\\pmb { A } = \\pmb { D } ^ { - \\frac 1 2 } \\bigg ( \\frac { \\mathsf { P } ( \\tilde { \\pmb { A } } ) + \\mathsf { P } ( \\tilde { \\pmb { A } } ) ^ { T } } { 2 } \\bigg ) \\pmb { D } ^ { - \\frac 1 2 }\n$$",
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+        "type": "text",
+        "text": "Here $\\mathsf { P }$ is a function with a non-negative range. In our experiments, when using an MLP generator, we apply the ReLU function to the elements of $\\tilde { A }$ . When using the fully-parameterized (FP) generator, applying ReLU results in a gradient flow problem as any edge whose corresponding value in $\\tilde { A }$ becomes less than or equal to zero stops receiving gradient updates. For this reason, for FP we apply the ELU function to the elements of $\\tilde { A }$ and then add a value of 1. The sub-expression $\\frac { \\mathsf { P } ( \\tilde { A } ) + \\mathsf { P } ( \\bar { \\tilde { A } } ) ^ { \\tilde { T } } } { 2 }$ makes the resulting matrix $\\mathsf { P } ( \\tilde { A } )$ symmetric. To understand the reason for taking the mean of $\\mathsf { P } ( \\tilde { A } )$ and $\\mathsf { P } ( \\tilde { \\mathbf { A } } ) ^ { T }$ , assume $\\tilde { A }$ is generated by $G _ { \\mathsf { M L P } }$ . If $v _ { j }$ is among the $k$ most similar nodes to $v _ { i }$ and vice versa, then the strength of the connection between $v _ { i }$ and $v _ { j }$ will remain the same. However, if, say, $v _ { j }$ is among the $k$ most similar nodes to $v _ { i }$ but $v _ { i }$ is not among the top $\\mathbf { k }$ for $v _ { j }$ , then taking the average of the similarities reduces the strength of the connection between $v _ { i }$ and $v _ { j }$ . Finally, once we have a symmetric adjacency with non-negative values, we compute the degree matrix $_ { D }$ for $\\frac { \\mathsf { P } ( \\tilde { A } ) + \\mathsf { P } ( \\tilde { A } ) ^ { T } } { 2 }$ and normalize $\\frac { \\mathsf { P } ( \\tilde { A } ) + \\mathsf { P } ( \\tilde { A } ) ^ { T } } { 2 }$ by multiplying it left and right with $D ^ { - { \\frac { 1 } { 2 } } }$ . ",
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+    {
+        "type": "text",
+        "text": "4.3 CLASSIFIER ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "The classifier is a function $\\mathsf { G N N } _ { \\mathsf { C } } : \\mathbb { R } ^ { n \\times f } \\times \\mathbb { R } ^ { n \\times n } \\to \\mathbb { R } ^ { n \\times | \\mathcal { C } | }$ with parameters $\\theta _ { \\mathsf { G N N } _ { \\mathsf { C } } }$ . It takes the node features $\\boldsymbol { X }$ and the generated adjacency $\\pmb { A }$ as input and provides for each node the logits for each class. $\\mathcal { C }$ corresponds to the classes and $| { \\mathcal { C } } |$ corresponds to the number of classes. We use a twolayer GCN for which $\\theta _ { \\mathsf { G N N } _ { \\mathsf { C } } } = \\{ W ^ { ( 1 ) } , W ^ { ( 2 ) } \\}$ and define our classifier as $\\mathsf { G N N } _ { \\mathsf { C } } ( A , X ; \\theta _ { \\mathsf { G N N } _ { \\mathsf { C } } } ) =$ $A { \\mathsf { R e L U } } ( A X W ^ { ( 1 ) } ) W ^ { ( 2 ) }$ but other GNN variants can be used as well (recall that $\\pmb { A }$ is normalized). The training loss $\\mathcal { L } _ { C }$ for the classification task is computed by taking the softmax of the logits to produce a probability distribution for each node and then computing the cross-entropy loss. ",
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+    {
+        "type": "text",
+        "text": "4.4 ADDING SELF-SUPERVISION ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "As explained in Section 1, in many domains, the number of labeled nodes may be insufficient for learning both the structure and the GNN parameters from data. To increase the amount of supervision for learning the structure, we propose a self-supervised approach based on denoising autoencoders (Vincent et al., 2008). Let $\\mathsf { \\bar { G } N N } _ { \\mathsf { D A E } } : \\mathbb { R } ^ { n \\times f } \\times \\mathbb { R } ^ { n \\times n } \\ \\overset { \\texttt { c . } } {  } \\mathbb { R } ^ { n \\times f }$ be a GNN that takes node features as well as a normalized adjacency produced by a generator as input and provides updated node features with the same dimension as input. We train $\\mathsf { G N N } _ { \\mathsf { D A E } }$ such that it receives a noisy version $\\tilde { X }$ of the features $\\boldsymbol { X }$ as input and produces the denoised features $\\boldsymbol { X }$ as output. Let idx represent the indices corresponding to the elements of $\\boldsymbol { X }$ to which we have added noise, and $X _ { i d x }$ represent the values at these indices. The aim of the training procedure is to minimize: ",
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+    {
+        "type": "equation",
+        "img_path": "images/4e2137549413ff01363d7e8a6b59f9fb07d6024ae0377d9de7f8b205355a4a32.jpg",
+        "text": "$$\n\\mathcal { L } _ { D A E } = \\mathsf { L } ( X _ { i d x } , \\mathsf { G N N } _ { \\mathsf { D A E } } ( \\tilde { X } , A ; \\theta _ { \\mathsf { G N N } _ { \\mathsf { D A E } } } ) _ { i d x } )\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "where $\\pmb { A }$ is the generated adjacency matrix and $\\mathsf { L }$ is a loss function. To add noise to the input features for datasets where features consist of binary vectors, in each iteration, $i d x$ consists of $r$ percent of the indices of $\\boldsymbol { X }$ whose values are ones and $r \\eta$ percent of the indices whose values are zeros, both selected uniformly at random in each epoch. Both $r$ and $\\eta$ (corresponding to the negative ratio) are hyperparameters. In this case, we add noise by setting the ones in the selected mask to zeros and L is the binary cross-entropy loss. For datasets where the input features are continuous numbers, idx consists of $r$ percent of the indices of $\\boldsymbol { X }$ selected uniformly at random in each epoch. We add noise by either replacing the values at idx with zeros or by adding independent Gaussian noises to each of the features. In this case, L is the mean-squared error loss. ",
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+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "To understand the cruciality of the proposed self-supervision, let us consider a scenario for training the model in Figure 1 (or a model with a similar architecture) but without the self-supervised task. As training proceeds, assume that two unlabeled nodes $v _ { i }$ and $v _ { j }$ are not directly connected to any labeled nodes. Then, since a two-layer GCN makes predictions for the nodes based on their two-hop neighbors, the edge between $v _ { i }$ and $v _ { j }$ receives no supervision1. Figure 2 provides an example of such a scenario. ",
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+    {
+        "type": "text",
+        "text": "As a quantitative example, in the original structures and train/validation/test splits of Cora and Citeseer, $8 0 . 4 \\%$ and $8 9 . 9 \\%$ of the nodes, and consequently $6 4 . 6 \\%$ and $8 0 . 8 \\%$ of pairs of nodes, are not connected to any labeled/train nodes. If no supervision is provided for some edges in the graph, after training the existence and weights of these edges may end up being set randomly (or the same as their initialization value), which may be problematic during testing. With the self-supervised task, however, although these edges may not receive supervision from the main task (i.e. from ${ \\mathsf { G C N } } _ { \\mathsf { C } } ,$ ), the supervision provided by the self-supervised task (i.e. from $G C N _ { \\mathsf { D A E } } ^ { - }$ ) helps learn an appropriate weight for them. ",
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+    },
+    {
+        "type": "image",
+        "img_path": "images/4923c24ed8b9e3535be51258c2a80a91cdbacee7a54088e137697d7c2e2de50d.jpg",
+        "image_caption": [
+            "Figure 2: The dashed edge receives no supervision when training a two-layer GCN as it is not in the two-hop neighborhood of any labeled node. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "4.5 SLAPS ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "Our final model, dubbed SLAPS, is trained to minimize $\\mathcal { L } = \\mathcal { L } _ { C } + \\lambda \\mathcal { L } _ { D A E }$ where $\\mathcal { L } _ { C }$ is the classification loss, $\\mathcal { L } _ { D A E }$ is the denoising autoencoder loss (see Equation 2), and $\\lambda$ is a hyperparameter controlling the relative importance of the two losses. ",
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+    {
+        "type": "text",
+        "text": "To verify the merit of the $\\mathsf { G N N } _ { \\mathsf { D A E } }$ for learning an adjacency matrix in isolation, we also consider a variant of SLAPS named $S L A P S _ { 2 s }$ that is trained in two stages. We first train the $\\mathsf { G N N } _ { \\mathsf { D A E } }$ model by minimizing the loss function described in Equation 2. Note that the loss function in Equation 2 depends on the parameters $\\theta _ { \\mathsf { G } }$ of the generator and the parameters $\\theta _ { \\mathsf { G N N } _ { \\mathsf { D A E } } }$ of the denoising autoencoder. After every $t$ epochs of training, we fix the adjacency matrix, train a classifier with the fixed adjacency matrix, and measure classification accuracy on the validation set. We select the epoch that produces the adjacency providing the best validation accuracy for the classifier. Note that in $S L \\bar { A } P S _ { 2 s }$ , the adjacency matrix is trained only based on $\\mathsf { G N N } _ { \\mathsf { D A E } }$ . ",
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+    {
+        "type": "text",
+        "text": "5 EXPERIMENTS ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "Baselines: We compare our proposal to several baselines with different properties. The first baseline is a multi-layer perceptron (MLP) which does not take the graph structure into account. We also compare against MLP-GAM\\* (Stretcu et al., 2019) which learns a fully-connected graph structure and uses this structure to supplement the loss function of the MLP toward predicting similar labels for neighboring nodes. Similar to Franceschi et al. (2019), we also consider a baseline named kNNGCN where we create a kNN graph based on the node features and feed this graph to a GCN. The graph structure remains fixed in this approach. We also compare with baselines that learn the graph structure from data including LDS (Franceschi et al., 2019), GRCN (Yu et al., 2020), DGCNN (Wang et al., 2019b), and IDGL (Chen et al., 2020). We feed a kNN graph to the models requiring an initial graph structure. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Datasets: We use three established benchmarks in the GNN literature namely Cora, Citeseer, and Pubmed (Sen et al., 2008) as well as a newly released dataset for node classification named ogbnarxiv (Hu et al., 2020a) that is orders of magnitude larger than the other three datasets and is more challenging due to the more realistic split of the data into train, validation, and test sets. For all the datasets, we only feed the node features to the models and not the graph structure. Following Franceschi et al. (2019), we also experiment with several classification (non-graph) datasets available in scikit-learn (Pedregosa et al., 2011) including Wine, Cancer, Digits, and 20News. Dataset statistics can be found in the Appendix. For Cora and Citeseer, the LDS model uses the train data for learning the parameters of their classification GCN, half of the validation for learning the parameters of the adjacency matrix (in their bi-level optimization setup, these are considered as hyperparameters), and the other half of the validation set for early stopping and tuning the other hyperparameters. Besides experimenting with the original setups of these two datasets, we also consider a setup that is closer (although not identical) to that of LDS: we use the train set and half of the validation set for training and the other half of validation for early stopping and hyperparameter tuning. We name the modified versions Cora390 and Citeseer370 respectively where the number proceeding the dataset name corresponds to the number of labels used for training. We also follow a similar procedure for the scikit-learn datasets. ",
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+    {
+        "type": "table",
+        "img_path": "images/ef5a8e05c2787a1c53fa11fe15d9749c873f32bf6d1b718019bd32ffb887bead.jpg",
+        "table_caption": [
+            "Table 1: Results of SLAPS and the baselines on established node classification benchmarks. $\\dagger$ indicates results have been taken from Franceschi et al. (2019). $^ \\ddag$ indicates results have been taken from Stretcu et al. (2019). Bold and underlined values indicate best and second-best mean performances respectively. OOM indicates out of memory. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Model</td><td>Generator</td><td>Cora</td><td>Citeseer</td><td>Cora390</td><td>Citeseer370</td><td>Pubmed</td><td>ogbn-arxiv</td></tr><tr><td>MLP</td><td rowspan=\"6\"></td><td>56.1 ± 1.6†</td><td>56.7 ± 1.7†</td><td>65.8±0.4</td><td>67.1±0.5</td><td>71.4±0.0</td><td>54.7± 0.1</td></tr><tr><td>MLP-GAM*</td><td>70.7</td><td>70.3t</td><td></td><td></td><td>71.9‡</td><td>一</td></tr><tr><td>kNN-GCN</td><td>66.5 ± 0.4†</td><td>68.3 ± 1.3†</td><td>72.5 ±0.5</td><td>71.8 ±0.8</td><td>70.4± 0.4</td><td>49.1 ± 0.3</td></tr><tr><td>LDS</td><td></td><td></td><td>71.5 ± 0.8†</td><td>71.5 ± 1.1†</td><td>OOM</td><td>OOM</td></tr><tr><td>GRCN</td><td>67.4 ± 0.3</td><td>67.3 ± 0.8</td><td>71.3± 0.9</td><td>70.9 ±0.7</td><td>67.3± 0.3</td><td>OOM</td></tr><tr><td>DGCNN IDGL</td><td>56.5 ± 1.2</td><td>55.1 ± 1.4</td><td>67.3 ± 0.7</td><td>66.6±0.8</td><td>70.1 ± 1.3</td><td>OOM</td></tr><tr><td>SLAPS</td><td>FP</td><td>70.9 ± 0.6</td><td>68.2±0.6 70.7±0.4</td><td>73.4 ± 0.5</td><td>72.7± 0.4</td><td>72.3 ± 0.4</td><td>OOM</td></tr><tr><td>SLAPS</td><td>MLP</td><td>72.4± 0.4</td><td>70.5 ± 1.1</td><td>76.6± 0.4</td><td>73.1± 0.6</td><td>0OM</td><td>OOM</td></tr><tr><td></td><td></td><td>72.8± 0.8</td><td></td><td>75.3 ± 1.0</td><td>73.0 ± 0.9</td><td>74.4 ± 0.6</td><td>56.6 ±0.1</td></tr><tr><td>SLAPS</td><td>MLP-D</td><td>73.4 ± 0.3</td><td>72.6±0.6</td><td>75.1 ± 0.5</td><td>73.9 ±0.4</td><td>73.1 ± 0.7</td><td>52.9 ±0.1</td></tr></table>",
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+    {
+        "type": "text",
+        "text": "5.1 COMPARATIVE RESULTS ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "The results of SLAPS and the baselines on the node classification benchmarks are reported in Table 1. Considering only the baselines first, we see that kNN-GCN significantly outperforms MLP on Cora and Citeseer but underperforms on Pubmed and ogbn-arxiv. This shows the importance of the similarity metric and the graph structure that is fed into GCN as a low-quality structure can harm model performance. LDS outperforms MLP but the fully parameterized adjacency matrix of LDS results in memory issues for Pubmed and ogbn-arxiv. As for GRCN, it was shown in the original paper that GRCN can revise a good initial adjacency matrix and provide a substantial boost in performance. However, as evidenced by the results, if the initial graph structure is somewhat poor, GRCN’s performance becomes on-par with kNN-GCN. IDGL is the best performing baseline but the iterative nature of it makes it slow to train and test. ",
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+    {
+        "type": "text",
+        "text": "SLAPS consistently outperforms the baselines on all datasets, in some cases by large margins. Among the generators, the winner is dataset-dependent with MLP-D mostly outperforming MLP on datasets with many features and MLP outperforming on datasets with small numbers of features. Using the software that was publicly released by the authors, all baselines that learn a graph structure fail on ogbn-arxiv and our implementation is the first that generalizes to such large graphs. ",
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+    {
+        "type": "text",
+        "text": "Table 2 reports the results for the scikit-learn datasets and compares with LDS and IDGL. On three out of four datasets, SLAPS outperforms the other two baselines. Among the datasets on which we can train SLAPS with the FP generator, 20news has the largest number of nodes. On this dataset, we observed that an FP generator suffers from overfitting and produces weaker results compared to other generators due to its large number of parameters. ",
+        "bbox": [
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+    {
+        "type": "table",
+        "img_path": "images/a8c11546e761f1b34f71868ab3f5e00bbada43fd37b09eaff91ff9bfd35b7fb3.jpg",
+        "table_caption": [
+            "Table 2: Results on classification datasets. $\\dagger$ indicates results have been taken from Franceschi et al. (2019). Bold and underlined values indicate best and second-best mean performances respectively. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Model</td><td>Generator</td><td>Wine</td><td>Cancer</td><td>Digits</td><td>20news</td></tr><tr><td>LDS IDGL</td><td></td><td>97.3 ± 0.4†</td><td>94.4± 1.9†</td><td>92.5± 0.7†</td><td>46.4±1.6†</td></tr><tr><td>SLAPS</td><td>FP</td><td>97.0 ± 0.7 96.6±0.4</td><td>94.2 ± 2.3 94.6± 0.3</td><td>92.5 ± 1.3</td><td>48.5 ± 0.6</td></tr><tr><td>SLAPS</td><td>MLP</td><td>96.3 ± 1.0</td><td>96.0 ± 0.8</td><td>94.4± 0.7 92.4± 0.6</td><td>44.4± 0.8 50.4± 0.7</td></tr><tr><td>SLAPS</td><td>MLP-D</td><td>96.5 ± 0.8</td><td></td><td></td><td></td></tr><tr><td></td><td></td><td></td><td>96.6 ± 0.2</td><td>93.2 ± 0.6</td><td>49.8 ± 0.9</td></tr></table>",
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+    {
+        "type": "text",
+        "text": "5.2 FURTHER ANALYSIS ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "$\\mathbf { S L A P S } _ { 2 s }$ : To provide more insight into the value provided by the self-supervision task on the learned adjacency, we conduct experiments with $\\mathrm { S L A P S } _ { 2 s }$ . Recall from Section 4.5 that in $\\mathrm { S L A P S } _ { 2 s }$ , the adjacency is learned only based on the self-supervision task and the node labels are only used for early stopping, hyperparameter tuning, and training $\\mathsf { G C N } _ { \\mathsf { C } }$ . Figure 3(a) shows the performance of SLAPS and $\\mathrm { S L A P S } _ { 2 s }$ on Cora and compares them with kNN-GCN. Although $\\mathrm { S L A P S } _ { 2 s }$ does not use the node labels in learning an adjacency matrix, it outperforms kNN-GCN $8 . 4 \\%$ improvement when using an FP generator). With an FP generator, $\\mathrm { S L A P S } _ { 2 s }$ even achieves competitive performance with SLAPS; this is mainly because FP does not leverage the supervision provided by $\\mathsf { G C N } _ { \\mathsf { C } }$ toward learning generalizable patterns that can be used for nodes other than those in the training set. These results corroborate the effectiveness of the self-supervision task for learning an adjacency matrix. Besides, the results show that learning the adjacency using both self-supervision and the task-specific node labels results in higher predictive accuracy. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "The value of λ: Figure 3(b) shows the performance of SLAPS2 on Cora and Citeseer with different values of $\\lambda$ . When $\\lambda = 0$ , corresponding to removing self-supervision, the model performance is somewhat poor. As soon as $\\lambda$ becomes positive, both models see a large boost in performance showing that self-supervision is crucial to the high performance of SLAPS. Increasing $\\lambda$ further provides larger boosts until it becomes so large that the self-supervision loss dominates the classification loss and the performance deteriorates. Note that with $\\lambda = 0$ , SLAPS with the MLP generator becomes a variant of the model proposed by Cosmo et al. (2020), but with a different similarity function. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Importance of $\\boldsymbol { k }$ in kNN: Figure 3(c) shows the performance of SLAPS on Cora for three graph generators as a function of $k$ in kNN. For all three cases, the value of $k$ plays a major role in model performance. The FP generator is the least sensitive because in FP, $k$ only affects the initialization of the adjacency matrix but then the model can change the number of neighbors of each node. For MLP and MLP-D, however, the number of neighbors of each node remains close to $k$ (but not necessarily equal as the adjacency processor can add or remove some edges) and the two generators become more sensitive to $k$ . For larger values of $k$ , the extra flexibility of the MLP generator enables removing some of the unwanted edges through the function P or reducing the weights of the unwanted edges resulting in MLP being less sensitive to large values of $k$ compared to MLP-D. ",
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+    {
+        "type": "text",
+        "text": "Symmetrization: To symmetrize the adjacency, in Equation 1 we took the average of $\\mathsf { P } ( \\tilde { A } )$ and $\\mathsf { P } ( \\tilde { \\mathbf { A } } ) ^ { T }$ . Here we also consider two other choices: 1) $\\mathsf { m a x } ( \\mathsf { P } ( \\tilde { \\boldsymbol { A } } ) , \\mathsf { P } ( \\tilde { \\boldsymbol { A } } ) ^ { T } )$ , and 2) not symmetrizing the adjacency (i.e. using $\\mathsf { P } ( \\tilde { A } ) )$ . Figure 3(d) compares these three choices on Cora and Citeseer with an MLP generator (other generators produced similar results). On both datasets, symmetrizing the adjacency provides a performance boost. Compared to mean symmetrization, max symmetrization performs slightly worse. This may be because max symmetrization does not distinguish between the case where both $v _ { i }$ and $v _ { j }$ are among the $k$ most similar nodes of each other and the case where only one of them is among the $k$ most similar nodes of the other. ",
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+    {
+        "type": "text",
+        "text": "Analysing the learned adjacency: Many graph-based semi-supervised classification models are based on the cluster assumption according to which nearby nodes are more likely to share the same label (Chapelle & Zien, 2005). To verify the quality of the adjacency matrix learned using SLAPS, for every pair of nodes in the test set, we compute the odds of the two nodes sharing the same label as a function of the normalized weight of the edge connecting them. Figure 3(e) represents the odds for different weight intervals. For both Cora and Citeseer, nodes connected with higher edge weights are more likely to share the same label compared to nodes with lower or zero edge weights. As a specific example, when $A _ { i j } ~ \\geq ~ 0 . 1$ , $v _ { i }$ and $v _ { j }$ are almost 2.5 times more likely to share the same label on Cora and almost 2.0 times more likely on Citeseer. Note that SLAPS may connect nodes based on a different criterion than the one used in the original datasets and so the learned adjacencies do not necessarily resemble the original structures. ",
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+        "page_idx": 7
+    },
+    {
+        "type": "image",
+        "img_path": "images/8c71bf8b447f2bec3821f9a1380d5933093b4c3ff3510bd92988cee66b2f9096.jpg",
+        "image_caption": [
+            "Figure 3: The performance of SLAPS (a) compared to $\\mathrm { S L A P S } _ { 2 s }$ on Cora with different generators, (b) with MLP graph generator on Cora and Citeseer as a function of $\\lambda$ , (c) with different graph generators on Cora as a function of $k$ in kNN, and (d) on Cora and Citeseer with different adjacency symmetrizations. (e) The odds of two nodes in the test set sharing the same label as a function of the edge weights learned by SLAPS. "
+        ],
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+        "type": "text",
+        "text": "",
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+    {
+        "type": "text",
+        "text": "6 CONCLUSION ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "In this paper, we proposed a model for learning the parameters of a graph neural network and the graph structure of the nodes simultaneously. We showed the effectiveness of our model using a comprehensive set of experiments and analyses. In the future, we would like to try more sophisticated graph generation models (e.g., GraphRNN (You et al., 2018) and GNF (Liu et al., 2019a)) and extend our approach to applications with a temporal aspect where node features are observed over time but their connections are not provided as input. ",
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+    },
+    {
+        "type": "text",
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+    {
+        "type": "text",
+        "text": "A APPENDIX ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Implementation Detail: We implemented our model in PyTorch (Paszke et al., 2017), used deep graph library (DGL) (Wang et al., 2019a) for the sparse operations, and used Adam (Kingma & Ba, 2014) as the optimizer. We performed early stopping and hyperparameter tuning based on the accuracy on the validation set for all datasets except Wine and Cancer. For these two datasets, validation accuracy reached 100 percent with many hyperparameter settings, making it difficult to select the best set of hyperparameters so instead, we used the validation cross-entropy loss. We fixed the maximum number of epochs to 2000. We use two-layer GCNs for both $G N N _ { \\mathrm { { C } } }$ and $\\mathsf { G N N } _ { \\mathsf { D A E } }$ as well as for baselines and two-layer MLPs throughout the paper (for experiments on ogbn-arxiv, although the original paper uses models with three layers and with batch normalization after each layer, to be consistent with our other experiments we used two layers and removed the normalization). We used two learning rates, one for $\\mathsf { G C N } _ { \\mathsf { C } }$ and one for the other parameters of the models. We tuned the two learning rates from the set $\\lbrace 0 . 0 1 , 0 . 0 0 1 \\rbrace$ . We added dropout layers with dropout probabilities of 0.5 after the first layer of the GNNs. We also added dropout to the adjacency matrix and tuned the values from the set $\\{ 0 . 2 5 , 0 . 5 \\}$ . We set the hidden dimension of $\\mathsf { G N N } _ { \\mathsf { C } }$ to 32 for all datasets except for ogbn-arxiv for which we set it to 256. We used cosine similarity for building the kNN graphs and tuned the value of $k$ from the set $\\{ 1 5 , 2 0 , 3 0 \\}$ . We tuned $\\lambda$ ( $\\lambda$ controls the relative importance of the two losses) from the set $\\{ 0 . 1 , 1 , 1 0 , 1 0 0 \\}$ . The code of our experiments will be available upon acceptance of the paper. ",
+        "bbox": [
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+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "For GRCN (Yu et al., 2020), DGCNN (Wang et al., 2019b), and IDGL (Chen et al., 2020), we used the code released by the authors and tuned the hyperparameters as suggested in the original papers. The results of LDS (Franceschi et al., 2019) are directly taken from the original paper. All the results for our model and the baselines are averaged over 10 runs. We report the mean and standard deviation. ",
+        "bbox": [
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+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "Dataset statistics: The statistics of the datasets used in the experiments can be found in Table 3. ",
+        "bbox": [
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+        "page_idx": 12
+    },
+    {
+        "type": "table",
+        "img_path": "images/f69353edb05ea3c2e25d96d90a2e122ac281d09b670b43a8c126b3d4a5b1aa07.jpg",
+        "table_caption": [
+            "Table 3: Dataset statistics. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Dataset</td><td>Nodes</td><td>Edges</td><td>Classes</td><td>Features</td><td>Label rate</td></tr><tr><td>Cora</td><td>2,708</td><td>5,429</td><td>7</td><td>1,433</td><td>0.052</td></tr><tr><td>Citeseer</td><td>3,327</td><td>4,732</td><td>6</td><td>3,703</td><td>0.036</td></tr><tr><td>Wine</td><td>178</td><td>0</td><td>3</td><td>13</td><td>0.112</td></tr><tr><td>Cancer</td><td>569</td><td>0</td><td>2</td><td>30</td><td>0.035</td></tr><tr><td>Digits</td><td>1,797</td><td>0</td><td>10</td><td>64</td><td>0.056</td></tr><tr><td>20news</td><td>9,607</td><td>0</td><td>10</td><td>236</td><td>0.021</td></tr><tr><td>Pubmed</td><td>19,717</td><td>44,338</td><td>3</td><td>500</td><td>0.003</td></tr><tr><td>ogbn-arxiv</td><td>169,343</td><td>1,166,243</td><td>40</td><td>128</td><td>0.537</td></tr></table>",
+        "bbox": [
+            259,
+            426,
+            738,
+            555
+        ],
+        "page_idx": 12
+    }
+]

parse/train/bMCfFepJXM/bMCfFepJXM.md

@@ -0,0 +1,375 @@
+# BRAC $+$ : GOING DEEPER WITH BEHAVIOR REGULARIZED OFFLINE REINFORCEMENT LEARNING
+
+Anonymous authors Paper under double-blind review
+
+# ABSTRACT
+
+Online interactions with the environment to collect data samples for training a Reinforcement Learning agent is not always feasible due to economic and safety concerns. The goal of Offline Reinforcement Learning (RL) is to address this problem by learning effective policies using previously collected datasets. Standard off-policy RL algorithms are prone to overestimations of the values of outof-distribution (less explored) actions and are hence unsuitable for Offline RL. Behavior regularization, which constraints the learned policy within the support set of the dataset, has been proposed to tackle the limitations of standard off-policy algorithms. In this paper, we improve the behavior regularized offline reinforcement learning and propose $B R A C +$ . We use an analytical upper bound on KL divergence as the behavior regularizor to reduce variance associated with sample based estimations. Additionally, we employ state-dependent Lagrange multipliers for the regularization term to avoid distributing KL divergence penalty across all states of the sampled batch. The proposed Lagrange multipliers allow more freedom of deviation to high probability (more explored) states leading to better rewards while simultaneously restricting low probability (less explored) states to prevent out-of-distribution actions. To prevent catastrophic performance degradation due to rare out-of-distribution actions, we add a gradient penalty term to the policy evaluation objective to penalize the gradient of the Q value w.r.t the out-of-distribution actions. By doing so, the Q values evaluated at the out-ofdistribution actions are bounded. On challenging offline RL benchmarks, ${ \mathrm { B R A C } } +$ outperforms the state-of-the-art model-free and model-based approaches.
+
+# 1 INTRODUCTION
+
+Reinforcement Learning (RL) has shown great success in a wide range of applications including board games (Silver et al., 2016), strategy games (Vinyals et al., 2019), energy systems (Zhang et al., 2019), robotics (Lin, 1992), recommendation systems (Choi et al., 2018), etc. The success of RL relies heavily on extensive online interactions with the environment for exploration. However, this is not always feasible in the real world as it can be expensive or dangerous (Levine et al., 2020).
+
+Offline RL, also known as batch RL, avoids online interactions with the environment by learning from a static dataset that is collected in an offline manner (Levine et al., 2020). While standard offpolicy RL algorithms (Mnih et al., 2013; Lillicrap et al., 2016; Haarnoja et al., 2018a) can, in theory, be employed to learn from an offline data, in practice, they perform poorly due to distributional shift between the behavior policy (probability distribution of actions conditioned on states as observed in the dataset) of the collected dataset and the learned policy (Levine et al., 2020). The distributional shift manifests itself in form of overestimation of the out-of-distribution (OOD) actions leading to erroneous Bellman backups.
+
+Prior works tackle this problem via behavior regularization (Fujimoto et al., 2018b; Kumar et al., 2019; Wu et al., 2019; Siegel et al., 2020). This ensures that the learned policy stays “close” to the behavior policy. This is achieved by adding a regularization term that calculates the $f$ -divergence between the learned policy and the behavior policy. Kernel Maximum Mean Discrepancy (MMD) (Gretton et al., 2007), Wasserstein distance and KL divergence are widely used (Wu et al., 2019). The regularization term is either fixed (Wu et al., 2019), or tuned via dual gradient descent (Kumar et al., 2019), or applied using a trust region objective (Siegel et al., 2020).
+
+In this paper, we propose improvements to the Behavior Regularized Actor Critic (BRAC) algorithm presented in (Wu et al., 2019). To obtain the same, we observe that sample based estimation of divergence measures is computationally expensive and prone to higher variance. Therefore, to reduce variance, we derive an analytical upper bound on the KL divergence measure as the regularization term in the objective function. Moreover, we show that current works that apply the regularization term i.e. the divergence measure, on the entire batch, end up distributing the penalty over all states in the batch in amounts inversely proportional to the state’s probability of occurrence in the batch. This needlessly restricts the deviation of highly explored states while allowing less explored ones to deviate farther leading to OOD actions. To address the same, we employ state dependent Lagrange multipliers for the regularization terms and automatically tune their strength using state-wise dual gradient descent. In addition, the performance of the learned agent trained using prior methods often deteriorates over the course of training. We found that if the learned Q function generalizes such that the gradient of the Q function w.r.t the OOD actions is monotonically increasing, behavior regularization fails to keep such actions within the support set. To mitigate this issue, we penalize the gradient of the Q function w.r.t the OOD actions by adding a gradient penalty term to the policy evaluation objective. This reduces the policy improvement at OOD actions to the problem of minimizing the divergence between the learned policy and the behavior policy.
+
+We call our improved algorithm ${ \mathrm { B R A C } } +$ following (Wu et al., 2019). Our experiments suggest that ${ \mathrm { B R A C } } +$ outperforms existing state-of-the-art model-free and model-based offline RL algorithms in various datasets on the D4RL benchmark (Fu et al., 2020).
+
+# 2 BACKGROUND
+
+Markov Decision Process RL algorithms aim to solve Markov Decision Process (MDP) with unknown dynamics. A Markov decision process (Sutton & Barto, 2018) is defined as a tuple $<$ ${ \mathcal { S } } , { \mathcal { A } } , R , P , \mu >$ , where $s$ is the set of states, $\mathcal { A }$ is the set of actions, $R ( s , a , s ^ { \prime } ) : \mathcal { S } \times \mathcal { A } \times \mathcal { S }  \mathbb { R }$ defines the intermediate reward when the agent transitions from state $s$ to $s ^ { \prime }$ by taking action $a$ , $P ( s ^ { \prime } | s , a ) : \mathcal { S } \times \mathcal { A } \times \mathcal { S } \to [ 0 , 1 ]$ defines the probability when the agent transitions from state $s$ to $s ^ { \prime }$ by taking action $a$ , $\mu : { \dot { \mathcal { S } } } \ { \dot {  } } \ [ 0 , 1 ]$ defines the starting state distribution. The objective of reinforcement learning is to select policy ${ \dot { \pi } } : \mu \to P ( A )$ to maximize the following objective:
+
+$$
+J ( \pi ) = \underset { s _ { 0 } \sim \mu , a _ { t } \sim \pi ( \cdot | s _ { t } ) } { \mathbb { E } } [ \sum _ { t = 0 } ^ { \infty } \gamma ^ { t } R ( s _ { t } , a _ { t } , s _ { t + 1 } ) ]
+$$
+
+Offline Reinforcement Learning The goal of offline RL is to learn policy $\pi _ { \theta }$ from a fixed dataset $\mathcal { D } = \{ ( s _ { i } , a _ { i } , s ^ { \prime } { } _ { i } , r _ { i } ) \} _ { i = 1 } ^ { N }$ consisting of single step transitions $\left\{ \left( s _ { i } , a _ { i } , s ^ { \prime } _ { i } , r _ { i } \right) \right\}$ . The dataset is assumed to be collected using a behavior policy $\pi _ { \beta }$ which denotes the conditional distribution $p ( a | s )$ observed in the dataset. Note that $\pi _ { \beta }$ may consist of multi-modal policy distribution. In principle, standard off-policy RL algorithms using a replay buffer (Mnih et al., 2013; Lillicrap et al., 2016; Haarnoja et al., 2018a) can directly learn from $\mathcal { D }$ . The key challenge resides in the policy evaluation step:
+
+$$
+Q _ { \psi } = \arg \operatorname* { m i n } _ { \psi } [ ( Q _ { \psi } ( s , a ) - ( r ( s , a ) + \gamma \mathbb { E } _ { a ^ { \prime } \sim \pi _ { \theta } } Q _ { \psi ^ { \prime } } ( s ^ { \prime } , a ^ { \prime } ) ) ) ] ^ { 2 } \qquad \mathrm { ( p o l i c y ~ e v a l u a t i o n ) }
+$$
+
+In this step, the target $\mathrm { Q }$ value depends on the learned policy. If the learned policy distribution $\pi _ { \theta }$ diverges from the data distribution $\pi _ { \beta }$ , it results in evaluation of target Q values using outof-distribution (OOD) actions. Such evaluations are prone to errors. Erroneous overestimation of values get exploited by the policy improvement, preventing the algorithm from learning useful policies. In order to avoid such cases, behavior regularization is adopted to force the learned policy to stay “close” to the behavior policy (Fujimoto et al., 2018b; Kumar et al., 2019; Wu et al., 2019).
+
+# 3 IMPROVING BEHAVIOR REGULARIZED OFFLINE REINFORCEMENT LEARNING
+
+In this section, we discuss and propose three non-trivial improvements to the Behavior Regularized Actor Critic (BRAC) offline reinforcement learning (Wu et al., 2019). BRAC (Wu et al., 2019)
+
+augments either the policy improvement or the policy evaluation step with a penalty term to constrain the policy within the support set of the behavior policy. For simplicity of illustration, we only consider the policy improvement step (Wu et al., 2019):
+
+$$
+\operatorname* { m a x } _ { \pi _ { \theta } } \mathbb { E } _ { ( s , a , r , s ^ { \prime } ) \sim \mathcal { D } } [ \mathbb { E } _ { a ^ { \prime \prime } \sim \pi _ { \theta } ( \cdot \vert s ) } [ Q _ { \psi } ( s , a ^ { \prime \prime } ) ] - \alpha \hat { D } ( \pi _ { \theta } ( \cdot \vert s ) , \pi _ { \beta } ( \cdot \vert s ) ) ]
+$$
+
+Here, $\hat { D }$ is a selected $f$ -divergence (Csiszar, 1972) measure used for behavior regularization. Zoom-´ ing into the regularization term, we get:
+
+$$
+\sum _ { s \in S } ( P _ { s } \alpha ) \hat { D } ( \pi _ { \theta } ( \cdot | s ) , \pi _ { \beta } ( \cdot | s ) )
+$$
+
+In the equation above, $s$ denotes the states that occur in a single sampled batch and $P _ { s }$ is the probability of state $s$ in the sampled batch. The first two improvements that we propose are targeted towards the following two terms in Equation 4: (1) The divergence term $\hat { D }$ which for each fixed $\pi _ { \theta }$ determines the impact actions have on the overall objective conditional on the state, and (2) the term $( P _ { s } \alpha )$ which determines the impact the “divergence penalty” of each state will have on the overall objective function. Note that this term is dependent upon the probability of the occurrence of state in the sampled batch. For (1), we propose to use an analytical upper bound of KL divergence to reduce variance. For (2), we propose to decouple the impact of “divergence penalty” of each state from the probability of its occurrence. Detailed explanation of the two improvements and the rationale behind them follows in the next two sub-sections.
+
+# 3.1 REGULARIZATION METHOD
+
+Behavior regularization is used to constrain the learned policy with the support set of the behavior policy. In other words, it ensures that the “difference” between the probability distributions of the learned policy $\pi _ { \boldsymbol { \theta } } ( \cdot | \boldsymbol { s } )$ and behavior policy $\pi _ { \beta } ( \cdot | s )$ is small. The following divergence measures are most widely used in the community:
+
+Kernel MMD Kernel Maximum Mean Discrepancy (MMD) (Gretton et al., 2007) was first introduced in (Kumar et al., 2019) to penalize the policy from diverging from the behavior policy:
+
+$$
+\mathbf { M M D } _ { k } ^ { 2 } ( \pi ( \cdot | s ) , \pi _ { b } ( \cdot | s ) ) = \underset { x , x ^ { \prime } \sim \pi ( \cdot | s ) } { \mathbb { E } } [ K ( x , x ^ { \prime } ) ] - 2 \underset { y \sim \pi ( \cdot | s ) } { \mathbb { E } } [ K ( x , y ) ] + \underset { y , y ^ { \prime } \sim \pi _ { b } ( \cdot | s ) } { \mathbb { E } } [ K ( y , y ^ { \prime } ) ]
+$$
+
+where $K$ is a kernel function. Symmetric kernel functions such as Laplacian and Gaussian kernels are typically used (Gretton et al., 2007).
+
+KL divergence For two probability distributions $P$ and $Q$ on some probability space $\chi$ , the KLDivergence from $Q$ to $P$ is given as
+
+$$
+\mathcal { D } _ { \mathrm { K L } } ( P , Q ) = \int _ { x \sim \chi } P ( x ) \log \frac { P ( x ) } { Q ( x ) } d x
+$$
+
+KL divergence is assymetric and both $\mathcal { D } _ { \mathrm { K L } } \big ( \pi _ { \theta } \big ( \cdot | s \big ) , \pi _ { b } \big ( \cdot | s \big ) \big )$ and $\mathcal { D } _ { \mathrm { K L } } ( \pi _ { b } ( \cdot | s ) , \pi _ { \theta } ( \cdot | s ) )$ are valid. However, $\mathcal { D } _ { \mathrm { K L } } ( \pi _ { b } ( \cdot | s ) , \pi _ { \theta } ( \cdot | s ) )$ is not suitable because it requires $\pi _ { \boldsymbol { \theta } } ( a \vert s ) \neq 0$ when $\dot { \pi _ { b } } ( a | s ) \neq 0$ , for each $a$ . It may assign $\pi _ { \theta } ( a | s ) > 0$ when $\pi _ { b } ( a | s ) = 0$ , producing out of distribution actions. Therefore, $\mathcal { D } _ { \mathrm { K L } } \big ( \pi _ { \theta } \big ( \cdot | s \big ) , \pi _ { b } \big ( \cdot | s \big ) \big )$ is used for regularization.
+
+# 3.1.1 ANALYTICAL KL DIVERGENCE UPPER BOUND
+
+All the existing behavior regularized methods estimate the divergence via samples (Kumar et al., 2019; Wu et al., 2019). While in theory it produces an unbiased estimator, it requires a large number of samples to reduce the variance.
+
+To stabilize the performance, the key idea is to have a low variance estimator for KL divergence. We obtain this by deriving an upper bound on the KL divergence between the learned policy $\pi _ { \theta }$ and the behavior policy $\pi _ { \beta }$ that can be computed analytically.
+
+Assume we learn $\pi _ { \beta }$ using a latent variable model (e.g. VAE (Kingma & Welling, 2014)) with latent variable $Z$ . According to the evidence lower bound (ELBO), we obtain $\log \pi _ { \beta } ( a | s ) \geq$ $\mathbb { E } _ { z \sim q ( z ) } [ \log p ( a | s , z ) ] - \mathcal { D } _ { \mathrm { K L } } ( \bar { q } ( z ) | | p ( z ) )$ , where $q ( z )$ is the approximated posterior distribution and $p ( z )$ is the prior. Then, the KL divergence is bounded by:
+
+$$
+\begin{array} { r l } & { \mathcal { D } _ { \mathrm { K L } } ( \pi _ { \theta } ( \cdot | s ) | | \pi _ { \beta } ( \cdot | s ) ) = \mathbb { E } _ { a \sim \pi _ { \theta } } [ \log \pi _ { \theta } ( a | s ) ] - \mathbb { E } _ { a \sim \pi _ { \theta } } [ \log \pi _ { \beta } ( a | s ) ] } \\ & { \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \mathbb { E } _ { a \sim \pi _ { \theta } } [ \log \pi _ { \beta } ( a | s ) ] - \mathbb { E } _ { \kappa \mathrm { L } } ( q ( z ) | | p ( z ) ) ] } \\ & { \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad ( \mathcal { T } _ { a \sim \pi _ { \theta } } [ \log ( a | s ) ] + \mathcal { D } _ { \mathrm { K L } } ( q ( z ) | | p ( z ) ) ] } \\ & { \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad ( \mathcal { T } _ { a \sim \pi _ { \theta } , z \sim q ( z ) } [ \mathcal { D } _ { \mathrm { K L } } ( \pi _ { \theta } ( \cdot | s ) | | p ( \cdot | s , z ) ) + \mathcal { D } _ { \mathrm { K L } } ( q ( z ) | | p ( z ) ) ] } \\ & { \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad ( \mathcal { T } _ { a \sim \pi _ { \theta } , z \sim q ( z ) } [ \mathcal { D } _ { \mathrm { K L } } ( \pi _ { \theta } ( \cdot | s ) | | p ( \cdot | s , z ) ) + \mathcal { D } _ { \mathrm { K L } } ( q ( z ) | | p ( z ) ) ] } \end{array}
+$$
+
+Thus, if we choose $p ( \cdot | s , z )$ to be a tractable distribution, Equation 7 can be computed analytically. Note that although $p ( \cdot | s , z )$ has a special form (e.g. Gaussian), the latent variable model can in theory represent any probability distribution without sacrificing the expressiveness. In practice, we choose $p ( \cdot | s , z )$ to be a Gaussian distribution.
+
+# 3.2 STATE DEPENDENT LAGRANGE MULTIPLIER
+
+From the term $( P _ { s } \alpha )$ in Equation 4, one can infer that the “penalty budget” of divergence measure gets distributed among the states in amounts inversely proportional to the probability of the occurrence of the states in the sampled batch. If we assume the batches are sampled uniformly, this implies that the states which are more numerous in the offline dataset $\mathcal { D }$ i.e. the states that have been explored more thoroughly are restricted from deviating from the behavior policy. Whereas the less numerous state that have had limited exploration enjoy more freedom for deviation. This is undesirable as restricting highly explored states is overly conservative while allowing less explored states to deviate can lead to OOD actions.
+
+To address this we add a state wise constraint for KL divergence as follows:
+
+$$
+\operatorname* { m a x } _ { \pi _ { \alpha } \Lambda } \mathbb { E } _ { s \sim \mathcal { D } } [ \mathbb { E } _ { a ^ { \prime } \sim \pi _ { \theta } ( \cdot | s ) } [ Q _ { \psi } ( s , a ^ { \prime } ) ] ] \mathrm { ~ s . t . ~ } \mathcal { D } _ { K L } ( \pi _ { \theta } ( \cdot | s ) , \pi _ { \beta } ( \cdot | s ) ) \leq \epsilon _ { \mathrm { K L } } , \forall s \quad \mathrm { ( p o l i c ) } .
+$$
+
+This translates to assignment of state dependent Lagrange multipliers when solved using dual gradient descent:
+
+$$
+\operatorname* { m a x } _ { \pi _ { \theta } } \mathbb { E } _ { s \sim \mathcal { D } } \big [ \mathbb { E } _ { a ^ { \prime } \sim \pi _ { \theta } ( \cdot \vert s ) } [ Q _ { \psi } ( s , a ^ { \prime } ) - \alpha ( s ) \mathcal { D } _ { \mathrm { K L } } ( \pi _ { \theta } ( \cdot \vert s ) , \pi _ { \beta } ( \cdot \vert s ) ) ] \big ]
+$$
+
+Maintaining a Lagrange multiplier for each state is impractical for large datasets. In practice, we parameterize $\alpha ( s )$ with a neural network.
+
+# 3.3 GRADIENT PENALIZED POLICY EVALUATION
+
+The fundamental challenge in offline reinforcement learning is to mitigate the impact of erroneous Q values that are evaluated at out-of-distribution actions and used in policy evaluation. Due to the limited representation capacity of neural networks, such actions are unavoidable for large datasets, even with state-dependent behavior regularization. CQL (Kumar et al., 2020) resolves this problem by optimizing a conservative lower bound of the Q value. The key idea in this paper is to bound the Q value at the out-of-distribution actions such that their values are not greater than the Q value of in-distribution actions. We achieve this by augmenting the policy evaluation step with a gradient penalty regularization term. To elaborate our approach, we first analyze the gradient of the policy improvement step:
+
+$$
+\nabla _ { \theta } J \approx \frac { 1 } { N } \sum _ { i = 1 } ^ { N } \nabla _ { a _ { i } } Q _ { \phi } ( s , a ) | _ { s = s _ { i } , a = a _ { i } } \nabla _ { \theta } \pi _ { \theta } ( a | s ) | _ { s = s _ { i } , a = a _ { i } } - \alpha ( s _ { i } ) \nabla _ { \theta } \mathcal { D } _ { \mathrm { K L } } ( \pi _ { \theta } ( \cdot | s ) , \pi _ { \beta } ( \cdot | s ) )
+$$
+
+If the current policy produces out-of-distribution actions and the Q network erroneously generalizes in such a way that the gradient of Q network is monotonically increasing, this leads to the unbounded value of the Q network and the failure of the behavior regularization. We created a toy example to illustrate this phenomenon in Appendix A. The analysis suggests that if we penalize the gradient of the Q network with respect to the out-of-distribution actions such that they are close to zero,
+
+# Algorithm 1 BRAC+: Improved Behavior Regularized Actor Critic
+
+1: Train the behavior policy $\pi _ { \beta }$ on the offline dataset $\mathcal { D } = \{ ( s _ { i } , a _ { i } , r _ { i } , s _ { i } ^ { ' } ) \} _ { i = 1 } ^ { N }$ via maximum   
+likelihood estimation: $\begin{array} { r } { \pi _ { \beta } = \arg \operatorname* { m a x } _ { \pi _ { \beta } } \sum _ { i = 1 } ^ { N } \log \pi _ { \beta } ( a _ { i } | s _ { i } ) } \end{array}$   
+2: Train initial policy: $\begin{array} { r } { \pi _ { \theta } = \arg \operatorname* { m i n } _ { \pi _ { \theta } \in \Pi } \mathcal { D } _ { \mathrm { K L } } ( \pi , \pi _ { \beta } ) } \end{array}$   
+3: Train initial Q network: $\begin{array} { r } { Q _ { \psi } = \arg \operatorname* { m i n } _ { \psi } [ ( Q _ { \psi } ( s , a ) - ( r ( s , a ) + \gamma \mathbb { E } _ { a ^ { \prime } \sim \pi _ { \theta } } Q _ { \psi ^ { \prime } } ( s ^ { \prime } , a ^ { \prime } ) ) ) ] ^ { 2 } } \end{array}$   
+4: for $e = 1 : E$ do   
+5: for $t = 1 : T$ do   
+6: Update Q network using Equation 10   
+7: Update the policy using Equation 7   
+8: Update $\alpha$ via dual gradient descent: $\alpha ( s )  \alpha ( s ) + \lambda _ { \alpha } ( \mathcal { D } _ { \mathrm { K L } } ( \pi _ { \theta } ( \cdot | s ) , \pi _ { \beta } ( \cdot | s ) ) - \epsilon _ { \mathrm { K L } } )$   
+9: Update $\beta$ via dual gradient descent: $\beta \gets \beta + \lambda _ { \beta } ( \mathcal { H } ( \pi _ { \theta } ( \cdot | s ) ) - \mathcal { H } _ { 0 } )$   
+10: Update the target network $\psi ^ { ' } = \tau \psi + ( 1 - \tau ) \psi ^ { ' }$   
+11: end for   
+12: end for
+
+performing policy improvement step simply reduces to minimizing the KL divergence. Inspired from (Gulrajani et al., 2017), we add a gradient penalty term to the policy evaluation step as:
+
+$$
+\begin{array} { r l } & { \underset { \psi } { \operatorname* { m i n } } \ : \mathbb { E } _ { ( s , a ) \sim \mathcal { D } } ( ( Q _ { \psi } ( s , a ) - r ( s , a ) + \gamma \mathbb { E } _ { a ^ { \prime } \sim \pi a \theta } Q _ { \psi ^ { \prime } } ( s ^ { \prime } , a ^ { \prime } ) ) ^ { 2 } } \\ & { \quad \quad \quad \quad \quad \quad \quad \quad \quad + \lambda \mathbb { E } _ { a ^ { \prime \prime } \sim \pi \pi _ { \theta } ( \cdot \vert s ) } ( \vert \vert \nabla _ { a ^ { \prime \prime } } Q _ { \psi } ( s , a ^ { \prime \prime } ) \vert \vert _ { 2 } f ( \mathcal { D } _ { K L } ( \pi _ { \theta } ( \cdot \vert s ) , \pi _ { \beta } ( \cdot \vert s ) ) - \epsilon _ { \mathrm { G P } } ) ) ) } \end{array}
+$$
+
+where $f$ is a non-decreasing function and $\epsilon _ { G P }$ is the threshold for gradient penalty. In this paper, we set $f$ to be the Heaviside step function (or indicator). Other variants such as its soft version (sigmoid function) is left for future work. Setting $\epsilon _ { G P } ~ = ~ \epsilon _ { K L }$ is too conservative because it prevents generalization. In practice, we set $\epsilon _ { G P }$ at epoch $t$ to be $\mu _ { K L } + \sigma _ { K L }$ , where $\mu _ { K L }$ and $\sigma _ { K L }$ is the mean and standard deviation of the KL divergence for all the states in the dataset at epoch $t - 1$ . In addition, we found that $\lambda = 0 . 1$ works well for all the tasks.
+
+# 4 RELATED WORK
+
+We briefly summarize prior works in offline RL and discuss their relationship with our approach. As discussed in Section 1, the fundamental challenge in learning from a static data is to avoid out-of-distribution actions (Levine et al., 2020). This requires solving two problems: 1) estimation of behavior policy, 2) quantification of out-of-distribution actions. We follow BCQ (Fujimoto et al., 2018b), BEAR (Kumar et al., 2019) and BRAC (Wu et al., 2019) by learning the behavior policy using a conditional VAE (Kingma & Welling, 2014). To avoid out-of-distribution actions, BCQ generates actions in the target values by perturbing the behavior policy. However, this is over-pessimistic in most of the cases. BRAC (Wu et al., 2019) constrains the policy using various sample-based $f$ -divergence measures including MMD, Wasserstein distance and KL divergence with penalized policy improvement or policy evaluation. BEAR (Kumar et al., 2019) is an instance of BRAC with penalized policy improvement using MMD (Gretton et al., 2007). Sample-based estimation is computationally expensive and suffers from high variance. In contrast, our method uses an analytical upper-bound of the KL divergence to constrain the distance between the learned policy and the behavior policy. It is both computationally efficient and has low variance. (Siegel et al., 2020) solves trust-region objective instead of using penalty. CQL (Kumar et al., 2020) avoids estimating the behavior policy by learning a conservative Q function that lower-bounds its true value. Hyperparameter search is another challenging problem in offline RL. (Lee et al., 2020) uses a gradient-based optimization of the hyperparameter using held-out data. MOPO (Yu et al., 2020) follows MBPO Janner et al. (2019) with additional reward penalty on unreliable model-generated transitions. MBOP (Argenson & Dulac-Arnold, 2020) learns the dynamics mode, the behavior policy and a truncated value function to perform online planning. (Kidambi et al., 2020) learns a surrogate MDP using the dataset, such that taking out-of-distribution actions transit to the terminal state. The out-of-distribution actions are detected using the agreement of model ensembles.
+
+# 5 EXPERIMENTS
+
+Our experiments1 aim to answer the following questions: 1) How does the performance of our improvements compare with state-of-the-art model-free and model-based offline RL methods? 2) How does the use of analytical variational upper bound on KL divergence for regularization term compare with sampled-based MMD? 3) How does the state-dependent Lagrange multiplier based regularization (state-wise regularizor) compare with global policy regularization? 4) Does the gradient penalized policy evaluation improve the stability during the training? To answer these questions, we evaluate our methods on a subset of the D4RL (Fu et al., 2020) benchmark. We consider three locomotion tasks (hopper, walker2d, and halfcheetah) and four types of datasets: 1) random (rand): collect the interactions of a run of random policy for 1M steps to create the dataset, 2) medium (med): collect the interactions of a run of medium quality policy for 1M steps as the dataset, 3) medium-expert (med-exp): run a medium quality policy and an expert quality policy for 1M steps, respectively, and combine their interactions to create the dataset, 4) mixed (medium-replay): train a policy using SAC (Haarnoja et al., 2018a) until the performance of the learned policy exceeds a pre-determined threshold, and take the replay buffer as the dataset. In addition, we consider more complex Adroit tasks (Rajeswaran et al., 2018) that requires controlling a 24-DoF robotic hand, using limited data from human demonstrations.
+
+We compare against state-of-the-art model-free and model-based baselines, including behavior cloning, BEAR (Kumar et al., 2019) that constrains the learned policy within the support of the behavior policy using sampled MMD, BRAC-p/v (Wu et al., 2019) that constrains the learned policy within the support of the behavior policy using various sample-based $f$ -divergences to penalize either the policy improvement (p) or the policy evaluation (v), $\operatorname { C Q L } ( { \mathcal { H } } )$ (Kumar et al., 2020) that learns a Q function that lower-bounds its true value. We also compare against model-based approaches including MOPO (Yu et al., 2020) that follows MBPO (Janner et al., 2019) with additional reward penalties and MBOP (Argenson & Dulac-Arnold, 2020) that learns an offline model to perform online planning.
+
+# 5.1 COMPARATIVE RESULTS
+
+Table 1: Results for OpenAI gym (Brockman et al., 2016) environments in the D4RL (Fu et al., 2020) datasets. For each task, we train for 1 million gradient steps and report the performance by running the policy obtained at the last epoch of the training for 100 episodes, averaged over 4 random seeds with standard deviation. Each number is the normalized score as proposed in (Fu et al., 2020). Please refer to (Fu et al., 2020) for results on more baselines.   
+
+<table><tr><td rowspan="2">Task Name</td><td colspan="4">Model-Free</td><td colspan="2">Model-Based</td></tr><tr><td>BEAR</td><td>BRAC-p/v</td><td>CQL(H)</td><td>BRAC+ (Ours)</td><td>MOPO</td><td>MBOP</td></tr><tr><td>halfcheetah-rand</td><td>25.1</td><td>24.1/31.2</td><td>35.4</td><td>26.4±1.0</td><td>31.9±2.8</td><td>6.3±4.0</td></tr><tr><td>walker2d-rand</td><td>7.3</td><td>-0.2/1.9</td><td>7.0</td><td>16.7± 2.3</td><td>13.0±2.6</td><td>8.1± 5.5</td></tr><tr><td>hopper-rand</td><td>11.4</td><td>11.0/12.2</td><td>10.8</td><td>12.5±0.3</td><td>13.3±1.6</td><td>10.8± 0.3</td></tr><tr><td>halfcheetah-med</td><td>41.7</td><td>43.8/46.3</td><td>44.4</td><td>46.6±0.6</td><td>40.2± 2.7</td><td>44.6±0.8</td></tr><tr><td>walker2d-med</td><td>59.1</td><td>77.5/81.1</td><td>79.2</td><td>75.1±3.5</td><td>14.0±10.1</td><td>41.0 ± 29.4</td></tr><tr><td>hopper-med</td><td>52.1</td><td>32.7/31.1</td><td>58.0</td><td>53.2±3.1</td><td>26.5±3.7</td><td>48.8± 26.8</td></tr><tr><td>halfcheetah-med-exp</td><td>53.4</td><td>44.2/41.9</td><td>62.4</td><td>61.2±2.8</td><td>57.9± 24.8</td><td>105.9± 17.8</td></tr><tr><td>walker2d-med-exp</td><td>40.1</td><td>76.9/81.6</td><td>98.7</td><td>95.3± 5.9</td><td>55.0± 19.1</td><td>70.2 ± 36.2</td></tr><tr><td>hopper-med-exp</td><td>96.3</td><td>1.9/0.8</td><td>111.0</td><td>112.9±0.1</td><td>51.7±42.9</td><td>55.1 ± 44.3</td></tr><tr><td>halfcheetah-mixed</td><td>38.6</td><td>45.4/47.7</td><td>46.2</td><td>46.1±0.2</td><td>54.0±2.6</td><td>42.3 ± 0.9</td></tr><tr><td>walker2d-mixed</td><td>19.2</td><td>-0.3/0.9</td><td>26.7</td><td>39.0±4.6</td><td>42.7±8.3</td><td>9.7 ± 5.3</td></tr><tr><td>hopper-mixed</td><td>33.7</td><td>0.6/0.6</td><td>48.6</td><td>72.7±18.9</td><td>92.5±6.3</td><td>12.4 ± 5.8</td></tr></table>
+
+Performance on multi-modal datasets We first compare the performance on multi-modal datasets i.e. med-exp and mixed datasets. Results shown in Table 1 suggest that our method outperforms various model-free baselines on most of the multi-modal datasets, especially on hopper-mix and walker2d-mix by up to $1 . 5 \mathrm { x }$ . Compared with BEAR (Kumar et al., 2019), our performance improvement arises from the advantage of the KL divergence over the kernel MMD (see discussions in Appendix B). The choice of parameters in $\operatorname { C Q L } ( \mathcal { H } )$ makes it too conservative to achieve higher performance.
+
+Table 2: Results for Adroit tasks with human demonstrations in the D4RL (Fu et al., 2020) datasets. The numbers are reported by following the same procedure as in Table 1 except we run the policy obtained at the last epoch of training for 1000 episodes due to large variance across different runs.   
+
+<table><tr><td>Task Name</td><td>BC</td><td>BEAR</td><td>BRAC-p/v</td><td>CQL(H)</td><td>CQL(p)</td><td>BRAC+ (Ours)</td></tr><tr><td> pen-human</td><td>34.4</td><td>-1.0</td><td>8.1/0.6</td><td>37.5</td><td>55.8</td><td>64.9± 1.6</td></tr><tr><td>hammer-human</td><td>1.5</td><td>0.3</td><td>0.3/0.2</td><td>4.4</td><td>2.1</td><td>3.9 ± 0.9</td></tr><tr><td>door-human</td><td>0.5</td><td>-0.3</td><td>-0.3/-0.3</td><td>9.9</td><td>9.1</td><td>11.5± 1.2</td></tr><tr><td>relocate-human</td><td>0.0</td><td>-0.3</td><td>-0.3/-0.3</td><td>0.20</td><td>0.35</td><td>0.20 ± 0.11</td></tr></table>
+
+Performance on single-modal datasets The performance of our method on single-modal (rand and med) dataset outperforms or matches with baseline methods as evident from Table 1 except halfcheetah-random dataset. We observe that the performance is very sensitive to the choice of target policy entropy. We hypothesize that the our choice of target policy entropy in the halfcheetahrandom task makes it hard to compose correct sub-optimal policies from a random collected dataset.
+
+![](images/c3e749d5069b0b1565520c3bc0b5e5d26cad7d2efabb6fab7f4282bf39bc0a5c.jpg)  
+Figure 1: Left: learning curve of pen-human-v0 task. Right: the average $\mathrm { Q }$ value of the first ensemble over the course of training. Both curves are smoothed by a factor of 20.
+
+Performance on datasets with human demonstrations The performance on Adroit tasks is shown in Table 2. These tasks are substantially harder than OpenAI gym tasks due to limited training data in a high dimensional observation and action space. Our method makes non-trivial improvement over the behavior cloning. Compared with the state-of-the-art approaches, our approach is superior on half of the tasks and matches the asymptotic performance on the remaining ones. Figure 1 shows that the Q value is bounded when the gradient penalized policy evaluation technique is employed. On the contrary, the Q value without the gradient penalized policy evaluation increases exponentially. Note that we use MMD with Laplacian kernel for Adroit tasks. We observe that the KL-based regularization struggles with datasets collected with narrow behavior distributions (have large density within a tiny space and almost zero density anywhere else). In such a case, the KL divergence is sensitive to tiny policy changes, making gradient-based optimization hard to converge.
+
+# 5.2 ABLATION STUDY
+
+To answer question (2), (3) and (4), we conduct a thorough ablation study on ${ \mathrm { B R A C } } +$ on various tasks with different data collection policies (hopper-mixed, walker2d-medium-expert, halfcheetahmedium).
+
+Sampled-based MMD vs. analytical upper bound KL The results of using sampled-based MMD versus analytical upper bound KL are shown in Figure 2a. The difference of the performance in the single-modal dataset (halfcheetah-medium) is negligible. However, the performance on multi-modal datasets is varied. Our toy example in Appendix B suggests that MMD and the backward KL divergence tends to cover all the “modes” in the behavior policy while the forward KL divergence tends to seek one of the “mode” in the behavior policy. Note that this argument is only valid if the learned policy is single-modal (e.g. Gaussian distribution). The superior performance in the walker2d-medium-expert when using forward KL regularization supports this argument. The results in hopper-mixed task seems to be contradictory. We hypothesize that since the analytical KL has low variance, the policy quickly adopts the out-of-distribution actions when available while in sample-based methods, such adoption is slower due to higher variance in the policy regularization.
+
+![](images/936da787ddd671fa414383deb9d728d42c0ce8d083411c0c19e2e2ca738bcbe8.jpg)  
+(c) Gradient penalty vs. No gradient penalty. State-wise Lagrange multiplier. Forward KL.   
+Figure 2: Figures of ablation study. Each setting is repeated for 4 random seeds. The curve is the mean and the shaded area is the standard deviation. The curves are smoothed by a factor of 20. The number of gradient steps per epoch is 2000. To make fair comparison, we only substitute KL divergence with MMD-based measurement with additional MMD-specific hyperparameter tuning. The other design choices are different from (Kumar et al., 2019). Details can be found in Appendix D.
+
+State-wise vs. global regularization The performance of using global and state-wise regularization on various tasks is shown in Figure 2b. It is noticeable that the state-wise regularization improves the performance in hopper-mixed and walker2d-medium-expert task, while the performance in halfcheetah-medium does not show much improvement. To understand the consequence of the state-wise regularization, we plot the histogram of the KL divergence of all the states in the dataset in Figure 3. The KL divergence histogram of using state-wise regularization is more concentrated around the threshold while the global regularization is more sporadic. Strictly enforcing the policy regularization around the threshold helps avoid out-of-distribution actions, that is often caused by a few states in the dataset.
+
+![](images/3992389a7bba94f23ff3c5baa970981f092f58a99c2dab5472bed071a685fb24.jpg)  
+Figure 3: Histogram of the KL divergence of all the states in the dataset with global and state-wise Lagrange multiplier.
+
+Gradient penalty vs. no gradient penalty Even with policy regularization, evaluating the Q value at the out-of-distribution actions can’t be fully avoided. Thus, it is important to bound the Q value at the out-of-distribution actions such that their values are not greater than the Q value of indistribution actions. Figure 2c shows the performance with and without gradient penalty in the policy evaluation step. While there is little difference in the hopper-mixed and the halfcheetah-medium task, the performance in the walker2d-medium-expert task stabilizes with the gradient penalized policy evaluation. On the contrary, the performance deteriorates over time without gradient penalty. Please refer to Appendix D for more analysis.
+
+# 6 DISCUSSIONS AND LIMITATIONS
+
+There are several limitations of our approach. The discussion in Appendix D suggests that although KL-regularized offline policy optimization is good at combining sub-optimal policies, they may stuck at local optimums; and they are hard to escape. Another drawback is that the threshold value is hard to set. In our experiments, we try a few numbers that are above the minimum possible KL threshold, which is obtained by training a policy that minimizes that KL divergence.
+
+Finally, we conjecture that the behavior-regularized approach is not sufficient to tackle offline RL problems since it fully ignores the state distribution. To see this, we can create a dataset that only adds a few trajectories from an expert policy to a dataset collected by a low-quality policy. If the low-quality policy doesn’t visit the “good” states in the expert policy (can’t combine sub-optimal policies), behavior-regularized approach leads to a policy that imitates the expert policy. Such imitation is likely to fail due to compounding errors (Ross et al., 2010). The right approach for this dataset is to completely ignore expert trajectories and combine sub-optimal policies in the low-quality regions. To achieve this, we need to consider the state distribution as the density of the “good” states is very low.
+
+# 7 CONCLUSION
+
+In this paper, we improved the behavior regularized offline reinforcement learning by proposing a low-variance upper bound of the KL divergence estimator to reduce variance, state-dependent Lagrange multiplier to allow more freedom of deviation to high probability states while restricting low probability states and gradient penalized policy evaluation such that the Q values of out-ofdistribution actions are not greater than those of in-distribution actions. Our experimental results on challenging benchmarks illustrate the benefits of our improvements.
+
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+
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+
+![](images/a1c91ac49d16ce5409884eebbeec19c1806634da6ec66fbfbb476987a6437c29.jpg)  
+Figure 4: Fitting a regression model with out of distribution data
+
+# A TOY EXAMPLE TO DEMONSTRATE BOUNDING THE VALUE OF THE OUT OF DISTRIBUTION INPUTS VIA GRADIENT PENALTY
+
+To demonstrate the effectiveness of the gradient penalty to enforce the bound of the predicted values of a regression model, we conduct experiments by fitting a regression model with in-distribution data while minimizing the norm of the gradient at out of distribution inputs. Specifically, we generate dataset $\{ ( x _ { i } , y _ { i } ) \} _ { i = 1 } ^ { N = 1 0 0 }$ as $x _ { i } \sim U ( - 0 . 8 , 0 . 8 )$ , $\begin{array} { r } { y _ { i } = 5 \sin ( \frac { \pi } { 2 } x _ { i } ) + \epsilon _ { i } } \end{array}$ , where $\epsilon _ { i } \sim \mathcal { N } ( 0 , 0 . 5 )$ . In addition, we generate t of distribution dataset $\{ \tilde { x } \} _ { j = 1 } ^ { M = 1 0 0 }$ as $\tilde { x } \sim U ( - 2 , - 0 . 8 ) \cup U ( 0 . 8 , 2 )$ . We $f$   
+hidden layer is 64 and the activation is RELU. We use Adam (Kingma & Ba, 2015) optimizer with learning rate 0.01. In addition to the standard MSE loss, we add a gradient penalty term inspired from (Gulrajani et al., 2017) such that the gradient at the out of distribution inputs is penalized. The overall loss function is:
+
+$$
+\mathcal { L } = \frac { 1 } { N } \sum _ { i = 1 } ^ { N } ( f ( x _ { i } ) - y _ { i } ) ^ { 2 } + \lambda \cdot \frac { 1 } { M } \sum _ { j = 1 } ^ { M } | | \nabla _ { \tilde { x } } f ( \tilde { x } _ { j } ) | | _ { 2 }
+$$
+
+In our experiments, we set $\lambda = 0 . 1$ . We fit $f$ with and without the gradient penalty term. Figure 4 shows the predicted value at both in-distribution and out-of-distribution inputs. The results suggest that: 1) both models generate good prediction for in-distribution inputs $( [ - 0 . 8 , 0 . 8 ] )$ , which is between the two red lines 2) both models generalize well at the out-of-distribution regions between the red line and the blue line $( [ - 1 , - 0 . 8 ]$ and [0.8, 1]). 3) both models erroneously generalize beyond the blue lines $( \lfloor - 2 , - 1 \rfloor$ and $\lfloor 1 , 2 \rfloor$ ). However, the value of the model trained without gradient penalty keep on increasing or decreasing while the model trained with gradient penalty has zero gradient. This suggests that with proper out-of-distribution regularization, the gradient direction at the out-of-distribution inputs point to the in-distribution regions.
+
+# B VISUALIZATION OF VARIOUS PROBABILITY DIVERGENCE MEASUREMENT
+
+To understand the impact of forward KL $( \mathcal { D } _ { \mathrm { K L } } ( \pi _ { \theta } | | \pi _ { b } ) ) $ ), backward KL $\left( \mathcal { D } _ { \mathrm { K L } } ( \pi _ { b } | | \pi _ { \theta } ) \right)$ ) and MMD distance in constraining the learned policy within the support set defined by the behavior policy, we create a toy example as shown in Figure 5. The black curves in the two figures represent the behavior distribution $\pi _ { b }$ . The orange, blue and green curves in the graph show forward KL, backward KL and MMD with Laplacian kernel between $\boldsymbol { \mathit { 1 0 } }$ and $\pi _ { \theta } ~ = ~ \mathcal { N } ( x , \sigma )$ , respectively, where $x$ is a variable between $[ - 1 0 , 1 0 ]$ and $\sigma$ is fixed. If we draw a horizontal line (red), the area below this line defines the support set. In the left figure, the behavior policy is single-modal and there is little difference between the three metrics. However, when the behavior policy is multi-modal, the forward KL divergence contains two bottoms, each corresponds to the peak probability density of the behavior policy. However, in backward KL and MMD distance, the policy with the smallest divergence actually has low probability density in the behavior policy. This is because minimizing the forward KL leads to “mode seeking” while minimizing the backward KL and MMD leads to “mode covering”. These two terms are heavily used when talking about generative models such as VAEs and GANs (Ke et al., 2019). When performing offline RL in multi-modal dataset, it is necessary to combine sub-optimal actions in the behavior policy, which is in fact “mode seeking”. Mode covering typically leads to out-of-distribution actions even when the divergence measurement is small (e.g. the area between the two modes are actually out-of-distribution) while mode seeking may be stuck at a local optimum (e.g. moving across different “modes” is hard).
+
+![](images/5a639b65eb020709bafc3ba11100d983cd249ee7f76e31c22d118ec56717e901.jpg)  
+Figure 5: The black curve in the two figures represents the behavior distribution $\pi _ { b }$ . The orange, blue and green curves in the graph show forward KL, backward KL and MMD with Laplacian kernel between $\pi _ { b }$ and $\pi _ { \boldsymbol { \theta } } = \mathcal { N } ( \boldsymbol { x } , \bar { \boldsymbol { \sigma } } )$ , respectively, where $x$ is a variable between $[ - 1 0 , 1 0 ]$ and $\sigma$ is fixed.
+
+In the left figure, $\pi _ { b } = \mathcal { N } ( 0 , 1 )$ . In the right figure, $\pi _ { b }$ is a mixture of two Gaussian distributions: $\mathcal { N } ( - 1 . 5 , 0 . 3 )$ and $\mathcal { N } ( 1 , 0 . 5 )$ . The weight of each component is 0.3 and 0.7, respectively. $\sigma$ is set to 0.2.
+
+# C MISSING BACKGROUND
+
+# C.1 VARIATIONAL AUTO-ENCODER
+
+A variational auto-encoder (VAE) (Kingma & Welling, 2014) is a generative model that aims to learn the data distribution $p ( X )$ given a set of observations $\{ x _ { i } \} _ { i = 1 } ^ { N }$ . While directly optimizing $p ( X )$ is intractable, we can optimize its evidence lower-bound (ELBO):
+
+$$
+\begin{array} { r } { \log p ( X ) \ge \mathbb { E } _ { z \sim q ( z ) } [ \log p ( X | z ) ] - \mathcal { D } _ { \mathrm { K L } } ( q ( z ) | | p ( z ) ) } \end{array}
+$$
+
+where $q ( Z )$ is the variational distribution and $p ( Z )$ is the prior. In VAE, $q ( Z )$ is $q ( Z | X )$ so that it is an auto-encoder. Optimizing Equation 12 using gradient descent requires back-propagate the gradient through a sample operator. Fortunately, if the latent variable is a multivariate Gaussian distribution, we can use re-parametrization trick. The tightness of the upper bound is the KL divergence between the approximated posterior distribution and the true posterior distribution.
+
+# D MISSING ABLATION STUDY
+
+For fair comparison between MMD and KL divergence in policy regularization, we only substitute KL divergence with MMD-based measurement with additional MMD-specific hyperparameter tuning. In the original implementation (Kumar et al., 2019), the author uses 4 ensembles of Q network and compute the Q value by a convex combination of both the minimum of the ensembles and the maximum of the ensembles. In our implementation, we follow the standard architecture in the online setting: we only use 2 ensembles of Q networks and compute the Q value as their minimum. In this work, we only compare against MMD with Laplacian kernels. The MMD-specific hyper-parameters are shown in Table 3.
+
+Difference in training the behavior policy (Kumar et al., 2019) learns the behavior policy as a $\beta$ -VAE (Higgins et al., 2017) with MSE reconstruction loss. This is equivalent to maximizing the log probability of a Gaussian distribution with fixed variance as the decoder output. However, the variational lower bound does not hold in $\beta$ -VAE that breaks our derivation. In this paper, we learn the behavior policy as a regular VAE. The decoder outputs a Gaussian distribution with inputconditioned mean and variance.
+
+Gradient penalized policy evaluation Figure 6 suggests that the L2 norm of the gradient at the out-of-distribution actions with the gradient penalty is much lower than that without the gradient penalty.
+
+![](images/1e7b6a34bc8b6db03c161a6bd33f24a4fbe50c88de544e7755a5d07be157b435.jpg)  
+Figure 6: Histogram of the L2 norm of the gradients for out-of-distribution actions
+
+Out-of-distribution generalization Lastly, we discuss the out-of-distribution generalization. In offline RL, we rate the learned policy with three levels. Level I policies are able to strictly follow the behavior policy. Level II policies are able to combine sub-optimal policies in the bahevior policy. Level III policies are able to generalize to out-of-distribution actions. To visualize the outof-distribution generalization, we plot the KL divergence of the learned policy on the testing state distribution in Figure 7. In the hopper-mixed and halfcheetah-medium task, there are periodic spikes, suggesting that the agents visit states which are not in the training distribution at test time. Although the results suggest that such generalization is correct, our approach is unable to explicitly quantify it. Thus, we leave it as future work.
+
+![](images/423e1a847c88d43168a412a1bcee0717bf24b6fbffd6ed2cce09d9a0542edad1.jpg)  
+Figure 7: The KL divergence between the learned policy and the behavior policy on the testing data distribution. The $x$ axis is the step in an episode.
+
+Performance sensitivity to the choices of target entropy During our experiments, we found that the performance using the KL divergence as the behavior regularization method is very sensitive to the choice of the target entropy. Policies trained with a larger entropy may not be able to capture the narrow expert distribution within the behavior policy distribution. This phenomenon is most typical in the halfcheetah-medium-expert dataset as depicted in Figure 8.
+
+![](images/e142754b28d4cb0fca25866e2b6b6daefced0991f27fffd724bcc1ecd202c59d.jpg)  
+Figure 8: The performance of the halfcheetah-medium-expert task on various choices of the target entropy.
+
+Table 3: MMD-specific Hyper-parameters   
+
+<table><tr><td>Hyper-parameter</td><td>hopper-medium-replay丨walker-medium-expert|</td><td></td><td>halfcheetah-medium</td></tr><tr><td>σ in Laplacian kernel</td><td>10</td><td>20</td><td>10</td></tr><tr><td>EMMD</td><td>0.2</td><td>0.2</td><td>0.3</td></tr><tr><td>Number of samples</td><td>10</td><td>10</td><td>10</td></tr></table>
+
+# E MISSING RELATED WORK
+
+For KL regularized policy improvement with fixed temperature, the optimal policy has a closed form solution (Wang et al., 2020). Although it avoids estimating the behavior policy, the issue with fixed temperature is that all the states are penalized equally. This leads to over-pessimistic for states with higher occurence in the dataset and over-optimistic for states with lower occurence. (Fox, 2019) presents a closed-form expression for the regularization coefficient that completely eliminates the bias in entropy-regularized value updates. However, the softmax operator introduced by the approach makes it hard to use in continuous action space.
+
+# F IMPLEMENTATION DETAILS
+
+Computation of the analytical KL upper bound To reduce the variance when computing the analytical KL upper bound, we sample $L$ latent variable $z$ and compute the average of the $L$ KL upper bounds. In our experiments, we set $L = 5$ . Note that it doesn’t reduce the bias of the upper bound.
+
+Reward scaling Any affine transformation of the reward function does not change the optimal policy of the MDP. In our experiments, we rescale the reward to $[ 0 , 1 ]$ as:
+
+$$
+r ^ { \prime } = ( r - r _ { m i n } ) / ( r _ { m a x } - r _ { m i n } )
+$$
+
+where $r _ { m a x }$ and $r _ { m i n }$ is the maximum and the minimum reward in the dataset.
+
+Entropy regularization The KL divergence is the sum of negative entropy of the learned policy plus the cross entropy between the learned policy and the behavior policy: $\bar { \mathcal { D } } _ { \mathrm { K L } } ( \pi _ { \theta } ( \cdot | s ) , \pi _ { \beta } ( \cdot | \bar { s } ) ) \stackrel { \cdot } { = }$ $\bar { - \mathcal { H } } ( \pi _ { \theta } ( \cdot | s ) ) + \mathcal { H } ( \bar { \pi _ { \theta } } ( \cdot | s ) , \pi _ { \beta } ( \cdot | s ) )$ . When the learned policy distribution violates the KL constraints, the KL divergence between the policy distribution and the behavior distribution is decreased by the optimizer. This is equivalent to increasing the entropy of the learned policy and decreasing the cross entropy between the learned policy and the behavior policy. In soft actor-critic (Haarnoja et al., 2018a), the minimum entropy of the learned policy is enforced to encourage exploration. However, due to the absence of exploration, stochastic policy with large entropy will sample out-ofdistribution actions when computing the target Q values in Equation 2. If such values are overestimated, the policy will exploit the erroneous Q values when performing the policy improvement in Equation 8 and lead to failure, which can’t be corrected without more data. Thus, we maintain the maximum entropy of the learned policy using the technique proposed in (Haarnoja et al., 2018b).
+
+Initialization If the dataset is collected using a narrow policy distribution in a high dimensional space (e.g. human demonstration), the constrained optimization problem using dual gradient descent finds it difficult to converge if random initialization is used for the policy network. To mitigate this issue, we start with a policy that has the minimum KL divergence with the behavior policy: $\begin{array} { r } { \pi _ { \theta } \ = \ \arg \operatorname* { m i n } _ { \pi _ { \theta } \in \Pi } \mathcal { D } _ { \mathrm { K L } } ( \pi , \pi _ { \beta } ) } \end{array}$ , where $\Pi$ represents a family of policy types. In this work, we consider $\Pi$ as Gaussian policies. Correspondingly, we initialize the Q network to $Q ^ { \pi _ { \theta } }$ .
+
+Policy network Our policy network is a 3-layer feed-forward neural network. The size of each hidden layer is 512. We apply RELU activation (Agarap, 2018) after each hidden layer. Following (Haarnoja et al., 2018a), the output is a Gaussian distribution with diagonal covariance matrix. We apply tanh to enforce the action bounds. The log-likelihood after applying the tanh function has a simple closed form solution. We refer to (Haarnoja et al., 2018a) Appendix C for more details.
+
+Q network Following (Haarnoja et al., 2018a; Fujimoto et al., 2018a; Wu et al., 2019), we train two independent Q network $\{ Q _ { \psi _ { 1 } } , Q _ { \psi _ { 2 } } \}$ to penalize uncertainty over the future states. We maintain a target Q network $\{ Q _ { \psi _ { 1 } ^ { \prime } } , Q _ { \psi _ { 2 } ^ { \prime } } \}$ with the same architecture and update the target weights using a weighted sum of the current Q network and the target Q network. When computing the target Q values, we simply take the minimum value of the two Q networks:
+
+$$
+Q _ { \psi ^ { \prime } } ( s ^ { \prime } , a ^ { \prime } ) = \operatorname* { m i n } _ { j = 1 , 2 } Q _ { \psi ^ { \prime } { } _ { j } } ( s ^ { \prime } , a ^ { \prime } )
+$$
+
+Each Q network is a 3-layer feed-forward neural network. The size of each hidden layer is 256. We apply RELU activation (Agarap, 2018) after each hidden layer.
+
+Behavior policy network Following the previous work (Fujimoto et al., 2018b; Kumar et al., 2019), we learn a conditional variational auto-encoder (Kingma & Welling, 2014) as our behavior policy network. The encoder takes a pair of states and actions, and outputs a Gaussian latent variable $Z$ . The decoder takes sampled latent code $z$ and states, and outputs a mixture of Gaussian distributions. Both the architecture of the encoder and the decoder is a 3-layer feed-forward neural network. The size of each hidden layer is 512. The activation is relu (Agarap, 2018). To avoid epistemic uncertainty, we train $B$ ensembles of behavior policy networks. At test time, we randomly select one model to perform the calculations. We found $B = 3$ is sufficient for all the experiments. We pre-train the the behavior policy network for $4 0 0 \mathrm { k }$ gradient steps.
+
+$\alpha$ network The $\alpha$ network takes in a state and outputs the Lagrange multiplier for the state. The architecture of the $\alpha$ network is a 3-layer feed-forward neural network with relu (Agarap, 2018) activation. The size of each hidden layer is 256. We use softplus activation after the output to ensure that all the values are positive.
+
+Table 4: Default hyper-parameters   
+
+<table><tr><td>Hyper-parameter</td><td>Value (Gym/Adroit)</td></tr><tr><td>Optimizer</td><td>Adam (Kingma &amp; Ba,2015)</td></tr><tr><td>Policy learning rate</td><td>5e-6/5e-8</td></tr><tr><td>Q network learning rate</td><td>3e-4</td></tr><tr><td>α learning rate</td><td>le-5/1e-7</td></tr><tr><td>batch size</td><td>100</td></tr><tr><td>Target update rate T</td><td>1e-3</td></tr><tr><td>Discount factor y</td><td>0.99</td></tr><tr><td>Initial β</td><td>10</td></tr><tr><td>β learning rate</td><td>1e-3</td></tr><tr><td>Steps per epoch T</td><td>2000</td></tr><tr><td>Number of epochs</td><td>500</td></tr></table>
+
+Table 5: Task-specific hyper-parameters for OpenAI gym tasks.   
+
+<table><tr><td>Task name</td><td>KL divergence threshold ∈kL</td><td>Maximum entropy Ho</td></tr><tr><td>halfcheetah-rand</td><td>9</td><td>-3</td></tr><tr><td>walker2d-rand</td><td>0.1</td><td>6</td></tr><tr><td>hopper-rand</td><td>3</td><td>-3</td></tr><tr><td>halfcheetah-med</td><td>4</td><td>-12</td></tr><tr><td>walker2d-med</td><td>2.1</td><td>-9</td></tr><tr><td>hopper-med</td><td>2.4</td><td>-6</td></tr><tr><td>halfcheetah-med-exp</td><td>11.5</td><td>-24</td></tr><tr><td>walker2d-med-exp</td><td>5</td><td>-12</td></tr><tr><td>hopper-med-exp</td><td>2.6</td><td>-6</td></tr><tr><td>halfcheetah-mixed</td><td>6</td><td>-12</td></tr><tr><td>walker2d-mixed</td><td>4</td><td>-6</td></tr><tr><td>hopper-mixed</td><td>3</td><td>-3</td></tr></table>
+
+Table 6: Task-specific hyper-parameters for Adroit tasks with human demonstrations.   
+
+<table><tr><td>Task name</td><td>MMD threshold ∈MMD</td><td>Minimum entropy Ho</td></tr><tr><td>pen-human</td><td>0.06</td><td>-200</td></tr><tr><td>hammer-human</td><td>0.1</td><td>-60</td></tr><tr><td>door-human</td><td>0.1</td><td>-60</td></tr><tr><td>relocate-human</td><td>0.1</td><td>-60</td></tr></table>

parse/train/bMCfFepJXM/bMCfFepJXM_content_list.json

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+    {
+        "type": "text",
+        "text": "BRAC $+$ : GOING DEEPER WITH BEHAVIOR REGULARIZED OFFLINE REINFORCEMENT LEARNING ",
+        "text_level": 1,
+        "bbox": [
+            176,
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Anonymous authors Paper under double-blind review ",
+        "bbox": [
+            184,
+            171,
+            398,
+            198
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "ABSTRACT ",
+        "text_level": 1,
+        "bbox": [
+            454,
+            234,
+            544,
+            251
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Online interactions with the environment to collect data samples for training a Reinforcement Learning agent is not always feasible due to economic and safety concerns. The goal of Offline Reinforcement Learning (RL) is to address this problem by learning effective policies using previously collected datasets. Standard off-policy RL algorithms are prone to overestimations of the values of outof-distribution (less explored) actions and are hence unsuitable for Offline RL. Behavior regularization, which constraints the learned policy within the support set of the dataset, has been proposed to tackle the limitations of standard off-policy algorithms. In this paper, we improve the behavior regularized offline reinforcement learning and propose $B R A C +$ . We use an analytical upper bound on KL divergence as the behavior regularizor to reduce variance associated with sample based estimations. Additionally, we employ state-dependent Lagrange multipliers for the regularization term to avoid distributing KL divergence penalty across all states of the sampled batch. The proposed Lagrange multipliers allow more freedom of deviation to high probability (more explored) states leading to better rewards while simultaneously restricting low probability (less explored) states to prevent out-of-distribution actions. To prevent catastrophic performance degradation due to rare out-of-distribution actions, we add a gradient penalty term to the policy evaluation objective to penalize the gradient of the Q value w.r.t the out-of-distribution actions. By doing so, the Q values evaluated at the out-ofdistribution actions are bounded. On challenging offline RL benchmarks, ${ \\mathrm { B R A C } } +$ outperforms the state-of-the-art model-free and model-based approaches. ",
+        "bbox": [
+            233,
+            266,
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "1 INTRODUCTION ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            599,
+            336,
+            614
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Reinforcement Learning (RL) has shown great success in a wide range of applications including board games (Silver et al., 2016), strategy games (Vinyals et al., 2019), energy systems (Zhang et al., 2019), robotics (Lin, 1992), recommendation systems (Choi et al., 2018), etc. The success of RL relies heavily on extensive online interactions with the environment for exploration. However, this is not always feasible in the real world as it can be expensive or dangerous (Levine et al., 2020). ",
+        "bbox": [
+            174,
+            631,
+            825,
+            700
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Offline RL, also known as batch RL, avoids online interactions with the environment by learning from a static dataset that is collected in an offline manner (Levine et al., 2020). While standard offpolicy RL algorithms (Mnih et al., 2013; Lillicrap et al., 2016; Haarnoja et al., 2018a) can, in theory, be employed to learn from an offline data, in practice, they perform poorly due to distributional shift between the behavior policy (probability distribution of actions conditioned on states as observed in the dataset) of the collected dataset and the learned policy (Levine et al., 2020). The distributional shift manifests itself in form of overestimation of the out-of-distribution (OOD) actions leading to erroneous Bellman backups. ",
+        "bbox": [
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+            708,
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Prior works tackle this problem via behavior regularization (Fujimoto et al., 2018b; Kumar et al., 2019; Wu et al., 2019; Siegel et al., 2020). This ensures that the learned policy stays “close” to the behavior policy. This is achieved by adding a regularization term that calculates the $f$ -divergence between the learned policy and the behavior policy. Kernel Maximum Mean Discrepancy (MMD) (Gretton et al., 2007), Wasserstein distance and KL divergence are widely used (Wu et al., 2019). The regularization term is either fixed (Wu et al., 2019), or tuned via dual gradient descent (Kumar et al., 2019), or applied using a trust region objective (Siegel et al., 2020). ",
+        "bbox": [
+            174,
+            827,
+            825,
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "In this paper, we propose improvements to the Behavior Regularized Actor Critic (BRAC) algorithm presented in (Wu et al., 2019). To obtain the same, we observe that sample based estimation of divergence measures is computationally expensive and prone to higher variance. Therefore, to reduce variance, we derive an analytical upper bound on the KL divergence measure as the regularization term in the objective function. Moreover, we show that current works that apply the regularization term i.e. the divergence measure, on the entire batch, end up distributing the penalty over all states in the batch in amounts inversely proportional to the state’s probability of occurrence in the batch. This needlessly restricts the deviation of highly explored states while allowing less explored ones to deviate farther leading to OOD actions. To address the same, we employ state dependent Lagrange multipliers for the regularization terms and automatically tune their strength using state-wise dual gradient descent. In addition, the performance of the learned agent trained using prior methods often deteriorates over the course of training. We found that if the learned Q function generalizes such that the gradient of the Q function w.r.t the OOD actions is monotonically increasing, behavior regularization fails to keep such actions within the support set. To mitigate this issue, we penalize the gradient of the Q function w.r.t the OOD actions by adding a gradient penalty term to the policy evaluation objective. This reduces the policy improvement at OOD actions to the problem of minimizing the divergence between the learned policy and the behavior policy. ",
+        "bbox": [
+            174,
+            103,
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "We call our improved algorithm ${ \\mathrm { B R A C } } +$ following (Wu et al., 2019). Our experiments suggest that ${ \\mathrm { B R A C } } +$ outperforms existing state-of-the-art model-free and model-based offline RL algorithms in various datasets on the D4RL benchmark (Fu et al., 2020). ",
+        "bbox": [
+            176,
+            347,
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "2 BACKGROUND ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            409,
+            326,
+            424
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Markov Decision Process RL algorithms aim to solve Markov Decision Process (MDP) with unknown dynamics. A Markov decision process (Sutton & Barto, 2018) is defined as a tuple $<$ ${ \\mathcal { S } } , { \\mathcal { A } } , R , P , \\mu >$ , where $s$ is the set of states, $\\mathcal { A }$ is the set of actions, $R ( s , a , s ^ { \\prime } ) : \\mathcal { S } \\times \\mathcal { A } \\times \\mathcal { S }  \\mathbb { R }$ defines the intermediate reward when the agent transitions from state $s$ to $s ^ { \\prime }$ by taking action $a$ , $P ( s ^ { \\prime } | s , a ) : \\mathcal { S } \\times \\mathcal { A } \\times \\mathcal { S } \\to [ 0 , 1 ]$ defines the probability when the agent transitions from state $s$ to $s ^ { \\prime }$ by taking action $a$ , $\\mu : { \\dot { \\mathcal { S } } } \\ { \\dot {  } } \\ [ 0 , 1 ]$ defines the starting state distribution. The objective of reinforcement learning is to select policy ${ \\dot { \\pi } } : \\mu \\to P ( A )$ to maximize the following objective: ",
+        "bbox": [
+            173,
+            439,
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "equation",
+        "img_path": "images/f19dc9a3e1b9ae15ef135390927c29082d3cea38d1df43b0c252b7cb7e077397.jpg",
+        "text": "$$\nJ ( \\pi ) = \\underset { s _ { 0 } \\sim \\mu , a _ { t } \\sim \\pi ( \\cdot | s _ { t } ) } { \\mathbb { E } } [ \\sum _ { t = 0 } ^ { \\infty } \\gamma ^ { t } R ( s _ { t } , a _ { t } , s _ { t + 1 } ) ]\n$$",
+        "text_format": "latex",
+        "bbox": [
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+            592
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Offline Reinforcement Learning The goal of offline RL is to learn policy $\\pi _ { \\theta }$ from a fixed dataset $\\mathcal { D } = \\{ ( s _ { i } , a _ { i } , s ^ { \\prime } { } _ { i } , r _ { i } ) \\} _ { i = 1 } ^ { N }$ consisting of single step transitions $\\left\\{ \\left( s _ { i } , a _ { i } , s ^ { \\prime } _ { i } , r _ { i } \\right) \\right\\}$ . The dataset is assumed to be collected using a behavior policy $\\pi _ { \\beta }$ which denotes the conditional distribution $p ( a | s )$ observed in the dataset. Note that $\\pi _ { \\beta }$ may consist of multi-modal policy distribution. In principle, standard off-policy RL algorithms using a replay buffer (Mnih et al., 2013; Lillicrap et al., 2016; Haarnoja et al., 2018a) can directly learn from $\\mathcal { D }$ . The key challenge resides in the policy evaluation step: ",
+        "bbox": [
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "equation",
+        "img_path": "images/b31a9b235e06cf7dca169aaec0f533d273e7f89f396c64137cbe02f78323eb02.jpg",
+        "text": "$$\nQ _ { \\psi } = \\arg \\operatorname* { m i n } _ { \\psi } [ ( Q _ { \\psi } ( s , a ) - ( r ( s , a ) + \\gamma \\mathbb { E } _ { a ^ { \\prime } \\sim \\pi _ { \\theta } } Q _ { \\psi ^ { \\prime } } ( s ^ { \\prime } , a ^ { \\prime } ) ) ) ] ^ { 2 } \\qquad \\mathrm { ( p o l i c y ~ e v a l u a t i o n ) }\n$$",
+        "text_format": "latex",
+        "bbox": [
+            214,
+            708,
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "In this step, the target $\\mathrm { Q }$ value depends on the learned policy. If the learned policy distribution $\\pi _ { \\theta }$ diverges from the data distribution $\\pi _ { \\beta }$ , it results in evaluation of target Q values using outof-distribution (OOD) actions. Such evaluations are prone to errors. Erroneous overestimation of values get exploited by the policy improvement, preventing the algorithm from learning useful policies. In order to avoid such cases, behavior regularization is adopted to force the learned policy to stay “close” to the behavior policy (Fujimoto et al., 2018b; Kumar et al., 2019; Wu et al., 2019). ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "3 IMPROVING BEHAVIOR REGULARIZED OFFLINE REINFORCEMENT LEARNING ",
+        "text_level": 1,
+        "bbox": [
+            176,
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "In this section, we discuss and propose three non-trivial improvements to the Behavior Regularized Actor Critic (BRAC) offline reinforcement learning (Wu et al., 2019). BRAC (Wu et al., 2019) ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "augments either the policy improvement or the policy evaluation step with a penalty term to constrain the policy within the support set of the behavior policy. For simplicity of illustration, we only consider the policy improvement step (Wu et al., 2019): ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/4b9148b79c78a204a759e2b24f047ee67bef674059b7d9c6ece703b049614d5a.jpg",
+        "text": "$$\n\\operatorname* { m a x } _ { \\pi _ { \\theta } } \\mathbb { E } _ { ( s , a , r , s ^ { \\prime } ) \\sim \\mathcal { D } } [ \\mathbb { E } _ { a ^ { \\prime \\prime } \\sim \\pi _ { \\theta } ( \\cdot \\vert s ) } [ Q _ { \\psi } ( s , a ^ { \\prime \\prime } ) ] - \\alpha \\hat { D } ( \\pi _ { \\theta } ( \\cdot \\vert s ) , \\pi _ { \\beta } ( \\cdot \\vert s ) ) ]\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Here, $\\hat { D }$ is a selected $f$ -divergence (Csiszar, 1972) measure used for behavior regularization. Zoom-´ ing into the regularization term, we get: ",
+        "bbox": [
+            174,
+            209,
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/7a84cd86b7d50b3721174e1289d32cd24fa78f7aa8b5863bc3d92cb09c530362.jpg",
+        "text": "$$\n\\sum _ { s \\in S } ( P _ { s } \\alpha ) \\hat { D } ( \\pi _ { \\theta } ( \\cdot | s ) , \\pi _ { \\beta } ( \\cdot | s ) )\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "In the equation above, $s$ denotes the states that occur in a single sampled batch and $P _ { s }$ is the probability of state $s$ in the sampled batch. The first two improvements that we propose are targeted towards the following two terms in Equation 4: (1) The divergence term $\\hat { D }$ which for each fixed $\\pi _ { \\theta }$ determines the impact actions have on the overall objective conditional on the state, and (2) the term $( P _ { s } \\alpha )$ which determines the impact the “divergence penalty” of each state will have on the overall objective function. Note that this term is dependent upon the probability of the occurrence of state in the sampled batch. For (1), we propose to use an analytical upper bound of KL divergence to reduce variance. For (2), we propose to decouple the impact of “divergence penalty” of each state from the probability of its occurrence. Detailed explanation of the two improvements and the rationale behind them follows in the next two sub-sections. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "3.1 REGULARIZATION METHOD ",
+        "text_level": 1,
+        "bbox": [
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+        "text": "Behavior regularization is used to constrain the learned policy with the support set of the behavior policy. In other words, it ensures that the “difference” between the probability distributions of the learned policy $\\pi _ { \\boldsymbol { \\theta } } ( \\cdot | \\boldsymbol { s } )$ and behavior policy $\\pi _ { \\beta } ( \\cdot | s )$ is small. The following divergence measures are most widely used in the community: ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Kernel MMD Kernel Maximum Mean Discrepancy (MMD) (Gretton et al., 2007) was first introduced in (Kumar et al., 2019) to penalize the policy from diverging from the behavior policy: ",
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+    },
+    {
+        "type": "equation",
+        "img_path": "images/cd2d80cd63dbf86e9cdd81fce18de0b47d986b79c671d0e0add7933df2fe16ae.jpg",
+        "text": "$$\n\\mathbf { M M D } _ { k } ^ { 2 } ( \\pi ( \\cdot | s ) , \\pi _ { b } ( \\cdot | s ) ) = \\underset { x , x ^ { \\prime } \\sim \\pi ( \\cdot | s ) } { \\mathbb { E } } [ K ( x , x ^ { \\prime } ) ] - 2 \\underset { y \\sim \\pi ( \\cdot | s ) } { \\mathbb { E } } [ K ( x , y ) ] + \\underset { y , y ^ { \\prime } \\sim \\pi _ { b } ( \\cdot | s ) } { \\mathbb { E } } [ K ( y , y ^ { \\prime } ) ]\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "where $K$ is a kernel function. Symmetric kernel functions such as Laplacian and Gaussian kernels are typically used (Gretton et al., 2007). ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "KL divergence For two probability distributions $P$ and $Q$ on some probability space $\\chi$ , the KLDivergence from $Q$ to $P$ is given as ",
+        "bbox": [
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+    {
+        "type": "equation",
+        "img_path": "images/6cd685fc5bdcbbde04fa80ea98162a97577c28bd5453c4d15bf3be1b06b32315.jpg",
+        "text": "$$\n\\mathcal { D } _ { \\mathrm { K L } } ( P , Q ) = \\int _ { x \\sim \\chi } P ( x ) \\log \\frac { P ( x ) } { Q ( x ) } d x\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "KL divergence is assymetric and both $\\mathcal { D } _ { \\mathrm { K L } } \\big ( \\pi _ { \\theta } \\big ( \\cdot | s \\big ) , \\pi _ { b } \\big ( \\cdot | s \\big ) \\big )$ and $\\mathcal { D } _ { \\mathrm { K L } } ( \\pi _ { b } ( \\cdot | s ) , \\pi _ { \\theta } ( \\cdot | s ) )$ are valid. However, $\\mathcal { D } _ { \\mathrm { K L } } ( \\pi _ { b } ( \\cdot | s ) , \\pi _ { \\theta } ( \\cdot | s ) )$ is not suitable because it requires $\\pi _ { \\boldsymbol { \\theta } } ( a \\vert s ) \\neq 0$ when $\\dot { \\pi _ { b } } ( a | s ) \\neq 0$ , for each $a$ . It may assign $\\pi _ { \\theta } ( a | s ) > 0$ when $\\pi _ { b } ( a | s ) = 0$ , producing out of distribution actions. Therefore, $\\mathcal { D } _ { \\mathrm { K L } } \\big ( \\pi _ { \\theta } \\big ( \\cdot | s \\big ) , \\pi _ { b } \\big ( \\cdot | s \\big ) \\big )$ is used for regularization. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "3.1.1 ANALYTICAL KL DIVERGENCE UPPER BOUND ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "All the existing behavior regularized methods estimate the divergence via samples (Kumar et al., 2019; Wu et al., 2019). While in theory it produces an unbiased estimator, it requires a large number of samples to reduce the variance. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "To stabilize the performance, the key idea is to have a low variance estimator for KL divergence. We obtain this by deriving an upper bound on the KL divergence between the learned policy $\\pi _ { \\theta }$ and the behavior policy $\\pi _ { \\beta }$ that can be computed analytically. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Assume we learn $\\pi _ { \\beta }$ using a latent variable model (e.g. VAE (Kingma & Welling, 2014)) with latent variable $Z$ . According to the evidence lower bound (ELBO), we obtain $\\log \\pi _ { \\beta } ( a | s ) \\geq$ $\\mathbb { E } _ { z \\sim q ( z ) } [ \\log p ( a | s , z ) ] - \\mathcal { D } _ { \\mathrm { K L } } ( \\bar { q } ( z ) | | p ( z ) )$ , where $q ( z )$ is the approximated posterior distribution and $p ( z )$ is the prior. Then, the KL divergence is bounded by: ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/9c311e6daa1f995fa43e12a5c05e932603e3c7d11f6647e99b24366338eaf3e0.jpg",
+        "text": "$$\n\\begin{array} { r l } & { \\mathcal { D } _ { \\mathrm { K L } } ( \\pi _ { \\theta } ( \\cdot | s ) | | \\pi _ { \\beta } ( \\cdot | s ) ) = \\mathbb { E } _ { a \\sim \\pi _ { \\theta } } [ \\log \\pi _ { \\theta } ( a | s ) ] - \\mathbb { E } _ { a \\sim \\pi _ { \\theta } } [ \\log \\pi _ { \\beta } ( a | s ) ] } \\\\ & { \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\mathbb { E } _ { a \\sim \\pi _ { \\theta } } [ \\log \\pi _ { \\beta } ( a | s ) ] - \\mathbb { E } _ { \\kappa \\mathrm { L } } ( q ( z ) | | p ( z ) ) ] } \\\\ & { \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad ( \\mathcal { T } _ { a \\sim \\pi _ { \\theta } } [ \\log ( a | s ) ] + \\mathcal { D } _ { \\mathrm { K L } } ( q ( z ) | | p ( z ) ) ] } \\\\ & { \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad ( \\mathcal { T } _ { a \\sim \\pi _ { \\theta } , z \\sim q ( z ) } [ \\mathcal { D } _ { \\mathrm { K L } } ( \\pi _ { \\theta } ( \\cdot | s ) | | p ( \\cdot | s , z ) ) + \\mathcal { D } _ { \\mathrm { K L } } ( q ( z ) | | p ( z ) ) ] } \\\\ & { \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad ( \\mathcal { T } _ { a \\sim \\pi _ { \\theta } , z \\sim q ( z ) } [ \\mathcal { D } _ { \\mathrm { K L } } ( \\pi _ { \\theta } ( \\cdot | s ) | | p ( \\cdot | s , z ) ) + \\mathcal { D } _ { \\mathrm { K L } } ( q ( z ) | | p ( z ) ) ] } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Thus, if we choose $p ( \\cdot | s , z )$ to be a tractable distribution, Equation 7 can be computed analytically. Note that although $p ( \\cdot | s , z )$ has a special form (e.g. Gaussian), the latent variable model can in theory represent any probability distribution without sacrificing the expressiveness. In practice, we choose $p ( \\cdot | s , z )$ to be a Gaussian distribution. ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "3.2 STATE DEPENDENT LAGRANGE MULTIPLIER ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "From the term $( P _ { s } \\alpha )$ in Equation 4, one can infer that the “penalty budget” of divergence measure gets distributed among the states in amounts inversely proportional to the probability of the occurrence of the states in the sampled batch. If we assume the batches are sampled uniformly, this implies that the states which are more numerous in the offline dataset $\\mathcal { D }$ i.e. the states that have been explored more thoroughly are restricted from deviating from the behavior policy. Whereas the less numerous state that have had limited exploration enjoy more freedom for deviation. This is undesirable as restricting highly explored states is overly conservative while allowing less explored states to deviate can lead to OOD actions. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "To address this we add a state wise constraint for KL divergence as follows: ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/67a260b9cd216ce101dc88c6442e968b7cc7cc01bab77b358d67684205133b4c.jpg",
+        "text": "$$\n\\operatorname* { m a x } _ { \\pi _ { \\alpha } \\Lambda } \\mathbb { E } _ { s \\sim \\mathcal { D } } [ \\mathbb { E } _ { a ^ { \\prime } \\sim \\pi _ { \\theta } ( \\cdot | s ) } [ Q _ { \\psi } ( s , a ^ { \\prime } ) ] ] \\mathrm { ~ s . t . ~ } \\mathcal { D } _ { K L } ( \\pi _ { \\theta } ( \\cdot | s ) , \\pi _ { \\beta } ( \\cdot | s ) ) \\leq \\epsilon _ { \\mathrm { K L } } , \\forall s \\quad \\mathrm { ( p o l i c ) } .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "This translates to assignment of state dependent Lagrange multipliers when solved using dual gradient descent: ",
+        "bbox": [
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+        "page_idx": 3
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+    {
+        "type": "equation",
+        "img_path": "images/b7de9ef8e56f4ef39fbd7836ff7e55dad91bae5a307f059e7e381da859601c1c.jpg",
+        "text": "$$\n\\operatorname* { m a x } _ { \\pi _ { \\theta } } \\mathbb { E } _ { s \\sim \\mathcal { D } } \\big [ \\mathbb { E } _ { a ^ { \\prime } \\sim \\pi _ { \\theta } ( \\cdot \\vert s ) } [ Q _ { \\psi } ( s , a ^ { \\prime } ) - \\alpha ( s ) \\mathcal { D } _ { \\mathrm { K L } } ( \\pi _ { \\theta } ( \\cdot \\vert s ) , \\pi _ { \\beta } ( \\cdot \\vert s ) ) ] \\big ]\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Maintaining a Lagrange multiplier for each state is impractical for large datasets. In practice, we parameterize $\\alpha ( s )$ with a neural network. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "3.3 GRADIENT PENALIZED POLICY EVALUATION ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "The fundamental challenge in offline reinforcement learning is to mitigate the impact of erroneous Q values that are evaluated at out-of-distribution actions and used in policy evaluation. Due to the limited representation capacity of neural networks, such actions are unavoidable for large datasets, even with state-dependent behavior regularization. CQL (Kumar et al., 2020) resolves this problem by optimizing a conservative lower bound of the Q value. The key idea in this paper is to bound the Q value at the out-of-distribution actions such that their values are not greater than the Q value of in-distribution actions. We achieve this by augmenting the policy evaluation step with a gradient penalty regularization term. To elaborate our approach, we first analyze the gradient of the policy improvement step: ",
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+    {
+        "type": "equation",
+        "img_path": "images/5492de82632953466d1e3fc4b0134796049a2c861891e41483934ab895bcd76b.jpg",
+        "text": "$$\n\\nabla _ { \\theta } J \\approx \\frac { 1 } { N } \\sum _ { i = 1 } ^ { N } \\nabla _ { a _ { i } } Q _ { \\phi } ( s , a ) | _ { s = s _ { i } , a = a _ { i } } \\nabla _ { \\theta } \\pi _ { \\theta } ( a | s ) | _ { s = s _ { i } , a = a _ { i } } - \\alpha ( s _ { i } ) \\nabla _ { \\theta } \\mathcal { D } _ { \\mathrm { K L } } ( \\pi _ { \\theta } ( \\cdot | s ) , \\pi _ { \\beta } ( \\cdot | s ) )\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "If the current policy produces out-of-distribution actions and the Q network erroneously generalizes in such a way that the gradient of Q network is monotonically increasing, this leads to the unbounded value of the Q network and the failure of the behavior regularization. We created a toy example to illustrate this phenomenon in Appendix A. The analysis suggests that if we penalize the gradient of the Q network with respect to the out-of-distribution actions such that they are close to zero, ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Algorithm 1 BRAC+: Improved Behavior Regularized Actor Critic ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "1: Train the behavior policy $\\pi _ { \\beta }$ on the offline dataset $\\mathcal { D } = \\{ ( s _ { i } , a _ { i } , r _ { i } , s _ { i } ^ { ' } ) \\} _ { i = 1 } ^ { N }$ via maximum   \nlikelihood estimation: $\\begin{array} { r } { \\pi _ { \\beta } = \\arg \\operatorname* { m a x } _ { \\pi _ { \\beta } } \\sum _ { i = 1 } ^ { N } \\log \\pi _ { \\beta } ( a _ { i } | s _ { i } ) } \\end{array}$   \n2: Train initial policy: $\\begin{array} { r } { \\pi _ { \\theta } = \\arg \\operatorname* { m i n } _ { \\pi _ { \\theta } \\in \\Pi } \\mathcal { D } _ { \\mathrm { K L } } ( \\pi , \\pi _ { \\beta } ) } \\end{array}$   \n3: Train initial Q network: $\\begin{array} { r } { Q _ { \\psi } = \\arg \\operatorname* { m i n } _ { \\psi } [ ( Q _ { \\psi } ( s , a ) - ( r ( s , a ) + \\gamma \\mathbb { E } _ { a ^ { \\prime } \\sim \\pi _ { \\theta } } Q _ { \\psi ^ { \\prime } } ( s ^ { \\prime } , a ^ { \\prime } ) ) ) ] ^ { 2 } } \\end{array}$   \n4: for $e = 1 : E$ do   \n5: for $t = 1 : T$ do   \n6: Update Q network using Equation 10   \n7: Update the policy using Equation 7   \n8: Update $\\alpha$ via dual gradient descent: $\\alpha ( s )  \\alpha ( s ) + \\lambda _ { \\alpha } ( \\mathcal { D } _ { \\mathrm { K L } } ( \\pi _ { \\theta } ( \\cdot | s ) , \\pi _ { \\beta } ( \\cdot | s ) ) - \\epsilon _ { \\mathrm { K L } } )$   \n9: Update $\\beta$ via dual gradient descent: $\\beta \\gets \\beta + \\lambda _ { \\beta } ( \\mathcal { H } ( \\pi _ { \\theta } ( \\cdot | s ) ) - \\mathcal { H } _ { 0 } )$   \n10: Update the target network $\\psi ^ { ' } = \\tau \\psi + ( 1 - \\tau ) \\psi ^ { ' }$   \n11: end for   \n12: end for ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "performing policy improvement step simply reduces to minimizing the KL divergence. Inspired from (Gulrajani et al., 2017), we add a gradient penalty term to the policy evaluation step as: ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/4e3a265e2cd1ef0e8c1bdb68c6f1a5aaef2facc1de8bdd08761ebaca97066aff.jpg",
+        "text": "$$\n\\begin{array} { r l } & { \\underset { \\psi } { \\operatorname* { m i n } } \\ : \\mathbb { E } _ { ( s , a ) \\sim \\mathcal { D } } ( ( Q _ { \\psi } ( s , a ) - r ( s , a ) + \\gamma \\mathbb { E } _ { a ^ { \\prime } \\sim \\pi a \\theta } Q _ { \\psi ^ { \\prime } } ( s ^ { \\prime } , a ^ { \\prime } ) ) ^ { 2 } } \\\\ & { \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad + \\lambda \\mathbb { E } _ { a ^ { \\prime \\prime } \\sim \\pi \\pi _ { \\theta } ( \\cdot \\vert s ) } ( \\vert \\vert \\nabla _ { a ^ { \\prime \\prime } } Q _ { \\psi } ( s , a ^ { \\prime \\prime } ) \\vert \\vert _ { 2 } f ( \\mathcal { D } _ { K L } ( \\pi _ { \\theta } ( \\cdot \\vert s ) , \\pi _ { \\beta } ( \\cdot \\vert s ) ) - \\epsilon _ { \\mathrm { G P } } ) ) ) } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "where $f$ is a non-decreasing function and $\\epsilon _ { G P }$ is the threshold for gradient penalty. In this paper, we set $f$ to be the Heaviside step function (or indicator). Other variants such as its soft version (sigmoid function) is left for future work. Setting $\\epsilon _ { G P } ~ = ~ \\epsilon _ { K L }$ is too conservative because it prevents generalization. In practice, we set $\\epsilon _ { G P }$ at epoch $t$ to be $\\mu _ { K L } + \\sigma _ { K L }$ , where $\\mu _ { K L }$ and $\\sigma _ { K L }$ is the mean and standard deviation of the KL divergence for all the states in the dataset at epoch $t - 1$ . In addition, we found that $\\lambda = 0 . 1$ works well for all the tasks. ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "4 RELATED WORK ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "We briefly summarize prior works in offline RL and discuss their relationship with our approach. As discussed in Section 1, the fundamental challenge in learning from a static data is to avoid out-of-distribution actions (Levine et al., 2020). This requires solving two problems: 1) estimation of behavior policy, 2) quantification of out-of-distribution actions. We follow BCQ (Fujimoto et al., 2018b), BEAR (Kumar et al., 2019) and BRAC (Wu et al., 2019) by learning the behavior policy using a conditional VAE (Kingma & Welling, 2014). To avoid out-of-distribution actions, BCQ generates actions in the target values by perturbing the behavior policy. However, this is over-pessimistic in most of the cases. BRAC (Wu et al., 2019) constrains the policy using various sample-based $f$ -divergence measures including MMD, Wasserstein distance and KL divergence with penalized policy improvement or policy evaluation. BEAR (Kumar et al., 2019) is an instance of BRAC with penalized policy improvement using MMD (Gretton et al., 2007). Sample-based estimation is computationally expensive and suffers from high variance. In contrast, our method uses an analytical upper-bound of the KL divergence to constrain the distance between the learned policy and the behavior policy. It is both computationally efficient and has low variance. (Siegel et al., 2020) solves trust-region objective instead of using penalty. CQL (Kumar et al., 2020) avoids estimating the behavior policy by learning a conservative Q function that lower-bounds its true value. Hyperparameter search is another challenging problem in offline RL. (Lee et al., 2020) uses a gradient-based optimization of the hyperparameter using held-out data. MOPO (Yu et al., 2020) follows MBPO Janner et al. (2019) with additional reward penalty on unreliable model-generated transitions. MBOP (Argenson & Dulac-Arnold, 2020) learns the dynamics mode, the behavior policy and a truncated value function to perform online planning. (Kidambi et al., 2020) learns a surrogate MDP using the dataset, such that taking out-of-distribution actions transit to the terminal state. The out-of-distribution actions are detected using the agreement of model ensembles. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "5 EXPERIMENTS ",
+        "text_level": 1,
+        "bbox": [
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+        "type": "text",
+        "text": "Our experiments1 aim to answer the following questions: 1) How does the performance of our improvements compare with state-of-the-art model-free and model-based offline RL methods? 2) How does the use of analytical variational upper bound on KL divergence for regularization term compare with sampled-based MMD? 3) How does the state-dependent Lagrange multiplier based regularization (state-wise regularizor) compare with global policy regularization? 4) Does the gradient penalized policy evaluation improve the stability during the training? To answer these questions, we evaluate our methods on a subset of the D4RL (Fu et al., 2020) benchmark. We consider three locomotion tasks (hopper, walker2d, and halfcheetah) and four types of datasets: 1) random (rand): collect the interactions of a run of random policy for 1M steps to create the dataset, 2) medium (med): collect the interactions of a run of medium quality policy for 1M steps as the dataset, 3) medium-expert (med-exp): run a medium quality policy and an expert quality policy for 1M steps, respectively, and combine their interactions to create the dataset, 4) mixed (medium-replay): train a policy using SAC (Haarnoja et al., 2018a) until the performance of the learned policy exceeds a pre-determined threshold, and take the replay buffer as the dataset. In addition, we consider more complex Adroit tasks (Rajeswaran et al., 2018) that requires controlling a 24-DoF robotic hand, using limited data from human demonstrations. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "We compare against state-of-the-art model-free and model-based baselines, including behavior cloning, BEAR (Kumar et al., 2019) that constrains the learned policy within the support of the behavior policy using sampled MMD, BRAC-p/v (Wu et al., 2019) that constrains the learned policy within the support of the behavior policy using various sample-based $f$ -divergences to penalize either the policy improvement (p) or the policy evaluation (v), $\\operatorname { C Q L } ( { \\mathcal { H } } )$ (Kumar et al., 2020) that learns a Q function that lower-bounds its true value. We also compare against model-based approaches including MOPO (Yu et al., 2020) that follows MBPO (Janner et al., 2019) with additional reward penalties and MBOP (Argenson & Dulac-Arnold, 2020) that learns an offline model to perform online planning. ",
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+    {
+        "type": "text",
+        "text": "5.1 COMPARATIVE RESULTS ",
+        "text_level": 1,
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+        "type": "table",
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+        "table_caption": [
+            "Table 1: Results for OpenAI gym (Brockman et al., 2016) environments in the D4RL (Fu et al., 2020) datasets. For each task, we train for 1 million gradient steps and report the performance by running the policy obtained at the last epoch of the training for 100 episodes, averaged over 4 random seeds with standard deviation. Each number is the normalized score as proposed in (Fu et al., 2020). Please refer to (Fu et al., 2020) for results on more baselines. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=\"2\">Task Name</td><td colspan=\"4\">Model-Free</td><td colspan=\"2\">Model-Based</td></tr><tr><td>BEAR</td><td>BRAC-p/v</td><td>CQL(H)</td><td>BRAC+ (Ours)</td><td>MOPO</td><td>MBOP</td></tr><tr><td>halfcheetah-rand</td><td>25.1</td><td>24.1/31.2</td><td>35.4</td><td>26.4±1.0</td><td>31.9±2.8</td><td>6.3±4.0</td></tr><tr><td>walker2d-rand</td><td>7.3</td><td>-0.2/1.9</td><td>7.0</td><td>16.7± 2.3</td><td>13.0±2.6</td><td>8.1± 5.5</td></tr><tr><td>hopper-rand</td><td>11.4</td><td>11.0/12.2</td><td>10.8</td><td>12.5±0.3</td><td>13.3±1.6</td><td>10.8± 0.3</td></tr><tr><td>halfcheetah-med</td><td>41.7</td><td>43.8/46.3</td><td>44.4</td><td>46.6±0.6</td><td>40.2± 2.7</td><td>44.6±0.8</td></tr><tr><td>walker2d-med</td><td>59.1</td><td>77.5/81.1</td><td>79.2</td><td>75.1±3.5</td><td>14.0±10.1</td><td>41.0 ± 29.4</td></tr><tr><td>hopper-med</td><td>52.1</td><td>32.7/31.1</td><td>58.0</td><td>53.2±3.1</td><td>26.5±3.7</td><td>48.8± 26.8</td></tr><tr><td>halfcheetah-med-exp</td><td>53.4</td><td>44.2/41.9</td><td>62.4</td><td>61.2±2.8</td><td>57.9± 24.8</td><td>105.9± 17.8</td></tr><tr><td>walker2d-med-exp</td><td>40.1</td><td>76.9/81.6</td><td>98.7</td><td>95.3± 5.9</td><td>55.0± 19.1</td><td>70.2 ± 36.2</td></tr><tr><td>hopper-med-exp</td><td>96.3</td><td>1.9/0.8</td><td>111.0</td><td>112.9±0.1</td><td>51.7±42.9</td><td>55.1 ± 44.3</td></tr><tr><td>halfcheetah-mixed</td><td>38.6</td><td>45.4/47.7</td><td>46.2</td><td>46.1±0.2</td><td>54.0±2.6</td><td>42.3 ± 0.9</td></tr><tr><td>walker2d-mixed</td><td>19.2</td><td>-0.3/0.9</td><td>26.7</td><td>39.0±4.6</td><td>42.7±8.3</td><td>9.7 ± 5.3</td></tr><tr><td>hopper-mixed</td><td>33.7</td><td>0.6/0.6</td><td>48.6</td><td>72.7±18.9</td><td>92.5±6.3</td><td>12.4 ± 5.8</td></tr></table>",
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+    {
+        "type": "text",
+        "text": "Performance on multi-modal datasets We first compare the performance on multi-modal datasets i.e. med-exp and mixed datasets. Results shown in Table 1 suggest that our method outperforms various model-free baselines on most of the multi-modal datasets, especially on hopper-mix and walker2d-mix by up to $1 . 5 \\mathrm { x }$ . Compared with BEAR (Kumar et al., 2019), our performance improvement arises from the advantage of the KL divergence over the kernel MMD (see discussions in Appendix B). The choice of parameters in $\\operatorname { C Q L } ( \\mathcal { H } )$ makes it too conservative to achieve higher performance. ",
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+        "table_caption": [
+            "Table 2: Results for Adroit tasks with human demonstrations in the D4RL (Fu et al., 2020) datasets. The numbers are reported by following the same procedure as in Table 1 except we run the policy obtained at the last epoch of training for 1000 episodes due to large variance across different runs. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Task Name</td><td>BC</td><td>BEAR</td><td>BRAC-p/v</td><td>CQL(H)</td><td>CQL(p)</td><td>BRAC+ (Ours)</td></tr><tr><td> pen-human</td><td>34.4</td><td>-1.0</td><td>8.1/0.6</td><td>37.5</td><td>55.8</td><td>64.9± 1.6</td></tr><tr><td>hammer-human</td><td>1.5</td><td>0.3</td><td>0.3/0.2</td><td>4.4</td><td>2.1</td><td>3.9 ± 0.9</td></tr><tr><td>door-human</td><td>0.5</td><td>-0.3</td><td>-0.3/-0.3</td><td>9.9</td><td>9.1</td><td>11.5± 1.2</td></tr><tr><td>relocate-human</td><td>0.0</td><td>-0.3</td><td>-0.3/-0.3</td><td>0.20</td><td>0.35</td><td>0.20 ± 0.11</td></tr></table>",
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+        "type": "text",
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+    {
+        "type": "text",
+        "text": "Performance on single-modal datasets The performance of our method on single-modal (rand and med) dataset outperforms or matches with baseline methods as evident from Table 1 except halfcheetah-random dataset. We observe that the performance is very sensitive to the choice of target policy entropy. We hypothesize that the our choice of target policy entropy in the halfcheetahrandom task makes it hard to compose correct sub-optimal policies from a random collected dataset. ",
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+    {
+        "type": "image",
+        "img_path": "images/c3e749d5069b0b1565520c3bc0b5e5d26cad7d2efabb6fab7f4282bf39bc0a5c.jpg",
+        "image_caption": [
+            "Figure 1: Left: learning curve of pen-human-v0 task. Right: the average $\\mathrm { Q }$ value of the first ensemble over the course of training. Both curves are smoothed by a factor of 20. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Performance on datasets with human demonstrations The performance on Adroit tasks is shown in Table 2. These tasks are substantially harder than OpenAI gym tasks due to limited training data in a high dimensional observation and action space. Our method makes non-trivial improvement over the behavior cloning. Compared with the state-of-the-art approaches, our approach is superior on half of the tasks and matches the asymptotic performance on the remaining ones. Figure 1 shows that the Q value is bounded when the gradient penalized policy evaluation technique is employed. On the contrary, the Q value without the gradient penalized policy evaluation increases exponentially. Note that we use MMD with Laplacian kernel for Adroit tasks. We observe that the KL-based regularization struggles with datasets collected with narrow behavior distributions (have large density within a tiny space and almost zero density anywhere else). In such a case, the KL divergence is sensitive to tiny policy changes, making gradient-based optimization hard to converge. ",
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+    {
+        "type": "text",
+        "text": "5.2 ABLATION STUDY ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "To answer question (2), (3) and (4), we conduct a thorough ablation study on ${ \\mathrm { B R A C } } +$ on various tasks with different data collection policies (hopper-mixed, walker2d-medium-expert, halfcheetahmedium). ",
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+    {
+        "type": "text",
+        "text": "Sampled-based MMD vs. analytical upper bound KL The results of using sampled-based MMD versus analytical upper bound KL are shown in Figure 2a. The difference of the performance in the single-modal dataset (halfcheetah-medium) is negligible. However, the performance on multi-modal datasets is varied. Our toy example in Appendix B suggests that MMD and the backward KL divergence tends to cover all the “modes” in the behavior policy while the forward KL divergence tends to seek one of the “mode” in the behavior policy. Note that this argument is only valid if the learned policy is single-modal (e.g. Gaussian distribution). The superior performance in the walker2d-medium-expert when using forward KL regularization supports this argument. The results in hopper-mixed task seems to be contradictory. We hypothesize that since the analytical KL has low variance, the policy quickly adopts the out-of-distribution actions when available while in sample-based methods, such adoption is slower due to higher variance in the policy regularization. ",
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+    {
+        "type": "image",
+        "img_path": "images/936da787ddd671fa414383deb9d728d42c0ce8d083411c0c19e2e2ca738bcbe8.jpg",
+        "image_caption": [
+            "(c) Gradient penalty vs. No gradient penalty. State-wise Lagrange multiplier. Forward KL. ",
+            "Figure 2: Figures of ablation study. Each setting is repeated for 4 random seeds. The curve is the mean and the shaded area is the standard deviation. The curves are smoothed by a factor of 20. The number of gradient steps per epoch is 2000. To make fair comparison, we only substitute KL divergence with MMD-based measurement with additional MMD-specific hyperparameter tuning. The other design choices are different from (Kumar et al., 2019). Details can be found in Appendix D. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "State-wise vs. global regularization The performance of using global and state-wise regularization on various tasks is shown in Figure 2b. It is noticeable that the state-wise regularization improves the performance in hopper-mixed and walker2d-medium-expert task, while the performance in halfcheetah-medium does not show much improvement. To understand the consequence of the state-wise regularization, we plot the histogram of the KL divergence of all the states in the dataset in Figure 3. The KL divergence histogram of using state-wise regularization is more concentrated around the threshold while the global regularization is more sporadic. Strictly enforcing the policy regularization around the threshold helps avoid out-of-distribution actions, that is often caused by a few states in the dataset. ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "image",
+        "img_path": "images/3992389a7bba94f23ff3c5baa970981f092f58a99c2dab5472bed071a685fb24.jpg",
+        "image_caption": [
+            "Figure 3: Histogram of the KL divergence of all the states in the dataset with global and state-wise Lagrange multiplier. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Gradient penalty vs. no gradient penalty Even with policy regularization, evaluating the Q value at the out-of-distribution actions can’t be fully avoided. Thus, it is important to bound the Q value at the out-of-distribution actions such that their values are not greater than the Q value of indistribution actions. Figure 2c shows the performance with and without gradient penalty in the policy evaluation step. While there is little difference in the hopper-mixed and the halfcheetah-medium task, the performance in the walker2d-medium-expert task stabilizes with the gradient penalized policy evaluation. On the contrary, the performance deteriorates over time without gradient penalty. Please refer to Appendix D for more analysis. ",
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "6 DISCUSSIONS AND LIMITATIONS ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "There are several limitations of our approach. The discussion in Appendix D suggests that although KL-regularized offline policy optimization is good at combining sub-optimal policies, they may stuck at local optimums; and they are hard to escape. Another drawback is that the threshold value is hard to set. In our experiments, we try a few numbers that are above the minimum possible KL threshold, which is obtained by training a policy that minimizes that KL divergence. ",
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Finally, we conjecture that the behavior-regularized approach is not sufficient to tackle offline RL problems since it fully ignores the state distribution. To see this, we can create a dataset that only adds a few trajectories from an expert policy to a dataset collected by a low-quality policy. If the low-quality policy doesn’t visit the “good” states in the expert policy (can’t combine sub-optimal policies), behavior-regularized approach leads to a policy that imitates the expert policy. Such imitation is likely to fail due to compounding errors (Ross et al., 2010). The right approach for this dataset is to completely ignore expert trajectories and combine sub-optimal policies in the low-quality regions. To achieve this, we need to consider the state distribution as the density of the “good” states is very low. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "7 CONCLUSION ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "In this paper, we improved the behavior regularized offline reinforcement learning by proposing a low-variance upper bound of the KL divergence estimator to reduce variance, state-dependent Lagrange multiplier to allow more freedom of deviation to high probability states while restricting low probability states and gradient penalized policy evaluation such that the Q values of out-ofdistribution actions are not greater than those of in-distribution actions. Our experimental results on challenging benchmarks illustrate the benefits of our improvements. ",
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
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+        "text": "Richard S. Sutton and Andrew G. Barto. Reinforcement Learning: An Introduction. A Bradford Book, Cambridge, MA, USA, 2018. ISBN 0262039249. ",
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+        "text": "Oriol Vinyals, Igor Babuschkin, Junyoung Chung, Michael Mathieu, Max Jaderberg, Wojtek Czarnecki, Andrew Dudzik, Aja Huang, Petko Georgiev, Richard Powell, Timo Ewalds, Dan Horgan, Manuel Kroiss, Ivo Danihelka, John Agapiou, Junhyuk Oh, Valentin Dalibard, David Choi, Laurent Sifre, Yury Sulsky, Sasha Vezhnevets, James Molloy, Trevor Cai, David Budden, Tom Paine, Caglar Gulcehre, Ziyu Wang, Tobias Pfaff, Toby Pohlen, Dani Yogatama, Julia Cohen, Katrina McKinney, Oliver Smith, Tom Schaul, Timothy Lillicrap, Chris Apps, Koray Kavukcuoglu, Demis Hassabis, and David Silver. AlphaStar: Mastering the Real-Time Strategy Game StarCraft II. https://deepmind.com/blog/ alphastar-mastering-real-time-strategy-game-starcraft-ii/, 2019. ",
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+        "text": "Ziyu Wang, A. Novikov, Konrad Zolna, Jost Tobias Springenberg, Scott Reed, B. Shahriari, N. Siegel, Josh Merel, Caglar Gulcehre, Nicolas Heess, and N. D. Freitas. Critic regularized regression. ArXiv, abs/2006.15134, 2020. ",
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+        "text": "Yifan Wu, George Tucker, and Ofir Nachum. Behavior regularized offline reinforcement learning, 2019. ",
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+            174,
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+        "text": "Tianhe Yu, Garrett Thomas, Lantao Yu, Stefano Ermon, James Zou, Sergey Levine, Chelsea Finn, and Tengyu Ma. Mopo: Model-based offline policy optimization, 2020. ",
+        "bbox": [
+            174,
+            895,
+            820,
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+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Chi Zhang, Sanmukh R. Kuppannagari, Rajgopal Kannan, and Viktor K. Prasanna. Building hvac scheduling using reinforcement learning via neural network based model approximation. In Proceedings of the 6th ACM International Conference on Systems for Energy-Efficient Buildings, Cities, and Transportation, BuildSys ’19, pp. 287–296, New York, NY, USA, 2019. Association for Computing Machinery. ISBN 9781450370059. doi: 10.1145/3360322.3360861. URL https://doi.org/10.1145/3360322.3360861. ",
+        "bbox": [
+            174,
+            103,
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+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "image",
+        "img_path": "images/a1c91ac49d16ce5409884eebbeec19c1806634da6ec66fbfbb476987a6437c29.jpg",
+        "image_caption": [
+            "Figure 4: Fitting a regression model with out of distribution data "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            243,
+            103,
+            758,
+            318
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "A TOY EXAMPLE TO DEMONSTRATE BOUNDING THE VALUE OF THE OUT OF DISTRIBUTION INPUTS VIA GRADIENT PENALTY ",
+        "text_level": 1,
+        "bbox": [
+            173,
+            372,
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+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "To demonstrate the effectiveness of the gradient penalty to enforce the bound of the predicted values of a regression model, we conduct experiments by fitting a regression model with in-distribution data while minimizing the norm of the gradient at out of distribution inputs. Specifically, we generate dataset $\\{ ( x _ { i } , y _ { i } ) \\} _ { i = 1 } ^ { N = 1 0 0 }$ as $x _ { i } \\sim U ( - 0 . 8 , 0 . 8 )$ , $\\begin{array} { r } { y _ { i } = 5 \\sin ( \\frac { \\pi } { 2 } x _ { i } ) + \\epsilon _ { i } } \\end{array}$ , where $\\epsilon _ { i } \\sim \\mathcal { N } ( 0 , 0 . 5 )$ . In addition, we generate t of distribution dataset $\\{ \\tilde { x } \\} _ { j = 1 } ^ { M = 1 0 0 }$ as $\\tilde { x } \\sim U ( - 2 , - 0 . 8 ) \\cup U ( 0 . 8 , 2 )$ . We $f$   \nhidden layer is 64 and the activation is RELU. We use Adam (Kingma & Ba, 2015) optimizer with learning rate 0.01. In addition to the standard MSE loss, we add a gradient penalty term inspired from (Gulrajani et al., 2017) such that the gradient at the out of distribution inputs is penalized. The overall loss function is: ",
+        "bbox": [
+            173,
+            421,
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+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "equation",
+        "img_path": "images/48cbbdcb118af8b75203fe4cd475a4b1870dfc365db205202792cd718cefb1e5.jpg",
+        "text": "$$\n\\mathcal { L } = \\frac { 1 } { N } \\sum _ { i = 1 } ^ { N } ( f ( x _ { i } ) - y _ { i } ) ^ { 2 } + \\lambda \\cdot \\frac { 1 } { M } \\sum _ { j = 1 } ^ { M } | | \\nabla _ { \\tilde { x } } f ( \\tilde { x } _ { j } ) | | _ { 2 }\n$$",
+        "text_format": "latex",
+        "bbox": [
+            325,
+            569,
+            673,
+            613
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "In our experiments, we set $\\lambda = 0 . 1$ . We fit $f$ with and without the gradient penalty term. Figure 4 shows the predicted value at both in-distribution and out-of-distribution inputs. The results suggest that: 1) both models generate good prediction for in-distribution inputs $( [ - 0 . 8 , 0 . 8 ] )$ , which is between the two red lines 2) both models generalize well at the out-of-distribution regions between the red line and the blue line $( [ - 1 , - 0 . 8 ]$ and [0.8, 1]). 3) both models erroneously generalize beyond the blue lines $( \\lfloor - 2 , - 1 \\rfloor$ and $\\lfloor 1 , 2 \\rfloor$ ). However, the value of the model trained without gradient penalty keep on increasing or decreasing while the model trained with gradient penalty has zero gradient. This suggests that with proper out-of-distribution regularization, the gradient direction at the out-of-distribution inputs point to the in-distribution regions. ",
+        "bbox": [
+            173,
+            619,
+            825,
+            746
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "B VISUALIZATION OF VARIOUS PROBABILITY DIVERGENCE MEASUREMENT ",
+        "text_level": 1,
+        "bbox": [
+            178,
+            767,
+            813,
+            782
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "To understand the impact of forward KL $( \\mathcal { D } _ { \\mathrm { K L } } ( \\pi _ { \\theta } | | \\pi _ { b } ) ) $ ), backward KL $\\left( \\mathcal { D } _ { \\mathrm { K L } } ( \\pi _ { b } | | \\pi _ { \\theta } ) \\right)$ ) and MMD distance in constraining the learned policy within the support set defined by the behavior policy, we create a toy example as shown in Figure 5. The black curves in the two figures represent the behavior distribution $\\pi _ { b }$ . The orange, blue and green curves in the graph show forward KL, backward KL and MMD with Laplacian kernel between $\\boldsymbol { \\mathit { 1 0 } }$ and $\\pi _ { \\theta } ~ = ~ \\mathcal { N } ( x , \\sigma )$ , respectively, where $x$ is a variable between $[ - 1 0 , 1 0 ]$ and $\\sigma$ is fixed. If we draw a horizontal line (red), the area below this line defines the support set. In the left figure, the behavior policy is single-modal and there is little difference between the three metrics. However, when the behavior policy is multi-modal, the forward KL divergence contains two bottoms, each corresponds to the peak probability density of the behavior policy. However, in backward KL and MMD distance, the policy with the smallest divergence actually has low probability density in the behavior policy. This is because minimizing the forward KL leads to “mode seeking” while minimizing the backward KL and MMD leads to “mode covering”. These two terms are heavily used when talking about generative models such as VAEs and GANs (Ke et al., 2019). When performing offline RL in multi-modal dataset, it is necessary to combine sub-optimal actions in the behavior policy, which is in fact “mode seeking”. Mode covering typically leads to out-of-distribution actions even when the divergence measurement is small (e.g. the area between the two modes are actually out-of-distribution) while mode seeking may be stuck at a local optimum (e.g. moving across different “modes” is hard). ",
+        "bbox": [
+            173,
+            797,
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+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "image",
+        "img_path": "images/5a639b65eb020709bafc3ba11100d983cd249ee7f76e31c22d118ec56717e901.jpg",
+        "image_caption": [
+            "Figure 5: The black curve in the two figures represents the behavior distribution $\\pi _ { b }$ . The orange, blue and green curves in the graph show forward KL, backward KL and MMD with Laplacian kernel between $\\pi _ { b }$ and $\\pi _ { \\boldsymbol { \\theta } } = \\mathcal { N } ( \\boldsymbol { x } , \\bar { \\boldsymbol { \\sigma } } )$ , respectively, where $x$ is a variable between $[ - 1 0 , 1 0 ]$ and $\\sigma$ is fixed. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            240,
+            109,
+            759,
+            261
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            173,
+            345,
+            825,
+            472
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "In the left figure, $\\pi _ { b } = \\mathcal { N } ( 0 , 1 )$ . In the right figure, $\\pi _ { b }$ is a mixture of two Gaussian distributions: $\\mathcal { N } ( - 1 . 5 , 0 . 3 )$ and $\\mathcal { N } ( 1 , 0 . 5 )$ . The weight of each component is 0.3 and 0.7, respectively. $\\sigma$ is set to 0.2. ",
+        "bbox": [
+            174,
+            478,
+            821,
+            520
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "C MISSING BACKGROUND",
+        "text_level": 1,
+        "bbox": [
+            176,
+            544,
+            410,
+            560
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "C.1 VARIATIONAL AUTO-ENCODER ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            577,
+            433,
+            592
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "A variational auto-encoder (VAE) (Kingma & Welling, 2014) is a generative model that aims to learn the data distribution $p ( X )$ given a set of observations $\\{ x _ { i } \\} _ { i = 1 } ^ { N }$ . While directly optimizing $p ( X )$ is intractable, we can optimize its evidence lower-bound (ELBO): ",
+        "bbox": [
+            176,
+            604,
+            823,
+            647
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "equation",
+        "img_path": "images/bc204ebf117900e818505662b26896a4cd24553abc728482a6d2c64c5c029cad.jpg",
+        "text": "$$\n\\begin{array} { r } { \\log p ( X ) \\ge \\mathbb { E } _ { z \\sim q ( z ) } [ \\log p ( X | z ) ] - \\mathcal { D } _ { \\mathrm { K L } } ( q ( z ) | | p ( z ) ) } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            325,
+            656,
+            671,
+            676
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "where $q ( Z )$ is the variational distribution and $p ( Z )$ is the prior. In VAE, $q ( Z )$ is $q ( Z | X )$ so that it is an auto-encoder. Optimizing Equation 12 using gradient descent requires back-propagate the gradient through a sample operator. Fortunately, if the latent variable is a multivariate Gaussian distribution, we can use re-parametrization trick. The tightness of the upper bound is the KL divergence between the approximated posterior distribution and the true posterior distribution. ",
+        "bbox": [
+            174,
+            684,
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+            756
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "D MISSING ABLATION STUDY ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            779,
+            441,
+            795
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "For fair comparison between MMD and KL divergence in policy regularization, we only substitute KL divergence with MMD-based measurement with additional MMD-specific hyperparameter tuning. In the original implementation (Kumar et al., 2019), the author uses 4 ensembles of Q network and compute the Q value by a convex combination of both the minimum of the ensembles and the maximum of the ensembles. In our implementation, we follow the standard architecture in the online setting: we only use 2 ensembles of Q networks and compute the Q value as their minimum. In this work, we only compare against MMD with Laplacian kernels. The MMD-specific hyper-parameters are shown in Table 3. ",
+        "bbox": [
+            174,
+            811,
+            825,
+            924
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "Difference in training the behavior policy (Kumar et al., 2019) learns the behavior policy as a $\\beta$ -VAE (Higgins et al., 2017) with MSE reconstruction loss. This is equivalent to maximizing the log probability of a Gaussian distribution with fixed variance as the decoder output. However, the variational lower bound does not hold in $\\beta$ -VAE that breaks our derivation. In this paper, we learn the behavior policy as a regular VAE. The decoder outputs a Gaussian distribution with inputconditioned mean and variance. ",
+        "bbox": [
+            173,
+            103,
+            825,
+            186
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "Gradient penalized policy evaluation Figure 6 suggests that the L2 norm of the gradient at the out-of-distribution actions with the gradient penalty is much lower than that without the gradient penalty. ",
+        "bbox": [
+            174,
+            208,
+            825,
+            250
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "image",
+        "img_path": "images/1e7b6a34bc8b6db03c161a6bd33f24a4fbe50c88de544e7755a5d07be157b435.jpg",
+        "image_caption": [
+            "Figure 6: Histogram of the L2 norm of the gradients for out-of-distribution actions "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            176,
+            267,
+            823,
+            417
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "Out-of-distribution generalization Lastly, we discuss the out-of-distribution generalization. In offline RL, we rate the learned policy with three levels. Level I policies are able to strictly follow the behavior policy. Level II policies are able to combine sub-optimal policies in the bahevior policy. Level III policies are able to generalize to out-of-distribution actions. To visualize the outof-distribution generalization, we plot the KL divergence of the learned policy on the testing state distribution in Figure 7. In the hopper-mixed and halfcheetah-medium task, there are periodic spikes, suggesting that the agents visit states which are not in the training distribution at test time. Although the results suggest that such generalization is correct, our approach is unable to explicitly quantify it. Thus, we leave it as future work. ",
+        "bbox": [
+            173,
+            486,
+            825,
+            611
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "image",
+        "img_path": "images/423e1a847c88d43168a412a1bcee0717bf24b6fbffd6ed2cce09d9a0542edad1.jpg",
+        "image_caption": [
+            "Figure 7: The KL divergence between the learned policy and the behavior policy on the testing data distribution. The $x$ axis is the step in an episode. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            176,
+            628,
+            820,
+            779
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "Performance sensitivity to the choices of target entropy During our experiments, we found that the performance using the KL divergence as the behavior regularization method is very sensitive to the choice of the target entropy. Policies trained with a larger entropy may not be able to capture the narrow expert distribution within the behavior policy distribution. This phenomenon is most typical in the halfcheetah-medium-expert dataset as depicted in Figure 8. ",
+        "bbox": [
+            174,
+            854,
+            825,
+            924
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "image",
+        "img_path": "images/e142754b28d4cb0fca25866e2b6b6daefced0991f27fffd724bcc1ecd202c59d.jpg",
+        "image_caption": [
+            "Figure 8: The performance of the halfcheetah-medium-expert task on various choices of the target entropy. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            174,
+            102,
+            821,
+            251
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "table",
+        "img_path": "images/cf69dc14b4ba863a643b6d166bc8b65b454bc7f4b9cba67a13ef8cd61faca6fa.jpg",
+        "table_caption": [
+            "Table 3: MMD-specific Hyper-parameters "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Hyper-parameter</td><td>hopper-medium-replay丨walker-medium-expert|</td><td></td><td>halfcheetah-medium</td></tr><tr><td>σ in Laplacian kernel</td><td>10</td><td>20</td><td>10</td></tr><tr><td>EMMD</td><td>0.2</td><td>0.2</td><td>0.3</td></tr><tr><td>Number of samples</td><td>10</td><td>10</td><td>10</td></tr></table>",
+        "bbox": [
+            173,
+            320,
+            826,
+            391
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "text",
+        "text": "E MISSING RELATED WORK ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            416,
+            424,
+            433
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "text",
+        "text": "For KL regularized policy improvement with fixed temperature, the optimal policy has a closed form solution (Wang et al., 2020). Although it avoids estimating the behavior policy, the issue with fixed temperature is that all the states are penalized equally. This leads to over-pessimistic for states with higher occurence in the dataset and over-optimistic for states with lower occurence. (Fox, 2019) presents a closed-form expression for the regularization coefficient that completely eliminates the bias in entropy-regularized value updates. However, the softmax operator introduced by the approach makes it hard to use in continuous action space. ",
+        "bbox": [
+            173,
+            448,
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+            546
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "text",
+        "text": "F IMPLEMENTATION DETAILS ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            565,
+            436,
+            582
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "text",
+        "text": "Computation of the analytical KL upper bound To reduce the variance when computing the analytical KL upper bound, we sample $L$ latent variable $z$ and compute the average of the $L$ KL upper bounds. In our experiments, we set $L = 5$ . Note that it doesn’t reduce the bias of the upper bound. ",
+        "bbox": [
+            173,
+            597,
+            825,
+            654
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "text",
+        "text": "Reward scaling Any affine transformation of the reward function does not change the optimal policy of the MDP. In our experiments, we rescale the reward to $[ 0 , 1 ]$ as: ",
+        "bbox": [
+            171,
+            667,
+            823,
+            698
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "equation",
+        "img_path": "images/7b065230b81c98ee8e5bba5098cf4ca7471265f68c10ac661fe218a2a28d60ab.jpg",
+        "text": "$$\nr ^ { \\prime } = ( r - r _ { m i n } ) / ( r _ { m a x } - r _ { m i n } )\n$$",
+        "text_format": "latex",
+        "bbox": [
+            390,
+            703,
+            607,
+            720
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "text",
+        "text": "where $r _ { m a x }$ and $r _ { m i n }$ is the maximum and the minimum reward in the dataset. ",
+        "bbox": [
+            176,
+            727,
+            687,
+            742
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "text",
+        "text": "Entropy regularization The KL divergence is the sum of negative entropy of the learned policy plus the cross entropy between the learned policy and the behavior policy: $\\bar { \\mathcal { D } } _ { \\mathrm { K L } } ( \\pi _ { \\theta } ( \\cdot | s ) , \\pi _ { \\beta } ( \\cdot | \\bar { s } ) ) \\stackrel { \\cdot } { = }$ $\\bar { - \\mathcal { H } } ( \\pi _ { \\theta } ( \\cdot | s ) ) + \\mathcal { H } ( \\bar { \\pi _ { \\theta } } ( \\cdot | s ) , \\pi _ { \\beta } ( \\cdot | s ) )$ . When the learned policy distribution violates the KL constraints, the KL divergence between the policy distribution and the behavior distribution is decreased by the optimizer. This is equivalent to increasing the entropy of the learned policy and decreasing the cross entropy between the learned policy and the behavior policy. In soft actor-critic (Haarnoja et al., 2018a), the minimum entropy of the learned policy is enforced to encourage exploration. However, due to the absence of exploration, stochastic policy with large entropy will sample out-ofdistribution actions when computing the target Q values in Equation 2. If such values are overestimated, the policy will exploit the erroneous Q values when performing the policy improvement in Equation 8 and lead to failure, which can’t be corrected without more data. Thus, we maintain the maximum entropy of the learned policy using the technique proposed in (Haarnoja et al., 2018b). ",
+        "bbox": [
+            173,
+            756,
+            825,
+            924
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "text",
+        "text": "Initialization If the dataset is collected using a narrow policy distribution in a high dimensional space (e.g. human demonstration), the constrained optimization problem using dual gradient descent finds it difficult to converge if random initialization is used for the policy network. To mitigate this issue, we start with a policy that has the minimum KL divergence with the behavior policy: $\\begin{array} { r } { \\pi _ { \\theta } \\ = \\ \\arg \\operatorname* { m i n } _ { \\pi _ { \\theta } \\in \\Pi } \\mathcal { D } _ { \\mathrm { K L } } ( \\pi , \\pi _ { \\beta } ) } \\end{array}$ , where $\\Pi$ represents a family of policy types. In this work, we consider $\\Pi$ as Gaussian policies. Correspondingly, we initialize the Q network to $Q ^ { \\pi _ { \\theta } }$ . ",
+        "bbox": [
+            173,
+            103,
+            825,
+            188
+        ],
+        "page_idx": 16
+    },
+    {
+        "type": "text",
+        "text": "Policy network Our policy network is a 3-layer feed-forward neural network. The size of each hidden layer is 512. We apply RELU activation (Agarap, 2018) after each hidden layer. Following (Haarnoja et al., 2018a), the output is a Gaussian distribution with diagonal covariance matrix. We apply tanh to enforce the action bounds. The log-likelihood after applying the tanh function has a simple closed form solution. We refer to (Haarnoja et al., 2018a) Appendix C for more details. ",
+        "bbox": [
+            173,
+            202,
+            825,
+            273
+        ],
+        "page_idx": 16
+    },
+    {
+        "type": "text",
+        "text": "Q network Following (Haarnoja et al., 2018a; Fujimoto et al., 2018a; Wu et al., 2019), we train two independent Q network $\\{ Q _ { \\psi _ { 1 } } , Q _ { \\psi _ { 2 } } \\}$ to penalize uncertainty over the future states. We maintain a target Q network $\\{ Q _ { \\psi _ { 1 } ^ { \\prime } } , Q _ { \\psi _ { 2 } ^ { \\prime } } \\}$ with the same architecture and update the target weights using a weighted sum of the current Q network and the target Q network. When computing the target Q values, we simply take the minimum value of the two Q networks: ",
+        "bbox": [
+            174,
+            286,
+            825,
+            358
+        ],
+        "page_idx": 16
+    },
+    {
+        "type": "equation",
+        "img_path": "images/91f888bc5e8404b3274b37b1735018515c811d620f9fb94f82da42430c78b448.jpg",
+        "text": "$$\nQ _ { \\psi ^ { \\prime } } ( s ^ { \\prime } , a ^ { \\prime } ) = \\operatorname* { m i n } _ { j = 1 , 2 } Q _ { \\psi ^ { \\prime } { } _ { j } } ( s ^ { \\prime } , a ^ { \\prime } )\n$$",
+        "text_format": "latex",
+        "bbox": [
+            392,
+            364,
+            606,
+            388
+        ],
+        "page_idx": 16
+    },
+    {
+        "type": "text",
+        "text": "Each Q network is a 3-layer feed-forward neural network. The size of each hidden layer is 256. We apply RELU activation (Agarap, 2018) after each hidden layer. ",
+        "bbox": [
+            174,
+            395,
+            823,
+            424
+        ],
+        "page_idx": 16
+    },
+    {
+        "type": "text",
+        "text": "Behavior policy network Following the previous work (Fujimoto et al., 2018b; Kumar et al., 2019), we learn a conditional variational auto-encoder (Kingma & Welling, 2014) as our behavior policy network. The encoder takes a pair of states and actions, and outputs a Gaussian latent variable $Z$ . The decoder takes sampled latent code $z$ and states, and outputs a mixture of Gaussian distributions. Both the architecture of the encoder and the decoder is a 3-layer feed-forward neural network. The size of each hidden layer is 512. The activation is relu (Agarap, 2018). To avoid epistemic uncertainty, we train $B$ ensembles of behavior policy networks. At test time, we randomly select one model to perform the calculations. We found $B = 3$ is sufficient for all the experiments. We pre-train the the behavior policy network for $4 0 0 \\mathrm { k }$ gradient steps. ",
+        "bbox": [
+            173,
+            438,
+            825,
+            565
+        ],
+        "page_idx": 16
+    },
+    {
+        "type": "text",
+        "text": "$\\alpha$ network The $\\alpha$ network takes in a state and outputs the Lagrange multiplier for the state. The architecture of the $\\alpha$ network is a 3-layer feed-forward neural network with relu (Agarap, 2018) activation. The size of each hidden layer is 256. We use softplus activation after the output to ensure that all the values are positive. ",
+        "bbox": [
+            174,
+            579,
+            825,
+            636
+        ],
+        "page_idx": 16
+    },
+    {
+        "type": "table",
+        "img_path": "images/30e2d602098560b912248512e1bd878e2d277cf0ae8755ec1d886c59e55e2dc9.jpg",
+        "table_caption": [
+            "Table 4: Default hyper-parameters "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Hyper-parameter</td><td>Value (Gym/Adroit)</td></tr><tr><td>Optimizer</td><td>Adam (Kingma &amp; Ba,2015)</td></tr><tr><td>Policy learning rate</td><td>5e-6/5e-8</td></tr><tr><td>Q network learning rate</td><td>3e-4</td></tr><tr><td>α learning rate</td><td>le-5/1e-7</td></tr><tr><td>batch size</td><td>100</td></tr><tr><td>Target update rate T</td><td>1e-3</td></tr><tr><td>Discount factor y</td><td>0.99</td></tr><tr><td>Initial β</td><td>10</td></tr><tr><td>β learning rate</td><td>1e-3</td></tr><tr><td>Steps per epoch T</td><td>2000</td></tr><tr><td>Number of epochs</td><td>500</td></tr></table>",
+        "bbox": [
+            307,
+            664,
+            691,
+            844
+        ],
+        "page_idx": 16
+    },
+    {
+        "type": "table",
+        "img_path": "images/a9193ce34de28d21bafa57d2c16f08ba15ae8f86857b3267c2c216a6055f1e32.jpg",
+        "table_caption": [
+            "Table 5: Task-specific hyper-parameters for OpenAI gym tasks. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Task name</td><td>KL divergence threshold ∈kL</td><td>Maximum entropy Ho</td></tr><tr><td>halfcheetah-rand</td><td>9</td><td>-3</td></tr><tr><td>walker2d-rand</td><td>0.1</td><td>6</td></tr><tr><td>hopper-rand</td><td>3</td><td>-3</td></tr><tr><td>halfcheetah-med</td><td>4</td><td>-12</td></tr><tr><td>walker2d-med</td><td>2.1</td><td>-9</td></tr><tr><td>hopper-med</td><td>2.4</td><td>-6</td></tr><tr><td>halfcheetah-med-exp</td><td>11.5</td><td>-24</td></tr><tr><td>walker2d-med-exp</td><td>5</td><td>-12</td></tr><tr><td>hopper-med-exp</td><td>2.6</td><td>-6</td></tr><tr><td>halfcheetah-mixed</td><td>6</td><td>-12</td></tr><tr><td>walker2d-mixed</td><td>4</td><td>-6</td></tr><tr><td>hopper-mixed</td><td>3</td><td>-3</td></tr></table>",
+        "bbox": [
+            232,
+            238,
+            766,
+            450
+        ],
+        "page_idx": 17
+    },
+    {
+        "type": "table",
+        "img_path": "images/49f3d5ca4c68fe2386652ebfed83eb4eca06bb2baeb7e7002cdfaf728de56f80.jpg",
+        "table_caption": [
+            "Table 6: Task-specific hyper-parameters for Adroit tasks with human demonstrations. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Task name</td><td>MMD threshold ∈MMD</td><td>Minimum entropy Ho</td></tr><tr><td>pen-human</td><td>0.06</td><td>-200</td></tr><tr><td>hammer-human</td><td>0.1</td><td>-60</td></tr><tr><td>door-human</td><td>0.1</td><td>-60</td></tr><tr><td>relocate-human</td><td>0.1</td><td>-60</td></tr></table>",
+        "bbox": [
+            271,
+            717,
+            725,
+            800
+        ],
+        "page_idx": 17
+    }
+]

parse/train/gIHd-5X324/gIHd-5X324.md

@@ -0,0 +1,320 @@
+# RETHINKING SOFT LABELS FOR KNOWLEDGE DISTILLATION: A BIAS-VARIANCE TRADEOFF PERSPECTIVE
+
+Helong Zhou1∗, Liangchen $\mathbf { S o n g ^ { 2 * \dagger } }$ , Jiajie Chen1∗ , Ye Zhou1, Guoli Wang13, Junsong Yuan2, Qian Zhang1 1Horizon Robotics 2University at Buffalo 3Tsinghua University {helong.zhou,jiajie.chen,ye.zhou,guoli.wang}@horizon.ai {lsong8,jsyuan}@buffalo.edu, qian01.zhang@horizon.ai
+
+# ABSTRACT
+
+Knowledge distillation is an effective approach to leverage a well-trained network or an ensemble of them, named as the teacher, to guide the training of a student network. The outputs from the teacher network are used as soft labels for supervising the training of a new network. Recent studies (Muller et al., 2019; Yuan ¨ et al., 2020) revealed an intriguing property of the soft labels that making labels soft serves as a good regularization to the student network. From the perspective of statistical learning, regularization aims to reduce the variance, however how bias and variance change is not clear for training with soft labels. In this paper, we investigate the bias-variance tradeoff brought by distillation with soft labels. Specifically, we observe that during training the bias-variance tradeoff varies sample-wisely. Further, under the same distillation temperature setting, we observe that the distillation performance is negatively associated with the number of some specific samples, which are named as regularization samples since these samples lead to bias increasing and variance decreasing. Nevertheless, we empirically find that completely filtering out regularization samples also deteriorates distillation performance. Our discoveries inspired us to propose the novel weighted soft labels to help the network adaptively handle the sample-wise biasvariance tradeoff. Experiments on standard evaluation benchmarks validate the effectiveness of our method. Our code is available at https://github.com/ bellymonster/Weighted-Soft-Label-Distillation.
+
+# 1 INTRODUCTION
+
+For deep neural networks (Goodfellow et al., 2016), knowledge distillation (KD) (Ba & Caruana, 2014; Hinton et al., 2015) refers to the technique that uses well-trained networks to guide the training of another network. Typically, the well-trained network is named as the teacher network while the network to be trained is named as the student network. For distillation, the predictions from the teacher network are leveraged and referred to as the soft labels (Balan et al., 2015; Muller et al., ¨ 2019). Soft labels generated by the teacher network have been proven effective in large-scale empirical studies (Liang et al., 2019; Tian et al., 2020; Zagoruyko & Komodakis, 2017; Romero et al., 2015) as well as recent theoretical studies (Phuong & Lampert, 2019).
+
+However, the reason why soft labels are beneficial to the student network is still not well explained. Giving a clear theoretical explanation is challenging: The optimization details of a deep network with the common one-hot labels are still not well-studied (Nagarajan & Kolter, 2019), not to mention training with the soft labels. Nevertheless, two recent studies (Muller et al., 2019; Yuan et al., ¨ 2020) shed light on the intuitions about how the soft labels work. Specifically, label smoothing, which is a special case of soft labels based training, is shown to regularize the activations of the penultimate layer to the network (Muller et al., 2019). The regularization property of soft labels is ¨ further explored in (Yuan et al., 2020). They hypothesize that in KD, one main reason why the soft labels work is the regularization introduced by soft labels. Based on the assumption, the authors design a teacher-free distillation method by turning the predictions of the student network into soft labels.
+
+Considering that soft labels are targets for distillation, the evidence of the regularization brought by soft labels drives us to rethink soft labels for KD: Soft labels are both supervisory signals and regularizers. Meanwhile, it is known that there is a tradeoff between fitting the data and imposing regularizations, i.e., the bias-variance dilemma (Kohavi & Wolpert, 1996; Bishop, 2006), but it is unclear how bias and variance change for distillation with soft labels. Since the bias-variance tradeoff is an important issue in statistical learning, we investigate whether the bias-variance tradeoff exists for soft labels and how the tradeoff affects distillation performance.
+
+We first compare the bias and variance decomposition of direct training with that of distillation with soft labels, noticing that distillation results in a larger bias error and a smaller variance. Then, we rewrite distillation loss into the form of a regularization loss adding the direct training loss. Through inspecting the gradients of the two terms during training, we notice that for soft labels, the biasvariance tradeoff varies sample-wisely. Moreover, by looking into a conclusion from (Muller et al., ¨ 2019), we observe that under the same temperature setting, the distillation performance is negatively associated with the number of some certain samples. These samples lead to bias increase and variance decrease and we name them as regularization samples. To investigate how regularization samples affect distillation, we first examine if we can design ad hoc filters for soft labels to avoid training with regularization samples. But completely filtering out regularization samples also deteriorates distillation performance, leading us to speculate that regularization samples are not well handled by standard KD. In the light of these findings, we propose weighted soft labels for distillation to handle the sample-wise bias-variance tradeoff, by adaptively assigning a lower weight to regularization samples and a larger weight to the others. To sum up, our contributions are:
+
+• For knowledge distillation, we analyze how the soft labels work from a perspective of biasvariance tradeoff.   
+• We discover that the bias-variance tradeoff varies sample-wisely. Also, we discover that if we fix the distillation temperature, the number of regularization samples is negatively associated with the distillation performance.   
+• We design straightforward schemes to alleviate negative impacts from regularization samples and then propose the novel weighted soft labels for distillation. Experiments on large scale datasets validate the effectiveness of the proposed weighted soft labels.
+
+# 2 RELATED WORKS
+
+Knowledge distillation. Hinton et al. (2015) proposed to distill outputs from large and cumbersome models into smaller and faster models, which is named as knowledge distillation. The outputs for large networks are averaged and formulated as soft labels. Also, other kinds of soft labels have been widely used for training deep neural networks (Szegedy et al., 2016; Pereyra et al., 2017). Treating soft labels as regularizers were pointed out in (Hinton et al., 2015) since a lot of helpful information can be carried in soft labels. More recently, Muller et al. (2019) showed the adverse ¨ effect of label smoothing upon distillation. It is a thought-provoking discovery for the reason that both label smoothing and distillation are exploiting the regularization property behind soft labels. Yuan et al. (2020) further investigated the regularization property of soft labels and then proposed a teacher free distillation scheme.
+
+Distillation loss. One of our main contributions is that we improve the distillation loss. For adaptively adjusting the distillation loss, Tang et al. (2019) pays attention to hard-to-learn and hard-tomimic samples, and the latter is weighted based on the prediction gap between teacher and student. However, it does not consider that the teacher may give an incorrect guide to the student, under which the prediction gap is still large and such a method may lead to the performance being hurt. Saputra et al. (2019) transfers teacher’s guidance only on the samples where the performance of the teacher surpasses the student, while Wen et al. (2019) deals with the incorrect guidance by probability shifting strategy. Our approach is different from the above methods, in terms of motivations as well as the proposed solutions.
+
+Bias-variance tradeoff. Bias-variance tradeoff is a well-studied topic in machine learning (Kohavi & Wolpert, 1996; Domingos, 2000; Valentini & Dietterich, 2004; Bishop, 2006) and for neural networks (Geman et al., 1992; Neal et al., 2018; Belkin et al., 2019; Yang et al., 2020). Existing methods are mainly concerned with the variance brought by the choice of network models. Our perspective is different from the previous methods since we focus on the behavior of samples during training. In our work, based on the results from Heskes (1998), we present the decomposition of distillation loss, which is defined by Kullback-Leibler divergence. Besides, our main contribution is not to study how to theoretically analyze the tradeoff, but how to adaptively tune the sample-wise tradeoff during training.
+
+# 3 BIAS-VARIANCE TRADEOFF FOR SOFT LABELS
+
+Soft labels play the role of supervisory signals and regularizations at the same time, which inspires us to rethink soft labels from the perspective of the bias-variance tradeoff. We begin our analysis with some mathematical descriptions. For a sample $\mathbf { x }$ labeled as $i$ -th class, let the ground-truth label be a one-hot vector y where $y _ { i } = 1$ and other entries are 0. Then for $x$ and softmax output temperature $\tau$ , the soft label predicted by the teacher network is denoted as $\hat { y } _ { \tau } ^ { t }$ and the output from the student is denoted as $\hat { y } _ { \tau } ^ { s }$ . The soft label $\hat { y } _ { \tau } ^ { t }$ is then used for training the student by the distillation loss, i.e. $\begin{array} { r } { L _ { \mathrm { k d } } = - \tau ^ { 2 } \sum _ { k } \hat { y } _ { k , \tau } ^ { t } \log \hat { y } _ { k , \tau } ^ { s } , } \end{array}$ , where $\hat { y } _ { k , \tau } ^ { s } , \hat { y } _ { k , \tau } ^ { t }$ means the $k$ -th element of the student’s output $\hat { y } _ { \tau } ^ { s }$ and the teacher’s output $\hat { y } _ { \tau } ^ { t }$ , respectively. With the above notations, the cross-entropy loss for training with one-hot labels is $L _ { \mathrm { c e } } = - y _ { k } \log \hat { y } _ { k , 1 } ^ { s }$ .
+
+![](images/1c5419e10bd9eacecac217de5a6662221e71b22dfee40c4412a7a936700c0514.jpg)  
+Figure 1: Bias and variance.
+
+We now present the bias-variance decomposition for $L _ { \mathrm { c e } }$ and $L _ { \mathrm { k d } }$ , based on the definition and notations from Heskes (1998). First, we denote the train dataset as $\mathcal { D }$ and the output distribution on a sample $\mathbf { x }$ of the network trained without distillation as $\hat { \mathbf { y } } _ { \mathrm { c e } } = f _ { \mathrm { c e } } ( \mathbf { x } ; \mathcal { D } )$ . For the network trained with distillation, the model also depends on the teacher network, so we define the output on $\mathbf { x }$ as $\hat { \mathbf { y } } _ { \mathrm { k d } } = f _ { \mathrm { k d } } ( \mathbf { x } ; \mathcal { D } , \mathcal { T } )$ , where $\tau$ is the selected teacher network. Then, let the averaged output of $\hat { \mathbf { y } } _ { \mathrm { k d } }$ and $\hat { \mathbf { y } } _ { \mathrm { c e } }$ be $\bar { \bf y } _ { \mathrm { k d } }$ and $\bar { \mathbf { y } } _ { \mathrm { c e } }$ , that is,
+
+$$
+\bar { \mathbf { y } } _ { \mathrm { c e } } = \frac { 1 } { Z _ { \mathrm { c e } } } \exp ( \mathbb { E } _ { T } [ \log \hat { \mathbf { y } } _ { \mathrm { c e } } ] ) , \quad \bar { \mathbf { y } } _ { \mathrm { k d } } = \frac { 1 } { Z _ { \mathrm { k d } } } \exp ( \mathbb { E } _ { \mathcal { D } , T } [ \log \hat { \mathbf { y } } _ { \mathrm { k d } } ] ) ,
+$$
+
+where $Z _ { \mathrm { c e } } , Z _ { \mathrm { k d } }$ are two normalization constant. Then according to Heskes (1998), we have the following decomposition for the expected error on the sample $\mathbf { x }$ and $\mathbf { y } = t ( \mathbf { x } )$ is the ground truth label:
+
+$$
+\begin{array} { r l } & { \mathrm { e r r o r c e } = \mathbb { E } _ { \mathbf { x } , \mathcal { D } } \left[ - \mathbf { y } \log \hat { \mathbf { y } } _ { \mathrm { c e } } \right] = \mathbb { E } _ { \mathbf { x } , \mathcal { D } } \left[ - \mathbf { y } \log \mathbf { y } + \mathbf { y } \log \frac { \mathbf { y } } { \bar { \mathbf { y } } _ { \mathrm { c e } } } + \mathbf { y } \log \frac { \bar { \mathbf { y } } _ { \mathrm { c e } } } { \hat { \mathbf { y } } _ { \mathrm { c e } } } \right] } \\ & { \qquad = \mathbb { E } _ { \mathbf { x } } \big [ - \mathbf { y } \log \mathbf { y } \big ] + \mathbb { E } _ { \mathbf { x } } \left[ \mathbf { y } \log \frac { \mathbf { y } } { \bar { \mathbf { y } } _ { \mathrm { c e } } } \right] + \mathbb { E } _ { \mathcal { D } } \left[ \mathbb { E } _ { \mathbf { x } } \left[ \mathbf { y } \log \frac { \bar { \mathbf { y } } _ { \mathrm { c e } } } { \hat { \mathbf { y } } _ { \mathrm { c e } } } \right] \right] } \\ & { \qquad = \mathbb { E } _ { \mathbf { x } } \big [ - \mathbf { y } \log \mathbf { y } \big ] + D _ { \mathrm { K L } } ( \mathbf { y } , \bar { \mathbf { y } } _ { \mathrm { c e } } ) + \mathbb { E } _ { \mathcal { D } } \big [ D _ { \mathrm { K L } } ( \bar { \mathbf { y } } _ { \mathrm { c e } } , \hat { \mathbf { y } } _ { \mathrm { c e } } ) \big ] } \\ & { \qquad = \mathrm { i n t i n s i c ~ n o i s e ~ \mathrm { n i s ~ } } + \mathrm { \mathrm { ~ s i a s ~ } } + \mathrm { v a r i a n c e } , } \end{array}
+$$
+
+where $D _ { \mathrm { K L } }$ is the Kullback-Leibler divergence. The derivation of the variance term is based on the facts that $\frac { \log \bar { \mathbf { y } } _ { \mathrm { c e } } } { \mathbb { E } _ { \mathcal { D } } [ \log \hat { \mathbf { y } } _ { \mathrm { c e } } ] }$ is a constant and $\mathbb { E } _ { \mathbf { x } } [ \mathbf { y } ] = \mathbb { E } _ { \mathbf { x } } [ \bar { \mathbf { y } } _ { \mathrm { c e } } ] = 1$ . Detailed derivations can be found from Eq. (4) in Heskes (1998). Next, we analyze the bias-variance decomposition of $L _ { \mathrm { k d } }$ . As mentioned above, when training with soft labels, extra randomness is introduced for the selection of a teacher network. In Fig. 1, we illustrate the corresponding bias and variance for the selection process of a set of soft labels, which are generated by a teacher network. In this case, a high variance model indicates the model (grey point) is closer to the one-hot trained model (black point), while a low variance model indicates that the model is closer to other possible models trained with soft labels (red points). Although for KD there are more sources introducing randomness, the overall variance brought by $L _ { \mathrm { k d } }$ is not necessarily higher than $L _ { \mathrm { c e } }$ . In fact, existing empirical results strongly suggest that the overall variance is smaller with KD. For example, students trained with soft labels are better calibrated than one-hot baselines (Muller et al., 2019) and KD makes the predictions of ¨ students more consistent when facing adversarial noise (Papernot et al., 2016). Here, we present these empirical evidence as an assumption:
+
+Assumption 1 The variance brought by $K D$ is smaller than direct training, that is, $\mathbb { E } _ { \mathcal { D } , \mathcal { T } } [ \bar { D } _ { \mathrm { K L } } ( \bar { \bf y } _ { \mathrm { k d } } , \hat { \bf y } _ { \mathrm { k d } } ) ] \leqslant \mathbb { E } _ { \mathcal { D } } [ D _ { \mathrm { K L } } ( \bar { \bf y } _ { \mathrm { c e } } , \hat { \bf y } _ { \mathrm { c e } } ) ]$ .
+
+Similar to Eq. (2), we write the decomposition for $L _ { \mathrm { k d } }$ as
+
+$$
+\begin{array} { r l r l r } { \mathrm { e r r o r } _ { \mathbf { k d } } = \mathbb { E } _ { \mathbf { x } } [ - \mathbf { y } \log \mathbf { y } ] + } & { } & { \qquad } & { } & { \qquad + \mathbb { E } _ { \mathcal { D } , \mathcal { T } } [ D _ { \mathrm { K L } } ( \bar { \mathbf { y } } _ { \mathbf { k d } } , \hat { \mathbf { y } } _ { \mathbf { k d } } ) ] . } \end{array}
+$$
+
+An observation here is that $\bar { \mathbf { y } } _ { \mathrm { c e } }$ converges to one-hot labels while $\bar { \bf y } _ { \mathrm { k d } }$ converges to soft labels, so $\bar { \bf y } _ { \mathrm { c e } }$ is closer to the one-hot ground-truth distribution $\mathbf { y }$ than $\bar { \bf y } _ { \mathrm { k d } }$ , i.e., $\mathbb { E } _ { \mathbf { x } } \left[ \mathbf { y } \mathrm { l o g } \left( \frac { \bar { \mathbf { y } } _ { \mathrm { c e } } } { \bar { \mathbf { y } } _ { \mathrm { k d } } } \right) \right] \geqslant 0$ . If we rewrite $L _ { \mathrm { k d } }$ as $L _ { \mathrm { k d } } = L _ { \mathrm { k d } } - L _ { \mathrm { c e } } + L _ { \mathrm { c e } }$ , then $L _ { \mathrm { k d } } \mathrm { ~ - ~ } L _ { \mathrm { c e } }$ causes that the bias increases by $\mathbb { E } _ { \mathbf { x } } \left[ \mathbf { y } \log \left( \frac { \bar { \mathbf { y } } _ { \mathrm { c e } } } { \bar { \mathbf { y } } _ { \mathrm { k d } } } \right) \right]$ and the variance decreases by $\mathbb { E } _ { \mathcal { D } } [ D _ { \mathrm { K L } } ( \bar { \bf y } _ { \mathrm { c e } } , \hat { \bf y } _ { \mathrm { c e } } ) ] - \mathbb { E } _ { \mathcal { D } , \tau } [ D _ { \mathrm { K L } } ( \bar { \bf y } _ { \mathrm { k d } } , \hat { \bf y } _ { \mathrm { k d } } ) ]$ .
+
+From the above analysis, we separate $L _ { \mathrm { k d } }$ into two terms, and $L _ { \mathrm { k d } } - L _ { \mathrm { c e } }$ leads to variance reduction, and $L _ { \mathrm { c e } }$ leads to bias reduction. In the following sections, we first analyze how $L _ { \mathrm { k d } } - L _ { \mathrm { c e } }$ links to the bias-variance tradeoff during training. Then we analyze the changes in the relative importance between bias reduction and variance reduction during training with soft labels.
+
+# 3.1 THE BIAS-VARIANCE TRADEOFF DURING TRAINING
+
+It is known that bias reduction and variance reduction are often in conflict and we cannot minimize bias and variance together. However, if we consider the change of bias and variance during the training process, the importance of tuning the tradeoff also changes during training. Specifically, shortly after the training of the network starts, the bias error dominates the total error and the variance is less important. As training goes on, gradients of reducing the bias error (induced by $L _ { \mathrm { c e , \ d } }$ ) and reducing the variance (induced by $L _ { \mathrm { k d } } - L _ { \mathrm { c e } } )$ can be of the same scale for some samples, then we need to balance the tradeoff because reducing one term is likely to increase another one. Therefore for soft labels, we need to handle the bias-variance tradeoff in a sample-wise manner and take the training process into consideration.
+
+To study the bias-variance tradeoff during training, we consider the gradients of bias and variance reduction. Let $_ { z }$ be the logits output of the student on input $x$ and $z _ { i }$ is $i$ -th element of it, then we are interested in ∂(Lkd−Lce)∂z . For simplifying analysis, we are concerned with the gradients on the ground-truth related logit, that is, the sample $x$ is labeled as $i$ -th class. Mathematically, for the gradients of variance reduction, we have
+
+$$
+\frac { \partial ( L _ { \mathrm { k d } } - L _ { \mathrm { c e } } ) } { \partial z _ { i } } = \tau ( \hat { y } _ { i , \tau } ^ { s } - \hat { y } _ { i , \tau } ^ { t } ) - ( \hat { y } _ { i , 1 } ^ { s } - y _ { i } ) = \tau \left( \frac { e ^ { z _ { i } / \tau } } { \sum _ { k } e ^ { z _ { k } / \tau } } - \hat { y } _ { i , \tau } ^ { t } \right) - \left( \frac { e ^ { z _ { i } } } { \sum _ { k } e ^ { z _ { k } } } - y _ { i } \right) ,
+$$
+
+where $\hat { y } _ { i , \tau } ^ { t }$ denotes the $i$ -th element of the teacher’s prediction, i.e., $\hat { y } _ { \tau } ^ { t }$ . The term $L _ { \mathrm { k d } } - L _ { \mathrm { c e } }$ is easy to understand when $\tau = 1$ since the gradient now becomes $y _ { i } - \hat { y } _ { i , 1 } ^ { t }$ . Meanwhile, for the bias reduction, we have $\begin{array} { r } { \frac { \partial L _ { \mathrm { c e } } } { \partial z _ { i } } = \hat { y } _ { i , 1 } ^ { s } - y _ { i } } \end{array}$ , so ∂Lce and ∂(Lkd−Lce) always have different signs, leading to a tradeoff. If $\frac { \partial L _ { \mathrm { c e } } } { \partial z _ { i } }$ is much higher than ∂(Lkd−Lce)∂z , the bias reduction dominates the overall optimization direction. Instead, i f ∂(Lkd−Lce)∂z becomes higher, the sample is used for variance reduction. Interestingly, we discover that under a fixed distillation temperature, the final performance is worse when more training samples are used for variance reduction, which will be introduced in the next section.
+
+# 3.2 REGULARIZATION SAMPLES
+
+Our analysis starts with a conclusion from Muller et al. (2019): ¨ if a teacher network is trained with label smoothing, knowledge distillation into a student network is much less effective. Inspired by the phenomenon, we gather the impact of bias and variance during training with different distillation settings. Let $\begin{array} { r } { a = \frac { \partial L _ { \mathrm { c e } } } { \partial z _ { i } } } \end{array}$ and $\begin{array} { r } { b = \frac { \partial ( L _ { \mathrm { k d } } - L _ { \mathrm { c e } } ) } { \partial z _ { i } } } \end{array}$ , then as introduced before, we use $a$ and $b$ to represent the impact of bias and variance, respectively. If we have $\vert b \vert > \vert a \vert$ for a sample, we name the sample as a regularization sample since the variance dominates the optimization direction. From the collected data, we find that the number of regularization samples is closely related to distillation performance.
+
+Table 1: We count the number of regularization samples with different distillation settings on CIFAR-100. The teacher-student network pair is WRN-40-2 (Zagoruyko & Komodakis, 2017) and WRN-16-2. Results are averaged over 5 repeated runs. The temperature column means the temperature for distillation and the label smoothing column means whether the teacher network is trained with label smoothing trick.   
+
+<table><tr><td colspan="3">Teacher: 76.55 w/ label smoothing,75.61 w/o label smoothing Baseline Top-1 Acc Student: 73.26</td></tr><tr><td>Temperature</td><td>Label smoothing?</td><td>Student Top-1 Acc Number of regularization samples</td></tr><tr><td rowspan="2">T=2</td><td>X 74.79</td><td>15379</td></tr><tr><td>√</td><td>74.62 25235</td></tr><tr><td rowspan="2">T=4</td><td>74.92</td><td>17709</td></tr><tr><td>X</td><td>74.59 24775</td></tr><tr><td rowspan="2">T=6</td><td>X</td><td>75.10 17408 74.46</td></tr><tr><td></td><td>24538</td></tr></table>
+
+![](images/22a82c829b2d71362ee6e3c5f07064ed52655b4b1c07453c4e1f09dd220ae21e.jpg)  
+Figure 2: The number of regularization samples with respect to training epochs. The distillation settings are the same as the settings in Tab. 1.
+
+In Tab. 1, we present the count of regularization samples for a student network trained by knowledge distillation. For distillation with a temperature higher than 1, which is the common setting, we observe that if the teacher network is trained with label smoothing, more samples will be involved in variance reduction. Also, distillation from a teacher trained with label smoothing performs worse, which is consistent with Muller et al. (2019). Therefore, we conclude that for distillation with soft ¨ labels, the regularization samples during training affect the final distillation performance.
+
+Moreover, we plot the number of regularization samples with respect to different training epochs in Fig. 2. As demonstrated in the plots, the number of such samples increases much faster when using the teacher trained with label smoothing for distillation. For regularization samples, the gap of their number between with and without label smoothing becomes larger for more training epochs. These observations verify our motivation that the bias-variance tradeoff varies sample-wisely and evolves during the training process.
+
+From the above results, we conclude that bias-variance tradeoff for soft labels varies sample-wisely, therefore the strategy for tuning the tradeoff should also be sample-wise. In the next section, we set up ad hoc filters for soft labels and further investigate how regularization samples affect distillation.
+
+# 3.3 HOW REGULARIZATION SAMPLES AFFECT DISTILLATION
+
+The results presented in the last section suggest that we should avoid training with regularization samples. Hence, we design two straightforward solutions and then find that totally filtering out regularization samples deteriorates the distillation performance.
+
+![](images/34eca95f156a069671552d2bd0584f7579088fea9aa05428f3da165524827517.jpg)  
+Figure 3: Computational graph of knowledge distillation with our proposed weighted soft labels.
+
+Table 2: Study of the impact on distillation for regularization samples. Loss function presented here is for the loss on a specific sample. Results are classification Top-1 accuracy. We follow the settings used in Tab. 1 and set $\tau = 4$ . Results are averaged over 5 runs.
+
+<table><tr><td rowspan="2"></td><td colspan="2">Teacher: 75.61；Student with direct training: 73.26;</td><td colspan="2">Student with standard KD: 74.92</td></tr><tr><td>Loss function</td><td>Student performance</td><td>Performance gap</td><td></td></tr><tr><td>Setting Mask KD loss on</td><td>Lce+Ld</td><td>73.51</td><td>to direct training +0.25</td><td>to KD -1.41</td></tr><tr><td>the label related logit Excluding regularization samples</td><td>{Lce,if |a|&lt;|b|</td><td>74.59</td><td>+1.33</td><td>-0.33</td></tr><tr><td rowspan="2">Only on regularization samples</td><td>Lce+Lkd,if ｜a| ≥|b</td><td></td><td></td><td></td></tr><tr><td>{Lce,if |a| ≥|b| Lce+Lkd,if ｜a|&lt;|bl</td><td>73.86</td><td>+0.60</td><td>-1.06</td></tr></table>
+
+The first experiment we conduct is to manually resolve the conflicting gradient on the label related logit, as defined in section 3.2. Specifically, we apply a mask to the distillation loss $L _ { \mathrm { k d } }$ such that $\begin{array} { r } { \frac { \partial { \cal L } _ { \mathrm { k d } } } { \partial z _ { i } } = 0 } \end{array}$ where $i$ is the label. Consequently, the loss for this sample now becomes $L _ { \mathrm { k d } } ^ { * } =$ $\begin{array} { r } { \sum _ { k \neq i } \hat { y } _ { k , \tau } ^ { t } \log \hat { y } _ { k , \tau } ^ { s } } \end{array}$ . The motivation behind the masked distillation loss is that we only transfer the knowledge of resemblances among the labels. Another experiment is to figure out what role in distillation those regularization samples will play. To investigate this, we carry out knowledge distillation on two subsets of samples: 1) $L _ { \mathrm { k d } }$ is not valid on regularization samples, and 2) $L _ { \mathrm { k d } }$ is valid only on regularization samples.
+
+The results of the two experiments are presented in Tab. 2. We can observe that all of the three approaches are not as good as the baseline knowledge distillation performance, but are better than the direct training baseline. First, since masking $L _ { \mathrm { k d } }$ loss on the label related logit results in worse performance compared to standard KD, we cannot resolve the tradeoff by applying a mask on the ground truth related logit. Then, from the second experiment, we can see that filtering out regularization samples deteriorates the distillation performance. Moreover, the result of the third experiment is higher than the direct training baseline, indicating that regularization samples are still valuable for distillation. The above results motivate us to think that regularization samples are not fully exploited by standard KD and we can tune the tradeoff to fulfill the potential of regularization samples.
+
+# 4 WEIGHTED SOFT LABELS
+
+From the last section, we realize that the bias-variance tradeoff varies sample-wisely during training and under fixed distillation settings, the number of regularization samples is negatively associated with the final distillation performance. Yet, discarding regularization samples deteriorates distillation performance and distilling knowledge from these samples is better than the direct training baseline. The above evidence inspires us to lower the weight of regularization samples.
+
+Recall that regularization samples are defined by the relative value of $a$ and $b$ , we propose to assign importance weight to a sample according to $a$ and $b$ . However, since $L _ { \mathrm { k d } }$ is computed with the hyperparameter temperature, $a$ and $b$ are correlated with the temperature and thus bring difficulty to tuning the hyperparameter. To make the weighting scheme independent of the temperature hyperparameter, we compare $a$ and $b$ with temperature $\tau = 1$ . Note that when $\tau = 1$ , $a = \hat { y } _ { i , 1 } ^ { s } - y _ { i }$ and $b = y _ { i } - \hat { y } _ { i , 1 } ^ { t }$ , so we compare $\hat { y } _ { i , 1 } ^ { s }$ and $\hat { y } _ { i , 1 } ^ { t }$ instead. Finally, in the light of previous works that assign sample-wise weights (Lin et al., 2017; Tang et al., 2019), we propose weighted soft labels for knowledge distillation, which is formally defined as
+
+Table 3: Top-1 classification accuracy results on CIFAR-100. Comparison results are quoted from Tian et al. (2020). We report our results over 5 repeated runs.   
+
+<table><tr><td rowspan="2">Teacher</td><td colspan="5">Same architecture style</td><td colspan="3">Different architecture style</td></tr><tr><td>WRN-40-2</td><td>resnet56</td><td>resnet110</td><td>resnet110</td><td>resnet32x4</td><td>resnet32x4 ShuffleNetV1</td><td>resnet32x4</td><td>WRN-40-2 ShufleNetV1</td></tr><tr><td>Student Teacher</td><td>WRN-40-1 75.61</td><td>resnet20 72.34</td><td>resnet20 74.31</td><td>resnet32 74.31</td><td>resnet8x4 79.42</td><td>79.42</td><td>ShuffleNetV2 79.42</td><td>75.61</td></tr><tr><td>Student</td><td>71.98</td><td>69.06</td><td>69.06</td><td>71.14</td><td>72.50</td><td>70.5</td><td>71.82</td><td>70.5</td></tr><tr><td></td><td></td><td></td><td>68.99</td><td>71.06</td><td>73.50</td><td>73.59</td><td>73.54</td><td>73.73</td></tr><tr><td>FitNet AT</td><td>72.24 72.77</td><td>69.21 70.55</td><td>70.22</td><td>72.31</td><td>73.44</td><td>71.73</td><td>72.73</td><td>73.32</td></tr><tr><td>SP</td><td>72.43</td><td>69.67</td><td>70.04</td><td>72.69</td><td>72.94</td><td>73.48</td><td>74.56</td><td>74.52</td></tr><tr><td>CC</td><td>72.21</td><td>69.63</td><td>69.48</td><td>71.48</td><td>72.97</td><td>71.14</td><td>71.29</td><td>71.38</td></tr><tr><td>VID</td><td>73.30</td><td>70.38</td><td>70.16</td><td>72.61</td><td>73.09</td><td>73.38</td><td>73.40</td><td>73.61</td></tr><tr><td>RKD</td><td>72.22</td><td>69.61</td><td>69.25</td><td>71.82</td><td>71.90</td><td>72.28</td><td>73.21</td><td>72.21</td></tr><tr><td>PKT</td><td>73.45</td><td>70.34</td><td>70.25</td><td>72.61</td><td>73.64</td><td>74.10</td><td>74.69</td><td>73.89</td></tr><tr><td>AB</td><td>72.38</td><td>69.47</td><td>69.53</td><td>70.98</td><td>73.17</td><td>73.55</td><td>74.31</td><td>73.34</td></tr><tr><td>FT</td><td>71.59</td><td>69.84</td><td>70.22</td><td>72.37</td><td>72.86</td><td>71.75</td><td>72.50</td><td>72.03</td></tr><tr><td>FSP</td><td>n/a</td><td>69.95</td><td>70.11</td><td>71.89</td><td>72.62</td><td>n/a</td><td>n/a</td><td>n/a</td></tr><tr><td>NST</td><td>72.24</td><td>69.60</td><td>69.53</td><td>71.96</td><td>73.30</td><td>74.12</td><td>74.68</td><td>74.89</td></tr><tr><td>KD</td><td>73.54</td><td>70.66</td><td>70.67</td><td>73.08</td><td>73.33</td><td>74.07</td><td>74.45</td><td>74.83</td></tr><tr><td>CRD</td><td>74.14</td><td>71.16</td><td>71.46</td><td>73.48</td><td>75.51</td><td>75.11</td><td>75.65</td><td>76.05</td></tr><tr><td>Ours</td><td>74.48</td><td>72.15</td><td>72.19</td><td>74.12</td><td>76.05</td><td>75.46</td><td>75.93</td><td>76.21</td></tr></table>
+
+$$
+L _ { \mathrm { w s l } } = \left( 1 - \exp \left( - \frac { \log \hat { y } _ { i , 1 } ^ { s } } { \log \hat { y } _ { i , 1 } ^ { t } } \right) \right) L _ { \mathrm { k d } } = \left( 1 - \exp \left( - \frac { L _ { \mathrm { c e } } ^ { s } } { L _ { \mathrm { c e } } ^ { t } } \right) \right) L _ { \mathrm { k d } } ,
+$$
+
+where $i$ is the ground truth class of the sample. The above equation means that a weighting factor is assigned to each sample’s $L _ { \mathrm { k d } }$ according to the predictions of the teacher and the student. In this way, if compared to the teacher, a student network is relatively better trained on a sample, we have $\hat { y } _ { i , 1 } ^ { s } > \hat { y } _ { i , 1 } ^ { t }$ , then a smaller weight is assigned to this sample. In Fig. 3, the whole computational graph of knowledge distillation with the proposed weighted soft labels is demonstrated. Finally, we add $L _ { \mathrm { w s l } }$ and $L _ { \mathrm { c e } }$ together to supervise the network, i.e., ${ \cal L } _ { \mathrm { t o t a l } } = { \cal L } _ { \mathrm { c e } } + \alpha { \cal L } _ { \mathrm { w s l } }$ , where $\alpha$ is a balancing hyperparameter.
+
+# 5 EXPERIMENTS
+
+To evaluate our weighted soft labels comprehensively, we first conduct experiments with various teacher-student pair settings on CIFAR-100 (Krizhevsky et al., 2009). Next, we compare our method with current state-of-the-art distillation methods on ImageNet (Deng et al., 2009). To validate the effectiveness of our method in terms of handling the bias-variance tradeoff, we conduct ablation experiments by applying weighted soft labels on different subsets.
+
+# 5.1 DATASET AND HYPERPARAMETER SETTINGS
+
+The datasets used in our experiments are CIFAR-100 (Krizhevsky et al., 2009) and ImageNet (Deng et al., 2009). CIFAR-100 contains 50K training and 10K test images of size $3 2 \times 3 2$ . ImageNet contains 1.2 million training and 50K validation images. Except the loss function, training settings like learning rate or training epochs are the same with Tian et al. (2020) for CIFAR-100 and Heo et al. (2019) for ImageNet. For distillation, we set the temperature $\tau = 4$ for CIFAR and $\tau = 2$ for ImageNet. For loss function, we set $\alpha = 2 . 2 5$ for distillation on CIFAR and $\alpha = 2 . 5$ for ImageNet via grid search. The teacher network is well-trained previously and fixed during training.
+
+For comparison, the following recent state-of-the-art methods are chosen: FitNet (Romero et al., 2015), AT (Zagoruyko & Komodakis, 2017), SP (Tung & Mori, 2019), CC (Peng et al., 2019), VID (Ahn et al., 2019), RKD (Park et al., 2019), PKT (Passalis & Tefas, 2018), AB (Heo et al., 2019),
+
+Table 4: Top-1 and Top-5 classification accuracy results on ImageNet validation set. All training hyperparameter like learning rate and training epochs are in accordance with (Heo et al., 2019).   
+
+<table><tr><td colspan="2">Teacher: 1 :ResNet-34 →Student:ResNet-18</td></tr><tr><td>Method</td><td>Top-1 Acc Top-5 Acc</td></tr><tr><td>Teacher Student</td><td>73.31 91.42 89.07</td></tr><tr><td>KD</td><td>69.75 70.67 90.04</td></tr><tr><td>AT</td><td>71.03 90.04</td></tr><tr><td>NST</td><td>70.29 89.53</td></tr><tr><td>FSP</td><td>70.58 89.61</td></tr><tr><td>RKD</td><td>70.40 89.78</td></tr><tr><td>Overhaul</td><td>71.03 90.15</td></tr><tr><td>CRD</td><td>71.17 90.13</td></tr><tr><td>Ours</td><td>72.04 90.70</td></tr></table>
+
+FT (Kim et al., 2018), FSP (Yim et al., 2017), NST (Huang & Wang, 2017), Overhaul (Heo et al., 2019) and CRD (Tian et al., 2020).
+
+<table><tr><td colspan="2">Teacher:ResNet-50 → Student: MobileNet-v1</td></tr><tr><td>Method</td><td>Top-1 Acc Top-5 Acc</td></tr><tr><td>Teacher</td><td>76.16 92.87</td></tr><tr><td>Student</td><td>68.87 88.76</td></tr><tr><td>KD</td><td>70.49 89.92</td></tr><tr><td>AT</td><td>70.18 89.68</td></tr><tr><td>FT</td><td>69.88 89.5</td></tr><tr><td>AB</td><td>68.89 88.71</td></tr><tr><td>RKD</td><td>68.50 88.32</td></tr><tr><td>Overhaul</td><td>71.33 90.33</td></tr><tr><td>CRD</td><td>69.07 88.94</td></tr><tr><td>Ours</td><td>71.52 90.34</td></tr></table>
+
+# 5.2 MODEL COMPRESSION
+
+Results on CIFAR-100 In Tab. 3, we present the Top-1 classification accuracy of our method and comparison methods. The results of comparison methods are quoted from Tian et al. (2020). Teacher-student pairs of the same and different architecture styles are considered. For pairs of same architecture style, we use wide residual networks (Zagoruyko & Komodakis, 2017) and residual networks (He et al., 2016). For pairs of different architecture style, residual networks and ShuffleNet (Zhang et al., 2018) pairs are chosen for experiments. As shown in the table, for distillation with both same and different architecture style, our method reached new state-of-the-art results. Specifically, our method outperforms standard KD by a large margin, which verifies the effectiveness of our method.
+
+Results on ImageNet In Tab. 4, we compare our method with current SOTA methods on ImageNet. Note that for the $\mathrm { R e s N e t } 3 4  \mathrm { R e s N e t } - 1 8$ distillation setting, the result of CRD is trained 10 more extra epochs while ours is the same as other methods. For ResNet- $5 0 $ MobileNet-v1 distillation setting, NST and FSP are not chosen for comparison as the two methods require too large GPU memories, so we include the accuracy of FT and AB reported in Heo et al. (2019) for comparison. Our results outperform all the existing methods, verifying the practical value of our method.
+
+# 5.3 ABLATION STUDIES
+
+Weighted soft labels on different subsets. Recall that we propose weighted soft labels for tuning samplewise bias-variance tradeoff, it is still unclear whether the improvements come from a well-handled samplewise bias-variance tradeoff. To investigate this issue, we compare the performance gain of weighted soft labels on different training subsets. Similar to the settings used in Tab. 2, we apply weighted soft labels on two different subsets: only the regularization samples and excluding regularization samples. In Tab. 5, we show the results on subsets of only regularization samples and excluding regularization samples. From the significant improvements, we can see that our method can not only improve performance on the RS subset, the improvements on excluding RS subset is also significant. We conclude that weighted soft labels can tune sample-wise bias-variance tradeoff globally and lead to an improved distillation performance.
+
+Table 5: Performance on different subsets with soft labels and our weighted soft labels. RS means regularization samples. Results are averaged over 5 runs.   
+
+<table><tr><td>Subsets</td><td>Standard KD</td><td>Weighted</td></tr><tr><td>Only on RS</td><td>73.86</td><td>74.46</td></tr><tr><td>Excluding RS</td><td>74.59</td><td>75.35</td></tr></table>
+
+Distillation with label smoothing trained teacher. Our exploration of bias-variance tradeoff starts with the conclusion made in Muller et al. (2019): a teacher net- ¨ work trained with the label smoothing trick is less effective for distillation. It is worthwhile to study whether the conclusion remains true for distillation with our weighted soft labels. As discussed before, we hold the opinion that too many regularization samples make the distillation less effective. Since our weighted soft label is proposed to mitigate the negative effects of the regularization samples, with the same settings from Tab. 1, we conduct comparison experiments in Tab. 6 to see if the negative effects still exist. It is evident that weighted soft labels significantly improve the distillation performance, especially for distillation from the teacher trained with label smoothing. Besides, using the teacher trained with label smoothing still performs worse than that without label smoothing, which again verifies the conclusion drawn by Muller et al. ¨ (2019).
+
+Table 6: Distillation using weighted soft labels and teacher trained with label smoothing (denoted as $L S ? _ { \mathrm { \Gamma } }$ ). Results are averaged over 5 runs.   
+
+<table><tr><td>T</td><td>LS？</td><td>Standard KD</td><td>Weighted</td></tr><tr><td>4</td><td>X</td><td>74.92</td><td>75.78</td></tr><tr><td>4</td><td>√</td><td>74.59</td><td>75.60</td></tr><tr><td>6</td><td>X</td><td>75.10</td><td>75.74</td></tr><tr><td>6</td><td>√</td><td>74.46</td><td>75.42</td></tr></table>
+
+# 6 CONCLUSION
+
+Recent studies (Muller et al., 2019; Yuan et al., 2020) point out that one important reason behind ¨ the effectiveness of distillation is the regularization effect brought by being soft. In this paper, we rethink the soft labels for distillation from a bias-variance tradeoff perspective. The tradeoff varies sample-wisely and we propose weighted soft labels to handle the tradeoff, of which the effectiveness is verified with experiments on standard evaluation benchmarks.
+
+# ACKNOWLEDGEMENTS
+
+This work is supported in part by a gift grant from Horizon Robotics and National Science Foundation Grant CNS-1951952. We thank Yichen Gong, Chuan Tian, Jiemin Fang and Yuzhu Sun for the discussion and assistance.
+
+# REFERENCES
+
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+
+![](images/66eb17c3d57a429523bedca877ffb5586df54a5dac477bdcf2cb7db5e1546152.jpg)  
+Figure 4: Visualization of the resemblances introduced by soft label regularizers: (a) VGG-19 (Teacher) $\to \operatorname { V G G - 1 6 }$ (Student), (b) ResNet-50 (Teacher) $ \mathrm { R e s N e t } { - 1 8 }$ (Student). And semantic similarity between label names: (c) LCH similarity (Pedersen et al., 2004), (d) WUP similarity (Pedersen et al., 2004). Darker areas denote larger values.
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+
+# A APPENDIX
+
+# A.1 VISUALIZATION OF THE RESEMBLANCES INTRODUCED BY SOFT LABEL REGULARIZER
+
+In Sthat $\frac { \partial ( L _ { \mathrm { k d } } - L _ { \mathrm { c e } } ) } { \partial z _ { i } }$ proposequals $y _ { i } - \hat { y } _ { i , 1 } ^ { t }$ . ∂(Lkd−Lce)∂z during the training process. When τ = 1, we show $y _ { i , \tau } ^ { t }$ the output from the teacher network mputed by a linear mapping of the activations in the teacher’s penultimate layer, the regularization indicates that the student should follow the learned the resemblances between classes (Hinton et al., 2015; Muller et al., 2019). Still, two questions are unclear: 1) what the resemblances are and 2) whether ¨ the regularization still indicates resemblances if $\tau$ is set to 4, a widely adopted hyperparameter (Tian et al., 2020).
+
+Towards answering the questions, we visualize the value of gradient vector $\frac { \partial ( L _ { \mathrm { k d } } - L _ { \mathrm { c e } } ) } { \partial z }$ concerning each class. Specifically, on ImageNet (Deng et al., 2009) training set and $\tau = 4$ , we calculate the average value of $\frac { \partial ( L _ { \mathrm { k d } } - L _ { \mathrm { c e } } ) } { \partial z }$ for each class. Let $M$ be the matrix of values with the $i j$ -th entry $M _ { i j }$ means averaged ∂(Lkd−Lce)∂z for class j. Since Pi Mij = 0, diagonal elements are ignored for visualization. The results are visualized in Fig. 4. We find that plotting the common correlation matrix heatmap is ambiguous, because the matrix to be visualized is of large size $( 1 0 0 0 \times 1 0 0 0 )$ with a large variance. By treating each entry $M _ { i j }$ as a vertex and then constructing a mesh for the matrix, we apply subdivision (Loop, 1987) to the mesh for smoothing the extreme points and finally rendering the mesh by ray-tracing package PlotOptiX. We can observe the several facts from the figures: 1) Comparing the sub-figure (a) and (b), we can see that for distillation resemblances implied by regularizers are similar across different teacher-student pairs. 2) Comparing (ab) with (cd), we can see that the resemblances are consistent with the semantic similarity of image class names.
+
+Table 7: Intermediate states between excluding and only on regularization samples   
+(a) CIFAR100 (WRN-40-2→WRN-40-1 with KD)   
+
+<table><tr><td colspan="4">Percentage of excluded regularization samples.</td></tr><tr><td>100%</td><td>75%</td><td>50% 74.72</td><td>25% 74.87</td></tr><tr><td>74.59</td><td>74.63</td><td></td><td></td></tr><tr><td colspan="4"> Percentage of adding non-regularization samples</td></tr><tr><td>0%</td><td>25%</td><td>50%</td><td>75%</td></tr><tr><td>73.86</td><td>74.12</td><td>74.47</td><td>74.71</td></tr></table>
+
+(b) CIFAR100 (WRN-40-2→WRN-40-1 with weighted soft labels)   
+
+<table><tr><td colspan="4">Percentage of excluded regularization samples</td></tr><tr><td>100%</td><td>75%</td><td>50%</td><td>25%</td></tr><tr><td>75.35</td><td>75.48</td><td>75.61</td><td>75.72</td></tr><tr><td colspan="4">Percentage of : f adding non-regularization samples</td></tr><tr><td>0%</td><td>25%</td><td>50%</td><td>75%</td></tr><tr><td>74.46</td><td>74.79</td><td>75.18</td><td>75.53</td></tr></table>
+
+In a word, for $\tau = 4$ , the variance reduction brought by soft labels still implies resemblances among labels, which are consistent with the semantic distance of class names. In the next section, we will analyze how bias-variance tradeoff changes when training with soft labels.
+
+# A.2 INTERMEDIATE STATES BETWEEN EXCLUDING AND ONLY ON REGULARIZATION SAMPLES
+
+To further investigate the phenomenon about regularization samples, we conduct experiments to show the intermediate states between excluding and only on regularization samples. Two settings are considered here: First, we gradually exclude regularization samples during training, from excluding all regularization samples to excluding $2 5 \%$ regularization samples; Second, we keep all regularization samples and then gradually add non-regularization samples. Since we judge a sample is regularization or not according to the training loss, we cannot pre-define a sample set such that a certain percentage samples are kept or dropped. Therefore, we propose to conduct these experiments by assigning a probability to whether backward the loss computed with regularization samples. For example, if during training, a sample is marked as regularization sample according to the value of $a$ and $b$ , we backward the loss of this sample by a probability $p = 0 . 5$ . In this way, we can get the performance of excluding $7 5 \%$ regularization samples. In Tab. 7, we first present result with KD in (a) and then present result with weighted soft labels applied in (b). We can observe that weighted soft labels are indeed balancing the sample-wise, not on dataset scale, bias and variance.
+
+# A.3 COMBINING WITH RKD (PARK ET AL., 2019).
+
+To investigate how the weighted soft labels can be applied to the variants of KD, we conduct an experiment of combining RKD (Park et al., 2019) with our weighted soft labels. Relational knowledge distillation measures the L2 distance of features between two samples or the angle formed by three samples as knowledge to transfer. In other words, the knowledge in RKD is measured by the relations between sample pairs. It is no longer sample-independent, which is different from the weighted soft labels applied to KD which can assign the weights sample-wisely. We currently take the averaged weighting factors of the involved sample pairs when calculating the distance/angle matrix. The results on CIFAR-100 are presented in Tab. 8 (averaged over 5 runs). As can be observed from the table, the weighted soft label applied to RKD still brings improvements, though not that big compared with WSL applied to KD. Also, we believe that it is an important future direction to explore the applications to more variants of KD.
+
+Table 8: Combining weighted soft labels with RKD (Park et al., 2019).   
+
+<table><tr><td>Distillation settings</td><td>WRN-40-2 →WRN-16-2</td><td>WRN-40-2 →WRN-40-1</td><td>resnet56→resnet20</td></tr><tr><td>Teacher</td><td>75.61</td><td>75.61</td><td>72.34</td></tr><tr><td>Student</td><td>73.26</td><td>71.98</td><td>69.06</td></tr><tr><td>RDK</td><td>74.12</td><td>73.34</td><td>70.25</td></tr><tr><td>WSL + RKD</td><td>74.65</td><td>73.89</td><td>70.73</td></tr></table>
+
+Table 9: Comparison to other weighting forms. (Setting: CIFAR100, WRN-40-2 WRN-40-1)   
+
+<table><tr><td>α</td><td>2.0</td><td>3.0</td><td>4.0</td></tr><tr><td>Sigmoid baseline</td><td>74.13</td><td>73.97</td><td>73.29</td></tr><tr><td>Ours</td><td>74.38</td><td>74.12</td><td>73.46</td></tr></table>
+
+# A.4 OTHER VARIANTS OF WEIGHTING.
+
+In the work, the weighting scheme is defined as $\begin{array} { r } { \left( 1 - \exp \left( - \frac { L _ { \mathrm { c e } } ^ { s } } { L _ { \mathrm { c e } } ^ { t } } \right) \right) } \end{array}$ , which is inspired by Lin et al. (2017); Tang et al. (2019). The basic idea is to convert $\frac { L _ { \mathrm { c e } } ^ { s } } { L _ { \mathrm { c e } } ^ { t } }$ into a value in $[ 0 , 1 ]$ , so that the weights of regularization samples are lower than those non-regularization samples. A straightforward baseline is that we can use the Sigmoid function to convert $\frac { L _ { \mathrm { c e } } ^ { s } } { L _ { \mathrm { c e } } ^ { t } }$ into a value in $[ 0 , 1 ]$ . Note that L cet is always L ce bigger than 1, so the weight needs scaling and can be defined as $\frac { 2 } { 1 + \exp ( - \frac { L _ { \mathrm { c e } } ^ { s } } { L _ { \mathrm { c e } } ^ { t } } ) } - 1$ . In Tab. 9, we present the comparison between adopted weighting form and the Sigmoid baseline. We can see that as long as we can adaptively tune the sample-wise bias-variance tradeoff, the performance is better than KD, i.e., without weighted soft labels.Therefore, although the proposed weighting form is not mathematically optimal, the not-too-big or not-too-small weights for these regularization examples are not hard to tune. These results verify our main contribution that there is sample-wise biasvariance tradeoff and we need to assign weights to the regularization examples.
+
+# A.5 ABLATION ON $\alpha$
+
+In Tab. 10 We first tune the value of $\alpha$ on CIFAR100, with four values $\{ 1 , 2 , 3 , 4 \}$ tested. Then we test with three values in [2, 3] in (b). Finally, we tune $\alpha$ on ImageNet in (c). As a conclusion, the results are not very sensitive to $\alpha$ and the cost of searching $\alpha$ in our work is not expensive.
+
+# A.6 RESULTS ON MULTINLI
+
+To further validate our method, we conduct experiments on an NLP dataset MultiNLI (Williams et al., 2018). In this setting, the teacher is BERT-base-cased with 12 layers, 768 Hidden and 108M params. The student is T3 with 3 layers, 768 Hidden and 44M params. Besides, we follow the training setting in Sun et al. (2019). In Tab. 11, we present the result comparisons of standard KD and our weighted soft labels.
+
+Table 10: Ablation on $\alpha$   
+
+<table><tr><td colspan="5">(a) CIFAR100 (WRN-40-2→WRN-40-1)</td></tr><tr><td>α</td><td>1</td><td>2</td><td>3</td><td>4</td></tr><tr><td>Top1</td><td>73.67</td><td>74.38</td><td>74.12</td><td>73.46</td></tr></table>
+
+(b) CIFAR100 (WRN-40-2 WRN-40-1)   
+
+<table><tr><td>α</td><td>2.25</td><td>2.5</td><td>2.75</td></tr><tr><td>Top1</td><td>74.48</td><td>74.34</td><td>74.21</td></tr></table>
+
+(c) ImageNet (ResNet-34 ResNet-18)
+
+<table><tr><td>α</td><td>2</td><td>2.25</td><td>2.5</td></tr><tr><td>Top1</td><td>71.91</td><td>71.96</td><td>72.04</td></tr></table>
+
+Table 11: Results on MultiNLI.   
+
+<table><tr><td></td><td>Teacher (BERT-12)</td><td>Student (BERT-3)</td><td>KD (BERT-3)</td><td>Ours (BERT-3)</td></tr><tr><td>Results reported by Sun et al. (2019)</td><td>83.7</td><td>74.8</td><td>75.4</td><td>-</td></tr><tr><td>Our replications</td><td>83.57</td><td>75.06</td><td>75.50</td><td>76.28</td></tr></table>

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+    {
+        "type": "text",
+        "text": "RETHINKING SOFT LABELS FOR KNOWLEDGE DISTILLATION: A BIAS-VARIANCE TRADEOFF PERSPECTIVE ",
+        "text_level": 1,
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+        "text": "Helong Zhou1∗, Liangchen $\\mathbf { S o n g ^ { 2 * \\dagger } }$ , Jiajie Chen1∗ , Ye Zhou1, Guoli Wang13, Junsong Yuan2, Qian Zhang1 1Horizon Robotics 2University at Buffalo 3Tsinghua University {helong.zhou,jiajie.chen,ye.zhou,guoli.wang}@horizon.ai {lsong8,jsyuan}@buffalo.edu, qian01.zhang@horizon.ai ",
+        "bbox": [
+            186,
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "ABSTRACT ",
+        "text_level": 1,
+        "bbox": [
+            454,
+            279,
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+        ],
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+    },
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+        "type": "text",
+        "text": "Knowledge distillation is an effective approach to leverage a well-trained network or an ensemble of them, named as the teacher, to guide the training of a student network. The outputs from the teacher network are used as soft labels for supervising the training of a new network. Recent studies (Muller et al., 2019; Yuan ¨ et al., 2020) revealed an intriguing property of the soft labels that making labels soft serves as a good regularization to the student network. From the perspective of statistical learning, regularization aims to reduce the variance, however how bias and variance change is not clear for training with soft labels. In this paper, we investigate the bias-variance tradeoff brought by distillation with soft labels. Specifically, we observe that during training the bias-variance tradeoff varies sample-wisely. Further, under the same distillation temperature setting, we observe that the distillation performance is negatively associated with the number of some specific samples, which are named as regularization samples since these samples lead to bias increasing and variance decreasing. Nevertheless, we empirically find that completely filtering out regularization samples also deteriorates distillation performance. Our discoveries inspired us to propose the novel weighted soft labels to help the network adaptively handle the sample-wise biasvariance tradeoff. Experiments on standard evaluation benchmarks validate the effectiveness of our method. Our code is available at https://github.com/ bellymonster/Weighted-Soft-Label-Distillation. ",
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+    {
+        "type": "text",
+        "text": "1 INTRODUCTION ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "For deep neural networks (Goodfellow et al., 2016), knowledge distillation (KD) (Ba & Caruana, 2014; Hinton et al., 2015) refers to the technique that uses well-trained networks to guide the training of another network. Typically, the well-trained network is named as the teacher network while the network to be trained is named as the student network. For distillation, the predictions from the teacher network are leveraged and referred to as the soft labels (Balan et al., 2015; Muller et al., ¨ 2019). Soft labels generated by the teacher network have been proven effective in large-scale empirical studies (Liang et al., 2019; Tian et al., 2020; Zagoruyko & Komodakis, 2017; Romero et al., 2015) as well as recent theoretical studies (Phuong & Lampert, 2019). ",
+        "bbox": [
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "However, the reason why soft labels are beneficial to the student network is still not well explained. Giving a clear theoretical explanation is challenging: The optimization details of a deep network with the common one-hot labels are still not well-studied (Nagarajan & Kolter, 2019), not to mention training with the soft labels. Nevertheless, two recent studies (Muller et al., 2019; Yuan et al., ¨ 2020) shed light on the intuitions about how the soft labels work. Specifically, label smoothing, which is a special case of soft labels based training, is shown to regularize the activations of the penultimate layer to the network (Muller et al., 2019). The regularization property of soft labels is ¨ further explored in (Yuan et al., 2020). They hypothesize that in KD, one main reason why the soft labels work is the regularization introduced by soft labels. Based on the assumption, the authors design a teacher-free distillation method by turning the predictions of the student network into soft labels. ",
+        "bbox": [
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+        "page_idx": 0
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+    {
+        "type": "text",
+        "text": "",
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+        "page_idx": 1
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+    {
+        "type": "text",
+        "text": "Considering that soft labels are targets for distillation, the evidence of the regularization brought by soft labels drives us to rethink soft labels for KD: Soft labels are both supervisory signals and regularizers. Meanwhile, it is known that there is a tradeoff between fitting the data and imposing regularizations, i.e., the bias-variance dilemma (Kohavi & Wolpert, 1996; Bishop, 2006), but it is unclear how bias and variance change for distillation with soft labels. Since the bias-variance tradeoff is an important issue in statistical learning, we investigate whether the bias-variance tradeoff exists for soft labels and how the tradeoff affects distillation performance. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "We first compare the bias and variance decomposition of direct training with that of distillation with soft labels, noticing that distillation results in a larger bias error and a smaller variance. Then, we rewrite distillation loss into the form of a regularization loss adding the direct training loss. Through inspecting the gradients of the two terms during training, we notice that for soft labels, the biasvariance tradeoff varies sample-wisely. Moreover, by looking into a conclusion from (Muller et al., ¨ 2019), we observe that under the same temperature setting, the distillation performance is negatively associated with the number of some certain samples. These samples lead to bias increase and variance decrease and we name them as regularization samples. To investigate how regularization samples affect distillation, we first examine if we can design ad hoc filters for soft labels to avoid training with regularization samples. But completely filtering out regularization samples also deteriorates distillation performance, leading us to speculate that regularization samples are not well handled by standard KD. In the light of these findings, we propose weighted soft labels for distillation to handle the sample-wise bias-variance tradeoff, by adaptively assigning a lower weight to regularization samples and a larger weight to the others. To sum up, our contributions are: ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "• For knowledge distillation, we analyze how the soft labels work from a perspective of biasvariance tradeoff.   \n• We discover that the bias-variance tradeoff varies sample-wisely. Also, we discover that if we fix the distillation temperature, the number of regularization samples is negatively associated with the distillation performance.   \n• We design straightforward schemes to alleviate negative impacts from regularization samples and then propose the novel weighted soft labels for distillation. Experiments on large scale datasets validate the effectiveness of the proposed weighted soft labels. ",
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+    {
+        "type": "text",
+        "text": "2 RELATED WORKS ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "Knowledge distillation. Hinton et al. (2015) proposed to distill outputs from large and cumbersome models into smaller and faster models, which is named as knowledge distillation. The outputs for large networks are averaged and formulated as soft labels. Also, other kinds of soft labels have been widely used for training deep neural networks (Szegedy et al., 2016; Pereyra et al., 2017). Treating soft labels as regularizers were pointed out in (Hinton et al., 2015) since a lot of helpful information can be carried in soft labels. More recently, Muller et al. (2019) showed the adverse ¨ effect of label smoothing upon distillation. It is a thought-provoking discovery for the reason that both label smoothing and distillation are exploiting the regularization property behind soft labels. Yuan et al. (2020) further investigated the regularization property of soft labels and then proposed a teacher free distillation scheme. ",
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+    {
+        "type": "text",
+        "text": "Distillation loss. One of our main contributions is that we improve the distillation loss. For adaptively adjusting the distillation loss, Tang et al. (2019) pays attention to hard-to-learn and hard-tomimic samples, and the latter is weighted based on the prediction gap between teacher and student. However, it does not consider that the teacher may give an incorrect guide to the student, under which the prediction gap is still large and such a method may lead to the performance being hurt. Saputra et al. (2019) transfers teacher’s guidance only on the samples where the performance of the teacher surpasses the student, while Wen et al. (2019) deals with the incorrect guidance by probability shifting strategy. Our approach is different from the above methods, in terms of motivations as well as the proposed solutions. ",
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+        "text": "Bias-variance tradeoff. Bias-variance tradeoff is a well-studied topic in machine learning (Kohavi & Wolpert, 1996; Domingos, 2000; Valentini & Dietterich, 2004; Bishop, 2006) and for neural networks (Geman et al., 1992; Neal et al., 2018; Belkin et al., 2019; Yang et al., 2020). Existing methods are mainly concerned with the variance brought by the choice of network models. Our perspective is different from the previous methods since we focus on the behavior of samples during training. In our work, based on the results from Heskes (1998), we present the decomposition of distillation loss, which is defined by Kullback-Leibler divergence. Besides, our main contribution is not to study how to theoretically analyze the tradeoff, but how to adaptively tune the sample-wise tradeoff during training. ",
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+    {
+        "type": "text",
+        "text": "3 BIAS-VARIANCE TRADEOFF FOR SOFT LABELS ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "Soft labels play the role of supervisory signals and regularizations at the same time, which inspires us to rethink soft labels from the perspective of the bias-variance tradeoff. We begin our analysis with some mathematical descriptions. For a sample $\\mathbf { x }$ labeled as $i$ -th class, let the ground-truth label be a one-hot vector y where $y _ { i } = 1$ and other entries are 0. Then for $x$ and softmax output temperature $\\tau$ , the soft label predicted by the teacher network is denoted as $\\hat { y } _ { \\tau } ^ { t }$ and the output from the student is denoted as $\\hat { y } _ { \\tau } ^ { s }$ . The soft label $\\hat { y } _ { \\tau } ^ { t }$ is then used for training the student by the distillation loss, i.e. $\\begin{array} { r } { L _ { \\mathrm { k d } } = - \\tau ^ { 2 } \\sum _ { k } \\hat { y } _ { k , \\tau } ^ { t } \\log \\hat { y } _ { k , \\tau } ^ { s } , } \\end{array}$ , where $\\hat { y } _ { k , \\tau } ^ { s } , \\hat { y } _ { k , \\tau } ^ { t }$ means the $k$ -th element of the student’s output $\\hat { y } _ { \\tau } ^ { s }$ and the teacher’s output $\\hat { y } _ { \\tau } ^ { t }$ , respectively. With the above notations, the cross-entropy loss for training with one-hot labels is $L _ { \\mathrm { c e } } = - y _ { k } \\log \\hat { y } _ { k , 1 } ^ { s }$ . ",
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+        "img_path": "images/1c5419e10bd9eacecac217de5a6662221e71b22dfee40c4412a7a936700c0514.jpg",
+        "image_caption": [
+            "Figure 1: Bias and variance. "
+        ],
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+        "text": "We now present the bias-variance decomposition for $L _ { \\mathrm { c e } }$ and $L _ { \\mathrm { k d } }$ , based on the definition and notations from Heskes (1998). First, we denote the train dataset as $\\mathcal { D }$ and the output distribution on a sample $\\mathbf { x }$ of the network trained without distillation as $\\hat { \\mathbf { y } } _ { \\mathrm { c e } } = f _ { \\mathrm { c e } } ( \\mathbf { x } ; \\mathcal { D } )$ . For the network trained with distillation, the model also depends on the teacher network, so we define the output on $\\mathbf { x }$ as $\\hat { \\mathbf { y } } _ { \\mathrm { k d } } = f _ { \\mathrm { k d } } ( \\mathbf { x } ; \\mathcal { D } , \\mathcal { T } )$ , where $\\tau$ is the selected teacher network. Then, let the averaged output of $\\hat { \\mathbf { y } } _ { \\mathrm { k d } }$ and $\\hat { \\mathbf { y } } _ { \\mathrm { c e } }$ be $\\bar { \\bf y } _ { \\mathrm { k d } }$ and $\\bar { \\mathbf { y } } _ { \\mathrm { c e } }$ , that is, ",
+        "bbox": [
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+    {
+        "type": "equation",
+        "img_path": "images/88e9f869c129672ef4870e397c652785769d7ab85487799c8fdcad7d0e9b1e30.jpg",
+        "text": "$$\n\\bar { \\mathbf { y } } _ { \\mathrm { c e } } = \\frac { 1 } { Z _ { \\mathrm { c e } } } \\exp ( \\mathbb { E } _ { T } [ \\log \\hat { \\mathbf { y } } _ { \\mathrm { c e } } ] ) , \\quad \\bar { \\mathbf { y } } _ { \\mathrm { k d } } = \\frac { 1 } { Z _ { \\mathrm { k d } } } \\exp ( \\mathbb { E } _ { \\mathcal { D } , T } [ \\log \\hat { \\mathbf { y } } _ { \\mathrm { k d } } ] ) ,\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "where $Z _ { \\mathrm { c e } } , Z _ { \\mathrm { k d } }$ are two normalization constant. Then according to Heskes (1998), we have the following decomposition for the expected error on the sample $\\mathbf { x }$ and $\\mathbf { y } = t ( \\mathbf { x } )$ is the ground truth label: ",
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+    {
+        "type": "equation",
+        "img_path": "images/671c1936a98a07d1a55c443eec22b06985c28951bf7e14cc54289fda3cdcc1fe.jpg",
+        "text": "$$\n\\begin{array} { r l } & { \\mathrm { e r r o r c e } = \\mathbb { E } _ { \\mathbf { x } , \\mathcal { D } } \\left[ - \\mathbf { y } \\log \\hat { \\mathbf { y } } _ { \\mathrm { c e } } \\right] = \\mathbb { E } _ { \\mathbf { x } , \\mathcal { D } } \\left[ - \\mathbf { y } \\log \\mathbf { y } + \\mathbf { y } \\log \\frac { \\mathbf { y } } { \\bar { \\mathbf { y } } _ { \\mathrm { c e } } } + \\mathbf { y } \\log \\frac { \\bar { \\mathbf { y } } _ { \\mathrm { c e } } } { \\hat { \\mathbf { y } } _ { \\mathrm { c e } } } \\right] } \\\\ & { \\qquad = \\mathbb { E } _ { \\mathbf { x } } \\big [ - \\mathbf { y } \\log \\mathbf { y } \\big ] + \\mathbb { E } _ { \\mathbf { x } } \\left[ \\mathbf { y } \\log \\frac { \\mathbf { y } } { \\bar { \\mathbf { y } } _ { \\mathrm { c e } } } \\right] + \\mathbb { E } _ { \\mathcal { D } } \\left[ \\mathbb { E } _ { \\mathbf { x } } \\left[ \\mathbf { y } \\log \\frac { \\bar { \\mathbf { y } } _ { \\mathrm { c e } } } { \\hat { \\mathbf { y } } _ { \\mathrm { c e } } } \\right] \\right] } \\\\ & { \\qquad = \\mathbb { E } _ { \\mathbf { x } } \\big [ - \\mathbf { y } \\log \\mathbf { y } \\big ] + D _ { \\mathrm { K L } } ( \\mathbf { y } , \\bar { \\mathbf { y } } _ { \\mathrm { c e } } ) + \\mathbb { E } _ { \\mathcal { D } } \\big [ D _ { \\mathrm { K L } } ( \\bar { \\mathbf { y } } _ { \\mathrm { c e } } , \\hat { \\mathbf { y } } _ { \\mathrm { c e } } ) \\big ] } \\\\ & { \\qquad = \\mathrm { i n t i n s i c ~ n o i s e ~ \\mathrm { n i s ~ } } + \\mathrm { \\mathrm { ~ s i a s ~ } } + \\mathrm { v a r i a n c e } , } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "where $D _ { \\mathrm { K L } }$ is the Kullback-Leibler divergence. The derivation of the variance term is based on the facts that $\\frac { \\log \\bar { \\mathbf { y } } _ { \\mathrm { c e } } } { \\mathbb { E } _ { \\mathcal { D } } [ \\log \\hat { \\mathbf { y } } _ { \\mathrm { c e } } ] }$ is a constant and $\\mathbb { E } _ { \\mathbf { x } } [ \\mathbf { y } ] = \\mathbb { E } _ { \\mathbf { x } } [ \\bar { \\mathbf { y } } _ { \\mathrm { c e } } ] = 1$ . Detailed derivations can be found from Eq. (4) in Heskes (1998). Next, we analyze the bias-variance decomposition of $L _ { \\mathrm { k d } }$ . As mentioned above, when training with soft labels, extra randomness is introduced for the selection of a teacher network. In Fig. 1, we illustrate the corresponding bias and variance for the selection process of a set of soft labels, which are generated by a teacher network. In this case, a high variance model indicates the model (grey point) is closer to the one-hot trained model (black point), while a low variance model indicates that the model is closer to other possible models trained with soft labels (red points). Although for KD there are more sources introducing randomness, the overall variance brought by $L _ { \\mathrm { k d } }$ is not necessarily higher than $L _ { \\mathrm { c e } }$ . In fact, existing empirical results strongly suggest that the overall variance is smaller with KD. For example, students trained with soft labels are better calibrated than one-hot baselines (Muller et al., 2019) and KD makes the predictions of ¨ students more consistent when facing adversarial noise (Papernot et al., 2016). Here, we present these empirical evidence as an assumption: ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Assumption 1 The variance brought by $K D$ is smaller than direct training, that is, $\\mathbb { E } _ { \\mathcal { D } , \\mathcal { T } } [ \\bar { D } _ { \\mathrm { K L } } ( \\bar { \\bf y } _ { \\mathrm { k d } } , \\hat { \\bf y } _ { \\mathrm { k d } } ) ] \\leqslant \\mathbb { E } _ { \\mathcal { D } } [ D _ { \\mathrm { K L } } ( \\bar { \\bf y } _ { \\mathrm { c e } } , \\hat { \\bf y } _ { \\mathrm { c e } } ) ]$ . ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Similar to Eq. (2), we write the decomposition for $L _ { \\mathrm { k d } }$ as ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/39ce04504256db0bd658e8e2a66a900286dc4ccf82a8a4563d2fa5551aa0b04d.jpg",
+        "text": "$$\n\\begin{array} { r l r l r } { \\mathrm { e r r o r } _ { \\mathbf { k d } } = \\mathbb { E } _ { \\mathbf { x } } [ - \\mathbf { y } \\log \\mathbf { y } ] + } & { } & { \\qquad } & { } & { \\qquad + \\mathbb { E } _ { \\mathcal { D } , \\mathcal { T } } [ D _ { \\mathrm { K L } } ( \\bar { \\mathbf { y } } _ { \\mathbf { k d } } , \\hat { \\mathbf { y } } _ { \\mathbf { k d } } ) ] . } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "An observation here is that $\\bar { \\mathbf { y } } _ { \\mathrm { c e } }$ converges to one-hot labels while $\\bar { \\bf y } _ { \\mathrm { k d } }$ converges to soft labels, so $\\bar { \\bf y } _ { \\mathrm { c e } }$ is closer to the one-hot ground-truth distribution $\\mathbf { y }$ than $\\bar { \\bf y } _ { \\mathrm { k d } }$ , i.e., $\\mathbb { E } _ { \\mathbf { x } } \\left[ \\mathbf { y } \\mathrm { l o g } \\left( \\frac { \\bar { \\mathbf { y } } _ { \\mathrm { c e } } } { \\bar { \\mathbf { y } } _ { \\mathrm { k d } } } \\right) \\right] \\geqslant 0$ . If we rewrite $L _ { \\mathrm { k d } }$ as $L _ { \\mathrm { k d } } = L _ { \\mathrm { k d } } - L _ { \\mathrm { c e } } + L _ { \\mathrm { c e } }$ , then $L _ { \\mathrm { k d } } \\mathrm { ~ - ~ } L _ { \\mathrm { c e } }$ causes that the bias increases by $\\mathbb { E } _ { \\mathbf { x } } \\left[ \\mathbf { y } \\log \\left( \\frac { \\bar { \\mathbf { y } } _ { \\mathrm { c e } } } { \\bar { \\mathbf { y } } _ { \\mathrm { k d } } } \\right) \\right]$ and the variance decreases by $\\mathbb { E } _ { \\mathcal { D } } [ D _ { \\mathrm { K L } } ( \\bar { \\bf y } _ { \\mathrm { c e } } , \\hat { \\bf y } _ { \\mathrm { c e } } ) ] - \\mathbb { E } _ { \\mathcal { D } , \\tau } [ D _ { \\mathrm { K L } } ( \\bar { \\bf y } _ { \\mathrm { k d } } , \\hat { \\bf y } _ { \\mathrm { k d } } ) ]$ . ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "From the above analysis, we separate $L _ { \\mathrm { k d } }$ into two terms, and $L _ { \\mathrm { k d } } - L _ { \\mathrm { c e } }$ leads to variance reduction, and $L _ { \\mathrm { c e } }$ leads to bias reduction. In the following sections, we first analyze how $L _ { \\mathrm { k d } } - L _ { \\mathrm { c e } }$ links to the bias-variance tradeoff during training. Then we analyze the changes in the relative importance between bias reduction and variance reduction during training with soft labels. ",
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+    {
+        "type": "text",
+        "text": "3.1 THE BIAS-VARIANCE TRADEOFF DURING TRAINING ",
+        "text_level": 1,
+        "bbox": [
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+        "type": "text",
+        "text": "It is known that bias reduction and variance reduction are often in conflict and we cannot minimize bias and variance together. However, if we consider the change of bias and variance during the training process, the importance of tuning the tradeoff also changes during training. Specifically, shortly after the training of the network starts, the bias error dominates the total error and the variance is less important. As training goes on, gradients of reducing the bias error (induced by $L _ { \\mathrm { c e , \\ d } }$ ) and reducing the variance (induced by $L _ { \\mathrm { k d } } - L _ { \\mathrm { c e } } )$ can be of the same scale for some samples, then we need to balance the tradeoff because reducing one term is likely to increase another one. Therefore for soft labels, we need to handle the bias-variance tradeoff in a sample-wise manner and take the training process into consideration. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "To study the bias-variance tradeoff during training, we consider the gradients of bias and variance reduction. Let $_ { z }$ be the logits output of the student on input $x$ and $z _ { i }$ is $i$ -th element of it, then we are interested in ∂(Lkd−Lce)∂z . For simplifying analysis, we are concerned with the gradients on the ground-truth related logit, that is, the sample $x$ is labeled as $i$ -th class. Mathematically, for the gradients of variance reduction, we have ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/24cd52bda9978aafd424aea85569176ede2dbe2b6a63def88c93cb3bd6453767.jpg",
+        "text": "$$\n\\frac { \\partial ( L _ { \\mathrm { k d } } - L _ { \\mathrm { c e } } ) } { \\partial z _ { i } } = \\tau ( \\hat { y } _ { i , \\tau } ^ { s } - \\hat { y } _ { i , \\tau } ^ { t } ) - ( \\hat { y } _ { i , 1 } ^ { s } - y _ { i } ) = \\tau \\left( \\frac { e ^ { z _ { i } / \\tau } } { \\sum _ { k } e ^ { z _ { k } / \\tau } } - \\hat { y } _ { i , \\tau } ^ { t } \\right) - \\left( \\frac { e ^ { z _ { i } } } { \\sum _ { k } e ^ { z _ { k } } } - y _ { i } \\right) ,\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "where $\\hat { y } _ { i , \\tau } ^ { t }$ denotes the $i$ -th element of the teacher’s prediction, i.e., $\\hat { y } _ { \\tau } ^ { t }$ . The term $L _ { \\mathrm { k d } } - L _ { \\mathrm { c e } }$ is easy to understand when $\\tau = 1$ since the gradient now becomes $y _ { i } - \\hat { y } _ { i , 1 } ^ { t }$ . Meanwhile, for the bias reduction, we have $\\begin{array} { r } { \\frac { \\partial L _ { \\mathrm { c e } } } { \\partial z _ { i } } = \\hat { y } _ { i , 1 } ^ { s } - y _ { i } } \\end{array}$ , so ∂Lce and ∂(Lkd−Lce) always have different signs, leading to a tradeoff. If $\\frac { \\partial L _ { \\mathrm { c e } } } { \\partial z _ { i } }$ is much higher than ∂(Lkd−Lce)∂z , the bias reduction dominates the overall optimization direction. Instead, i f ∂(Lkd−Lce)∂z becomes higher, the sample is used for variance reduction. Interestingly, we discover that under a fixed distillation temperature, the final performance is worse when more training samples are used for variance reduction, which will be introduced in the next section. ",
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+    {
+        "type": "text",
+        "text": "3.2 REGULARIZATION SAMPLES ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Our analysis starts with a conclusion from Muller et al. (2019): ¨ if a teacher network is trained with label smoothing, knowledge distillation into a student network is much less effective. Inspired by the phenomenon, we gather the impact of bias and variance during training with different distillation settings. Let $\\begin{array} { r } { a = \\frac { \\partial L _ { \\mathrm { c e } } } { \\partial z _ { i } } } \\end{array}$ and $\\begin{array} { r } { b = \\frac { \\partial ( L _ { \\mathrm { k d } } - L _ { \\mathrm { c e } } ) } { \\partial z _ { i } } } \\end{array}$ , then as introduced before, we use $a$ and $b$ to represent the impact of bias and variance, respectively. If we have $\\vert b \\vert > \\vert a \\vert$ for a sample, we name the sample as a regularization sample since the variance dominates the optimization direction. From the collected data, we find that the number of regularization samples is closely related to distillation performance. ",
+        "bbox": [
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+    {
+        "type": "table",
+        "img_path": "images/d793d4ba5df44af0918c116651f669642f7fd205727622b29ab83f335d633bac.jpg",
+        "table_caption": [
+            "Table 1: We count the number of regularization samples with different distillation settings on CIFAR-100. The teacher-student network pair is WRN-40-2 (Zagoruyko & Komodakis, 2017) and WRN-16-2. Results are averaged over 5 repeated runs. The temperature column means the temperature for distillation and the label smoothing column means whether the teacher network is trained with label smoothing trick. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td colspan=\"3\">Teacher: 76.55 w/ label smoothing,75.61 w/o label smoothing Baseline Top-1 Acc Student: 73.26</td></tr><tr><td>Temperature</td><td>Label smoothing?</td><td>Student Top-1 Acc Number of regularization samples</td></tr><tr><td rowspan=\"2\">T=2</td><td>X 74.79</td><td>15379</td></tr><tr><td>√</td><td>74.62 25235</td></tr><tr><td rowspan=\"2\">T=4</td><td>74.92</td><td>17709</td></tr><tr><td>X</td><td>74.59 24775</td></tr><tr><td rowspan=\"2\">T=6</td><td>X</td><td>75.10 17408 74.46</td></tr><tr><td></td><td>24538</td></tr></table>",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "image",
+        "img_path": "images/22a82c829b2d71362ee6e3c5f07064ed52655b4b1c07453c4e1f09dd220ae21e.jpg",
+        "image_caption": [
+            "Figure 2: The number of regularization samples with respect to training epochs. The distillation settings are the same as the settings in Tab. 1. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "In Tab. 1, we present the count of regularization samples for a student network trained by knowledge distillation. For distillation with a temperature higher than 1, which is the common setting, we observe that if the teacher network is trained with label smoothing, more samples will be involved in variance reduction. Also, distillation from a teacher trained with label smoothing performs worse, which is consistent with Muller et al. (2019). Therefore, we conclude that for distillation with soft ¨ labels, the regularization samples during training affect the final distillation performance. ",
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+    {
+        "type": "text",
+        "text": "Moreover, we plot the number of regularization samples with respect to different training epochs in Fig. 2. As demonstrated in the plots, the number of such samples increases much faster when using the teacher trained with label smoothing for distillation. For regularization samples, the gap of their number between with and without label smoothing becomes larger for more training epochs. These observations verify our motivation that the bias-variance tradeoff varies sample-wisely and evolves during the training process. ",
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+    {
+        "type": "text",
+        "text": "From the above results, we conclude that bias-variance tradeoff for soft labels varies sample-wisely, therefore the strategy for tuning the tradeoff should also be sample-wise. In the next section, we set up ad hoc filters for soft labels and further investigate how regularization samples affect distillation. ",
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "3.3 HOW REGULARIZATION SAMPLES AFFECT DISTILLATION",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "The results presented in the last section suggest that we should avoid training with regularization samples. Hence, we design two straightforward solutions and then find that totally filtering out regularization samples deteriorates the distillation performance. ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "image",
+        "img_path": "images/34eca95f156a069671552d2bd0584f7579088fea9aa05428f3da165524827517.jpg",
+        "image_caption": [
+            "Figure 3: Computational graph of knowledge distillation with our proposed weighted soft labels. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Table 2: Study of the impact on distillation for regularization samples. Loss function presented here is for the loss on a specific sample. Results are classification Top-1 accuracy. We follow the settings used in Tab. 1 and set $\\tau = 4$ . Results are averaged over 5 runs. ",
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+    {
+        "type": "table",
+        "img_path": "images/94eaddea662788b6d2603a6316e7fa91a9e9d2c584df3b7899e227beac9944b8.jpg",
+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=\"2\"></td><td colspan=\"2\">Teacher: 75.61；Student with direct training: 73.26;</td><td colspan=\"2\">Student with standard KD: 74.92</td></tr><tr><td>Loss function</td><td>Student performance</td><td>Performance gap</td><td></td></tr><tr><td>Setting Mask KD loss on</td><td>Lce+Ld</td><td>73.51</td><td>to direct training +0.25</td><td>to KD -1.41</td></tr><tr><td>the label related logit Excluding regularization samples</td><td>{Lce,if |a|&lt;|b|</td><td>74.59</td><td>+1.33</td><td>-0.33</td></tr><tr><td rowspan=\"2\">Only on regularization samples</td><td>Lce+Lkd,if ｜a| ≥|b</td><td></td><td></td><td></td></tr><tr><td>{Lce,if |a| ≥|b| Lce+Lkd,if ｜a|&lt;|bl</td><td>73.86</td><td>+0.60</td><td>-1.06</td></tr></table>",
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+    {
+        "type": "text",
+        "text": "The first experiment we conduct is to manually resolve the conflicting gradient on the label related logit, as defined in section 3.2. Specifically, we apply a mask to the distillation loss $L _ { \\mathrm { k d } }$ such that $\\begin{array} { r } { \\frac { \\partial { \\cal L } _ { \\mathrm { k d } } } { \\partial z _ { i } } = 0 } \\end{array}$ where $i$ is the label. Consequently, the loss for this sample now becomes $L _ { \\mathrm { k d } } ^ { * } =$ $\\begin{array} { r } { \\sum _ { k \\neq i } \\hat { y } _ { k , \\tau } ^ { t } \\log \\hat { y } _ { k , \\tau } ^ { s } } \\end{array}$ . The motivation behind the masked distillation loss is that we only transfer the knowledge of resemblances among the labels. Another experiment is to figure out what role in distillation those regularization samples will play. To investigate this, we carry out knowledge distillation on two subsets of samples: 1) $L _ { \\mathrm { k d } }$ is not valid on regularization samples, and 2) $L _ { \\mathrm { k d } }$ is valid only on regularization samples. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "The results of the two experiments are presented in Tab. 2. We can observe that all of the three approaches are not as good as the baseline knowledge distillation performance, but are better than the direct training baseline. First, since masking $L _ { \\mathrm { k d } }$ loss on the label related logit results in worse performance compared to standard KD, we cannot resolve the tradeoff by applying a mask on the ground truth related logit. Then, from the second experiment, we can see that filtering out regularization samples deteriorates the distillation performance. Moreover, the result of the third experiment is higher than the direct training baseline, indicating that regularization samples are still valuable for distillation. The above results motivate us to think that regularization samples are not fully exploited by standard KD and we can tune the tradeoff to fulfill the potential of regularization samples. ",
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+    {
+        "type": "text",
+        "text": "4 WEIGHTED SOFT LABELS",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "From the last section, we realize that the bias-variance tradeoff varies sample-wisely during training and under fixed distillation settings, the number of regularization samples is negatively associated with the final distillation performance. Yet, discarding regularization samples deteriorates distillation performance and distilling knowledge from these samples is better than the direct training baseline. The above evidence inspires us to lower the weight of regularization samples. ",
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+    {
+        "type": "text",
+        "text": "Recall that regularization samples are defined by the relative value of $a$ and $b$ , we propose to assign importance weight to a sample according to $a$ and $b$ . However, since $L _ { \\mathrm { k d } }$ is computed with the hyperparameter temperature, $a$ and $b$ are correlated with the temperature and thus bring difficulty to tuning the hyperparameter. To make the weighting scheme independent of the temperature hyperparameter, we compare $a$ and $b$ with temperature $\\tau = 1$ . Note that when $\\tau = 1$ , $a = \\hat { y } _ { i , 1 } ^ { s } - y _ { i }$ and $b = y _ { i } - \\hat { y } _ { i , 1 } ^ { t }$ , so we compare $\\hat { y } _ { i , 1 } ^ { s }$ and $\\hat { y } _ { i , 1 } ^ { t }$ instead. Finally, in the light of previous works that assign sample-wise weights (Lin et al., 2017; Tang et al., 2019), we propose weighted soft labels for knowledge distillation, which is formally defined as ",
+        "bbox": [
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+    },
+    {
+        "type": "table",
+        "img_path": "images/988ff4615f0aae92fd42a0bb258929339b59ca3a3e6c58e5455bef5b8c721c48.jpg",
+        "table_caption": [
+            "Table 3: Top-1 classification accuracy results on CIFAR-100. Comparison results are quoted from Tian et al. (2020). We report our results over 5 repeated runs. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=\"2\">Teacher</td><td colspan=\"5\">Same architecture style</td><td colspan=\"3\">Different architecture style</td></tr><tr><td>WRN-40-2</td><td>resnet56</td><td>resnet110</td><td>resnet110</td><td>resnet32x4</td><td>resnet32x4 ShuffleNetV1</td><td>resnet32x4</td><td>WRN-40-2 ShufleNetV1</td></tr><tr><td>Student Teacher</td><td>WRN-40-1 75.61</td><td>resnet20 72.34</td><td>resnet20 74.31</td><td>resnet32 74.31</td><td>resnet8x4 79.42</td><td>79.42</td><td>ShuffleNetV2 79.42</td><td>75.61</td></tr><tr><td>Student</td><td>71.98</td><td>69.06</td><td>69.06</td><td>71.14</td><td>72.50</td><td>70.5</td><td>71.82</td><td>70.5</td></tr><tr><td></td><td></td><td></td><td>68.99</td><td>71.06</td><td>73.50</td><td>73.59</td><td>73.54</td><td>73.73</td></tr><tr><td>FitNet AT</td><td>72.24 72.77</td><td>69.21 70.55</td><td>70.22</td><td>72.31</td><td>73.44</td><td>71.73</td><td>72.73</td><td>73.32</td></tr><tr><td>SP</td><td>72.43</td><td>69.67</td><td>70.04</td><td>72.69</td><td>72.94</td><td>73.48</td><td>74.56</td><td>74.52</td></tr><tr><td>CC</td><td>72.21</td><td>69.63</td><td>69.48</td><td>71.48</td><td>72.97</td><td>71.14</td><td>71.29</td><td>71.38</td></tr><tr><td>VID</td><td>73.30</td><td>70.38</td><td>70.16</td><td>72.61</td><td>73.09</td><td>73.38</td><td>73.40</td><td>73.61</td></tr><tr><td>RKD</td><td>72.22</td><td>69.61</td><td>69.25</td><td>71.82</td><td>71.90</td><td>72.28</td><td>73.21</td><td>72.21</td></tr><tr><td>PKT</td><td>73.45</td><td>70.34</td><td>70.25</td><td>72.61</td><td>73.64</td><td>74.10</td><td>74.69</td><td>73.89</td></tr><tr><td>AB</td><td>72.38</td><td>69.47</td><td>69.53</td><td>70.98</td><td>73.17</td><td>73.55</td><td>74.31</td><td>73.34</td></tr><tr><td>FT</td><td>71.59</td><td>69.84</td><td>70.22</td><td>72.37</td><td>72.86</td><td>71.75</td><td>72.50</td><td>72.03</td></tr><tr><td>FSP</td><td>n/a</td><td>69.95</td><td>70.11</td><td>71.89</td><td>72.62</td><td>n/a</td><td>n/a</td><td>n/a</td></tr><tr><td>NST</td><td>72.24</td><td>69.60</td><td>69.53</td><td>71.96</td><td>73.30</td><td>74.12</td><td>74.68</td><td>74.89</td></tr><tr><td>KD</td><td>73.54</td><td>70.66</td><td>70.67</td><td>73.08</td><td>73.33</td><td>74.07</td><td>74.45</td><td>74.83</td></tr><tr><td>CRD</td><td>74.14</td><td>71.16</td><td>71.46</td><td>73.48</td><td>75.51</td><td>75.11</td><td>75.65</td><td>76.05</td></tr><tr><td>Ours</td><td>74.48</td><td>72.15</td><td>72.19</td><td>74.12</td><td>76.05</td><td>75.46</td><td>75.93</td><td>76.21</td></tr></table>",
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+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+    },
+    {
+        "type": "equation",
+        "img_path": "images/40d9023f47bfe7f486308595ed945bdb41ece209eb9ccc973a5a51db78c2de7b.jpg",
+        "text": "$$\nL _ { \\mathrm { w s l } } = \\left( 1 - \\exp \\left( - \\frac { \\log \\hat { y } _ { i , 1 } ^ { s } } { \\log \\hat { y } _ { i , 1 } ^ { t } } \\right) \\right) L _ { \\mathrm { k d } } = \\left( 1 - \\exp \\left( - \\frac { L _ { \\mathrm { c e } } ^ { s } } { L _ { \\mathrm { c e } } ^ { t } } \\right) \\right) L _ { \\mathrm { k d } } ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            269,
+            473,
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "where $i$ is the ground truth class of the sample. The above equation means that a weighting factor is assigned to each sample’s $L _ { \\mathrm { k d } }$ according to the predictions of the teacher and the student. In this way, if compared to the teacher, a student network is relatively better trained on a sample, we have $\\hat { y } _ { i , 1 } ^ { s } > \\hat { y } _ { i , 1 } ^ { t }$ , then a smaller weight is assigned to this sample. In Fig. 3, the whole computational graph of knowledge distillation with the proposed weighted soft labels is demonstrated. Finally, we add $L _ { \\mathrm { w s l } }$ and $L _ { \\mathrm { c e } }$ together to supervise the network, i.e., ${ \\cal L } _ { \\mathrm { t o t a l } } = { \\cal L } _ { \\mathrm { c e } } + \\alpha { \\cal L } _ { \\mathrm { w s l } }$ , where $\\alpha$ is a balancing hyperparameter. ",
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+    },
+    {
+        "type": "text",
+        "text": "5 EXPERIMENTS ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "To evaluate our weighted soft labels comprehensively, we first conduct experiments with various teacher-student pair settings on CIFAR-100 (Krizhevsky et al., 2009). Next, we compare our method with current state-of-the-art distillation methods on ImageNet (Deng et al., 2009). To validate the effectiveness of our method in terms of handling the bias-variance tradeoff, we conduct ablation experiments by applying weighted soft labels on different subsets. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "5.1 DATASET AND HYPERPARAMETER SETTINGS ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "The datasets used in our experiments are CIFAR-100 (Krizhevsky et al., 2009) and ImageNet (Deng et al., 2009). CIFAR-100 contains 50K training and 10K test images of size $3 2 \\times 3 2$ . ImageNet contains 1.2 million training and 50K validation images. Except the loss function, training settings like learning rate or training epochs are the same with Tian et al. (2020) for CIFAR-100 and Heo et al. (2019) for ImageNet. For distillation, we set the temperature $\\tau = 4$ for CIFAR and $\\tau = 2$ for ImageNet. For loss function, we set $\\alpha = 2 . 2 5$ for distillation on CIFAR and $\\alpha = 2 . 5$ for ImageNet via grid search. The teacher network is well-trained previously and fixed during training. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "For comparison, the following recent state-of-the-art methods are chosen: FitNet (Romero et al., 2015), AT (Zagoruyko & Komodakis, 2017), SP (Tung & Mori, 2019), CC (Peng et al., 2019), VID (Ahn et al., 2019), RKD (Park et al., 2019), PKT (Passalis & Tefas, 2018), AB (Heo et al., 2019), ",
+        "bbox": [
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "table",
+        "img_path": "images/36655c59b60f3a149a9ff42be711661bfdc1c778712beceb3e919375581d021d.jpg",
+        "table_caption": [
+            "Table 4: Top-1 and Top-5 classification accuracy results on ImageNet validation set. All training hyperparameter like learning rate and training epochs are in accordance with (Heo et al., 2019). "
+        ],
+        "table_footnote": [
+            "FT (Kim et al., 2018), FSP (Yim et al., 2017), NST (Huang & Wang, 2017), Overhaul (Heo et al., 2019) and CRD (Tian et al., 2020). "
+        ],
+        "table_body": "<table><tr><td colspan=\"2\">Teacher: 1 :ResNet-34 →Student:ResNet-18</td></tr><tr><td>Method</td><td>Top-1 Acc Top-5 Acc</td></tr><tr><td>Teacher Student</td><td>73.31 91.42 89.07</td></tr><tr><td>KD</td><td>69.75 70.67 90.04</td></tr><tr><td>AT</td><td>71.03 90.04</td></tr><tr><td>NST</td><td>70.29 89.53</td></tr><tr><td>FSP</td><td>70.58 89.61</td></tr><tr><td>RKD</td><td>70.40 89.78</td></tr><tr><td>Overhaul</td><td>71.03 90.15</td></tr><tr><td>CRD</td><td>71.17 90.13</td></tr><tr><td>Ours</td><td>72.04 90.70</td></tr></table>",
+        "bbox": [
+            178,
+            140,
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+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "table",
+        "img_path": "images/9fa149aecdff1ef52d33d393a32331d0c110bb29dfb9a3ec549ceaa321f204bf.jpg",
+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td colspan=\"2\">Teacher:ResNet-50 → Student: MobileNet-v1</td></tr><tr><td>Method</td><td>Top-1 Acc Top-5 Acc</td></tr><tr><td>Teacher</td><td>76.16 92.87</td></tr><tr><td>Student</td><td>68.87 88.76</td></tr><tr><td>KD</td><td>70.49 89.92</td></tr><tr><td>AT</td><td>70.18 89.68</td></tr><tr><td>FT</td><td>69.88 89.5</td></tr><tr><td>AB</td><td>68.89 88.71</td></tr><tr><td>RKD</td><td>68.50 88.32</td></tr><tr><td>Overhaul</td><td>71.33 90.33</td></tr><tr><td>CRD</td><td>69.07 88.94</td></tr><tr><td>Ours</td><td>71.52 90.34</td></tr></table>",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "5.2 MODEL COMPRESSION ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Results on CIFAR-100 In Tab. 3, we present the Top-1 classification accuracy of our method and comparison methods. The results of comparison methods are quoted from Tian et al. (2020). Teacher-student pairs of the same and different architecture styles are considered. For pairs of same architecture style, we use wide residual networks (Zagoruyko & Komodakis, 2017) and residual networks (He et al., 2016). For pairs of different architecture style, residual networks and ShuffleNet (Zhang et al., 2018) pairs are chosen for experiments. As shown in the table, for distillation with both same and different architecture style, our method reached new state-of-the-art results. Specifically, our method outperforms standard KD by a large margin, which verifies the effectiveness of our method. ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Results on ImageNet In Tab. 4, we compare our method with current SOTA methods on ImageNet. Note that for the $\\mathrm { R e s N e t } 3 4  \\mathrm { R e s N e t } - 1 8$ distillation setting, the result of CRD is trained 10 more extra epochs while ours is the same as other methods. For ResNet- $5 0 $ MobileNet-v1 distillation setting, NST and FSP are not chosen for comparison as the two methods require too large GPU memories, so we include the accuracy of FT and AB reported in Heo et al. (2019) for comparison. Our results outperform all the existing methods, verifying the practical value of our method. ",
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "5.3 ABLATION STUDIES ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Weighted soft labels on different subsets. Recall that we propose weighted soft labels for tuning samplewise bias-variance tradeoff, it is still unclear whether the improvements come from a well-handled samplewise bias-variance tradeoff. To investigate this issue, we compare the performance gain of weighted soft labels on different training subsets. Similar to the settings used in Tab. 2, we apply weighted soft labels on two different subsets: only the regularization samples and excluding regularization samples. In Tab. 5, we show the results on subsets of only regularization samples and excluding regularization samples. From the significant improvements, we can see that our method can not only improve performance on the RS subset, the improvements on excluding RS subset is also significant. We conclude that weighted soft labels can tune sample-wise bias-variance tradeoff globally and lead to an improved distillation performance. ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "table",
+        "img_path": "images/9fe7f2964c0b3a47abb8283db41cf9c2d2e7f87a74614f96372ab0368a191208.jpg",
+        "table_caption": [
+            "Table 5: Performance on different subsets with soft labels and our weighted soft labels. RS means regularization samples. Results are averaged over 5 runs. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Subsets</td><td>Standard KD</td><td>Weighted</td></tr><tr><td>Only on RS</td><td>73.86</td><td>74.46</td></tr><tr><td>Excluding RS</td><td>74.59</td><td>75.35</td></tr></table>",
+        "bbox": [
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+        "page_idx": 7
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+    {
+        "type": "text",
+        "text": "",
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+    {
+        "type": "text",
+        "text": "Distillation with label smoothing trained teacher. Our exploration of bias-variance tradeoff starts with the conclusion made in Muller et al. (2019): a teacher net- ¨ work trained with the label smoothing trick is less effective for distillation. It is worthwhile to study whether the conclusion remains true for distillation with our weighted soft labels. As discussed before, we hold the opinion that too many regularization samples make the distillation less effective. Since our weighted soft label is proposed to mitigate the negative effects of the regularization samples, with the same settings from Tab. 1, we conduct comparison experiments in Tab. 6 to see if the negative effects still exist. It is evident that weighted soft labels significantly improve the distillation performance, especially for distillation from the teacher trained with label smoothing. Besides, using the teacher trained with label smoothing still performs worse than that without label smoothing, which again verifies the conclusion drawn by Muller et al. ¨ (2019). ",
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "table",
+        "img_path": "images/74f7c0fbe1a05aa888eb35e66ac4c1324fb5484ac2a7e5af13eec9bf98b5f298.jpg",
+        "table_caption": [
+            "Table 6: Distillation using weighted soft labels and teacher trained with label smoothing (denoted as $L S ? _ { \\mathrm { \\Gamma } }$ ). Results are averaged over 5 runs. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>T</td><td>LS？</td><td>Standard KD</td><td>Weighted</td></tr><tr><td>4</td><td>X</td><td>74.92</td><td>75.78</td></tr><tr><td>4</td><td>√</td><td>74.59</td><td>75.60</td></tr><tr><td>6</td><td>X</td><td>75.10</td><td>75.74</td></tr><tr><td>6</td><td>√</td><td>74.46</td><td>75.42</td></tr></table>",
+        "bbox": [
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+        "page_idx": 8
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "6 CONCLUSION ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Recent studies (Muller et al., 2019; Yuan et al., 2020) point out that one important reason behind ¨ the effectiveness of distillation is the regularization effect brought by being soft. In this paper, we rethink the soft labels for distillation from a bias-variance tradeoff perspective. The tradeoff varies sample-wisely and we propose weighted soft labels to handle the tradeoff, of which the effectiveness is verified with experiments on standard evaluation benchmarks. ",
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+    {
+        "type": "text",
+        "text": "ACKNOWLEDGEMENTS ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "This work is supported in part by a gift grant from Horizon Robotics and National Science Foundation Grant CNS-1951952. We thank Yichen Gong, Chuan Tian, Jiemin Fang and Yuzhu Sun for the discussion and assistance. ",
+        "bbox": [
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+        "image_caption": [
+            "Figure 4: Visualization of the resemblances introduced by soft label regularizers: (a) VGG-19 (Teacher) $\\to \\operatorname { V G G - 1 6 }$ (Student), (b) ResNet-50 (Teacher) $ \\mathrm { R e s N e t } { - 1 8 }$ (Student). And semantic similarity between label names: (c) LCH similarity (Pedersen et al., 2004), (d) WUP similarity (Pedersen et al., 2004). Darker areas denote larger values. "
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+    {
+        "type": "text",
+        "text": "A APPENDIX ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            674,
+            297,
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+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "A.1 VISUALIZATION OF THE RESEMBLANCES INTRODUCED BY SOFT LABEL REGULARIZER ",
+        "text_level": 1,
+        "bbox": [
+            171,
+            705,
+            812,
+            719
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "In Sthat $\\frac { \\partial ( L _ { \\mathrm { k d } } - L _ { \\mathrm { c e } } ) } { \\partial z _ { i } }$ proposequals $y _ { i } - \\hat { y } _ { i , 1 } ^ { t }$ . ∂(Lkd−Lce)∂z during the training process. When τ = 1, we show $y _ { i , \\tau } ^ { t }$ the output from the teacher network mputed by a linear mapping of the activations in the teacher’s penultimate layer, the regularization indicates that the student should follow the learned the resemblances between classes (Hinton et al., 2015; Muller et al., 2019). Still, two questions are unclear: 1) what the resemblances are and 2) whether ¨ the regularization still indicates resemblances if $\\tau$ is set to 4, a widely adopted hyperparameter (Tian et al., 2020). ",
+        "bbox": [
+            173,
+            727,
+            825,
+            835
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Towards answering the questions, we visualize the value of gradient vector $\\frac { \\partial ( L _ { \\mathrm { k d } } - L _ { \\mathrm { c e } } ) } { \\partial z }$ concerning each class. Specifically, on ImageNet (Deng et al., 2009) training set and $\\tau = 4$ , we calculate the average value of $\\frac { \\partial ( L _ { \\mathrm { k d } } - L _ { \\mathrm { c e } } ) } { \\partial z }$ for each class. Let $M$ be the matrix of values with the $i j$ -th entry $M _ { i j }$ means averaged ∂(Lkd−Lce)∂z for class j. Since Pi Mij = 0, diagonal elements are ignored for visualization. The results are visualized in Fig. 4. We find that plotting the common correlation matrix heatmap is ambiguous, because the matrix to be visualized is of large size $( 1 0 0 0 \\times 1 0 0 0 )$ with a large variance. By treating each entry $M _ { i j }$ as a vertex and then constructing a mesh for the matrix, we apply subdivision (Loop, 1987) to the mesh for smoothing the extreme points and finally rendering the mesh by ray-tracing package PlotOptiX. We can observe the several facts from the figures: 1) Comparing the sub-figure (a) and (b), we can see that for distillation resemblances implied by regularizers are similar across different teacher-student pairs. 2) Comparing (ab) with (cd), we can see that the resemblances are consistent with the semantic similarity of image class names. ",
+        "bbox": [
+            173,
+            843,
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+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "table",
+        "img_path": "images/4edeabcaac3cde92aef41b634620d4be0a0145774b1212dc8b97e163b9e77ad4.jpg",
+        "table_caption": [
+            "Table 7: Intermediate states between excluding and only on regularization samples ",
+            "(a) CIFAR100 (WRN-40-2→WRN-40-1 with KD) "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td colspan=\"4\">Percentage of excluded regularization samples.</td></tr><tr><td>100%</td><td>75%</td><td>50% 74.72</td><td>25% 74.87</td></tr><tr><td>74.59</td><td>74.63</td><td></td><td></td></tr><tr><td colspan=\"4\"> Percentage of adding non-regularization samples</td></tr><tr><td>0%</td><td>25%</td><td>50%</td><td>75%</td></tr><tr><td>73.86</td><td>74.12</td><td>74.47</td><td>74.71</td></tr></table>",
+        "bbox": [
+            328,
+            147,
+            669,
+            256
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "table",
+        "img_path": "images/1c092f9599c1b51059b6602fd357db4eaa9168af367e313c27f9a561f695e54b.jpg",
+        "table_caption": [
+            "(b) CIFAR100 (WRN-40-2→WRN-40-1 with weighted soft labels) "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td colspan=\"4\">Percentage of excluded regularization samples</td></tr><tr><td>100%</td><td>75%</td><td>50%</td><td>25%</td></tr><tr><td>75.35</td><td>75.48</td><td>75.61</td><td>75.72</td></tr><tr><td colspan=\"4\">Percentage of : f adding non-regularization samples</td></tr><tr><td>0%</td><td>25%</td><td>50%</td><td>75%</td></tr><tr><td>74.46</td><td>74.79</td><td>75.18</td><td>75.53</td></tr></table>",
+        "bbox": [
+            328,
+            289,
+            668,
+            397
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            174,
+            426,
+            825,
+            539
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "In a word, for $\\tau = 4$ , the variance reduction brought by soft labels still implies resemblances among labels, which are consistent with the semantic distance of class names. In the next section, we will analyze how bias-variance tradeoff changes when training with soft labels. ",
+        "bbox": [
+            174,
+            546,
+            825,
+            587
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "A.2 INTERMEDIATE STATES BETWEEN EXCLUDING AND ONLY ON REGULARIZATION SAMPLES ",
+        "text_level": 1,
+        "bbox": [
+            173,
+            606,
+            767,
+            632
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "To further investigate the phenomenon about regularization samples, we conduct experiments to show the intermediate states between excluding and only on regularization samples. Two settings are considered here: First, we gradually exclude regularization samples during training, from excluding all regularization samples to excluding $2 5 \\%$ regularization samples; Second, we keep all regularization samples and then gradually add non-regularization samples. Since we judge a sample is regularization or not according to the training loss, we cannot pre-define a sample set such that a certain percentage samples are kept or dropped. Therefore, we propose to conduct these experiments by assigning a probability to whether backward the loss computed with regularization samples. For example, if during training, a sample is marked as regularization sample according to the value of $a$ and $b$ , we backward the loss of this sample by a probability $p = 0 . 5$ . In this way, we can get the performance of excluding $7 5 \\%$ regularization samples. In Tab. 7, we first present result with KD in (a) and then present result with weighted soft labels applied in (b). We can observe that weighted soft labels are indeed balancing the sample-wise, not on dataset scale, bias and variance. ",
+        "bbox": [
+            173,
+            645,
+            825,
+            824
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "A.3 COMBINING WITH RKD (PARK ET AL., 2019). ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            842,
+            537,
+            856
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "To investigate how the weighted soft labels can be applied to the variants of KD, we conduct an experiment of combining RKD (Park et al., 2019) with our weighted soft labels. Relational knowledge distillation measures the L2 distance of features between two samples or the angle formed by three samples as knowledge to transfer. In other words, the knowledge in RKD is measured by the relations between sample pairs. It is no longer sample-independent, which is different from the weighted soft labels applied to KD which can assign the weights sample-wisely. We currently take the averaged weighting factors of the involved sample pairs when calculating the distance/angle matrix. The results on CIFAR-100 are presented in Tab. 8 (averaged over 5 runs). As can be observed from the table, the weighted soft label applied to RKD still brings improvements, though not that big compared with WSL applied to KD. Also, we believe that it is an important future direction to explore the applications to more variants of KD. ",
+        "bbox": [
+            174,
+            867,
+            823,
+            924
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "table",
+        "img_path": "images/502600c3d7a5b77db8a76189cd07976f092ca82b07ebe865ad744f6107933868.jpg",
+        "table_caption": [
+            "Table 8: Combining weighted soft labels with RKD (Park et al., 2019). "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Distillation settings</td><td>WRN-40-2 →WRN-16-2</td><td>WRN-40-2 →WRN-40-1</td><td>resnet56→resnet20</td></tr><tr><td>Teacher</td><td>75.61</td><td>75.61</td><td>72.34</td></tr><tr><td>Student</td><td>73.26</td><td>71.98</td><td>69.06</td></tr><tr><td>RDK</td><td>74.12</td><td>73.34</td><td>70.25</td></tr><tr><td>WSL + RKD</td><td>74.65</td><td>73.89</td><td>70.73</td></tr></table>",
+        "bbox": [
+            178,
+            126,
+            818,
+            208
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "table",
+        "img_path": "images/a1c1772d4caa4a5a643a6f0f48b4d72a9ca6a63cd74031aa6deed691f58e9952.jpg",
+        "table_caption": [
+            "Table 9: Comparison to other weighting forms. (Setting: CIFAR100, WRN-40-2 WRN-40-1) "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>α</td><td>2.0</td><td>3.0</td><td>4.0</td></tr><tr><td>Sigmoid baseline</td><td>74.13</td><td>73.97</td><td>73.29</td></tr><tr><td>Ours</td><td>74.38</td><td>74.12</td><td>73.46</td></tr></table>",
+        "bbox": [
+            346,
+            252,
+            650,
+            311
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            173,
+            342,
+            825,
+            440
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "A.4 OTHER VARIANTS OF WEIGHTING. ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            457,
+            454,
+            472
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "In the work, the weighting scheme is defined as $\\begin{array} { r } { \\left( 1 - \\exp \\left( - \\frac { L _ { \\mathrm { c e } } ^ { s } } { L _ { \\mathrm { c e } } ^ { t } } \\right) \\right) } \\end{array}$ , which is inspired by Lin et al. (2017); Tang et al. (2019). The basic idea is to convert $\\frac { L _ { \\mathrm { c e } } ^ { s } } { L _ { \\mathrm { c e } } ^ { t } }$ into a value in $[ 0 , 1 ]$ , so that the weights of regularization samples are lower than those non-regularization samples. A straightforward baseline is that we can use the Sigmoid function to convert $\\frac { L _ { \\mathrm { c e } } ^ { s } } { L _ { \\mathrm { c e } } ^ { t } }$ into a value in $[ 0 , 1 ]$ . Note that L cet is always L ce bigger than 1, so the weight needs scaling and can be defined as $\\frac { 2 } { 1 + \\exp ( - \\frac { L _ { \\mathrm { c e } } ^ { s } } { L _ { \\mathrm { c e } } ^ { t } } ) } - 1$ . In Tab. 9, we present the comparison between adopted weighting form and the Sigmoid baseline. We can see that as long as we can adaptively tune the sample-wise bias-variance tradeoff, the performance is better than KD, i.e., without weighted soft labels.Therefore, although the proposed weighting form is not mathematically optimal, the not-too-big or not-too-small weights for these regularization examples are not hard to tune. These results verify our main contribution that there is sample-wise biasvariance tradeoff and we need to assign weights to the regularization examples. ",
+        "bbox": [
+            173,
+            479,
+            826,
+            667
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "A.5 ABLATION ON $\\alpha$ ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            685,
+            334,
+            699
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "In Tab. 10 We first tune the value of $\\alpha$ on CIFAR100, with four values $\\{ 1 , 2 , 3 , 4 \\}$ tested. Then we test with three values in [2, 3] in (b). Finally, we tune $\\alpha$ on ImageNet in (c). As a conclusion, the results are not very sensitive to $\\alpha$ and the cost of searching $\\alpha$ in our work is not expensive. ",
+        "bbox": [
+            174,
+            710,
+            825,
+            753
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "A.6 RESULTS ON MULTINLI ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            770,
+            387,
+            784
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "To further validate our method, we conduct experiments on an NLP dataset MultiNLI (Williams et al., 2018). In this setting, the teacher is BERT-base-cased with 12 layers, 768 Hidden and 108M params. The student is T3 with 3 layers, 768 Hidden and 44M params. Besides, we follow the training setting in Sun et al. (2019). In Tab. 11, we present the result comparisons of standard KD and our weighted soft labels. ",
+        "bbox": [
+            174,
+            796,
+            825,
+            866
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "table",
+        "img_path": "images/b13a40331c3cbb9a470921c9cf0e32904f241583c1d2ae42e0b0acb38849f306.jpg",
+        "table_caption": [
+            "Table 10: Ablation on $\\alpha$ "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td colspan=\"5\">(a) CIFAR100 (WRN-40-2→WRN-40-1)</td></tr><tr><td>α</td><td>1</td><td>2</td><td>3</td><td>4</td></tr><tr><td>Top1</td><td>73.67</td><td>74.38</td><td>74.12</td><td>73.46</td></tr></table>",
+        "bbox": [
+            359,
+            255,
+            638,
+            313
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "table",
+        "img_path": "images/efa7c504b87df4a8aa54cc4ddd566ac4361b73eb2c149fbe4a28fe60f50203f2.jpg",
+        "table_caption": [
+            "(b) CIFAR100 (WRN-40-2 WRN-40-1) "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>α</td><td>2.25</td><td>2.5</td><td>2.75</td></tr><tr><td>Top1</td><td>74.48</td><td>74.34</td><td>74.21</td></tr></table>",
+        "bbox": [
+            380,
+            347,
+            632,
+            387
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "(c) ImageNet (ResNet-34 ResNet-18) ",
+        "bbox": [
+            369,
+            405,
+            629,
+            419
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "table",
+        "img_path": "images/5e852e00282277dd89b900bca1f2ecd9095b6e37c0574f3c761a85c546fd85e0.jpg",
+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>α</td><td>2</td><td>2.25</td><td>2.5</td></tr><tr><td>Top1</td><td>71.91</td><td>71.96</td><td>72.04</td></tr></table>",
+        "bbox": [
+            380,
+            421,
+            624,
+            462
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "table",
+        "img_path": "images/b28488f5cf7c83b8c9e13e61c672f1574291eebbd5bf25412d84b340a75e3736.jpg",
+        "table_caption": [
+            "Table 11: Results on MultiNLI. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td></td><td>Teacher (BERT-12)</td><td>Student (BERT-3)</td><td>KD (BERT-3)</td><td>Ours (BERT-3)</td></tr><tr><td>Results reported by Sun et al. (2019)</td><td>83.7</td><td>74.8</td><td>75.4</td><td>-</td></tr><tr><td>Our replications</td><td>83.57</td><td>75.06</td><td>75.50</td><td>76.28</td></tr></table>",
+        "bbox": [
+            176,
+            750,
+            820,
+            797
+        ],
+        "page_idx": 14
+    }
+]

parse/train/nMefdZyJ7ie/nMefdZyJ7ie.md

@@ -0,0 +1,215 @@
+# NEURAL DISJUNCTIVE NORMAL FORM: VERTICALLY INTEGRATING LOGIC WITH DEEP LEARNING FOR CLASSIFICATION
+
+Anonymous authors Paper under double-blind review
+
+# ABSTRACT
+
+We present Neural Disjunctive Normal Form (Neural DNF), a hybrid neurosymbolic classifier that vertically integrates propositional logic with a deep neural network. Here, we aim at a vertical integration of logic and deep learning: we utilize the ability of deep neural networks as feature extractors to extract intermediate representation from data, and then a Disjunctive Normal Form (DNF) module to perform logical rule-based classification; we also seek this integration to be tight that these two normally-incompatible modules can be learned in an end-to-end manner, for which we propose the BOAT algorithm.
+
+Compared with standard deep classifiers which use a linear model or variants of additive model as the classification head, Neural DNF provides a new choice of model based on logic rules. It offers interpretability via an explicit symbolic representation, strong model expressity, and a different type of model inductive bias. Neural DNF is particularly suited for certain tasks that require some logical composition and provides extra interpretability.
+
+# 1 INTRODUCTION
+
+In the recent years, the emphasis of machine learning, particularly deep learning, has been to increase the capacity to accurately model complex patterns, but typically such flexibility results in blackboxes that are too complex for humans to understand. While the ability of deep learning on feature extraction and pattern recognition is widely recognized, the blackbox nature makes it hard for human to interpret the underlying mechanism; furthermore, this lack of interpretability leads to difficulties for human to interact with or manipulate the model, when trying to debug undesired behaviors or to improve model by incorporating human knowledge.
+
+A possible solution to this limitation of deep learning, as discussed and promoted by many researchers (Marcus, 2020; Bengio, 2019; Yi et al., 2019; Mao et al., 2019; Hudson and Manning, 2019; Penkov, 2019) , is to develop hybird neuro-symbolic models. While there are many different approaches on integrating neural and symbolic models, one of these approaches called Vertical Integration1, aims at exploiting the best of both worlds in a straight-forward way: it utilizes deep learning to learn high-level features from raw sensory data (which deep learning are good at), and utilizes symbolic models to reason about the high-level features (which symbolic models are good at). More formally, the vertical integration approach can be represented as a two stage model $f = g \circ \phi$ whose prediction on a sample $x$ is given by $\hat { y } = f ( x ) = g \bar { ( } \phi ( x ) )$ where the first-stage $\phi$ is a neural network feature extractor and the second-stage $g$ is the symbolic model processing the extracted features into final prediction. Vertical integration has its biological inspiration: we know that certain areas in the brain are used to process input signals (Grill-Spector and Malach, 2004) while others are responsible for logical reasoning (Shokri-Kojori et al., 2012). But from a more practical perspective, vertical integration promises a solution to demystify the blackbox by decomposing the prediction task into two stages, handled by different models; and a better-demystified interpretable model can enable easier human interactions and manipulations on the model.
+
+The choice of symbolic model to integrate naturally depends on the task. SAT-Net (Wang et al., 2019a) utilizes a symbolic SAT solver layer on top of a convolutional neural network (CNN) for solving a satisfiability-based task of visual Sudoku; Donadello et al. (2017) utilizes first-order fuzzy logic on top of a Fast-RCNN to extract structured semantic descriptions from images; DeepProbLog (Manhaeve et al., 2018) builds a probabilistic logic program on top of a CNN within a domainspecified grammar template for tasks like visual digit addition and sorting. In this paper, since we deal with the most basic task of binary classification, we choose propositional logic in Disjunctive Normal Form (DNF), also known as ‘decision rules’ or ‘rule set’. As a well-studied symbolic model, DNF has been established as being general and interpretable. DNF performs a simple and transparent ‘OR-of-ANDs’ prediction: if at least one AND clause (a conjunction of conditions) is satisfied, it predicts the positive class; otherwise negative. DNF is interpretable not only because its symbolic structure is intuitive to follow, but also that each conjunctive clause of DNF can be viewed as a separate IF-THEN rule, providing ‘smaller-than-global’ interpretations (Rudin, 2019). DNF is a general rule format, as any propositional logic formula has an equivalent DNF formula, and thus any rule-based binary classifier including decision set/list/tree can be expressed by a DNF.
+
+We seek Neural DNF’s vertical integration to be tight, i.e. we wish to optimize the deep feature extractor and the DNF end-to-end. However, the main technical challenge here is that the learning algorithms for rule-based models and deep learning models are generally incompatible, making end-to-end learning not directly possible. In order to address this challenge, we propose BOAT (BiOptimizer learning with Adaptively-Temperatured noise) to train $\phi$ and $g$ jointly (see fig. 1): we use standard continuous-parameter optimizer Adam (Kingma and Ba, 2014) to optimize the first-stage neural network $\phi$ and modify a binary-parameter optimizer from Helwegen et al. (2019) to optimize the parameters of the second-stage rule-based $g$ . As the novel key ingredient of BOAT, we propose adaptively-temperatured noise to perform weight perturbation which enables the learning of the rulebased $g$ . Our experiments demonstrate that learning constantly fails without such noise. Compared with standard deep learning classifiers which has a fully-connected linear layer as $g$ , Neural DNF offers (1) a logic-based inductive bias and (2) better interpretability at competitive accuracy.
+
+![](images/f754695a37752b3695a0207a11aef7e966cf16a0868b5295926649e8a5124821.jpg)  
+Figure 1: Overview of Neural DNF and the proposed BOAT Algorithm
+
+The rule-based Neural DNF is particularly suited for classification tasks that require the logical compositions of certain concepts. We demonstrate that with experiments on a toy dataset called 2D-XOR, an extension of the XOR problem which has received special interest in the literature (Minsky and Papert, 1969). We consider 2D-XOR as a minimal example to show the benefits of Neural DNF’s vertical integration, because 2D-XOR requires a model to both learn the right highlevel features (concepts) from the raw data and the right XOR function. We further apply Neural DNF to image datasets in two scenarios: In the first scenario, we use a regular deep network as feature extractor with no further constraints. We show that Neural DNF can successfully learn both the feature extractor and the logical rules for classification, and achieve competitive accuracy. Note that in this case there is no guarantee that the extracted features are meaningful to human. In the second scenario, we constrain the feature extractor to produce human-aligned interpretable features by enforcing an auxiliary concept loss based on human concept annotations. In this scenario, Neural DNF becomes highly interpretable that the interpretability enables human to easily interact with and manipulate the learned model, such as performing human-intervention on the extracted features to improve accuracy, or slightly tweaking the model to recognize an imaginary class that does not exist in the dataset. In conclusion, our experiments show that Neural DNF achieves accuracy comparable that of blackbox deep learning models while offering an interpretable symbolic DNF representation, that makes human easier to interact with or manipulate the model.
+
+# 2 NEURAL DISJUNCTIVE NORMAL FORM
+
+We consider a standard supervised learning setting of classification: given a dataset of $D =$ $\{ ( x , y ) \} _ { 1 } ^ { | D | }$ , we wish to learn a classification function $f$ . For simplicity, we now assume $y$ to be binary labels $( y \in \{ 0 , 1 \} )$ and will extend to multi-class later. We use the two-stage formulation : the classifier function $f = g \circ \phi$ is a composition of two functions $\phi$ and $g$ , where $\phi$ is a neural network feature extractor, taking raw data $x$ as input and returning a set of intermediate representations $\{ c _ { 1 } , c _ { 2 } , \dotsc , c _ { K } \} \in \{ 0 , \bar { 1 } \} ^ { K }$ where $K$ is a predefined number. We use 0 and 1 to represent False and True. We call each $c$ a concept predicate to emphasize that $c$ has a Boolean interpretation: it indicates the presence or the absence of a concept. A Disjunctive Normal Form module $g$ is used for the actual classification. The overall classification is given by ${ \hat { y } } = f ( x ) = g ( \phi ( x ) )$ .
+
+The logical rule-based $g$ is formulated as a Disjunctive Normal Form (DNF). $g$ takes a set of Boolean predicates as input feature and makes binary prediction. $g$ can be more intuitively interpreted as a decision set, consisting of a set of if-then rules: $g$ predicts the positive class if at least one of the rules is satisfied and predicts the negative class otherwise. We define a rule $r _ { i }$ to be a conjunction of one or more conditions (literals): $r _ { i } = b _ { i 1 } \wedge b _ { i 2 } \wedge . . .$ where $b _ { j }$ for $j \in \{ 1 , 2 , \cdot \cdot \cdot 2 K \}$ can be a predicate $c$ or its negation $\neg c$ . A DNF is a disjunction of one or more rules: $r _ { 1 } \vee r _ { 2 } \vee . . . \vee r _ { n }$ .
+
+Rule learning algorithm for DNF can be viewed as a subset selection problem: first, a pool of all candidate rules is constructed and then ‘learning’ corresponds to finding a good subset of rules as the learned DNF. Let $N$ to be the size of pool of rules and $K$ the number of predicates, we formulate using a binary matrix $W _ { 2 K \times N }$ and a binary vector $S _ { N }$ . $W _ { 2 K \times N }$ represents the pool of candidate rules: each column represents a rule and the non-zero elements of a column indicate the conditions of that rule. $S _ { N }$ can be viewed as a membership vector that determines which rules are selected as the learned DNF. Given input concept predicates ${ \pmb { c } } = \{ { \boldsymbol { c } } \} _ { 1 } ^ { K }$ , $g$ computes the Boolean function as:
+
+$$
+{ \hat { y } } = g ( c ) = \bigvee _ { S _ { j } = 1 } ^ { N } \bigwedge _ { W _ { i , j } = 1 } b _ { i } \qquad { \mathrm { w h e r e ~ } } \{ b \} _ { i } ^ { 2 K } = \{ c _ { 1 } , \neg c _ { 1 } , c _ { 2 } , \neg c _ { 2 } , \dotsc , c _ { k } , \neg c _ { k } \}
+$$
+
+Note that eq. (1) is used during training; after training, $W _ { 2 K \times N }$ is discarded and only the selected rules are stored. Conventional methods of rule learning (Lakkaraju et al., 2016; Wang et al., 2017) usually construct a fixed $W$ by rule pre-mining; and then ‘learning’ corresponds to the discrete optimization of the membership vector $_ { s }$ . However, this is not compatible if we wish to jointly optimize $g$ with the neural network $\phi$ end-to-end. In order to do so, we make two essential modifications: (I) we make both $W _ { 2 K \times N }$ and $S _ { N }$ learnable parameters; (II) we introduce a differentiable replacement2 of the logical operation of eq. (1) so that gradients can be properly backpropagated:
+
+$$
+r _ { j } = \bigwedge _ { W _ { i , j } = 1 } b _ { i }
+$$
+
+$$
+r _ { j } = \prod _ { i } ^ { 2 k } F _ { \mathrm { c o n j } } ( b _ { i } , W _ { i , j } ) , \mathrm { w h e r e } F _ { \mathrm { c o n j } } ( b , w ) = 1 - w ( 1 - b )
+$$
+
+$$
+\hat { y } = 1 - \prod _ { j } ^ { N } ( 1 - F _ { \mathrm { d i s j } } ( r _ { j } , S _ { j } ) ) , \mathrm { w h e r e } F _ { \mathrm { d i s j } } ( r , s ) = r \cdot s
+$$
+
+eqs. (2) and (3) computes the DNF function exactly as eq. (1), but note that since $W$ and $\pmb { S }$ are binary, optimizing $\{ W , S \}$ with mini-batch gradient descent is non-trivial. The above formulation can be easily extended to multi-class settings by using a different $W$ and $\pmb { S }$ for each class (thus we have a different DNF ‘tower’ for each class) while the concept predicates being shared across classes; in test time when more than one classes are predicted as positive, some tie-breaking procedure can be employed. In this paper, we simply use lazy tie-breaking by selecting the first encountered positive class in ascending order (e.g. when class 1, 4, 7 are all predicted as positive, we select class 1).
+
+The neural network feature extractor $\phi$ processes the raw input $x$ into a set of intermediate representations $\{ c _ { 1 } , c _ { 2 } , \ldots , c _ { k } \}$ called concept predicates. The neural network $\phi$ ’s output $\tilde { c _ { i } }$ is realvalued. To ensure the extracted $c$ can be processed by the DNF $g , c$ needs to be binary. We use a binary step function to discretize $\tilde { c _ { i } }$ into Boolean predicates: $c _ { i } = 1$ if $\tilde { c _ { i } } > 0$ and $c _ { i } = 0$ otherwise. However, since gradient through this step function is zero almost anywhere and thus prevents training, we utilize the Improved SemHash (Kaiser and Bengio, 2018). It does not need any manual annealing of temperature (Bulat et al., 2019; Hansen et al., 2019) or additional loss functions and has been demonstrated to be a robust discretization technique in various domains (Kaiser and Bengio, 2018; Chen et al., 2019b; Kaiser et al., 2019). Ideally, the intermediate output $^ c$ should be the highlevel ‘natural’ features that are aligned with human-comprehensible concepts. Quoted from Melis and Jaakkola (2018), for medical image processing, the concept predicate $^ c$ should indicate tissue ruggedness, irregularity, elongation, etc, and are ‘the first aspects that doctors look for when diagnosing’. However, this is ill-defined and indeed a very much more challenging task. We explore two options of $\phi$ on some image datasets: in section 4.2. we choose to use a convolutional neural network and do not put any extra constraints; in section 4.3, we enforce the feature extractor to produce interpretable representations by enforcing an auxiliary concept loss based on human annotations.
+
+Objective function of Neural DNF is given as ${ \cal L } \ = \ \mathbb { L } _ { l o s s } + \lambda _ { g } \mathbb { R } _ { g } ( W , S )$ where $\begin{array} { r l } { \mathbb { L } _ { l o s s } } & { { } = } \end{array}$ ${ \begin{array} { r } { { \frac { 1 } { | D | } } \sum _ { ( x , y ) \in D } L ( y , g _ { W } ( \phi _ { \theta } ( x ) ) ) } \end{array} }$ is the classification loss (using MSE or BCE) and $\mathbb { R } _ { g }$ is the regularization for $g$ . For a simple regularization of $g$ , we want the number of rules and the length of rules to be small. Note that as only rules selected by $\pmb { S }$ are actually used, we can use a grouped L1-norm by $\begin{array} { r } { \mathbb { R } _ { g } ( W , S ) = \lambda _ { g } \sum _ { j } ^ { \bar { N } } | S _ { j } | _ { 1 } | W . , _ { j } | _ { 1 } } \end{array}$ . Note that in the multi-class setting, our extension of Neural DNF is simply treating a multi-class classification as multiple one-versus-all binary classifications. In this case, the objective function sums up the classification loss for all the classes.
+
+# 3 THE BOAT ALGORITHM FOR LEARNING NEURAL DNF
+
+In this section, we introduce Bi-Optimizer learning with Adaptively-Temperatured noise(BOAT), the learning algorithm for Neural DNF. BOAT utilizes two optimizers: a standard deep learning optimizer Adam (Kingma and Ba, 2014) that optimizes the continuous parameters $\theta$ of the neural network $\phi$ and a binary-parameter optimizer adopted from [Helwegen et al., 2019] that optimizes the binary parameters $\{ W , S \}$ of the $\mathrm { D N F } g$ . As the key ingredient of BOAT, during training the binary parameters $W$ and $_ { s }$ are perturbed by noise whose magnitude is controlled by the noise temperature $\sigma$ (eq. (4)). We emphasize that (1) the introduced noise is necessary as otherwise learning of $\{ W , S \}$ constantly fails even in very simple cases (see appendix fig 1) and (2) the noise temperature $\sigma$ is also a learnable continuous parameter, which can be optimized together with other continuous parameters by Adam. Making the temperature learnable avoids the tedious tuning of temperature schedules.
+
+Joint optimization of $_ { g _ { W } , s }$ and $\phi _ { \theta }$ is non-trivial, because although the overall model is differentiable, $\{ W , S \}$ consists of binary values $( \{ 0 , 1 \} )$ and thus standard deep learning optimizers designed for continuous parameters cannot be directly applied. One alternative, denoted as DNF-Real (Payani and Fekri, 2019; Wang et al., 2019b), is to simply use real-valued weight $\{ \tilde { W } , \tilde { S } \}$ transformed using sigmoid/tanh functions as a surrogate and then use Adam as the optimizer. After training, real-valued weights are thresholded to binary values, and performance drop can occur. In practice, we find DNF-Real to be optimization-friendly just like any other real-valued DNNs but it is not guaranteed to result in binary-valued parameters. Another alternative, which we denote as DNF-STE, is adopted from binary neural network research (Courbariaux et al., 2016; Darabi et al., 2018). It maintains real-valued latent parameters which are binarized in forward pass computation. In backward computation the gradients are updated to the latent real-valued parameters using the straight-through estimator (STE) (Bengio et al., 2013). This technique is widely used and works quite well for large-scale binary neural networks. However, for a small-sized DNF $g$ , DNF-STE is very sensitive to initialization and hyperparameters and can easily be stuck in local minima.
+
+The introduced BOAT combines the best of both approaches: (1) It is optimization-friendly and not very sensitive to initialization and hyperparameters. This is especially important as $g$ is not overparameterized; (2) It naturally optimizes binary parameters. A pseudocode is provided in appendix.
+
+Modified Bop: BOAT uses a modified version of the Bop optimizer (Helwegen et al., 2019) to optimize the binary-valued parameter $\{ W , S \}$ given gradient as learning signals. We make minor modifications to suit Bop (originally for $\{ - 1 , 1 \} )$ into the case of $\{ 0 , \bar { 1 } \}$ . The Modified Bop optimizer introduced in this paper uses gradient as the learning signal and flips the value of $w \in$ $\{ 0 , 1 \}$ only if the gradient signal $m$ exceeds a predefined accepting threshold $\tau$ :
+
+$$
+w = { \left\{ \begin{array} { l l } { 1 - w , } & { { \mathrm { i f ~ } } | m | > \tau } \\ { w , } & { { \mathrm { o t h e r w i s e } } . } \end{array} \right. }
+$$
+
+where $m$ is the gradient-based learning signal computed in the backward pass. A non-zero $\tau$ is introduced to avoid rapid back-and-forth flips of the binary parameter and we find it helpful to stabilize the learning because $m$ is of high variance. To obtain consistent learning signals, instead of using vanilla gradient $\nabla$ as $m$ , the exponential moving average of gradients is used:
+
+$$
+m = \gamma m + ( 1 - \gamma ) \nabla
+$$
+
+where $\gamma$ is the exponential decay rate and $\nabla$ is the mini-batch gradient. We use $\gamma = 1 - 1 0 ^ { - 5 }$ and $\tau = 1 0 ^ { - 6 }$ as default value while the trick for tuning $\gamma$ and $\tau$ can be found in original Bop paper.
+
+Key ingredient of BOAT: Adaptively-temperatured Noise: The reason we introduce adaptivelytemperatured noise is that simply using the introduced Modified Bop shares the same drawback as using DNF-STE: optimization is very sensitive to initialization and hyperparameters and can be stuck in local minima very easily. When stuck in local minima, the gradients w.r.t $W$ and $_ { s }$ effectively become zero, and thus any further updates for $W$ and $\pmb { S }$ are disabled. We suspect the reason is that even the DNF function eqs. (2) and (3) is well defined on $[ 0 , 1 ]$ , since the choice of values of $W$ and $\pmb { S }$ can only take $\{ 0 , 1 \}$ , the loss surface is non-smooth and thus the optimization becomes hard. To overcome this, we propose to perturb the binary weights $w$ during training by adding noise in the forward pass such that the perturbed $\tilde { w }$ lies in $[ \dot { 0 } , 1 ]$ . We believe the introduced noise smoothes the loss surface and helps the optimization. Specifically, for every entry $w$ in $W$ and $\pmb { S }$ , we utilize a noise temperature parameter $\sigma _ { w } \in [ 0 , 0 . 5 ]$ to perturb $w$ with noise as follows:
+
+$$
+\tilde { w } = \left\{ \begin{array} { l l } { 1 - \sigma _ { w } \cdot \epsilon } & { \mathrm { i f } w = 1 } \\ { 0 + \sigma _ { w } \cdot \epsilon } & { \mathrm { i f } w = 0 } \end{array} , \mathrm { w h e r e } \epsilon \sim \mathrm { U n i f o r m } ( 0 , 1 ) \right.
+$$
+
+During training the perturbed weight $\tilde { w }$ is used in the forward pass computation of the objective function; in test time, we simply disable this perturbation. To force $\sigma _ { w }$ lies in range $[ 0 , 0 . 5 ]$ , we clip $\sigma _ { w }$ by $\sigma _ { w } = m i n ( 0 . 5 , m a x ( \sigma _ { w } , 0 ) )$ after evey mini-batch update. Note that $\sigma _ { w }$ is not globally shared: we have a $\sigma _ { w }$ for every $w$ in $W$ and $_ { s }$ (so in total $2 K * N + N )$ . We make $\sigma _ { w }$ also a learnable parameter. We initialize $\sigma _ { w } = \sigma _ { 0 }$ and optimize $\sigma _ { w }$ by Adam as well. The choice of initial value $\sigma _ { 0 }$ requires heuristic: with a too large $\sigma _ { 0 }$ optimization becomes slower (appendix fig 1c ), and $\sigma$ cannot be too small: in the extreme case with zero noise the optimization of $W , S$ will constantly fail (appendix fig 1b). We find values in [0.1, 0.3] all work fine and we use $\sigma _ { 0 } = 0 . 2$ as the default initial value. Remark: We evaluate BOAT with its alternatives for learning the DNF alone in appendix sec E.1. We can also apply same noise for DNF-STE, but it converges slower (appendix fig 2). We conjecture the reason to be the acceptance threshold $\tau$ which effectively filters out noisy learning signals so that rapid flipping is prevented, suggested as main advantage of Bop.
+
+# 4 EXPERIMENTS
+
+# 4.1 ON A 2D TOY DATASET.
+
+Here we first apply Neural DNF on a 2D toy dataset 2D-XOR (fig. 2). 2D-XOR is generated by first drawing four isotropic 2D Gaussian clusters and mark them as red square, blue square, blue circle and red circle (in clockwise order). We use a XOR-like label assigningment, labeling red squares and blue circles as positive and the rest two clusters as negative. We consider 2D-XOR as an extension of the historically important XOR problem, and its difficulty is that the feature of being ‘red’/‘nonred’ and ‘square’/‘non-square’ is not provided as input directly but needs to be learned in a 2-d input space. This makes the learning harder since it requires to learn both the correct intermediate feature and the correct logical decision function. We use a one fully-connected layer network as $\phi$ to produce two concept predicates $c _ { 1 } , c _ { 2 }$ and can visualize in as two lines in the 2-d space. We expect the learned Neural DNF to find the two correct concept predicates that represent shape and color, respectively, and the correct classification function (‘color is red and shape is square or color is not red and shape is not square’). As shown in fig. 2, we visualize the decision boundary of $c _ { 1 }$ and $c _ { 2 }$ by two lines and we find $c _ { 1 } , c _ { 2 }$ do separate red-against-blue and square-against-circle as expected. We view fig. 2 as a minimal working example of Neural DNF demonstrating that it is useful when (1) the feature used for logic-based function are not provided but needs to be learned and (2) the underlying classification process consists of subtypes (each described as one rule) and requires some logical compositions.
+
+![](images/df74a23710be10dfc88d3cdd0cbd8e47d6e9d564a540c654b15425bd2133f90c.jpg)  
+Figure 2: Neural DNF on 2D-XOR
+
+# 4.2 ON IMAGE DATASETS. SCENARIO 1
+
+In this section we apply Neural DNF with a CNN as feature extractor without any further contraints. and then demonstrate empirically that given the same first-stage feature extractor architecture, Neural DNF sacrifices slight loss of accuracy (table 1) while gaining more faithful interpretations (fig. 3). We first evaluate the accuracy of Neural DNF on several image datasets table 1. Using the same architecture for $\phi$ , the models we compare are (1) Neural DNF (2) a standard DNN (i.e. a linear model as $g$ ) and (3) self-explaining neural network (SENN) (Melis and Jaakkola, 2018), which uses a neural network to generate the coefficients of a linear model conditioned on the input and claims to have better interpretability. Since binarization is not required for DNN or SENN, we also evaluate DNN and SENN with/without it. We observe that all models perform similarly well in terms of test accuracy across datasets, while Neural DNF loses accuracy only slightly. Comparing DNN and SENN with/without binarization, we can see that both DNN and SENN loses accuracy slightly because of the binarization. So the loss of Neural DNF ’s accuracy can be explained from two sources: (1) the binarization (Improved SemHash) and (2) the use of rules as the classifier instead of the more well-studied linear (additive) models. We suspect this is also partially because Neural DNF treats multi-class settomg as multiple one-versus-all classifications and does not produce probabilistic outputs like DNN and SENN. However, this should not be viewed as a weakness as long as the accuracy loss is slight. We believe that this issue can be alleviated that given a sufficiently flexible $\phi$ , the accuracy loss of Neural DNF can be negligible like the case of simple dataset MNIST.
+
+Table 1: Test Accuracy on some image datasets   
+
+<table><tr><td></td><td>MNIST</td><td>KMNIST</td><td>SVHN</td><td>CIFAR10</td></tr><tr><td>NeuralDNF</td><td>99.08%</td><td>95.43%</td><td>90.13%</td><td>67.91%</td></tr><tr><td>standard DNN</td><td>99.12%</td><td>95.86%</td><td>90.74%</td><td>70.43%</td></tr><tr><td>standard DNN(without Binarization)</td><td>99.11%</td><td>96.02%</td><td>91.45%</td><td>71.45%</td></tr><tr><td>SENN</td><td>98.48%</td><td>92.64%</td><td>90.79%</td><td>71.09%</td></tr><tr><td>SENN (without Binarization)</td><td>98.50%</td><td>91.46%</td><td>92.15%</td><td>72.32%</td></tr></table>
+
+Now regarding the interpretability, the problem here is, we can in principle inspect3 the meaning of concept and rules of Neural DNF, but since we have no constraints on $\phi$ , the extracted features are only highly discriminative and is not guaranteed to be aligned with human perceptible concepts. A symbolic DNF that operates on non-interpretable representations is still non-interpretable. We will revisit this interpretable representation problem in next section; but despite tje first-stage, here we can still compare the interpretability of the second-stage model. Despite many qualitive arguments for favouring the interpretability of the symbolic DNF (appendix B.3), we can further quantitively evaluate the faithfulness of models’ interpretability, a critical critera that measures how faithful the explanation for a particular prediction are to the underlying computation of prediction. We adopt the faithfulness metric proposed by (Melis and Jaakkola, 2018) which measures the correlation of the change of the prediction (class probabilities) and the change of the explanation (relevance scores of features) upon perturbation of test examples.4 We report the faithfulness metric on test set for DNN, SENN and Neural DNF in fig. 3. For DNN, we use the coefficients of final linear layer as relevance scores. Alternatively, for DNN we can also utilize some representitive post-hoc interpretation methods (also known as attribution methods): Guided Backprop (GB) (Springenberg et al., 2014), gradient SHAP (Lundberg and Lee, 2017), Integrated Gradient (IG) (Sundararajan et al., 2017) and DeepLIFT (Shrikumar et al., 2017). SENN can use its self-generated coefficients as relevance scores (Melis and Jaakkola, 2018). For Neural DNF, we use the difference analysis (Robnik-Sikonja and ˇ Kononenko, 2008), a model-agnostic relevance score assignment method to assign scores. As shown in fig. 3, Neural DNF consistently achieves the highest faithfulness estimates for all datasets and its faithfulness has the smallest variance. This means Neural DNF’s explanations are highly faithful in a robust way across different test samples. SENN is more faithful than DNN; and for the post-hoc methods while GB performs very similarly with DNN-coef, IG/SHAP/DeepLIFT drastically improves the faithfulness. This is because the latter three methods compute the relevance w.r.t to a baseline reference, a concept that is now considered very essential (Sturmfels et al., 2020). However, we emphasize that even these sophisticated post-hoc methods do not reach the same level of faithfulness as Neural DNF, in particular not on MNIST and KMNIST. We believe that Neural DNF’s extremely high faithfulness is the direct result of the symbolic nature of DNF.
+
+![](images/9b8ca1388678a485c438c8763f9c671a43cbd1759022ec6e8ff94786fbaab64e.jpg)  
+Figure 3: Evaluating the faithfulness of explanations on test set
+
+# 4.3 ON IMAGE DATASETS, SCENARIO 2
+
+Here we test a new scenario where we have concept annotations so we can train the feature extractor to produce human-aligned interpretable representations. Our goal is that if we can align the extracted features with human understanding, then we can achieve a highly interpretable model which human can easily interact with and manipulate. We use the CUB dataset (Wah et al., 2011) which has 200 class and 112 binary annotataed concepts (preprocessed by Koh et al. (2020)). In Koh et al. (2020) the authors propose the concept bottleneck model, a similar two-stage model using a Inception-V3 based feature extractor $\phi$ and a linear layer $g$ . Given concepts the annotations, a concept prediction loss can be applied so that the extracted faetures are constrained to align with annotated values. Koh et al. (2020) also propose the test-time human intervention: since the extracted concepts can be wrong, human can check and correct extracted concepts so that test accuracy can be improved significantly through this interaction. We follow Koh et al. (2020) and replace the second-stage $g$ with a DNF. Koh et al. (2020) evaluate several strategies for training the two-stage model. Here we use the independent training strategy: we train $\phi$ to predict concepts and train $g$ to predict class label using concepts; only in test time we stack $\phi$ and $g$ together. The reason of choosing independent training instead of joint training is counterintuive, however, this is because we find that the extracted concepts after joint training are less well-aligned with human annotations, and it then makes the human intervention less effective than independent trained models. In other words, based on noninterpretable features, any human interaction/manipulation will be unreliable. This phenomenon is consistent with Koh et al. (2020), and we think novel architectures that extracts interpretable features without the need of annotations can solve it, in that case joint training should give better results. In table 2 we report the accuracy of Neural DNF, the concept bottleneck model and blackbox DNN. Neural DNF achieves less test accuracy compared with the concept bottleneck model by a large margin, we suspect it might because Neural DNF is dealing with too many (200-class) one-versusall prediction and does not produce probabilistic outputs like other models do. But after intervention both Neural DNF and the concept bottleneck model achiveve perfect accuracy. This indicates that (1) the accuracy by applying human intervention can be improved significantly which is the benefits of having an highly interpretable model; (2) the bottleneck of accuracy is the feature extractor: the classification of CUB can be effectively solved given a perfect feature extractor.
+
+Table 2: Test Accuracy on CUB dataset   
+
+<table><tr><td></td><td>test acc</td><td>test acc with human intervention</td></tr><tr><td>NeuralDNF</td><td>61.94%</td><td>100.00%</td></tr><tr><td>Concept bottleneck model</td><td>72.38%</td><td>100.00%</td></tr><tr><td>Blackbox DNN</td><td>74.69%</td><td>N/A</td></tr></table>
+
+While losing quite much in accuracy, the symbolic nature enables the Neural DNF to do the things the concept bottleneck model cannot. We show in fig. 4 (top) a correctly predicted sample for the class of ‘Chestnut sided Warbler’ and in fig. 4 (middle) an incorretly predicted sample and how human intervention can correct the prediction. DNF rules are a very human-readable format, that the rules are very sparse involving only a few concepts and the logical operation is intuitive to understand. The DNF rules also alleviate the burden of human intervention compared to the concept bottleneck, as only a few concepts are used in the decision rule, a human user can only check and correct these concepts indicated in the rules and does not have to go through every concepts which the concept bottleneck model requires. Also, the symbolic nature of DNF enables us to incorporate knowledge and manipulate the models easily in a way that the concept bottleneck model (with a linear $g$ ) cannot offer. In in fig. 4 (bottom), we provide an illustration on how we can tweak the Neural DNF to predict an imaginary class ”blue-crown Chestnut sided Warbler” that does not exist in the training dataset (not even in real world). Taking the rule for ‘Chestnut sided Warbler’, we can simply replacing the condition of ‘yellow crown is True’ with ‘blue crown is True’. Note that ’blue crown’ is already in the training set and one of the concepts $\phi$ can extract. Of course we can go further that we can train and append new feature extractors for new concepts and then play with new concepts. There are, however, limitations because of the expressity of prositional logic.
+
+![](images/097c97e7d1852a6f28ab32dc687f627ba096a602a067a13b697ea426ba0a2f9f.jpg)  
+Figure 4: (top) illustration of a correct prediction of Neural DNF for class ‘Chestnut sided Warbler’; (middle) illustration of human interaction: an incorrect prediction of Neural DNF for class ‘Chestnut sided Warbler’ can be corrected by human intervention; (bottom) illustration of manipulating a learned Neural DNF to classify an imaginary class ‘Blue-crown Chestnut sided Warbler’.
+
+# 5 CONCLUSION AND FUTURE WORK
+
+In this paper, we have presented Neural DNF, a vertically integrated neuro-symbolic classifier. Neural DNF takes a step towards the fundamental problem of integrating continuous perception and logical reasoning. Neural DNF adopts a two-stage model that utlizes a DNN to extract features, called concept predicates, and a propositional logic DNF module to make classification based on the concepts. We propose the BOAT algorithm for joint learning of the DNF module and the feature extractor DNN. In order to avoid laborious manual tuning, we employ an Improved SemHash method to binarize the extracted features, to obtain concept predicates, because it does not need to manually tune any extra parameters or add further losses. We also introduce adaptive temperatured-noise, which is a minimal modification of the binary-parameter optimizer Bop that enables the effective optimization of the parameters of the second-stage rule-based model.
+
+This paper suggests several directions for future research. In our experiments in Neural DNF , we have used concept annotations and the concept-bottleneck DNN architecture to extract interpretable features. However, in practice there is often no prior knowledge of ‘concept annotations’ available. Therefore, novel deep learning architectures that can extract human-understandable concepts from data without the need of concept annotations should be investigated. This may require domainspecific knowledge representations and loss functions (Melis and Jaakkola, 2018; Chen et al., 2019a; Biffi et al., 2018; Kim and Canny, 2017; Johnson et al., 2016). Another future direction is the use of more powerful languages than propositional logic, for example, the more general inductive logic programming. Since BOAT requires only minimal changes of the standard deep learning optimization, we believe it can be potentially applied to models with a more powerful second-stage rule-based module, or to other models with mixed binary-continuous parameters.
+
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+        "type": "text",
+        "text": "We present Neural Disjunctive Normal Form (Neural DNF), a hybrid neurosymbolic classifier that vertically integrates propositional logic with a deep neural network. Here, we aim at a vertical integration of logic and deep learning: we utilize the ability of deep neural networks as feature extractors to extract intermediate representation from data, and then a Disjunctive Normal Form (DNF) module to perform logical rule-based classification; we also seek this integration to be tight that these two normally-incompatible modules can be learned in an end-to-end manner, for which we propose the BOAT algorithm. ",
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+        "type": "text",
+        "text": "Compared with standard deep classifiers which use a linear model or variants of additive model as the classification head, Neural DNF provides a new choice of model based on logic rules. It offers interpretability via an explicit symbolic representation, strong model expressity, and a different type of model inductive bias. Neural DNF is particularly suited for certain tasks that require some logical composition and provides extra interpretability. ",
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+        "type": "text",
+        "text": "1 INTRODUCTION ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "In the recent years, the emphasis of machine learning, particularly deep learning, has been to increase the capacity to accurately model complex patterns, but typically such flexibility results in blackboxes that are too complex for humans to understand. While the ability of deep learning on feature extraction and pattern recognition is widely recognized, the blackbox nature makes it hard for human to interpret the underlying mechanism; furthermore, this lack of interpretability leads to difficulties for human to interact with or manipulate the model, when trying to debug undesired behaviors or to improve model by incorporating human knowledge. ",
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+        "type": "text",
+        "text": "A possible solution to this limitation of deep learning, as discussed and promoted by many researchers (Marcus, 2020; Bengio, 2019; Yi et al., 2019; Mao et al., 2019; Hudson and Manning, 2019; Penkov, 2019) , is to develop hybird neuro-symbolic models. While there are many different approaches on integrating neural and symbolic models, one of these approaches called Vertical Integration1, aims at exploiting the best of both worlds in a straight-forward way: it utilizes deep learning to learn high-level features from raw sensory data (which deep learning are good at), and utilizes symbolic models to reason about the high-level features (which symbolic models are good at). More formally, the vertical integration approach can be represented as a two stage model $f = g \\circ \\phi$ whose prediction on a sample $x$ is given by $\\hat { y } = f ( x ) = g \\bar { ( } \\phi ( x ) )$ where the first-stage $\\phi$ is a neural network feature extractor and the second-stage $g$ is the symbolic model processing the extracted features into final prediction. Vertical integration has its biological inspiration: we know that certain areas in the brain are used to process input signals (Grill-Spector and Malach, 2004) while others are responsible for logical reasoning (Shokri-Kojori et al., 2012). But from a more practical perspective, vertical integration promises a solution to demystify the blackbox by decomposing the prediction task into two stages, handled by different models; and a better-demystified interpretable model can enable easier human interactions and manipulations on the model. ",
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+        "type": "text",
+        "text": "The choice of symbolic model to integrate naturally depends on the task. SAT-Net (Wang et al., 2019a) utilizes a symbolic SAT solver layer on top of a convolutional neural network (CNN) for solving a satisfiability-based task of visual Sudoku; Donadello et al. (2017) utilizes first-order fuzzy logic on top of a Fast-RCNN to extract structured semantic descriptions from images; DeepProbLog (Manhaeve et al., 2018) builds a probabilistic logic program on top of a CNN within a domainspecified grammar template for tasks like visual digit addition and sorting. In this paper, since we deal with the most basic task of binary classification, we choose propositional logic in Disjunctive Normal Form (DNF), also known as ‘decision rules’ or ‘rule set’. As a well-studied symbolic model, DNF has been established as being general and interpretable. DNF performs a simple and transparent ‘OR-of-ANDs’ prediction: if at least one AND clause (a conjunction of conditions) is satisfied, it predicts the positive class; otherwise negative. DNF is interpretable not only because its symbolic structure is intuitive to follow, but also that each conjunctive clause of DNF can be viewed as a separate IF-THEN rule, providing ‘smaller-than-global’ interpretations (Rudin, 2019). DNF is a general rule format, as any propositional logic formula has an equivalent DNF formula, and thus any rule-based binary classifier including decision set/list/tree can be expressed by a DNF. ",
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+        "type": "text",
+        "text": "We seek Neural DNF’s vertical integration to be tight, i.e. we wish to optimize the deep feature extractor and the DNF end-to-end. However, the main technical challenge here is that the learning algorithms for rule-based models and deep learning models are generally incompatible, making end-to-end learning not directly possible. In order to address this challenge, we propose BOAT (BiOptimizer learning with Adaptively-Temperatured noise) to train $\\phi$ and $g$ jointly (see fig. 1): we use standard continuous-parameter optimizer Adam (Kingma and Ba, 2014) to optimize the first-stage neural network $\\phi$ and modify a binary-parameter optimizer from Helwegen et al. (2019) to optimize the parameters of the second-stage rule-based $g$ . As the novel key ingredient of BOAT, we propose adaptively-temperatured noise to perform weight perturbation which enables the learning of the rulebased $g$ . Our experiments demonstrate that learning constantly fails without such noise. Compared with standard deep learning classifiers which has a fully-connected linear layer as $g$ , Neural DNF offers (1) a logic-based inductive bias and (2) better interpretability at competitive accuracy. ",
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+    {
+        "type": "image",
+        "img_path": "images/f754695a37752b3695a0207a11aef7e966cf16a0868b5295926649e8a5124821.jpg",
+        "image_caption": [
+            "Figure 1: Overview of Neural DNF and the proposed BOAT Algorithm "
+        ],
+        "image_footnote": [],
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+        "text": "The rule-based Neural DNF is particularly suited for classification tasks that require the logical compositions of certain concepts. We demonstrate that with experiments on a toy dataset called 2D-XOR, an extension of the XOR problem which has received special interest in the literature (Minsky and Papert, 1969). We consider 2D-XOR as a minimal example to show the benefits of Neural DNF’s vertical integration, because 2D-XOR requires a model to both learn the right highlevel features (concepts) from the raw data and the right XOR function. We further apply Neural DNF to image datasets in two scenarios: In the first scenario, we use a regular deep network as feature extractor with no further constraints. We show that Neural DNF can successfully learn both the feature extractor and the logical rules for classification, and achieve competitive accuracy. Note that in this case there is no guarantee that the extracted features are meaningful to human. In the second scenario, we constrain the feature extractor to produce human-aligned interpretable features by enforcing an auxiliary concept loss based on human concept annotations. In this scenario, Neural DNF becomes highly interpretable that the interpretability enables human to easily interact with and manipulate the learned model, such as performing human-intervention on the extracted features to improve accuracy, or slightly tweaking the model to recognize an imaginary class that does not exist in the dataset. In conclusion, our experiments show that Neural DNF achieves accuracy comparable that of blackbox deep learning models while offering an interpretable symbolic DNF representation, that makes human easier to interact with or manipulate the model. ",
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+        "type": "text",
+        "text": "2 NEURAL DISJUNCTIVE NORMAL FORM ",
+        "text_level": 1,
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+        "text": "We consider a standard supervised learning setting of classification: given a dataset of $D =$ $\\{ ( x , y ) \\} _ { 1 } ^ { | D | }$ , we wish to learn a classification function $f$ . For simplicity, we now assume $y$ to be binary labels $( y \\in \\{ 0 , 1 \\} )$ and will extend to multi-class later. We use the two-stage formulation : the classifier function $f = g \\circ \\phi$ is a composition of two functions $\\phi$ and $g$ , where $\\phi$ is a neural network feature extractor, taking raw data $x$ as input and returning a set of intermediate representations $\\{ c _ { 1 } , c _ { 2 } , \\dotsc , c _ { K } \\} \\in \\{ 0 , \\bar { 1 } \\} ^ { K }$ where $K$ is a predefined number. We use 0 and 1 to represent False and True. We call each $c$ a concept predicate to emphasize that $c$ has a Boolean interpretation: it indicates the presence or the absence of a concept. A Disjunctive Normal Form module $g$ is used for the actual classification. The overall classification is given by ${ \\hat { y } } = f ( x ) = g ( \\phi ( x ) )$ . ",
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+        "text": "The logical rule-based $g$ is formulated as a Disjunctive Normal Form (DNF). $g$ takes a set of Boolean predicates as input feature and makes binary prediction. $g$ can be more intuitively interpreted as a decision set, consisting of a set of if-then rules: $g$ predicts the positive class if at least one of the rules is satisfied and predicts the negative class otherwise. We define a rule $r _ { i }$ to be a conjunction of one or more conditions (literals): $r _ { i } = b _ { i 1 } \\wedge b _ { i 2 } \\wedge . . .$ where $b _ { j }$ for $j \\in \\{ 1 , 2 , \\cdot \\cdot \\cdot 2 K \\}$ can be a predicate $c$ or its negation $\\neg c$ . A DNF is a disjunction of one or more rules: $r _ { 1 } \\vee r _ { 2 } \\vee . . . \\vee r _ { n }$ . ",
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+        "type": "text",
+        "text": "Rule learning algorithm for DNF can be viewed as a subset selection problem: first, a pool of all candidate rules is constructed and then ‘learning’ corresponds to finding a good subset of rules as the learned DNF. Let $N$ to be the size of pool of rules and $K$ the number of predicates, we formulate using a binary matrix $W _ { 2 K \\times N }$ and a binary vector $S _ { N }$ . $W _ { 2 K \\times N }$ represents the pool of candidate rules: each column represents a rule and the non-zero elements of a column indicate the conditions of that rule. $S _ { N }$ can be viewed as a membership vector that determines which rules are selected as the learned DNF. Given input concept predicates ${ \\pmb { c } } = \\{ { \\boldsymbol { c } } \\} _ { 1 } ^ { K }$ , $g$ computes the Boolean function as: ",
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+        "type": "equation",
+        "img_path": "images/76c29d97093b4c826677aae12d8c2e5c7e3cd50797500bf49f39b576ac97554a.jpg",
+        "text": "$$\n{ \\hat { y } } = g ( c ) = \\bigvee _ { S _ { j } = 1 } ^ { N } \\bigwedge _ { W _ { i , j } = 1 } b _ { i } \\qquad { \\mathrm { w h e r e ~ } } \\{ b \\} _ { i } ^ { 2 K } = \\{ c _ { 1 } , \\neg c _ { 1 } , c _ { 2 } , \\neg c _ { 2 } , \\dotsc , c _ { k } , \\neg c _ { k } \\}\n$$",
+        "text_format": "latex",
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+        "type": "text",
+        "text": "Note that eq. (1) is used during training; after training, $W _ { 2 K \\times N }$ is discarded and only the selected rules are stored. Conventional methods of rule learning (Lakkaraju et al., 2016; Wang et al., 2017) usually construct a fixed $W$ by rule pre-mining; and then ‘learning’ corresponds to the discrete optimization of the membership vector $_ { s }$ . However, this is not compatible if we wish to jointly optimize $g$ with the neural network $\\phi$ end-to-end. In order to do so, we make two essential modifications: (I) we make both $W _ { 2 K \\times N }$ and $S _ { N }$ learnable parameters; (II) we introduce a differentiable replacement2 of the logical operation of eq. (1) so that gradients can be properly backpropagated: ",
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+        "img_path": "images/93faf93c54c978eeb566059dc1719d28c60c5a2279bd8994d3dfd0e711ce1057.jpg",
+        "text": "$$\nr _ { j } = \\bigwedge _ { W _ { i , j } = 1 } b _ { i }\n$$",
+        "text_format": "latex",
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+        "img_path": "images/416c322aceecfd0b42a865aa783e3b496ca7953a6647909b861b5b43421ee1e6.jpg",
+        "text": "$$\nr _ { j } = \\prod _ { i } ^ { 2 k } F _ { \\mathrm { c o n j } } ( b _ { i } , W _ { i , j } ) , \\mathrm { w h e r e } F _ { \\mathrm { c o n j } } ( b , w ) = 1 - w ( 1 - b )\n$$",
+        "text_format": "latex",
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+        "type": "equation",
+        "img_path": "images/e7681101c56e518c85da89b07d36e71c1dd25a0ce9219a3a32cc6445e3ab1f7b.jpg",
+        "text": "$$\n\\hat { y } = 1 - \\prod _ { j } ^ { N } ( 1 - F _ { \\mathrm { d i s j } } ( r _ { j } , S _ { j } ) ) , \\mathrm { w h e r e } F _ { \\mathrm { d i s j } } ( r , s ) = r \\cdot s\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "eqs. (2) and (3) computes the DNF function exactly as eq. (1), but note that since $W$ and $\\pmb { S }$ are binary, optimizing $\\{ W , S \\}$ with mini-batch gradient descent is non-trivial. The above formulation can be easily extended to multi-class settings by using a different $W$ and $\\pmb { S }$ for each class (thus we have a different DNF ‘tower’ for each class) while the concept predicates being shared across classes; in test time when more than one classes are predicted as positive, some tie-breaking procedure can be employed. In this paper, we simply use lazy tie-breaking by selecting the first encountered positive class in ascending order (e.g. when class 1, 4, 7 are all predicted as positive, we select class 1). ",
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+        "type": "text",
+        "text": "The neural network feature extractor $\\phi$ processes the raw input $x$ into a set of intermediate representations $\\{ c _ { 1 } , c _ { 2 } , \\ldots , c _ { k } \\}$ called concept predicates. The neural network $\\phi$ ’s output $\\tilde { c _ { i } }$ is realvalued. To ensure the extracted $c$ can be processed by the DNF $g , c$ needs to be binary. We use a binary step function to discretize $\\tilde { c _ { i } }$ into Boolean predicates: $c _ { i } = 1$ if $\\tilde { c _ { i } } > 0$ and $c _ { i } = 0$ otherwise. However, since gradient through this step function is zero almost anywhere and thus prevents training, we utilize the Improved SemHash (Kaiser and Bengio, 2018). It does not need any manual annealing of temperature (Bulat et al., 2019; Hansen et al., 2019) or additional loss functions and has been demonstrated to be a robust discretization technique in various domains (Kaiser and Bengio, 2018; Chen et al., 2019b; Kaiser et al., 2019). Ideally, the intermediate output $^ c$ should be the highlevel ‘natural’ features that are aligned with human-comprehensible concepts. Quoted from Melis and Jaakkola (2018), for medical image processing, the concept predicate $^ c$ should indicate tissue ruggedness, irregularity, elongation, etc, and are ‘the first aspects that doctors look for when diagnosing’. However, this is ill-defined and indeed a very much more challenging task. We explore two options of $\\phi$ on some image datasets: in section 4.2. we choose to use a convolutional neural network and do not put any extra constraints; in section 4.3, we enforce the feature extractor to produce interpretable representations by enforcing an auxiliary concept loss based on human annotations. ",
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+        "type": "text",
+        "text": "",
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+    {
+        "type": "text",
+        "text": "Objective function of Neural DNF is given as ${ \\cal L } \\ = \\ \\mathbb { L } _ { l o s s } + \\lambda _ { g } \\mathbb { R } _ { g } ( W , S )$ where $\\begin{array} { r l } { \\mathbb { L } _ { l o s s } } & { { } = } \\end{array}$ ${ \\begin{array} { r } { { \\frac { 1 } { | D | } } \\sum _ { ( x , y ) \\in D } L ( y , g _ { W } ( \\phi _ { \\theta } ( x ) ) ) } \\end{array} }$ is the classification loss (using MSE or BCE) and $\\mathbb { R } _ { g }$ is the regularization for $g$ . For a simple regularization of $g$ , we want the number of rules and the length of rules to be small. Note that as only rules selected by $\\pmb { S }$ are actually used, we can use a grouped L1-norm by $\\begin{array} { r } { \\mathbb { R } _ { g } ( W , S ) = \\lambda _ { g } \\sum _ { j } ^ { \\bar { N } } | S _ { j } | _ { 1 } | W . , _ { j } | _ { 1 } } \\end{array}$ . Note that in the multi-class setting, our extension of Neural DNF is simply treating a multi-class classification as multiple one-versus-all binary classifications. In this case, the objective function sums up the classification loss for all the classes. ",
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+    {
+        "type": "text",
+        "text": "3 THE BOAT ALGORITHM FOR LEARNING NEURAL DNF ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "In this section, we introduce Bi-Optimizer learning with Adaptively-Temperatured noise(BOAT), the learning algorithm for Neural DNF. BOAT utilizes two optimizers: a standard deep learning optimizer Adam (Kingma and Ba, 2014) that optimizes the continuous parameters $\\theta$ of the neural network $\\phi$ and a binary-parameter optimizer adopted from [Helwegen et al., 2019] that optimizes the binary parameters $\\{ W , S \\}$ of the $\\mathrm { D N F } g$ . As the key ingredient of BOAT, during training the binary parameters $W$ and $_ { s }$ are perturbed by noise whose magnitude is controlled by the noise temperature $\\sigma$ (eq. (4)). We emphasize that (1) the introduced noise is necessary as otherwise learning of $\\{ W , S \\}$ constantly fails even in very simple cases (see appendix fig 1) and (2) the noise temperature $\\sigma$ is also a learnable continuous parameter, which can be optimized together with other continuous parameters by Adam. Making the temperature learnable avoids the tedious tuning of temperature schedules. ",
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+        "type": "text",
+        "text": "Joint optimization of $_ { g _ { W } , s }$ and $\\phi _ { \\theta }$ is non-trivial, because although the overall model is differentiable, $\\{ W , S \\}$ consists of binary values $( \\{ 0 , 1 \\} )$ and thus standard deep learning optimizers designed for continuous parameters cannot be directly applied. One alternative, denoted as DNF-Real (Payani and Fekri, 2019; Wang et al., 2019b), is to simply use real-valued weight $\\{ \\tilde { W } , \\tilde { S } \\}$ transformed using sigmoid/tanh functions as a surrogate and then use Adam as the optimizer. After training, real-valued weights are thresholded to binary values, and performance drop can occur. In practice, we find DNF-Real to be optimization-friendly just like any other real-valued DNNs but it is not guaranteed to result in binary-valued parameters. Another alternative, which we denote as DNF-STE, is adopted from binary neural network research (Courbariaux et al., 2016; Darabi et al., 2018). It maintains real-valued latent parameters which are binarized in forward pass computation. In backward computation the gradients are updated to the latent real-valued parameters using the straight-through estimator (STE) (Bengio et al., 2013). This technique is widely used and works quite well for large-scale binary neural networks. However, for a small-sized DNF $g$ , DNF-STE is very sensitive to initialization and hyperparameters and can easily be stuck in local minima. ",
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+        "text": "The introduced BOAT combines the best of both approaches: (1) It is optimization-friendly and not very sensitive to initialization and hyperparameters. This is especially important as $g$ is not overparameterized; (2) It naturally optimizes binary parameters. A pseudocode is provided in appendix. ",
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+        "text": "Modified Bop: BOAT uses a modified version of the Bop optimizer (Helwegen et al., 2019) to optimize the binary-valued parameter $\\{ W , S \\}$ given gradient as learning signals. We make minor modifications to suit Bop (originally for $\\{ - 1 , 1 \\} )$ into the case of $\\{ 0 , \\bar { 1 } \\}$ . The Modified Bop optimizer introduced in this paper uses gradient as the learning signal and flips the value of $w \\in$ $\\{ 0 , 1 \\}$ only if the gradient signal $m$ exceeds a predefined accepting threshold $\\tau$ : ",
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+        "type": "equation",
+        "img_path": "images/fee4e997679ae627a1ca3c8bf052e177eca67e87a8069bc9690ae6259c25fc35.jpg",
+        "text": "$$\nw = { \\left\\{ \\begin{array} { l l } { 1 - w , } & { { \\mathrm { i f ~ } } | m | > \\tau } \\\\ { w , } & { { \\mathrm { o t h e r w i s e } } . } \\end{array} \\right. }\n$$",
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+    {
+        "type": "text",
+        "text": "where $m$ is the gradient-based learning signal computed in the backward pass. A non-zero $\\tau$ is introduced to avoid rapid back-and-forth flips of the binary parameter and we find it helpful to stabilize the learning because $m$ is of high variance. To obtain consistent learning signals, instead of using vanilla gradient $\\nabla$ as $m$ , the exponential moving average of gradients is used: ",
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+    {
+        "type": "equation",
+        "img_path": "images/83ebf9de15514d92a6411bbbc63573142eb520fca18fac62ee610fa1d131f425.jpg",
+        "text": "$$\nm = \\gamma m + ( 1 - \\gamma ) \\nabla\n$$",
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+    {
+        "type": "text",
+        "text": "where $\\gamma$ is the exponential decay rate and $\\nabla$ is the mini-batch gradient. We use $\\gamma = 1 - 1 0 ^ { - 5 }$ and $\\tau = 1 0 ^ { - 6 }$ as default value while the trick for tuning $\\gamma$ and $\\tau$ can be found in original Bop paper. ",
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+        "type": "text",
+        "text": "Key ingredient of BOAT: Adaptively-temperatured Noise: The reason we introduce adaptivelytemperatured noise is that simply using the introduced Modified Bop shares the same drawback as using DNF-STE: optimization is very sensitive to initialization and hyperparameters and can be stuck in local minima very easily. When stuck in local minima, the gradients w.r.t $W$ and $_ { s }$ effectively become zero, and thus any further updates for $W$ and $\\pmb { S }$ are disabled. We suspect the reason is that even the DNF function eqs. (2) and (3) is well defined on $[ 0 , 1 ]$ , since the choice of values of $W$ and $\\pmb { S }$ can only take $\\{ 0 , 1 \\}$ , the loss surface is non-smooth and thus the optimization becomes hard. To overcome this, we propose to perturb the binary weights $w$ during training by adding noise in the forward pass such that the perturbed $\\tilde { w }$ lies in $[ \\dot { 0 } , 1 ]$ . We believe the introduced noise smoothes the loss surface and helps the optimization. Specifically, for every entry $w$ in $W$ and $\\pmb { S }$ , we utilize a noise temperature parameter $\\sigma _ { w } \\in [ 0 , 0 . 5 ]$ to perturb $w$ with noise as follows: ",
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+        "text": "$$\n\\tilde { w } = \\left\\{ \\begin{array} { l l } { 1 - \\sigma _ { w } \\cdot \\epsilon } & { \\mathrm { i f } w = 1 } \\\\ { 0 + \\sigma _ { w } \\cdot \\epsilon } & { \\mathrm { i f } w = 0 } \\end{array} , \\mathrm { w h e r e } \\epsilon \\sim \\mathrm { U n i f o r m } ( 0 , 1 ) \\right.\n$$",
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+    },
+    {
+        "type": "text",
+        "text": "During training the perturbed weight $\\tilde { w }$ is used in the forward pass computation of the objective function; in test time, we simply disable this perturbation. To force $\\sigma _ { w }$ lies in range $[ 0 , 0 . 5 ]$ , we clip $\\sigma _ { w }$ by $\\sigma _ { w } = m i n ( 0 . 5 , m a x ( \\sigma _ { w } , 0 ) )$ after evey mini-batch update. Note that $\\sigma _ { w }$ is not globally shared: we have a $\\sigma _ { w }$ for every $w$ in $W$ and $_ { s }$ (so in total $2 K * N + N )$ . We make $\\sigma _ { w }$ also a learnable parameter. We initialize $\\sigma _ { w } = \\sigma _ { 0 }$ and optimize $\\sigma _ { w }$ by Adam as well. The choice of initial value $\\sigma _ { 0 }$ requires heuristic: with a too large $\\sigma _ { 0 }$ optimization becomes slower (appendix fig 1c ), and $\\sigma$ cannot be too small: in the extreme case with zero noise the optimization of $W , S$ will constantly fail (appendix fig 1b). We find values in [0.1, 0.3] all work fine and we use $\\sigma _ { 0 } = 0 . 2$ as the default initial value. Remark: We evaluate BOAT with its alternatives for learning the DNF alone in appendix sec E.1. We can also apply same noise for DNF-STE, but it converges slower (appendix fig 2). We conjecture the reason to be the acceptance threshold $\\tau$ which effectively filters out noisy learning signals so that rapid flipping is prevented, suggested as main advantage of Bop. ",
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+    {
+        "type": "text",
+        "text": "4 EXPERIMENTS ",
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+    {
+        "type": "text",
+        "text": "4.1 ON A 2D TOY DATASET. ",
+        "text_level": 1,
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+        "text": "Here we first apply Neural DNF on a 2D toy dataset 2D-XOR (fig. 2). 2D-XOR is generated by first drawing four isotropic 2D Gaussian clusters and mark them as red square, blue square, blue circle and red circle (in clockwise order). We use a XOR-like label assigningment, labeling red squares and blue circles as positive and the rest two clusters as negative. We consider 2D-XOR as an extension of the historically important XOR problem, and its difficulty is that the feature of being ‘red’/‘nonred’ and ‘square’/‘non-square’ is not provided as input directly but needs to be learned in a 2-d input space. This makes the learning harder since it requires to learn both the correct intermediate feature and the correct logical decision function. We use a one fully-connected layer network as $\\phi$ to produce two concept predicates $c _ { 1 } , c _ { 2 }$ and can visualize in as two lines in the 2-d space. We expect the learned Neural DNF to find the two correct concept predicates that represent shape and color, respectively, and the correct classification function (‘color is red and shape is square or color is not red and shape is not square’). As shown in fig. 2, we visualize the decision boundary of $c _ { 1 }$ and $c _ { 2 }$ by two lines and we find $c _ { 1 } , c _ { 2 }$ do separate red-against-blue and square-against-circle as expected. We view fig. 2 as a minimal working example of Neural DNF demonstrating that it is useful when (1) the feature used for logic-based function are not provided but needs to be learned and (2) the underlying classification process consists of subtypes (each described as one rule) and requires some logical compositions. ",
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+        "image_caption": [
+            "Figure 2: Neural DNF on 2D-XOR "
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+    {
+        "type": "text",
+        "text": "4.2 ON IMAGE DATASETS. SCENARIO 1 ",
+        "text_level": 1,
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+        "text": "In this section we apply Neural DNF with a CNN as feature extractor without any further contraints. and then demonstrate empirically that given the same first-stage feature extractor architecture, Neural DNF sacrifices slight loss of accuracy (table 1) while gaining more faithful interpretations (fig. 3). We first evaluate the accuracy of Neural DNF on several image datasets table 1. Using the same architecture for $\\phi$ , the models we compare are (1) Neural DNF (2) a standard DNN (i.e. a linear model as $g$ ) and (3) self-explaining neural network (SENN) (Melis and Jaakkola, 2018), which uses a neural network to generate the coefficients of a linear model conditioned on the input and claims to have better interpretability. Since binarization is not required for DNN or SENN, we also evaluate DNN and SENN with/without it. We observe that all models perform similarly well in terms of test accuracy across datasets, while Neural DNF loses accuracy only slightly. Comparing DNN and SENN with/without binarization, we can see that both DNN and SENN loses accuracy slightly because of the binarization. So the loss of Neural DNF ’s accuracy can be explained from two sources: (1) the binarization (Improved SemHash) and (2) the use of rules as the classifier instead of the more well-studied linear (additive) models. We suspect this is also partially because Neural DNF treats multi-class settomg as multiple one-versus-all classifications and does not produce probabilistic outputs like DNN and SENN. However, this should not be viewed as a weakness as long as the accuracy loss is slight. We believe that this issue can be alleviated that given a sufficiently flexible $\\phi$ , the accuracy loss of Neural DNF can be negligible like the case of simple dataset MNIST. ",
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+        "table_caption": [
+            "Table 1: Test Accuracy on some image datasets "
+        ],
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+        "table_body": "<table><tr><td></td><td>MNIST</td><td>KMNIST</td><td>SVHN</td><td>CIFAR10</td></tr><tr><td>NeuralDNF</td><td>99.08%</td><td>95.43%</td><td>90.13%</td><td>67.91%</td></tr><tr><td>standard DNN</td><td>99.12%</td><td>95.86%</td><td>90.74%</td><td>70.43%</td></tr><tr><td>standard DNN(without Binarization)</td><td>99.11%</td><td>96.02%</td><td>91.45%</td><td>71.45%</td></tr><tr><td>SENN</td><td>98.48%</td><td>92.64%</td><td>90.79%</td><td>71.09%</td></tr><tr><td>SENN (without Binarization)</td><td>98.50%</td><td>91.46%</td><td>92.15%</td><td>72.32%</td></tr></table>",
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+        "type": "text",
+        "text": "Now regarding the interpretability, the problem here is, we can in principle inspect3 the meaning of concept and rules of Neural DNF, but since we have no constraints on $\\phi$ , the extracted features are only highly discriminative and is not guaranteed to be aligned with human perceptible concepts. A symbolic DNF that operates on non-interpretable representations is still non-interpretable. We will revisit this interpretable representation problem in next section; but despite tje first-stage, here we can still compare the interpretability of the second-stage model. Despite many qualitive arguments for favouring the interpretability of the symbolic DNF (appendix B.3), we can further quantitively evaluate the faithfulness of models’ interpretability, a critical critera that measures how faithful the explanation for a particular prediction are to the underlying computation of prediction. We adopt the faithfulness metric proposed by (Melis and Jaakkola, 2018) which measures the correlation of the change of the prediction (class probabilities) and the change of the explanation (relevance scores of features) upon perturbation of test examples.4 We report the faithfulness metric on test set for DNN, SENN and Neural DNF in fig. 3. For DNN, we use the coefficients of final linear layer as relevance scores. Alternatively, for DNN we can also utilize some representitive post-hoc interpretation methods (also known as attribution methods): Guided Backprop (GB) (Springenberg et al., 2014), gradient SHAP (Lundberg and Lee, 2017), Integrated Gradient (IG) (Sundararajan et al., 2017) and DeepLIFT (Shrikumar et al., 2017). SENN can use its self-generated coefficients as relevance scores (Melis and Jaakkola, 2018). For Neural DNF, we use the difference analysis (Robnik-Sikonja and ˇ Kononenko, 2008), a model-agnostic relevance score assignment method to assign scores. As shown in fig. 3, Neural DNF consistently achieves the highest faithfulness estimates for all datasets and its faithfulness has the smallest variance. This means Neural DNF’s explanations are highly faithful in a robust way across different test samples. SENN is more faithful than DNN; and for the post-hoc methods while GB performs very similarly with DNN-coef, IG/SHAP/DeepLIFT drastically improves the faithfulness. This is because the latter three methods compute the relevance w.r.t to a baseline reference, a concept that is now considered very essential (Sturmfels et al., 2020). However, we emphasize that even these sophisticated post-hoc methods do not reach the same level of faithfulness as Neural DNF, in particular not on MNIST and KMNIST. We believe that Neural DNF’s extremely high faithfulness is the direct result of the symbolic nature of DNF. ",
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+        "image_caption": [
+            "Figure 3: Evaluating the faithfulness of explanations on test set "
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+    {
+        "type": "text",
+        "text": "4.3 ON IMAGE DATASETS, SCENARIO 2 ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "Here we test a new scenario where we have concept annotations so we can train the feature extractor to produce human-aligned interpretable representations. Our goal is that if we can align the extracted features with human understanding, then we can achieve a highly interpretable model which human can easily interact with and manipulate. We use the CUB dataset (Wah et al., 2011) which has 200 class and 112 binary annotataed concepts (preprocessed by Koh et al. (2020)). In Koh et al. (2020) the authors propose the concept bottleneck model, a similar two-stage model using a Inception-V3 based feature extractor $\\phi$ and a linear layer $g$ . Given concepts the annotations, a concept prediction loss can be applied so that the extracted faetures are constrained to align with annotated values. Koh et al. (2020) also propose the test-time human intervention: since the extracted concepts can be wrong, human can check and correct extracted concepts so that test accuracy can be improved significantly through this interaction. We follow Koh et al. (2020) and replace the second-stage $g$ with a DNF. Koh et al. (2020) evaluate several strategies for training the two-stage model. Here we use the independent training strategy: we train $\\phi$ to predict concepts and train $g$ to predict class label using concepts; only in test time we stack $\\phi$ and $g$ together. The reason of choosing independent training instead of joint training is counterintuive, however, this is because we find that the extracted concepts after joint training are less well-aligned with human annotations, and it then makes the human intervention less effective than independent trained models. In other words, based on noninterpretable features, any human interaction/manipulation will be unreliable. This phenomenon is consistent with Koh et al. (2020), and we think novel architectures that extracts interpretable features without the need of annotations can solve it, in that case joint training should give better results. In table 2 we report the accuracy of Neural DNF, the concept bottleneck model and blackbox DNN. Neural DNF achieves less test accuracy compared with the concept bottleneck model by a large margin, we suspect it might because Neural DNF is dealing with too many (200-class) one-versusall prediction and does not produce probabilistic outputs like other models do. But after intervention both Neural DNF and the concept bottleneck model achiveve perfect accuracy. This indicates that (1) the accuracy by applying human intervention can be improved significantly which is the benefits of having an highly interpretable model; (2) the bottleneck of accuracy is the feature extractor: the classification of CUB can be effectively solved given a perfect feature extractor. ",
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+        "type": "table",
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+        "table_caption": [
+            "Table 2: Test Accuracy on CUB dataset "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td></td><td>test acc</td><td>test acc with human intervention</td></tr><tr><td>NeuralDNF</td><td>61.94%</td><td>100.00%</td></tr><tr><td>Concept bottleneck model</td><td>72.38%</td><td>100.00%</td></tr><tr><td>Blackbox DNN</td><td>74.69%</td><td>N/A</td></tr></table>",
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+    {
+        "type": "text",
+        "text": "While losing quite much in accuracy, the symbolic nature enables the Neural DNF to do the things the concept bottleneck model cannot. We show in fig. 4 (top) a correctly predicted sample for the class of ‘Chestnut sided Warbler’ and in fig. 4 (middle) an incorretly predicted sample and how human intervention can correct the prediction. DNF rules are a very human-readable format, that the rules are very sparse involving only a few concepts and the logical operation is intuitive to understand. The DNF rules also alleviate the burden of human intervention compared to the concept bottleneck, as only a few concepts are used in the decision rule, a human user can only check and correct these concepts indicated in the rules and does not have to go through every concepts which the concept bottleneck model requires. Also, the symbolic nature of DNF enables us to incorporate knowledge and manipulate the models easily in a way that the concept bottleneck model (with a linear $g$ ) cannot offer. In in fig. 4 (bottom), we provide an illustration on how we can tweak the Neural DNF to predict an imaginary class ”blue-crown Chestnut sided Warbler” that does not exist in the training dataset (not even in real world). Taking the rule for ‘Chestnut sided Warbler’, we can simply replacing the condition of ‘yellow crown is True’ with ‘blue crown is True’. Note that ’blue crown’ is already in the training set and one of the concepts $\\phi$ can extract. Of course we can go further that we can train and append new feature extractors for new concepts and then play with new concepts. There are, however, limitations because of the expressity of prositional logic. ",
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+    {
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+        "img_path": "images/097c97e7d1852a6f28ab32dc687f627ba096a602a067a13b697ea426ba0a2f9f.jpg",
+        "image_caption": [
+            "Figure 4: (top) illustration of a correct prediction of Neural DNF for class ‘Chestnut sided Warbler’; (middle) illustration of human interaction: an incorrect prediction of Neural DNF for class ‘Chestnut sided Warbler’ can be corrected by human intervention; (bottom) illustration of manipulating a learned Neural DNF to classify an imaginary class ‘Blue-crown Chestnut sided Warbler’. "
+        ],
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+    {
+        "type": "text",
+        "text": "5 CONCLUSION AND FUTURE WORK ",
+        "text_level": 1,
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+        "text": "In this paper, we have presented Neural DNF, a vertically integrated neuro-symbolic classifier. Neural DNF takes a step towards the fundamental problem of integrating continuous perception and logical reasoning. Neural DNF adopts a two-stage model that utlizes a DNN to extract features, called concept predicates, and a propositional logic DNF module to make classification based on the concepts. We propose the BOAT algorithm for joint learning of the DNF module and the feature extractor DNN. In order to avoid laborious manual tuning, we employ an Improved SemHash method to binarize the extracted features, to obtain concept predicates, because it does not need to manually tune any extra parameters or add further losses. We also introduce adaptive temperatured-noise, which is a minimal modification of the binary-parameter optimizer Bop that enables the effective optimization of the parameters of the second-stage rule-based model. ",
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+        "type": "text",
+        "text": "This paper suggests several directions for future research. In our experiments in Neural DNF , we have used concept annotations and the concept-bottleneck DNN architecture to extract interpretable features. However, in practice there is often no prior knowledge of ‘concept annotations’ available. Therefore, novel deep learning architectures that can extract human-understandable concepts from data without the need of concept annotations should be investigated. This may require domainspecific knowledge representations and loss functions (Melis and Jaakkola, 2018; Chen et al., 2019a; Biffi et al., 2018; Kim and Canny, 2017; Johnson et al., 2016). Another future direction is the use of more powerful languages than propositional logic, for example, the more general inductive logic programming. Since BOAT requires only minimal changes of the standard deep learning optimization, we believe it can be potentially applied to models with a more powerful second-stage rule-based module, or to other models with mixed binary-continuous parameters. ",
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+    {
+        "type": "text",
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