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+# Information-Theoretic Safe Exploration with Gaussian Processes
+
+Alessandro G. Bottero $^{1,2}$ , Carlos E. Luis $^{1,2}$ , Julia Vinogradska $^{1}$ , Felix Berkenkamp $^{1}$ , Jan Peters $^{2}$ $^{1}$ Bosch Center for Artificial Intelligence, Germany  
+ $^{2}$ Technische Universität Darmstadt, Germany  
+Alessandro Giacomo. Bottero@de.bosch.com
+
+# Abstract
+
+We consider a sequential decision making task where we are not allowed to evaluate parameters that violate an a priori unknown (safety) constraint. A common approach is to place a Gaussian process prior on the unknown constraint and allow evaluations only in regions that are safe with high probability. Most current methods rely on a discretization of the domain and cannot be directly extended to the continuous case. Moreover, the way in which they exploit regularity assumptions about the constraint introduces an additional critical hyperparameter. In this paper, we propose an information-theoretic safe exploration criterion that directly exploits the GP posterior to identify the most informative safe parameters to evaluate. Our approach is naturally applicable to continuous domains and does not require additional hyperparameters. We theoretically analyze the method and show that we do not violate the safety constraint with high probability and that we explore by learning about the constraint up to arbitrary precision. Empirical evaluations demonstrate improved data-efficiency and scalability.
+
+# 1 Introduction
+
+In sequential decision making problems, we iteratively select parameters in order to optimize a given performance criterion. However, real-world applications such as robotics (Berkenkamp et al., 2021), mechanical systems (Schillinger et al., 2017) or medicine (Sui et al., 2015) are often subject to additional safety constraints that we cannot violate during the exploration process (Dulac-Arnold et al., 2019). Since it is a priori unknown which parameters lead to constraint violations, we need to actively and carefully learn about the constraints without violating them. That is, we need to learn about the safety of parameters by only evaluating parameters that are currently known to be safe.
+
+Existing methods by Schreiter et al. (2015); Sui et al. (2015) tackle this problem by placing a Gaussian process (GP) prior over the constraint and only evaluate parameters that do not violate the constraint with high probability. To learn about the safety of parameters, they evaluate the parameter with the largest posterior variance. This process is made more efficient by SAFEOPT, which restricts its safe set expansion exploration component to parameters that are close to the boundary of the current set of safe parameters (Sui et al., 2015) at the cost of an additional tuning hyperparameter (Lipschitz constant). However, uncertainty about the constraint is only a proxy objective that only indirectly learns about the safety of parameters. Consequently, data-efficiency could be improved with an exploration criterion that directly maximizes the information gained about the safety of parameters.
+
+Our contribution In this paper, we propose Information-Theoretic Safe Exploration (ISE), a safe exploration algorithm that directly exploits the information gain about the safety of parameters in order to expand the region of the parameter space that we can classify as safe with high confidence. By directly optimizing for safe information gain, ISE is more data-efficient than existing approaches without manually restricting evaluated parameters to be on the boundary of the safe set, particularly
+
+in scenarios where the posterior variance alone is not enough to identify good evaluation candidates, as in the case of heteroskedastic observation noise. This exploration criterion also means that we do not require additional hyperparameters beyond the GP posterior and that ISE is directly applicable to continuous domains. We theoretically analyze our method and prove that it learns about the safety of reachable parameters to arbitrary precision.
+
+Related work Information-based selection criteria with Gaussian processes models are successfully used in the context of unconstrained Bayesian optimization (BO, Shahriari et al. (2016); Bubeck and Cesa-Bianchi (2012)), where the goal is to find the parameters that maximize an a priori unknown function. Hennig and Schuler (2012); Henrández-Lobato et al. (2014); Wang and Jegelka (2017) select parameters that provide the most information about the optimal parameters, while Fröhlich et al. (2020) consider the information under noisy parameters. The success of these information-based approaches also relies on the superior data efficiency that they demonstrated. We draw inspiration from these methods when defining an information-based criterion w.r.t. the safety of parameters to guide safe exploration.
+
+In the presence of constraints that the final solution needs to satisfy, but which we can violate during exploration, Gelbart et al. (2014) propose to combine typical BO acquisition functions with the probability of satisfying the constraint. Instead, Gotovos et al. (2013) propose an uncertainty-based criterion that learns about the feasible region of parameters. When we are not allowed to ever evaluate unsafe parameters, safe exploration is a necessary sub-routine of BO algorithms to learn about the safety of parameters. To safely explore, Schreiter et al. (2015) globally learn about the constraint by evaluating the most uncertain parameters. SAFEOPT by Sui et al. (2015) extends this to joint exploration and optimization and makes it more efficient by explicitly restricting safe exploration to the boundary of the safe set. Sui et al. (2018) proposes STAGEOPT, which additionally separates the exploration and optimization phases. Both of these algorithms assume access to a Lipschitz constant to define parameters close to the boundary of the safe set, which is a difficult tuning parameter in practice. These methods have been extended to multiple constraints by Berkenkamp et al. (2021), while Kirschner et al. (2019) scale them to higher dimensions with LINEBO, which explores in low-dimensional sub-spaces. To improve computational costs, Duivenvoorden et al. (2017) suggest a continuous approximation to SAFEOPT without providing exploration guarantees. All of these methods rely on function uncertainty to drive exploration, while we directly maximize the information gained about the safety of parameters.
+
+Safe exploration also arises in the context of Markov decision processes (MDP), (Moldovan and Abbeel, 2012; Hans et al., 2008). In particular, Turchetta et al. (2016, 2019) traverse the MDP to learn about the safety of parameters using methods that, at their core, explore using the same ideas as SAFEOPT and STAGEOPT to select parameters to evaluate. Consequently, our proposed method for safe exploration is also directly applicable to their setting.
+
+# 2 Problem Statement
+
+In this section, we introduce the problem and notation that we use throughout the paper. We are given an unknown and expensive to evaluate safety constraint  $f: \mathcal{X} \to \mathbb{R}$  s.t. parameters that satisfy  $f(\boldsymbol{x}) \geq 0$  are classified as safe, while others are unsafe. To start exploring safely, we also have access to an initial safe parameter  $\boldsymbol{x}_0$  that satisfies the safety constraint,  $f(\boldsymbol{x}_0) \geq 0$ . We sequentially select safe parameters  $\boldsymbol{x}_n \in \mathcal{X}$  where to evaluate  $f$  in order to learn about the safety of parameters beyond  $\boldsymbol{x}_0$ . At each iteration  $n$ , we obtain a noisy observation of  $y_n := f(\boldsymbol{x}_n) + \nu_n$  that is corrupted by additive homoscedastic Gaussian noise  $\nu_n \sim \mathcal{N}(0, \sigma_\nu^2)$ . We illustrate the task in Figure 1a, where starting from  $\boldsymbol{x}_0$  we aim to safely explore the domain so that we ultimately classify as safe all the safe parameters that are reachable from  $\boldsymbol{x}_0$ .
+
+As  $f$  is unknown and the evaluations  $y_{n}$  are noisy, it is not feasible to select parameters that are safe with certainty and we provide high-probability safety guarantees instead. To this end, we assume that the safety constraint  $f$  has bounded norm in the Reproducing Kernel Hilbert Space (RKHS) (Schölkopf and Smola, 2002)  $\mathcal{H}_k$  associated to some kernel  $k: \mathcal{X} \times \mathcal{X} \to \mathbb{R}$  with  $k(\boldsymbol{x},\boldsymbol{x}^{\prime}) \leq 1$ . This assumption allows us to model  $f$  as a Gaussian process (GP) (Srinivas et al., 2010).
+
+Gaussian Processes A GP is a stochastic process specified by a mean function  $\mu : \mathcal{X} \to \mathbb{R}$  and a kernel  $k$  (Rasmussen and Williams, 2006). It defines a probability distribution over real-valued functions on  $\mathcal{X}$ , such that any finite collection of function values at parameters  $[\pmb{x}_1, \dots, \pmb{x}_n]$  is
+
+![](images/af155e23aa5ccdd62391251bc4ff2eb96d1073b6ad08e917cd6a85cb4f8c92f5.jpg)  
+(a) Problem components.
+
+![](images/bbffb2487fa72ce03817f662715b55e267c781116199ad3bbab55f110da20f79.jpg)  
+(b) ISE mutual information.  
+Figure 1: In (a) we illustrate the safe exploration task. Based on the unknown safety constraint  $f$ , we are only allowed to evaluate safe parameters  $x$  with values  $f(x) \geq 0$  above the safety threshold (dashed line). Starting from a safe seed  $x_0$  a safe exploration strategy needs to discover the largest reachable safe region of the parameter space containing  $x_0$ . In (b) we show the mutual information  $I_n(\{x, y\}; \Psi(z))$  in green for different  $x$  inside the safe set and for a fixed  $z$  outside (red dashed line). ISE maximizes this mutual information jointly over  $x$  and  $z$ .
+
+distributed as a multivariate normal distribution. The GP prior can then be conditioned on (noisy) function evaluations  $\mathcal{D}_n = \{(x_i, y_i)\}_{i=1}^n$ . If the noise is Gaussian, then the resulting posterior is also a GP and with posterior mean and variance
+
+$$
+\mu_ {n} (\boldsymbol {x}) = \mu (\boldsymbol {x}) + \boldsymbol {k} (\boldsymbol {x}) ^ {\top} (\boldsymbol {K} + \boldsymbol {I} \sigma_ {\nu} ^ {2}) ^ {- 1} (\boldsymbol {y} - \boldsymbol {\mu}),
+$$
+
+$$
+\sigma_ {n} ^ {2} (\boldsymbol {x}) = k (\boldsymbol {x}, \boldsymbol {x}) - \boldsymbol {k} (\boldsymbol {x}) ^ {\top} \left(\boldsymbol {K} + \boldsymbol {I} \sigma_ {\nu} ^ {2}\right) ^ {- 1} \boldsymbol {k} (\boldsymbol {x}), \tag {1}
+$$
+
+where  $\pmb{\mu} \coloneqq [\mu(\pmb{x}_1), \dots, \mu(\pmb{x}_n)]$  is the mean vector at parameters  $\pmb{x}_i \in \mathcal{D}_n$  and  $[\pmb{y}]_i \coloneqq y(\pmb{x}_i)$  the corresponding vector of observations. We have  $\left[\pmb{k}(\pmb{x})\right]_i \coloneqq k(\pmb{x}, \pmb{x}_i)$ , the kernel matrix has entries  $[K]_{ij} \coloneqq k(\pmb{x}_i, \pmb{x}_j)$ , and  $\pmb{I}$  is the identity matrix. In the following, we assume without loss of generality that the prior mean is identically zero:  $\mu(\pmb{x}) \equiv 0$ .
+
+Safe set Using the previous assumptions, we can construct high-probability confidence intervals on the function values  $f(\pmb{x})$ . Concretely, for any  $\delta > 0$  it is possible to find a sequence of positive numbers  $\{\beta_n\}$  such that  $f(\pmb{x}) \in [\mu_n(\pmb{x}) \pm \beta_n \sigma_n(\pmb{x})]$  with probability at least  $1 - \delta$ , jointly for all  $\pmb{x} \in \mathcal{X}$  and  $n \geq 1$ . For a proof and more details see (Chowdhury and Gopalan, 2017). We use these confidence intervals to define a safe set
+
+$$
+S _ {n} := \left\{\boldsymbol {x} \in \mathcal {X}: \mu_ {n} (\boldsymbol {x}) - \beta_ {n} \sigma_ {n} (\boldsymbol {x}) \geq 0 \right\} \cup \left\{\boldsymbol {x} _ {0} \right\}, \tag {2}
+$$
+
+which contains all parameters whose  $\beta_{n}$ -lower confidence bound is above the safety threshold and the initial safe parameter  $\pmb{x}_0$ . Consequently, we know that all parameters in  $S_{n}$  are safe,  $f(\pmb{x}) \geq 0$  for all  $\pmb{x} \in S_{n}$ , with probability at least  $1 - \delta$  jointly over all iterations  $n$ .
+
+Safe exploration Given the safe set  $S_{n}$ , the next question is which parameters in  $S_{n}$  to evaluate in order to efficiently expand it. Most existing safe exploration methods rely on uncertainty sampling over subsets of  $S_{n}$ . SAFEOPT-like approaches, for example, use the Lipschitz assumption on  $f$  to identify parameters in  $S_{n}$  that could expand the safe set and then select the parameter that has the biggest uncertainty among those. In the next sections, we present and analyze our safe exploration strategy, ISE, that instead uses an information gain measure to identify the parameters that allow us to efficiently learn about the safety of parameters outside of  $S_{n}$ .
+
+# 3 Information-Theoretic Safe Exploration
+
+We present Information-Theoretic Safe Exploration (ISE), which guides the safe exploration by using an information-theoretic criterion. Our goal is to design an exploration strategy that directly exploits the properties of GPs to learn about the safety of parameters outside of  $S_{n}$ . We draw inspiration from Hennig and Schuler (2012); Wang and Jegelka (2017) who exploit information-theoretic insights to design data-efficient BO acquisition functions for their respective optimization objectives.
+
+Information gain measure In our case, we want to evaluate  $f$  at safe parameters that are maximally informative about the safety of other parameters, in particular of those where we are uncertain
+
+Algorithm 1 Information-Theoretic Safe Exploration
+
+1: Input: GP prior  $(\mu_0, k, \sigma_\nu)$ , Safe seed  $x_0$  
+2: for  $n = 0, \dots, N$  do  
+3:  $x_{n + 1}\gets \arg \max_{\pmb {x}\in S_n}\max_{\pmb {z}\in \mathcal{X}}\hat{I}_n(\{\pmb {x},\pmb {y}\};\Psi (\pmb {z}))$  
+4:  $y_{n + 1}\gets f(\pmb{x}_{n + 1}) + \nu$  
+5: Update GP posterior with  $(\pmb{x}_{n + 1},y_{n + 1})$
+
+about whether they are safe or not. To this end, we need a corresponding measure of information gain. We define such a measure using the binary variable  $\Psi(x) = \mathbb{I}_{f(x) \geq 0}$ , which is equal to one iff  $f(x) \geq 0$ . Its entropy is given by
+
+$$
+H _ {n} [ \Psi (\boldsymbol {z}) ] = - p _ {n} ^ {-} (\boldsymbol {z}) \ln \left(p _ {n} ^ {-} (\boldsymbol {z})\right) - (1 - p _ {n} ^ {-} (\boldsymbol {z})) \ln (1 - p _ {n} ^ {-} (\boldsymbol {z})) \tag {3}
+$$
+
+where  $p_n^- (z)$  is the probability of  $z$  being unsafe:  $p_n^- (z) = \frac{1}{2} +\frac{1}{2}\mathrm{erf}\left(-\frac{1}{\sqrt{2}}\frac{\mu_n(z)}{\sigma_n(z)}\right)$ . The random variable  $\Psi (z)$  has high-entropy when we are uncertain whether a parameter is safe or not; that is, its entropy decreases monotonically as  $|\mu_n(z)|$  increases and the GP posterior moves away from the safety threshold. It also decreases monotonically as  $\sigma_{n}(z)$  decreases and we become more certain about the constraint. This behavior also implies that the entropy goes to zero as the confidence about the safety of  $z$  increases, as desired.
+
+Given our definition of  $\Psi$ , we consider the mutual information  $I\big(\{\pmb{x},y\} ;\Psi (\pmb {z})\big)$  between an observation  $y$  at a parameter  $\pmb{x}$  and the value of  $\Psi$  at another parameter  $\pmb{z}$ . Since  $\Psi$  is the indicator function of the safe regions of the parameter space, the quantity  $I_{n}\big(\{\pmb {x},y\} ;\Psi (\pmb {z})\big)$  measures how much information about the safety of  $\pmb{z}$  we gain by evaluating the safety constraint  $f$  at  $\pmb{x}$  at iteration  $n$ , averaged over all possible observed values  $y$ . This interpretation follows directly from the definition of mutual information:  $I_{n}\big(\{\pmb {x},y\} ;\Psi (\pmb {z})\big) = H_{n}\big[\Psi (\pmb {z})\big] - \mathbb{E}_{y}\Big[H_{n + 1}\big[\Psi (\pmb {z})\big|\{\pmb {x},y\} \Big]\Big]$ , where  $H_{n}[\Psi (z)]$  is the entropy of  $\Psi (z)$  according to the GP posterior at iteration  $n$ , while  $H_{n + 1}\big[\Psi (z)\big|\{\pmb {x},y\} \big]$  is its entropy at iteration  $n + 1$ , conditioned on a measurement  $y$  at  $\pmb{x}$  at iteration  $n$ . Intuitively,  $I_{n}\big(\{\pmb {x},y\} ;\Psi (\pmb {z})\big)$  is negligible whenever the confidence about the safety of  $\pmb{z}$  is high or, more generally, whenever an evaluation at  $\pmb{x}$  does not have the potential to substantially change our belief about the safety of  $\pmb{z}$ . The mutual information is large whenever an evaluation at  $\pmb{x}$  on average causes the confidence about the safety of  $\pmb{z}$  to increase significantly. As an example, in Figure 1 we plot  $I_{n}\big(\{\pmb {x},y\} ;\Psi (\pmb {z})\big)$  as a function of  $\pmb {x}\in S_n$  for a specific choice of  $\pmb{z}$  and for an RBF kernel. As one would expect, we see that the closer it gets to  $\pmb{z}$ , the bigger the mutual information becomes, and that it vanishes in the neighborhood of previously evaluated parameters, where the posterior variance is negligible.
+
+To compute  $I_{n}\big(\{\pmb {x},\pmb {y}\} ;\Psi (\pmb {z})\big)$ , we need to average (3) conditioned on an evaluation  $y$  over all possible values of  $y$ . However, the resulting integral is intractable given the expression of  $H_{n}[\Psi (\pmb {z})]$  in (3). In order to get a tractable result, we derive a close approximation of (3),
+
+$$
+H _ {n} \left[ \Psi (z) \right] \approx \hat {H} _ {n} \left[ \Psi (z) \right] \doteq \ln (2) \exp \left\{- \frac {1}{\pi \ln (2)} \left(\frac {\mu_ {n} (z)}{\sigma_ {n} (z)}\right) ^ {2} \right\}. \tag {4}
+$$
+
+The approximation in (4) is obtained by truncating the Taylor expansion of  $H_{n}[\Psi(z)]$  at the second order, and it recovers almost exactly its true behavior (see Appendix B for details). Since the posterior mean at  $z$  after an evaluation at  $x$  depends linearly on  $\mu_{n}(x)$ , and since the probability density of  $y$  depends exponentially on  $-\mu_{n}^{2}(x)$ , using (4) reduces the conditional entropy  $\mathbb{E}_{y}\left[\hat{H}_{n+1}\left[\Psi(z)\right|\{x,y\}\right]$  to a Gaussian integral with the exact solution
+
+$$
+\begin{array}{l} \mathbb {E} _ {y} \left[ \hat {H} _ {n + 1} [ \Psi (z) | \{x, y \} ] \right] = \\ \ln (2) \sqrt {\frac {\sigma_ {\nu} ^ {2} + \sigma_ {n} ^ {2} (\boldsymbol {x}) \left(1 - \rho_ {n} ^ {2} (\boldsymbol {x} , \boldsymbol {z})\right)}{\sigma_ {\nu} ^ {2} + \sigma_ {n} ^ {2} (\boldsymbol {x}) \left(1 + c _ {2} \rho_ {n} ^ {2} (\boldsymbol {x} , \boldsymbol {z})\right)}} \exp \left\{- c _ {1} \frac {\mu_ {n} ^ {2} (\boldsymbol {z})}{\sigma_ {n} ^ {2} (\boldsymbol {z})} \frac {\sigma_ {\nu} ^ {2} + \sigma_ {n} ^ {2} (\boldsymbol {x})}{\sigma_ {\nu} ^ {2} + \sigma_ {n} ^ {2} (\boldsymbol {x}) \left(1 + c _ {2} \rho_ {n} ^ {2} (\boldsymbol {x} , \boldsymbol {z})\right)} \right\}, \tag {5} \\ \end{array}
+$$
+
+where  $\rho_{n}(\pmb{x},\pmb{z})$  is the linear correlation coefficient between  $f(\pmb{x})$  and  $f(z)$ , and with  $c_{1}$  and  $c_{2}$  given by  $c_{1} := 1 / \ln(2)\pi$  and  $c_{2} := 2c_{1} - 1$ . This result allows us to analytically calculate the approximated
+
+![](images/c354268bec093c3e8b7fb9b383ae8417d2492c90f82ab6fce56400e54572b145.jpg)  
+(a) Iteration 1
+
+![](images/17336e677470549662f0d4f064b2df0e7ff235a357ac1c2a27fdae4974ed7be1.jpg)  
+(b) Iteration 3  
+Figure 2: Example run of ISE at different iterations. The GP's posterior mean (blue line) and confidence interval  $\mu_{n} \pm \beta_{n}\sigma_{n}$  (blue shaded) approximate the true safety constraint  $f$  (red line) based on selected data points (green crosses) and, together with the safety threshold at zero (orange line), identify the current safe set  $S_{n}$  (light blue bar, bottom). The vertical green line indicates the location of the next parameter  $x_{n+1}$  selected by (6). Initially, ISE evaluates parameters on the boundary of the safe set, but eventually also evaluates inside the safe set if that provides the most information. ISE quickly discover the largest reachable safe set (dark blue bar, bottom).
+
+![](images/ef604546033b14a53bd864b037d70814a29a95e8e3d966d4c442c798a7a9e755.jpg)  
+(c) Iteration 8
+
+mutual information  $\hat{I}_n\big(\{\pmb {x},y\} ;\Psi (\pmb {z})\big)\doteq \hat{H}_n\big[\Psi (\pmb {z})\big] - \mathbb{E}_y\Big[\hat{H}_{n + 1}\big[\Psi (\pmb {z})\big|\{\pmb {x},y\} \Big]\Big],$  which we use to define the ISE acquisition function, and which we analyze theoretically in Section 4.
+
+ISE acquisition function Now that we have defined a way to measure and compute the information gain about the safety of parameters, we can use it to design an exploration strategy that selects the next parameters to evaluate. The natural choice for such selection criterion is to select the parameter that maximizes the information gain; that is, we select  $\pmb{x}_{n + 1}$  according to
+
+$$
+\boldsymbol {x} _ {n + 1} \in \underset {\boldsymbol {x} \in S _ {n}} {\arg \max } \underset {\boldsymbol {z} \in \mathcal {X}} {\max } \hat {I} _ {n} \left(\left\{\boldsymbol {x}, y \right\}; \Psi (\boldsymbol {z})\right), \tag {6}
+$$
+
+where we jointly optimize over  $\mathbf{x}$  in the safe set  $S_{n}$  and an unconstrained second parameter  $\mathbf{z}$ . Evaluating  $f$  at  $\mathbf{x}_{n + 1}$  according to (6) maximizes the information gained about the safety of some parameter  $\mathbf{z} \in \mathcal{X}$ , so that it allows us to efficiently learn about parameters that are not yet known to be safe. While  $\mathbf{z}$  can lie in the whole domain, the parameters where we are the most uncertain about the safety constraint lie outside the safe set. By leaving  $\mathbf{z}$  unconstrained, we show in our theoretical analysis in Section 4 that, once we have learned about the safety of parameters outside the safe set, (6) resorts to learning about the constraint function also inside  $S_{n}$ . An overview of ISE can be found in Algorithm 1 and we show an example run of a one-dimensional illustration of the algorithm in Figure 2.
+
+# 4 Theoretical Results
+
+In this section, we study the expression for  $\hat{I}_n(\{\pmb{x},y\};\Psi (\pmb {z}))$  obtained using (4) and (5) and analyze the properties of the ISE exploration criterion (6). By construction of  $S_{n}$  in (2) and the assumptions on  $f$  in Section 2, we know that any parameter selected according to (6) is safe with high probability, see Appendix A for details. To show that we also learn about the safe set, we first need to define what it means to successfully explore starting from  $\pmb{x}_0$ . The main challenge is that it is difficult to analyze how a GP generalizes based on noisy observations, so that it is difficult to define a notion of convergence that is not dependent on the specific run. SAFEOPT avoids this issue by expanding the safe set not based on the GP, but only using the Lipschitz constant  $L$ . Contrary to their approach, we depend on the GP to generalize from the safe set. In this case, the natural notion of convergence is provided by the posterior variance. In particular, we say that at iteration  $n$  we have explored the safe set up to  $\varepsilon$ -accuracy if  $\sigma_n^2 (\pmb {x})\leq \varepsilon$  for all parameters  $\pmb{x}$  in  $S_{n}$ . In the following, we show that ISE asymptotically leads either to  $\varepsilon$ -accurate exploration of the safe set or to indefinite expansion of the safe set. In future work it will be interesting to further investigate the notion of generalization and to derive a similar convergence result as those obtained by Sui et al. (2015).
+
+Theorem 1. Assume that  $\pmb{x}_{n+1}$  is chosen according to (6), and that there exists  $\hat{n}$  such that for all  $n \geq \hat{n}$ $S_{n+1} \subseteq S_n$ . Moreover, assume that for all  $n \geq \hat{n}$ ,  $|\mu_n(\pmb{x})| \leq M$  for some  $M > 0$  for all  $\pmb{x} \in S_n$ . Then, for all  $\varepsilon > 0$  there exists  $N_\varepsilon$  such that  $\sigma_n^2(\pmb{x}) \leq \varepsilon$  for every  $\pmb{x} \in S_n$  if  $n \geq \hat{n} + N_\varepsilon$ .
+
+The smallest of such  $N_{\varepsilon}$  is given by
+
+$$
+N _ {\varepsilon} = \min  \left\{N \in \mathbb {N}: b ^ {- 1} \left(\frac {C \gamma_ {N}}{N}\right) \leq \varepsilon \right\}, \tag {7}
+$$
+
+where  $b(\eta) \coloneqq \ln(2)\exp\left\{-c_1\frac{M^2}{\eta}\right\}\left[1 - \sqrt{\frac{\sigma_\nu^2}{2c_1\eta + \sigma_\nu^2}}\right]$ ,  $\gamma_N = \max_{D \subset \mathcal{X}; |D| = N} I\big(\pmb{f}(D); \pmb{y}(D)\big)$  is the maximum information capacity of the chosen kernel (Srinivas et al., 2010; Contal et al., 2014), and  $C = \ln(2)/\sigma_\nu^2\ln\big(1 + \sigma_\nu^{-2}\big)$ .
+
+Proof. See Appendix A.
+
+![](images/78ebb40a185f27fd50951d16779d989b314b5cedde566c04c41863f2c498b616.jpg)
+
+Theorem 1 tells us that if at some point the set of safe parameters  $S_{n}$  stops expanding, then the posterior variance over the safe set vanishes eventually. The intuition behind Theorem 1 is that if there were a parameter  $x$  in the safe set whose posterior mean remained finite and whose posterior variance remained bounded from below, then an evaluation of  $f$  at such  $x$  would yield a non-negligible average information gain about the safety of  $x$ , so that, since  $x$  is in the safe set, at some point ISE will be forced to choose to evaluate  $x$ , reducing its posterior variance. This result guarantees that, should the safe set stop expanding, ISE will asymptotically explore the safe set up to an arbitrary  $\varepsilon$ -accuracy. In practice, we observe that ISE first focuses on reducing the uncertainty in areas of the safe set that are most informative about parameters whose classification is still uncertain (e.g. the boundary of the safe set), and only eventually turns to learning about the inside of the safe set. This behavior is what ultimately leads to the posterior variance to decay over the whole  $S_{n}$ . Therefore, even if in general it is not always possible to say whether or not the safe set will ever stop expanding, we can read Theorem 1 as an exploration guarantee for ISE, as it rules out the possibility that the proposed acquisition function forever leaves the uncertainty high in areas of the safe set that, if better understood, could lead to an expansion of the safe set.
+
+Theorem 1 requires a bound on the GP posterior mean function, which is always satisfied with high probability based on our assumptions about  $f$ . Specifically, we have that  $|\mu_n(\pmb{x})| \leq 2\beta_n$  with probability of at least  $1 - \delta$  for all  $n$  (see Appendix A for details). Therefore, it does not represent an additional restrictive assumption for  $f$ . Finally, we also note that the constant  $N_{\varepsilon}$  defined by (7) always exists since the function  $b$  is monotonically increasing, as long as  $\gamma_N$  grows sublinearly in  $N$ . Srinivas et al. (2010) prove that this is the case for commonly-used kernel and, more generally, it is a prerequisite for data-efficient learning with GP models.
+
+# 5 Discussion and Limitations
+
+ISE drives exploration of the parameter space by selecting the parameters to evaluate according to (6). An alternative but conceptually similar approach to this criterion would be to consider the parameter that yields the biggest information gain on average over the domain, i.e., substituting the inner max in (6) with an average over  $\mathcal{X}$ . The resulting integral, however, is intractable and would require further approximations. Moreover, the parameter found by solving (6) will also yield a high average information gain over the domain, due to the regularity of all involved objects.
+
+Being able to work in a continuous domain, ISE can deal with higher dimensional domains better than algorithms requiring a discrete parameter space. However, as noted in Section 4, finding  $x_{n+1}$  as in (6) means to solve a non-convex optimization problem with twice the dimension of the the parameter space, which can also become a computationally challenging problem as the dimension grows. In a high-dimensional setting, we follow LINEBO by Kirschner et al. (2019), which at each iteration selects a random one-dimensional subspace to which it restricts the optimization of the acquisition function.
+
+In Sections 2 and 3, we assumed the observation process to be homoskedastic. However, it needs not to be the case, and the results can be extended to the case of heteroskedastic Gaussian noise. The observation noise at a parameter  $x$  explicitly appears in the ISE acquisition function, since it crucially affects the amount of information that we can gain by evaluating the constraint  $f$  at  $x$ . On the contrary, STAGEOPT-like methods do not consider the observation noise in their acquisition functions. As a consequence, ISE can perform significantly better in an heteroskedastic setting, as we also show in Section 6.
+
+Lastly, we reiterate that the theoretical safety guarantees offered by ISE are derived under the assumption that  $f$  is a bounded norm element of the RKHS space associated with the GP's kernel. In applications, therefore, the choice of the kernel function becomes even more crucial when safety is an issue. For details on how to construct and choose kernels see (Garnett, 2022). The safety guarantees also depend on the choice of  $\beta_{n}$ . Typical expressions for  $\beta_{n}$  include the RKHS norm of the constraint  $f$  (Chowdhury and Gopalan, 2017; Fiedler et al., 2021), which is in general difficult to estimate in practice. Because of this, usually in practice a constant value of  $\beta_{n}$  is used instead.
+
+# 6 Experiments
+
+In this section we empirically evaluate ISE. Additional details about the experiments and setup can be found in Appendix C. As commonly done in the literature (see Section 5), we set  $\beta_{n} = 2$  for all experiments. This choice guarantees safety per iteration, rather than jointly for all  $n$  and it allows for a less conservative bound than the one needed for the joint guarantees.
+
+GP samples For the first part of the experiments, we evaluate ISE on constraint functions  $f$  that we obtain by sampling a GP prior at a finite number of points. This allows us to test ISE under the assumptions of the theory and we compare its performance to that of the exploration part of STAGEOPT (Sui et al., 2018). STAGEOPT is a modified version of SAFEOPT, in which the exploration and optimization parts are performed separately: first the SAFEOPT exploration strategy is used to expand the safe set as much as possible, then the objective function is optimized within the discovered safe set. We further modify the version of STAGEOPT that we use in the experiment by defining the safe set in the same way ISE does, i.e., by means of the GP posterior, as done, for example, also by Berkenkamp et al. (2016). We select 100 samples from a two-dimensional GP with RBF kernel, defined in  $[-2.5, 2.5] \times [-2.5, 2.5]$  and run ISE and STAGEOPT for 100 iterations for each sample. As STAGEOPT requires a discretization of the domain, we use this discretization to compare the sample efficiency of the two methods, by computing, at each iteration, what percentage of the discretized domain is classified as safe. Moreover, we also compare with the heuristic acquisition inspired by SAFEOPT proposed by Berkenkamp et al. (2016). This method works exactly as STAGEOPT, with the difference that the set of expanders is computed using directly the GP posterior, rather than the Lipschitz constant. More precisely, a parameter  $x$  is considered an expander if observing a value of  $\mu_n(x) + \beta_n\sigma_n(x)$  at  $x$  would enlarge the safe set. For the STAGEOPT run, we use the kernel metric to compute the set of potential expanders, for different values of the Lipschitz constant  $L$ . From the results shown in Figure 3a, we see not only that ISE performs as well or better than all tested instances of STAGEOPT, but also how the choice of  $L$  affects the performance of the latter. This plot makes it also evident how crucial the choice of the Lipschitz constant is for STAGEOPT and SAFEOPT-like algorithms in general. In Table 1, in Appendix C, we report the average percentage of safety violations per run achieved by ISE and STAGEOPT. As expected, we see that the percentage of safety violations is comparable among all algorithms.
+
+To show that for STAGEOPT exploration not only overestimating the Lipschitz constant, but also underestimating it can negatively impact performance, we consider the simple one-dimensional constraint function  $f(x) = e^{-x} + 0.05$  and run the safe exploration for multiple values of the Lipschitz constant. This function gets increasingly away from the safety threshold for  $x \to -\infty$ , while it asymptotically approaches the threshold for  $x \to \infty$ , so that a good exploration algorithm would, ideally, quickly classify as safe the domain region for  $x < 0$  and then keep exploring the boundary of the safe set for  $x > 0$ . The results plotted in Figure 3b show how both a too high and a too low Lipschitz constant can lead to sub-optimal exploration. In the case of a too small constant, this is because STAGEOPT considers expanders almost all parameters in the domain, leading to additional evaluations in the region for  $x < 0$  that are unlikely to cause expansion of the safe set. On the other hand, a too high value of the Lipschitz constant can lead to the set of expanders to be empty as soon as the posterior mean gets close to the safety threshold for  $x > 0$ .
+
+OpenAI Gym Control After investigating the performance of ISE under the hypothesis of the theory, we apply it to two classic control tasks from the OpenAI Gym framework (Brockman et al., 2016), with the goal of finding the set of parameters of a controller that satisfy some safety constraint. In particular we consider linear controllers for the inverted pendulum and cart pole tasks.
+
+For the inverted pendulum task, the linear controller is given by  $u_{t} = \alpha_{1}\theta_{t} + \alpha_{2}\dot{\theta}_{t}$ , where  $u_{t}$  is the control signal at time  $t$ , while  $\theta_t$  and  $\dot{\theta}_t$  are, respectively, the angular position and the angular velocity
+
+![](images/2af728a224d8192a79b7ddfec1098c5b51f4ff8cb86a74c380b8d75cb45299a7.jpg)  
+(a) Comparison with STAGEOPT for GP samples.
+
+![](images/410368a2913a0612fae726ed076797e338a5857a98677c1cb14760967b35db98.jpg)  
+(b) Comparison in 1D exponential example.  
+Figure 3: (a) Average expansion of safe set over 100 two-dimensional GP samples. The average percentage of the domain classified as safe is plotted as a function of  $n$  with its standard error. The lines for  $L = 0$  and  $L = 1$  overlap. We can see that ISE obtains an higher sample efficiency than the best instance of STAGEOPT and a comparable one with the heuristic acquisition function proposed by Berkenkamp et al. (2016). The plot also shows that STAGEOPT performance is heavily affected by the choice of  $L$ . In (b) the average percentage of the domain classified as safe in the one dimensional example for  $f(x) = e^{-x} + 0.05$  is plotted as function of the iteration  $n$  with its standard error, and it shows the detrimental effect of over- and underestimating  $L$ .
+
+of the pendulum. Starting from a position close to the upright equilibrium, the controller's task is the stabilization of the pendulum, subject to a safety constraint on the maximum velocity reached within one episode. For some given initial controller configuration  $\alpha_0 \coloneqq (\alpha_1^0, \alpha_2^0)$ , we want to explore the controller's parameter space avoiding configurations that lead the pendulum to swing with a too high velocity. We apply ISE to explore the  $\alpha$ -space with  $x_0 = \alpha_0$  and the safety constraint being the maximum angular velocity reached by the pendulum in an episode of fixed length. In this case, the safety threshold is not at zero, but rather at some finite value  $\dot{\theta}_M$ , and the safe parameters are those for which the maximum velocity is below  $\dot{\theta}_M$ . The formalism developed in the previous sections can be easily applied to this scenario if we consider  $f(\alpha) = -\left(\max_t \dot{\theta}_t(\alpha) - \dot{\theta}_M\right)$ . In Figure 4a we show the true safe set for this problem, while in Figures 4b-4d one can see how ISE safely explores the true safe set. These plots show how the ISE acquisition function (6) selects parameters that are close to the current safe set boundary and, hence, most informative about the safety of parameters outside of the safe set. This behavior eventually leads to the full true safe set to be classified as safe by the GP posterior, as Figure 4d shows.
+
+The cart pole task is similar to the inverted pendulum one, but the parameter space has three dimensions. The controller we consider is given by  $u_{t} = \alpha_{1}\theta_{t} + \alpha_{2}\dot{\theta}_{t} + \alpha_{3}\dot{s}_{t}$ , where  $\theta_{t}$  and  $\dot{\theta}_t$  are, respectively, the angular position and angular velocity of the pole at time  $t$ , while  $\dot{s}_t$  is the cart's velocity. We set the initial state to zero angular and linear velocity and with the pole close to the vertical position, with the controller's goal being to keep the pole stable in the upright position. A combination of the three parameters  $\alpha_{1},\alpha_{2}$  and  $\alpha_{3}$  is considered safe if the angle of the pole does not exceed a given threshold within the episode. Again, we can easily cast this safety constraint in terms of the formalism developed in the paper:  $f(\pmb {\alpha}) = -(\max_t\theta_t(\pmb {\alpha}) - \theta_M)$ , where  $\theta_{M}$  is the maximum allowed angle. Figure 5a shows the expansion of the cart pole  $\pmb{\alpha}$  space promoted by ISE, compared with STAGEOPT for different values of the Lipschitz constant. Both methods achieve a comparable sample efficiency and both lead to the classification as safe of the full true safe set.
+
+High dimensional domains Many interesting applications have a high dimensional parameter space. While SAFEOPT-like methods are difficult to apply already with dimension  $>3$  due to the discretization of the domain, ISE can perform well also in four or five dimensions. To see this, we apply ISE to the constraint function  $f(\pmb{x}) = e^{-\pmb{x}^2} + 2e^{-(\pmb{x} - \pmb{x}_1)^2} + 5e^{-(\pmb{x} - \pmb{x}_2)^2} - 0.2$ . Figure 5b shows the ISE performance in dimension 5. We see that ISE is able to promote the expansion of the safe set, leading to an increasingly bigger portion of the true safe set to be classified as safe.
+
+Heteroskedastic noise domains For even higher dimensions, we can follow a similar approach to LINEBO, limiting the optimization of the acquisition function to a randomly selected one-dimensional subspace of the domain. Moreover, as discussed in Section 5, it is also interesting to test ISE in the case of heteroskedastic observation noise, since the noise is a critical quantity for the ISE acquisition function, while it does not affect the selection criterion of STAGEOPT-like methods. Therefore, in
+
+![](images/87755448234fa7108d66e4f0b34aef8955c7726b6c29d25b578bb1b5fca3267a.jpg)
+
+![](images/e527b3326620714aae2891570521352dfd0dc6a940df149f996a3a67f0f7056d.jpg)
+
+![](images/8c3fefd837daa1c8b134ea5fd7aa6b3c4ec2009bd0c5f6edb29e344713047a75.jpg)  
+(a) True safe set  
+(c) Iteration 30
+
+![](images/92efe77b502d553384cb5e1bb9d52b4af7d8ee2013fdf1c13fd77edf4b3f9161.jpg)  
+(b) Iteration 10  
+(d) Iteration 50
+
+![](images/610500a54fdfe6f49d79a97fac0779eab81af83d16450e599453e4926c88ea53.jpg)  
+Figure 4: Safe exploration of the linear controller's parameter space in the inverted pendulum experiment. In (a) we see the true safe set, while in (b-d) we see the safe set (blue region) as identified by ISE at various iterations. The point marked by the green dot is  $\alpha_{n+1}$  as selected by (6), while the black cross is the initial safe seed  $\alpha_0$ .  
+(a) Exploration of the safe set for the cart pole task.  
+Figure 5: In (a) we plot the percentage of the true safe set of the cart pole task classified as safe by ISE and STAGEOPT, while in (b) we see the expansion of the five-dimensional safe set promoted by ISE, for the safety constraint  $f$  used in the high dimensional experiments.
+
+![](images/d9125405c58a8fc52d092df2d78040b0dde42764b32bb2b51aaefb9ddf2c42ac.jpg)  
+(b) Expansion of the safe set in dimension five.
+
+this experiment we combine a high dimensional problem with heteroskedastic noise. In particular, we apply a LINEBO version of ISE to the constraint function  $f(\pmb{x}) = \frac{1}{2} e^{-\pmb{x}^2} + e^{-(\pmb{x} \pm \pmb{x}_1)^2} + 3e^{-(\pmb{x} \pm \pmb{x}_2)^2} + 0.2$  in dimension nine and ten, with the safe seed being the origin. This function has two symmetric global optima at  $\pm x_2$  and we set two different noise levels in the two symmetric domain halves containing the optima. To assess the exploration performance, we use the simple regret, defined as the difference between the current safe optimum and the true safe optimum. As the results in Figure 6 show, ISE achieve a greater sample efficiency than the other STAGEOPT-like methods. Namely, for a given number of iterations, by explicitly exploiting knowledge about the observation noise, ISE is able to classify as safe regions of the domain further away from the origin, in which the constraint function assumes its largest values, resulting in a smaller regret. On the other hand, SAFEOPT-like methods only focus on the posterior variance, so that the higher observation noise causes them to remain stuck in a smaller neighborhood of the origin, resulting in bigger regret.
+
+![](images/480d746d6d2e837f717860d699081b0aa932c5653c0043126f45a5fd016ac773.jpg)  
+(a) Dimension 9.
+
+![](images/fe15de9b4abccd4b75c5c9c47c3338a42e66cadd1d3250a64a476d47e1b981c4.jpg)  
+(b) Dimension 10.  
+Figure 6: Example of high dimensional exploration, for  $d = 9$  and  $d = 10$ . After every ten iterations, we perform safe optimization with UCB acquisition within the current safe set and plot the simple regret  $r_n$  with respect to the safe optimum. (a) and (a) show, respectively, the average regret over 70 runs in dimension nine and ten, as a function of the number of iterations. We can see that this adapted version of ISE promotes expansion of the safe set, leading to classifying as safe regions where the latent function attains its largest value. The plots also show that ISE achieves a better sample efficiency than both STAGEOPT-like exploration and the STAGEOPT inspired heuristic acquisition.
+
+# 7 Conclusion and Societal Impact
+
+We have introduced Information-Theoretic Safe Exploration (ISE), a novel approach to safely explore a space in a sequential decision task where the safety constraint is a priori unknown. ISE efficiently and safely explores by evaluating only parameters that are safe with high probability and by choosing those parameters that yield the greatest information gain about the safety of other parameters. We theoretically analyzed ISE and showed that it leads to arbitrary reduction of the uncertainty in the largest reachable safe set containing the starting parameter. Our experiments support these theoretical results and demonstrate an increased sample efficiency and scalability of ISE compared to SAFEOPT-based approaches.
+
+In many safety sensitive applications the shape of the safety constraints is unknown, so that an important prerequisite for any kind of process is to identify what parameters are safe to evaluate. By providing a principled way to do this, the contributions of this paper allow to deal with safety in a broad range of applications, which can favor the usage of ML approaches also in safety sensitive settings. On the other hand, misuse of the proposed method cannot be prevented in general.
+
+# References
+
+Felix Berkenkamp, Angela P. Schoellig, and Andreas Krause. Safe controller optimization for quadrators with gaussian processes. In IEEE International Conference on Robotics and Automation (ICRA), 2016.  
+Felix Berkenkamp, Andreas Krause, and Angela P Schoellig. Bayesian optimization with safety constraints: safe and automatic parameter tuning in robotics. *Machine Learning Journal*, Special Issue on Robust Machine Learning, 2021.  
+Greg Brockman, Vicki Cheung, Ludwig Pettersson, Jonas Schneider, John Schulman, Jie Tang, and Wojciech Zaremba. Openai gym. arXiv preprint arXiv:1606.01540, 2016.  
+Sébastien Bubeck and Nicolò Cesa-Bianchi. Regret analysis of stochastic and nonstochastic multiarmed bandit problems. Foundations and Trends® in Machine Learning, 5(1):1-112, 2012.  
+Sayak Ray Chowdhury and Aditya Gopalan. On kernelized multi-armed bandits. In International Conference on Machine Learning (ICML), 2017.  
+Emile Contal, Vianney Perchet, and Nicolas Vayatis. Gaussian process optimization with mutual information. In International Conference on International Conference on Machine Learning (ICML), 2014.  
+Rikky R.P.R. Duivenvoorden, Felix Berkenkamp, Nicolas Carion, Andreas Krause, and Angela P. Schoellig. Constrained Bayesian optimization with particle swarms for safe adaptive controller tuning. In Proc. of the IFAC (International Federation of Automatic Control) World Congress, pages 12306-12313, 2017.  
+Gabriel Dulac-Arnold, Daniel J. Mankowitz, and Todd Hester. Challenges of real-world reinforcement learning. In International Conference on International Conference on Machine Learning (ICML), 2019.  
+Christian Fiedler, Carsten W. Scherer, and Sebastian Trimpe. Practical and rigorous uncertainty bounds for gaussian process regression. In AAAI Conference on Artificial Intelligence, 2021.  
+Lukas Fröhlich, Edgar Klenske, Julia Vinogradska, Christian Daniel, and Melanie Zeilinger. Noisy-input entropy search for efficient robust bayesian optimization. In International Conference on Artificial Intelligence and Statistics (AISTATS), 2020.  
+Roman Garnett. Bayesian Optimization. Cambridge University Press, 2022. in preparation.  
+Michael A. Gelbart, Jasper Snoek, and Ryan P. Adams. Bayesian optimization with unknown constraints. In Conference on Uncertainty in Artificial Intelligence (UAI), 2014.  
+Alkis Gotovos, Nathalie Casati, Gregory Hitz, and Andreas Krause. Active learning for level set estimation. In International Joint Conference on Artificial Intelligence (IJCAI), 2013.  
+Alexander Hans, Daniel Schneegass, Anton Schäfer, and Steffen Udluft. Safe exploration for reinforcement learning. In European Symposium on Artificial Neural Networks (ESANN), 2008.  
+Philipp Hennig and Christian J. Schuler. Entropy search for information-efficient global optimization. Journal of Machine Learning Research, 13:1809-1837, 2012.  
+Jose Miguel Henrández-Lobato, Matthew W. Hoffman, and Zoubin Ghahramani. Predictive entropy search for efficient global optimization of black-box functions. In International Conference on Neural Information Processing Systems (NIPS), 2014.  
+Johannes Kirschner, Mojmir Mutny, Nicole Hiller, Rasmus Ischebeck, and Andreas Krause. Adaptive and safe bayesian optimization in high dimensions via one-dimensional subspaces. In International Conference for Machine Learning (ICML), 2019.  
+Teodor Mihai Moldovan and Pieter Abbeel. Safe exploration in markov decision processes. In International Coference on International Conference on Machine Learning (ICML), 2012.
+
+Carl Edward Rasmussen and Christopher K. I. Williams. Gaussian processes for machine learning. MIT Press, Cambridge, MA, USA, 2006.  
+Mark Schillinger, Benjamin Hartmann, Patric Skalecki, Mona Meister, Duy Nguyen-Tuong, and Oliver Nelles. Safe active learning and safe bayesian optimization for tuning a pi-controller. IFAC-PapersOnLine, 50(1):5967-5972, 2017.  
+Bernhard Schölkopf and Alexander J. Smola. Learning with kernels: support vector machines, regularization, optimization, and beyond. MIT Press, Cambridge, MA, USA, 2002.  
+Jens Schreiter, Duy Nguyen-Tuong, Mona Eberts, Bastian Bischoff, Heiner Markert, and Marc Toussaint. Safe exploration for active learning with gaussian processes. In European Conference on Machine Learning and Knowledge Discovery in Databases (ECMLPKDD), 2015.  
+Bobak Shahriari, Kevin Swersky, Ziyu Wang, Ryan P. Adams, and Nando de Freitas. Taking the Human Out of the Loop: A Review of Bayesian Optimization. Proceedings of the IEEE, 104: 148-175, 2016.  
+Niranjan Srinivas, Andreas Krause, Sham Kakade, and Matthias Seeger. Gaussian process optimization in the bandit setting: No regret and experimental design. In International Conference on Machine Learning (ICML), 2010.  
+Yanan Sui, Alkis Gotovos, Joel W. Burdick, and Andreas Krause. Safe exploration for optimization with Gaussian processes. In International Conference on Machine Learning (ICML), 2015.  
+Yanan Sui, Vincent Zhuang, Joel Burdick, and Yisong Yue. Stagewise safe Bayesian optimization with Gaussian processes. In International Conference on Machine Learning (ICML), 2018.  
+Matteo Turchetta, Felix Berkenkamp, and Andreas Krause. Safe exploration in finite markov decision processes with gaussian processes. In Advances in Neural Information Processing Systems (NeurIPS), 2016.  
+Matteo Turchetta, Felix Berkenkamp, and Andreas Krause. Safe exploration for interactive machine learning. In Advances in Neural Information Processing Systems (NeurIPS), 2019.  
+Zi Wang and Stefanie Jegelka. Max-value entropy search for efficient bayesian optimization. In International Conference on Machine Learning (ICML), 2017.
+
+# Checklist
+
+1. For all authors...
+
+(a) Do the main claims made in the abstract and introduction accurately reflect the paper's contributions and scope? [Yes]  
+(b) Did you describe the limitations of your work? [Yes] Limitations are discussed in Section 5  
+(c) Did you discuss any potential negative societal impacts of your work? [Yes] See Section 7  
+(d) Have you read the ethics review guidelines and ensured that your paper conforms to them? [Yes]
+
+2. If you are including theoretical results...
+
+(a) Did you state the full set of assumptions of all theoretical results? [Yes] See Section 4.  
+(b) Did you include complete proofs of all theoretical results? [Yes] See Appendix A.
+
+3. If you ran experiments...
+
+(a) Did you include the code, data, and instructions needed to reproduce the main experimental results (either in the supplemental material or as a URL)? [Yes] The code is available at https://github.com/boschresearch/information-theoretic-safe-exploration. Experiment details to reproduce our results can be found in Appendix C.  
+(b) Did you specify all the training details (e.g., data splits, hyperparameters, how they were chosen)? [Yes] See Appendix C.  
+(c) Did you report error bars (e.g., with respect to the random seed after running experiments multiple times)? [Yes] They are in the respective plots.  
+(d) Did you include the total amount of compute and the type of resources used (e.g., type of GPUs, internal cluster, or cloud provider)? [Yes] See Appendix C.
+
+4. If you are using existing assets (e.g., code, data, models) or curating/releasing new assets...
+
+(a) If your work uses existing assets, did you cite the creators? [Yes] See Section 6 and Appendix C  
+(b) Did you mention the license of the assets? [Yes] See Appendix C  
+(c) Did you include any new assets either in the supplemental material or as a URL? [No] No new assets have been used nor introduced.  
+(d) Did you discuss whether and how consent was obtained from people whose data you're using/curating? [N/A]  
+(e) Did you discuss whether the data you are using/curating contains personally identifiable information or offensive content? [N/A]
+
+5. If you used crowdsourcing or conducted research with human subjects...
+
+(a) Did you include the full text of instructions given to participants and screenshots, if applicable? [N/A]  
+(b) Did you describe any potential participant risks, with links to Institutional Review Board (IRB) approvals, if applicable? [N/A]  
+(c) Did you include the estimated hourly wage paid to participants and the total amount spent on participant compensation? [N/A]

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+# Inherently Explainable Reinforcement Learning in Natural Language
+
+Xiangyu Peng†
+
+Chen Xing†
+
+Prafulla Kumar Choubey† Chien-Sheng Wu, Caiming Xiong
+
+$^{\dagger}$ Georgia Institute of Technology
+
+$\ddagger$ Allen Institute for AI
+
+{xpeng62,riedl}  $@$  gatech.edu，raja@allenai.org
+
+# Abstract
+
+We focus on the task of creating a reinforcement learning agent that is inherently explainable—with the ability to produce immediate local explanations by thinking out loud while performing a task and analyzing entire trajectories post-hoc to produce temporally extended explanations. This Hierarchically Explainable Reinforcement Learning agent (HEX-RL), operates in Interactive Fictions, text-based game environments in which an agent perceives and acts upon the world using textual natural language. These games are usually structured as puzzles or quests with long-term dependencies in which an agent must complete a sequence of actions to succeed—providing ideal environments in which to test an agent's ability to explain its actions. Our agent is designed to treat explainability as a first-class citizen, using an extracted symbolic knowledge graph-based (KG) state representation coupled with a Hierarchical Graph Attention mechanism that points to the facts in the internal graph representation that most influenced the choice of actions. Experiments show that this agent provides significantly improved explanations over strong baselines, as rated by human participants generally unfamiliar with the environment, while also matching state-of-the-art task performance.
+
+# 1 Introduction
+
+Explainable AI refers to artificial intelligence methods and techniques that provide human-understandable insights into how and why an AI system chooses actions or makes predictions. Such explanations are critical for ensuring reliability and improving trustworthiness by increasing user understanding of the underlying model. In this work we specifically focus on creating deep reinforcement learning (RL) agents that can explain their actions in sequential decision making environments through natural language.
+
+In contrast to the majority of contemporary work in the area which focuses on supervised machine learning problems which require singular instance level local explanations (You et al., 2016; Xu et al., 2015; Wang et al., 2017; Wegreffe and Marasovic, 2021), such environments—in which agents need to reason causally about actions over a long series
+
+Observation: Forest Path
+
+The path heads northsouth here.
+
+One particularly large tree with some low branches stands at the edge of the path...
+
+Action: go up
+
+Game Score: 0
+
+![](images/8e7681a63d848a057481286b433fab4410e6bace21ebc147394c6a615497d361.jpg)
+
+\* Explanation:
+
+I am in the Forest Path now. trees are interreactable
+
+Agent needs this action to: take egg at Up a tree
+
+Observation: Up a tree
+
+Beside you on the branch is a small birds nest In the birds nest is a large egg encrusted with precious jewels, scavenged by a childless son
+
+Action: take egg
+
+Game Score: 5
+
+![](images/643b8fcfe8e2d7336b39f4f217a95fcdaa4a034ef3a3dd37aaf93fd501390218.jpg)
+
+\* Explanation: egg is interactable
+
+Observation: Up a tree Taken.
+
+Figure 1: Excerpt from zork1 with immediate step-by-step explanations constructed from the KG represented by  $\hat{x}$  and temporally extended explanations by  $\hat{z}$ . Colors represent different categories of KG facts seen in Fig.2
+
+of steps—require an agent to take into account both environmentally grounded context as well as goals when producing explanations. Agents implicitly contain beliefs regarding the downstream effects—the changes to the world—that actions taken at the current timestep will have. This requires explanations in these environments to contain an additional temporally extended component taking the full trajectory's context into account—complementary to the immediate step-by-step explanations.
+
+Interactive Fiction (IF) games (Fig. 1) are partially observable environments where an agent perceives and acts upon a world using potentially incomplete textual natural language descriptions. They are structured as long puzzles and quests that require agents to reason about thousands of locations, characters, and objects over hundreds of steps, creating chains of dependencies that an agent must fulfill to complete the overall task. They provide ideal experimental test-beds for creating agents that can both reason in text and explain it.
+
+We introduce an approach to game playing agents—Hierarchically Explainable Reinforcement Learning (HEX-RL)—that is designed to be inherently explainable, in the sense that its internal state representation—i.e. belief state about the world—takes the form of a symbolic, human-interpretable knowledge graph (KG) that is built as the agent explores the world. The graph is encoded by a Graph Attention network (GAT) (Veličković et al., 2017) extended to contain a hierarchical graph attention mechanism that focuses on different sub-graphs in the overall KG representation. Each of these sub-graphs contains different information such as attributes of objects, objects the player has, objects in the room, current location, etc. Using these encoding networks in conjunction with the underlying world KG, the agent is able to create immediate explanations akin to a running commentary that points to the facts within this knowledge graph that most influence its current choice of actions when attempting to achieve the tasks in the game on a step-by-step basis.
+
+While graph attention can tell us which elements in the KG are attended to when maximizing expected reward from the current state, it cannot explain the intermediate, unrewarded dependencies that need to be satisfied to meet the long term task goals. For example, in the game zork1, the agent needs to pick up a lamp early on in the game—an unrewarded action—but the lamp is only used much later on to progress through a location without light. Thus, our agent additionally analyzes an overall episode trajectory—a sequence of knowledge graph states and actions from when the agent first starts in a world to either task completion or agent death—to find the intermediate set of states that are most important for completing the overall task. This information is used to generate a temporally extended explanation that condenses the immediate step-by-step explanations to only the most important steps required to fulfill dependencies for the task.
+
+Our contributions are as twofold: (1) we create an inherently explainable agent that uses an ever-updating knowledge-graph based state representation to generate step-by-step immediate explanations for executed actions as well as performing a post-hoc analysis to create temporal explanations; and (2) a thorough experimental study against strong baselines that shows that our agent generates significantly improved explanations for its actions when rated by human participants unfamiliar with the domain while not losing any task performance compared to the current state-of-the-art knowledge graph-based agents.
+
+# 2 Background and Related Work
+
+Interactive Fiction (IF) games are simulations featuring language-based state and action spaces. It provides a platform for exploring lifelong open-domain dialogue learning (Shuster et al., 2020) and action elimination with deep reinforcement learning (Zahavy et al., 2018). In this paper, we use IF games as our test-bed because they provide an ideal platform for collecting data, linking game states and actions to the corresponding natural language explanations. We use the definition of text-adventure games as seen in Côté et al. (2018) and Hausknecht et al. (2020). We take Jericho (Hausknecht et al., 2020), a framework for interacting with text games, as the interface connecting learning agents with interactive fiction games. A text game can be defined as a partially-observable Markov Decision Process:  $G = \langle S, P, A, O, \Omega, R, \gamma \rangle$ , representing the set of environment states, conditional transition probabilities between states, the vocabulary or words used to compose text
+
+![](images/a79987f02327c2ebc40f02461326689c316b2438470e6590f1c84e3b0b3acdf8.jpg)  
+Figure 2: Knowledge graph extraction and the (HEX-RL) agent's architecture at step  $t$ .
+
+commands, observations, observation conditional probabilities, reward function, and discount factor, respectively. The reinforcement learning agent is trained to learned a policy  $\pi_G(o) \to a$ .
+
+Knowledge Graphs for Text Games. Ammanabrolu et al. (2020) proposed Q*BERT, a reinforcement learning agent that learns a KG of the world by answering questions. Xu et al. (2020) introduce the SHA-KG, a stacked Hierarchical Graph Attention mechanism to construct an explicit representation of the reasoning process by exploiting the structure of the KG. Adhikari et al. (2020) present the Graph-Aided Transformer Agent (GATA) which learns to construct a KG during game play and improves zero-shot generalization on procedurally generated TextWorld games. Other works such as Murugesan et al. (2020) explore how to use KGs to endow agents with commonsense. While these works showcase the effectiveness of KGs on task performance and do not focus on how explainable their architectures are. We further note that these architectures do now allow for as fine-grained attention-based attribution as HEX-RL's architecture does—e.g. Q*BERT does not use relationship information in their policy and SHA-KG averages attention across large portions of the graph and is unable to point to specific triples in its KG representation to explain an action.
+
+Explainable Deep RL. Contemporary work on explaining deep reinforcement learning policies can be broadly categorized based on: (1) how the information is extracted, either via intrinsic motivation during training (Shu et al., 2017; Hein et al., 2017; Verma et al., 2018) or through post-hoc analysis (Rusu et al., 2015; Hayes and Shah, 2017; Juozapaitis et al., 2019; Madumal et al., 2020); and (2) the scope—either global (Zahavy et al., 2016; Hein et al., 2017; Verma et al., 2018; Liu et al., 2018) or local (Shu et al., 2017; Liu et al., 2018; Madumal et al., 2020; Guo et al., 2021). In our work, we create an agent that spans more than one of these categories providing immediately local explanations through extracted knowledge graph representations and post-hoc temporal explanations. Inspired by Madumal et al. (2020), we learn a graphical causal model which focuses on using relations between steps in a puzzle to generate temporal explanations instead of generating counterfactuals.
+
+# 3 Hierarchically Explainable RL
+
+Our work aims to generate (1) immediate step-by-step explanations of an agent's policy by capturing the importance of the current game state observation and (2) temporally extended explanations that take into context an entire trajectory via a post-hoc analysis. Formally, let  $\mathbf{X} = \{\mathbf{s}_t, \mathbf{a}_t\}_{t=1:T}$  be the set of game steps that compose a trajectory. Each game state  $s_t$  consists of a knowledge graph  $G_t$  representing all the information learned since the start of the game. This graph is further split into four sub-knowledge graphs each containing different, semantically related relationship types. This section first describes a graph attention based architecture that uses these sub-graphs to produce immediate explanations. We then describe how to filter the game states in a trajectory into a condensed set of the most important ones  $\mathbf{X}' \subset \mathbf{X}$  that best capture the underlying dependencies that need to be fulfilled to complete the task—enabling us to produce temporal explanations.
+
+Knowledge Graph State Representation. Building on Ammanabrolu et al. (2020), constructing the knowledge graph is treated as a question-answering task. KGs in these games take the form of RDF triples (Klyne 2004) of <subject, relation, object>—extracted from text observations and update as the agent explores the world. The agent answers questions about the environment such as, "What am I carrying?" or "What objects are around me?" A specially constructed dataset for question answering in text games—JerichoQA—is used to fine-tune ALBERT (Lan et al., 2019) to answer these questions (See Appendix A.3). The answers form a set of candidate graph vertices  $V_{t}$  for the current step and questions form the set of relations  $R_{t}$ . Both  $V_{t}$  and  $R_{t}$  are then combined with the graph at the previous step  $G_{t - 1}$  to update the agent's belief about the world state into  $G_{t}$ . The left side of Figure 2 showcases this.
+
+In an attempt to enable more fine grained explanation generation and inspired by Xu et al. (2020), we divide the knowledge graph  $G$  into multiple sub-graphs  $G^{atr}$ ,  $G^{inv}$ ,  $G^{obj}$ ,  $G^{loc}$ , each representing (1) attributes of objects, (2) objects the player has, (3) objects in the room, and (4) other information such as location (right side of Fig. 2) based on the corresponding relationship types extracted by the ALBERT-QA module. The union of all sub-graphs is equivalent of  $V_{t}$  and  $R_{t}$  extracted from the current game state. The full knowledge graph  $G_{t}$  captures the overall game state since the start of the game. The sub-graphs easily reflect different relationships of the current game state.
+
+Template Action Space. Agents output a language string into the game to describe the actions that they want to perform. To ensure tractability, this action space can be simplified down into templates. Templates consist of interchangeable verbs phrases (VP), optionally followed by prepositional phrases (VP PP), e.g. ([carry/take]_) and ([throw/discard/put] _ [against/on/down ] _), where the verbs and prepositions within [. are aliases. Actions are constructed from templates by filling in the template's blanks using words in the game's vocabulary. Size of action space is shown in Appendix A.1.
+
+# 3.1 Immediate Explanations
+
+Our immediate explanations consist of finding the subset of triplets in sub-graphs  $G^{atr}, G^{inv}, G^{obj}, G^{loc}$  c the action decision made at the current step—is capable of explaining the action. We introduce a deep RL architecture capable of this.
+
+Hierarchical Knowledge Graph Attention Architecture. At each step, a total score  $R_{t}$  and an observation  $o_{t}$  is received—consisting of  $\left(o_{t_{\mathrm{desc}}}, o_{t_{\mathrm{game}}}, o_{t_{\mathrm{inv}}}, a_{t-1}\right)$  corresponding to the room description, game feedback, inventory, and previous action and are processed using a GRU based encoder using the hidden state from the previous step, combining them into a single observation embedding  $\mathbf{o}_{t} \in \mathbb{R}^{d_{text} \times c}$  (bottom of Fig. 2).
+
+The full knowledge graph  $G_{t}$  is processed via Graph Attention Networks (GATs) (Veličković et al. 2017) followed by a linear layer to get the graph representation  $\mathbf{g}_{\mathbf{t}} \in \mathbb{R}^{d_{text}}$  (middle of Fig. 2). We compute LSTM attention between  $\mathbf{o}_{t}$  and  $\mathbf{g}_{\mathbf{t}}$  as:
+
+$$
+\boldsymbol {\alpha} _ {\mathrm {L S T M}} = \operatorname {s o f t m a x} \left(\boldsymbol {W} _ {1} \boldsymbol {h} _ {\mathrm {L S T M}} + \boldsymbol {b} _ {1}\right) \tag {1}
+$$
+
+$$
+\boldsymbol {h} _ {\mathrm {L S T M}} = \tanh  \left(\boldsymbol {W} _ {\mathrm {o}} \boldsymbol {o} _ {\mathrm {t}} \oplus \left(\boldsymbol {W} _ {\mathrm {g}} \mathbf {g} _ {\mathrm {t}} + \boldsymbol {b} _ {\mathrm {g}}\right)\right) \tag {2}
+$$
+
+where  $\oplus$  denotes the addition of a matrix and a vector.  $W_{1}\in \mathbb{R}^{d_{text}\times d_{text}}$ ,  $W_{\mathrm{g}}\in \mathbb{R}^{d_{text}\times d_{text}}$ ,  $W_{0}\in \mathbb{R}^{d_{text}\times d_{text}}$  are weights and  $b_{1}\in \mathbb{R}^{d_{text}}$ ,  $b_{0}\in \mathbb{R}^{d_{text}}$  are biases. The overall representation vector is updated as:
+
+$$
+\mathbf {q} _ {\mathrm {t}} = \mathbf {g} _ {\mathrm {t}} + \sum_ {i} ^ {c} \boldsymbol {\alpha} _ {\text {L S T M}, i} \odot \boldsymbol {o} _ {\mathrm {t}, i} \tag {3}
+$$
+
+where  $\odot$  denotes dot-product and  $c$  is the number of  $\pmb{o}_{\mathrm{t}}$  's components.
+
+Sub-graphs are also encoded by GATs to get the graph representation  $\mathbf{g}_{\mathbf{t}}^{\prime} \in \mathbb{R}^{d_{G_t,sub} \times m}$  (no. of subgraphs). The Hierarchical Graph Attention between  $\mathbf{q}_t \in \mathbb{R}^{d_{G_t,sub}2}$  and  $\mathbf{g}_{\mathbf{t}}^{\prime}$  is calculated by:
+
+$$
+\boldsymbol {\alpha} _ {\text {H i e r a r c h i c a l}} = \operatorname {s o f t m a x} \left(\boldsymbol {W} _ {\mathrm {H}} \boldsymbol {h} _ {\mathrm {H}} + \boldsymbol {b} _ {\mathrm {H}}\right) \tag {4}
+$$
+
+$$
+\boldsymbol {h} _ {\mathrm {H}} = \tanh  \left(\boldsymbol {W} _ {\mathrm {g} ^ {\prime}} \boldsymbol {g} _ {\mathrm {t}} ^ {\prime} \oplus \left(\boldsymbol {W} _ {\mathrm {q}} \boldsymbol {q} _ {\mathrm {t}} + \boldsymbol {b} _ {\mathrm {q}}\right)\right) \tag {5}
+$$
+
+where  $W_{\mathrm{H}} \in \mathbb{R}^{d_{G_t,sub} \times d_{G_t,sub}}$ ,  $W_{\mathrm{g'}} \in \mathbb{R}^{d_{G_t,sub} \times d_{G_t,sub}}$ ,  $W_{\mathrm{q}} \in \mathbb{R}^{d_{G_t,sub} \times d_{G_t,sub}}$  are weights and  $b_{\mathrm{H}} \in \mathbb{R}^{d_{G_t,sub}}$ ,  $b_{\mathrm{q}} \in \mathbb{R}^{d_{G_t,sub}}$  are biases. Then we get state representation, consisting of the textual observations full knowledge graph and sub-knowledge graph.
+
+$$
+\mathbf {v} _ {\mathbf {t}} = \mathbf {q} _ {\mathbf {t}} + \sum_ {i} ^ {s} \boldsymbol {\alpha} _ {\text {H i e r a c h i c a l}, i} \odot \mathbf {g} _ {\mathbf {t}, i} ^ {\prime} \tag {6}
+$$
+
+where  $s$  is the number of sub-graphs (4 in our paper). The full architecture can be found in Figure 2.
+
+The agent is trained via the Advantage Actor Critic (A2C) (Mnih et al., 2016) method to maximize long term expected reward in the game in a manner otherwise unchanged from Ammanabrolu et al. (2020) (See Appendix A.2). These attention values thus reflect the portions of the knowledge graphs that the agent must focus on to best achieve this goal of maximizing reward.
+
+Hierarchical Graph Attention Explanation. The graph attention  $\alpha_{\text{Hierarchical}}$  is used to capture the relative importance of game state observations and KG entities in influencing action choice. For each sub-graph, the graph attention,  $\alpha_{\text{Hierarchical},i} \in \mathbb{R}^{n_{\text{nodes}} \times m}$  is summed over all the channels  $m$  to obtain  $\alpha_{\text{Hierarchical},i}' \in \mathbb{R}^{n_{\text{nodes}} \times 1}$ , showing the importance of the KG nodes in the  $i$ th sub-graph. The top- $k$  valid entities (and corresponding edges) with highest absolute value of its attention form the set of knowledge graph triplets that best locally explain the action  $a_t$ .
+
+In order to make the explanation more readable for a human reader, we further transform knowledge graph triplets to natural language by template filling.
+
+We create templates for each type of sub-graphs  $G^{atr}$ ,  $G^{inv}$ ,  $G^{obj}$ ,  $G^{loc}$ .
+
+- <object, is, attribute>  $\rightarrow$  "Object is attribute"  
+- <player, has, object>  $\rightarrow$  "I have object"  
+- <object,in,location>→“Object is in location”  
+-  $\langle location1, direction, location2 \rangle \rightarrow$  "location 1 is in the direction of location 2", e.g.  $\langle forest, north, house \rangle$  is converted to "Forest is in the north of house"
+
+More examples can be found in Appendix A.4.
+
+# 3.2 Temporally Extended Explanations
+
+Graph attention tells us which entities in the KG are attended to when making a decision, but is not enough alone for explaining "why" actions are the right ones in the context of fulfilling dependencies that may potentially be unrewarded by the game—especially given the fact that there are potentially multiple ways of achieving the overall task. HEX-RL thus saves trajectories for hundreds of test time rollouts of the games, performed once a policy has been trained (Table1 and Appendix A.5). The game trajectories consist of all the game states, actions taken, predicted critic values, game scores, the knowledge graphs, and the immediate step level explanations generated as previously described. HEX-RL produces a temporal explanation by performing a post-hoc analysis on these game trajectories. The agent then analyzes and filters these trajectories in an attempt to find the subset of states that are most crucial to achieving the task as summarized in Figure3—then using that subset of states to generate temporal trajectory level explanations.
+
+Bayesian State Filter. We first train a Bayesian model to predict the conditional probability  $\mathbb{P}(A\mid B_i)$  of a game step  $(A)$  given any other possible game step  $(B_{i})$  in the game trajectories. More specifically, each game step is composed of 3 elements, game state  $o_t$ , action  $a_{t}$  and the current knowledge graph  $G_{t}$ . The key intuition here being that state, action pairs that appear in a certain ordering in multiple trajectories are more likely to dependant on each other.
+
+The set of game steps with the highest  $\mathbb{P}(A\mid B_i)$  is used to explain taking the action associated with game state  $A$ . For example, "take egg" ( $A$ ) is required to "open egg" ( $B$ ), and  $\mathbb{P}(A\mid B) = 1$ , hence "open egg" is used as a reason why action "take egg" must be taken first. The initial set of game states  $\mathbf{X}$  is filtered into  $\mathbf{X}_1$  by working backwards from the final goal state by finding the set of states that form the most likely chain of causal dependencies that lead to it. Details can be found in Appendix A.5 and A.6.
+
+Language Model Action Filter. Following this, we apply a GPT-2 (Radford et al., 2019) language model trained to generate actions based on transcripts of text games from human play-throughs to further filter out important states—known as the Contextual Action Language Model (CALM) (Yao et al., 2020). As this language model is trained on human transcripts, we hypothesize that it is able to further filter down the set of important states by finding the states that have corresponding actions that a human player would be more likely to perform—thus potentially leading to more natural explanations. CALM takes into observation  $o_{t}$ , action  $a_{t}$  and the following observation  $o_{t+1}$ , and predicts next valid actions  $a_{t+1}$ . In our work, we use CALM as a filter to look for the relations between a game step  $A$  and the explanation candidates  $B_{i} \in \mathbf{X}_{1}$ . We feed CALM with the prompt  $o_{A}, a_{A}, o_{B_{i}}$  to get an action candidate set. When the two game steps  $A$  and  $B_{i}$  are highly correlated, given  $o_{A}, a_{A}$  and  $o_{B_{i}}$ , CALM should successfully predict  $a_{B_{i}}$
+
+with high probability. The game steps  $B_{i}$ , whose associated action  $a_{B_i}$  is in this generated action candidates set, are saved as the next set of filtered important candidate game states  $(\mathbf{X}_2)$ .
+
+![](images/2c8578e3fb09b22fdc022e959aa60daaf3549bc71c27b6b78e0dc456a7124aad.jpg)  
+Figure 3: Temporal explanation pipeline for why the agent chose the action—"go north" at "Behind House".
+
+Semantic State-Action Filter. To better account for the irregularities of the puzzle like environment, we adopt a semantic filter to obtain the final important state set  $\mathbf{X}_3$ . Here, given  $A, B_i \in \mathbf{X}_2$ , states are further filtered on the basis of whether one of these scenarios occurs: (1)  $a_A$  and  $a_{B_i}$  contain the same entities, e.g. "take egg" and "open egg". (2)  $G_A$  and  $G_{B_i}$  share the same entities, e.g. "lamp" occurs in both observations. (3)  $A$  and  $B_i$  occur in the same location, e.g. after taking action  $a_A$ , the player enters "kitchen" and  $B$  occurs in "kitchen". (4) The state has a non-zero reward or a high absolute critic value, indicating that
+
+it is either a state important for achieving the goals of the game or it is a state to be avoided. The final set of important game states  $\mathbf{X}_3$  is used to synthesize post-hoc temporal explanations for why an action was performed in a particular state—as seen in Figure1—taking into account the overall context of the dependencies required to be satisfied and building on the immediate step level explanations for each given state in  $\mathbf{X}_3$ . Ablation studies pin-pointing the relative contributions of the different filters are found in Section4.4. We concluded that all three steps of the filtering process to identify important states are necessary for creating coherent temporal explanations that effectively take into account the context of the agent's goals.
+
+Table 1: Example state saved during game play.  
+
+<table><tr><td>STEP: 16</td></tr><tr><td>Text Observation:</td></tr><tr><td>Up a Tree</td></tr><tr><td>Beside you on the branch is a small birds nest.</td></tr><tr><td>In the birds nest is a large egg encrusted with jewels...</td></tr><tr><td>Knowledge graph:</td></tr><tr><td>-tree, in, forest), &lt;egg, is, interactable&gt;...</td></tr><tr><td>Action: take egg</td></tr><tr><td>Immediate explanation: egg is interactable</td></tr><tr><td>Game Score:5</td></tr><tr><td>Critic Value: 5.7457</td></tr></table>
+
+Goals of the game or it is a state to be avoided. The synthesize post-hoc temporal explanations for why has seen in Figure1—taking into account the overall satisfied and building on the immediate step level on studies pin-pointing the relative contributions to  $\mathbf{S}$ . We concluded that all three steps of the filtering strategy for creating coherent temporal explanations of the agent's goals.
+
+Table 2: Asymptotic scores on games by different methods across 5 independent runs. Eps. indicates normalized scores averaged across 100 episodes of testing which occurs at the end of the training period and Max indicates the maximum score seen by the agent over the same period. We present results on two training rewards for HEX-RL, game only and game and IM.  
+
+<table><tr><td rowspan="2">Experiment</td><td rowspan="2" colspan="2">LSTM-A2C</td><td rowspan="2" colspan="2">KG-A2C</td><td rowspan="2" colspan="2">SHA-KG</td><td rowspan="2" colspan="2">Q*BERT</td><td colspan="2">HEX-RL</td><td colspan="2">HEX-RL</td><td rowspan="2">Max</td></tr><tr><td colspan="2">Game Only</td><td colspan="2">Game and IM</td></tr><tr><td>Metric</td><td>Eps.</td><td>Max</td><td>Eps.</td><td>Max</td><td>Eps.</td><td>Max</td><td>Eps.</td><td>Max</td><td>Eps.</td><td>Max</td><td>Eps.</td><td>Max</td><td>-</td></tr><tr><td>zork1</td><td>27</td><td>31.2</td><td>34</td><td>35</td><td>33.6</td><td>34.5</td><td>35</td><td>35</td><td>29.8</td><td>40</td><td>30.2</td><td>40</td><td>350</td></tr><tr><td>library</td><td>8.2</td><td>10</td><td>14.3</td><td>19</td><td>10.0</td><td>15.8</td><td>18</td><td>18</td><td>16.0</td><td>19</td><td>13.8</td><td>21</td><td>30</td></tr><tr><td>detective</td><td>141</td><td>188</td><td>207.9</td><td>214</td><td>246.1</td><td>308</td><td>274</td><td>310</td><td>276.7</td><td>330</td><td>276.9</td><td>330</td><td>360</td></tr><tr><td>balances</td><td>10</td><td>10</td><td>10</td><td>10</td><td>9.8</td><td>10</td><td>10</td><td>10</td><td>10.0</td><td>10</td><td>10.0</td><td>10</td><td>51</td></tr><tr><td>pentari</td><td>50.4</td><td>55</td><td>50.7</td><td>56</td><td>48.2</td><td>51.3</td><td>50</td><td>56</td><td>34.6</td><td>55</td><td>44.7</td><td>60</td><td>70</td></tr><tr><td>ztuu</td><td>5</td><td>5</td><td>5</td><td>5</td><td>5</td><td>25</td><td>5</td><td>5</td><td>5.0</td><td>5</td><td>5.1</td><td>9</td><td>100</td></tr><tr><td>ludicorp</td><td>14.4</td><td>18</td><td>17.8</td><td>19</td><td>17.6</td><td>17.8</td><td>18</td><td>19</td><td>14.0</td><td>18</td><td>17.6</td><td>18</td><td>150</td></tr><tr><td>deephome</td><td>1</td><td>1</td><td>1</td><td>1</td><td>1</td><td>1</td><td>1</td><td>1</td><td>1.0</td><td>1</td><td>1.0</td><td>1</td><td>300</td></tr><tr><td>temple</td><td>8</td><td>8</td><td>7.6</td><td>8</td><td>7.9</td><td>6.9</td><td>8</td><td>8</td><td>8.0</td><td>8</td><td>7.6</td><td>8</td><td>35</td></tr><tr><td>% compl.</td><td>22.6</td><td>25.9</td><td>27.3</td><td>30.8</td><td>27.2</td><td>33.1</td><td>30.8</td><td>34.9</td><td>27.2</td><td>33.9</td><td>28.2</td><td>35.8</td><td>100</td></tr><tr><td>std dev</td><td>0.02</td><td>0.01</td><td>0.06</td><td>0.01</td><td>-</td><td>-</td><td>0.03</td><td>0.00</td><td>0.03</td><td>0.01</td><td>0.03</td><td>0.02</td><td>-</td></tr></table>
+
+# 4 Evaluation
+
+Our evaluation consists of three phases: (1) We show that HEX-RL has the comparable performance to state-of-art reinforcement learning agents on text games in Section 4.1. (2) Then in Section 4.2 we evaluate our immediate attention explanation model by comparing the explanations generated by HEX-RL and agents that do not use knowledge graphs (See Fig. 2 and Section 3.1). (3) In Section 4.3 we compare immediate to temporal explanations, focusing on the effects that including trajectory level context when evaluating explanations in the context of agent goals. (4) In Section 4.4 we conduct human participant ablation study evaluating the individual contributions of the filtration pipeline for generating temporal explanations seen in Figure 3.
+
+# 4.1 Task Performance Evaluation
+
+We compare HEX-RL with four strong state-of-art reinforcement learning agents—focusing on contemporary agents that use knowledge graphs—on an established test set of 9 games from the Jericho benchmark (Hausknecht et al. 2020).
+
+- LSTM-A2C is a baseline that only uses natural language observations as state representation that is encoded with an LSTM-based policy network.  
+- KG-A2C. Instead of training a question-answering system like Q*BERT to build knowledge graph state representation, KG-A2C (Ammanabrolu and Hausknecht, 2020) extracts knowledge graph triplets from the text observations using a rules based approach built on OpenIE (Angeli et al., 2015).  
+- SHA-KG is adapted from Xu et al. (2020) and uses a rules-based approach to construct a knowledge graph for the agent which is then fed into a Hierarchical Graph Attention network as in HEX-RL. This agent separates the sub-graphs out using a rules-based approach and makes no use of any graph edge relationship information.  
+- Q*BERT. [Ammanabrolu et al. (2020)] uses a similar method of creating the knowledge graph through question answering but does not use the hierarchical graph attention architecture combined with the sub-graphs.
+
+These baselines are all trained via the Advantage Actor Critic (A2C) (Mnih et al., 2016) method—further comparisons to other contemporary agents can be found in Appendix A.7. It is also worth noting that most contemporary state of the art deep RL agents for text games use recurrent neural policy networks as opposed to transformer networks due to their improved performance in this domain.
+
+HEX-RL Training. We trained HEX-RL on two reward types: (a) game only, which indicates that we only use score obtained from the game as reward. (2) game with intrinsic motivation (game
+
+and  $IM$ ), which contains an additional intrinsic motivation reward based on knowledge graph expansion as seen in Ammanabrolu et al. (2020)—where the agent is additionally rewarded for learning more about the world by finding new facts for knowledge graph (see Appendix A.8, A.9 and A.10).
+
+Table2 shows the performance of HEX-RL and the other four baselines. We can see that designing the HEX-RL agent to be inherently explainable through the use of Hierarchical Graph Attention and the sub-graphs improves the overall maximum score seen during training when compared to any of the other agents. In terms of the average score seen during the final 100 episodes, HEX-RL with intrinsic motivation outperforms all baselines with the exception of Q*BERT—there HEX-RL significantly outperforms Q*BERT on one game, is outperformed on two games, and comparable on the remaining six games. HEX-RL thus performs comparably to other state-of-the-art baselines in terms of overall task performance while also boasting the additional ability to explain its actions.
+
+# 4.2 Immediate Explanation Evaluation
+
+Having established that HEX-RL's performance while playing text games is comparable to other state-of-the-art agents, we attempt to answer the question of exactly how useful the knowledge graph based architecture is when generating immediate step-by-step explanations by comparing HEX-RL to a baseline that doesn't use knowledge graphs in a human participant study. Two models for step-by-step explanations are compared:
+
+- LSTM Attention explanations. Extracts the most important substring in the observations through LSTM attention  $\alpha_{LSTM}$  and then uses those words to create an explanation.  
+- Hierarchical Graph Attention explanations. Extracts KG triplets most influenced the choice of actions by Hierarchical Attention  $\alpha_{\text{Hierarchical}}$  and then transforming them into readable language explanations through templates.
+
+We recruited 40 participants—generally unfamiliar with the environment at hand—on a crowd sourcing platform. Each participant reads a randomly selected subset of 10 explanation pairs (drawn randomly from a pool totaling 60 explanation pairs), generated by Hierarchical Graph Attention and LSTM attention explanation on three games in the Jericho benchmark: zorkl, library, and balances. We choose three games with very different structures and genres as defined in Hausknecht et al. (2020). They each require a diverse set of action types and solutions to complete and thus provide a wide area of coverage when used as test beds for human evaluation of explanations. Then they are given the following metrics and asked to choose which explanation they prefer for that metric:
+
+- Confidence: This explanation makes you more confident that the agent made the right choice.  
+- Human-likeness: This explanation expresses more human-like thinking on the action choice.  
+- Understandability: This explanation makes you understand why the agent made the choice.
+
+Variations of these questions have been used to evaluate other explainable AI systems (eg. Ehsan et al. (2019)). At least 5 participants give their preference for each explanation pair. We take the majority preference from humans participants as the result. More details are shown in Appendix B.1
+
+Figure 4 shows the result of the human evaluation of attention explanations. Hierarchical graph attention explanation is preferred over LSTM attention explanation in all three dimensions. These results are statistically significant  $(p < 0.05)$  with fair inter-rater reliabilities. We also observe that these three dimensions are highly, positively correlated using Spearman's Rank Order Correlation.
+
+A slightly higher proportion of participants preferred the LSTM Attention explanations in the human-likeness dimension compared to the other two. The participants preferring LSTM Attention explanation stated that they found it intuitive but often incoherent and found the Hierarchical Graph Attention explanations to be more robotic. LSTM attention explanations are substrings of the human-written observation and thus have the potential to be more natural sounding than
+
+the templated Hierarchical Graph Attention explanations when they are coherent enough to be understood. The KG sacrifices a small amount of human-likeness in return for much greater overall coherence and accuracy. KGs with Hierarchical Graph Attention give us explanations that are more easily understood and inspire greater confidence in the agent's decisions.
+
+Example qualitative LSTM Attention and Hierarchical Graph Attention explanations can be found in Appendix C.2. As our system relies on graph hierarchical graph attention to generate immediate explanations, a well-trained knowledge graph representation module of the world knowledge is required. Most cases where the agent fails to provide satisfactory immediate explanations are either when: (1) the explanation is not directly linked to the one of the facts we choose to extract from the knowledge graph, such as object/location information; and (2) due to the error of knowledge graph extraction models themselves.
+
+# 4.3 Immediate vs. Temporal Explanations
+
+Having proved the effectiveness of the knowledge graph at the immediate step-by-step explanation level, we evaluate our method of producing temporal explanations and how they
+
+compare to the immediate explanations along two dimensions: (1) coherence; and (2) explanation accuracy when taken in the context of the agent's goals.
+
+Participants first read a trajectory of the game combined with step-by-step immediate explanations and the game goal, then indicate how much they agree with the statements on a Likert scale of 1 (strong disagree) to 5 (strong agree). Here, we add two metrics from the previous study:
+
+- Goal context: You are able to understand why the agent takes this particular sequence of actions given what you know about the goal.  
+- Readability: This explanation is easy to read.
+
+Figure 5 shows the average scores for each question for the immediate and temporal explanations. The temporal explanations achieve comparable performance to the immediate explanations on all metrics except for the the metric relating to goal context where they significantly out-perform the immediate explanations. A majority of participants stated that a condensed trajectory level explanation made the goals of the agent easier to understand than reading through each step level explanation. These results indicate that HEX-RL can generally successfully identify the most important states in a trajectory and use them to create temporal explanations that are on par with immediate explanations in terms of coherence but provide significantly more context
+
+in terms of explaining an agent's actions with respect to its task-based goals.
+
+![](images/548ac0053db855d234522bfda15210b80c00cabc8559697d94afddde45663ab1.jpg)  
+Figure 4: Human evaluation results showing the proportion of participants that prefer Hierarchical Graph Attention vs. LSTM Attention explanations,  $\ast \ast$  indicates  $p <   0.01$  ,  $\dagger$  indicates  $\kappa >0.2$  or fair agreement.  $\ddagger$  indicates  $\kappa >0.4$  or moderate agreement. Evaluation results on each game are shown in AppendixA.11
+
+![](images/5a51dee19003e9ed145b1909635188a507a7f8ea8c80f1e202c6580f7d482fe3.jpg)  
+Figure 5: Human judgment results on immediate and temporal explanation, * indicates  $p < 0.05$ . Error bars indicate a confidence level of  $95\%$ .
+
+Cases where the agent does not provide a temporally coherent and goal-driven explanation revolve around failures—particularly by the Bayesian State Filter—in detecting the most important states in the trajectory. A qualitative analysis (as seen in Appendix C.3) suggests that this occurs in cases where there are a large number of branching paths that lead to the same end state. Thus, the quality of the generated temporal explanations appears to be inversely proportional to the relative complexity of the game as measured by its branching factor.
+
+# 4.4 Temporal Explanation Ablation Study
+
+Having established the overall effectiveness of the filters in HEX-RL that create the temporal explanations, we perform pair-wise ablation studies to pinpoint the relative contributions of the different filters seen in Fig.3 We first compare explanations generated using a set of important states filtered from the trajectory using the Bayes model compared to Bayes+CALM explanation. This how applying the language model action filter affects the quality of the temporal explanations. As before, we recruited 30 participants on a crowd sourcing platform. Each participant reads a randomly selected subset of explanation pairs, comprised of temporal explanations filtered by Bayes and Bayes+CALM models. Figure 6a shows that after applying the CALM model to filter explanation candidates, generated explanations are significantly preferred on the "Confidence" and "Understandability" dimensions.
+
+Similarly, we then conducted another ablation study to validate the contribution of semantic filter by comparing the Bayes+CALM filtering method to the full HEX-RL using Bayes+CALM+Semantic filters. The experiment setup is the same as the previous ablation study. Figure  $\boxed{6b}$  shows that Bayes+CALM+Semantic performs significantly better than Bayes+CALM on all three dimensions.
+
+We additionally observe that these three metrics are highly, positively correlated using Spearman's Rank Order Correlation in both of these ablation studies. When asked to justify their choices, participants indicated that the full HEX-RL system with Bayes+CALM+Semantic filters provided temporal explanations that they felt was more understandable than alternatives. These results indicate that all three steps of the filtering process to identify important states are necessary for creating coherent temporal explanations that effectively take into account the context of the agent's goals.
+
+![](images/b8eff92813be5efc899820ccf6d7d8488a64439b70ab15ff5f606adb83fa3316.jpg)  
+(a) Bayes vs. Bayes + CALM explanation
+
+![](images/e2ec370abef699e7274678cfdb7ad97d88a89a3ecb70aab981a1a35011e9ab4a.jpg)  
+(b) Bayes + CALM vs. Bayes + CALM + Semantic explanation  
+Figure 6: Human evaluation results on ablation study, * indicates  $p < 0.05$ , † indicates  $\kappa > 0.2$  or fair agreement.
+
+# 5 Conclusions
+
+Explaining deep RL policies for sequential decision making problems in natural language is a sparsely studied problem despite a steadily growing need. An oft given reason for this phenomenon is that deep RL methods perform better without the additional burden of being explainable. In an attempt to encourage work in this area, we create the Hierarchically Explainable Reinforcement Learning (HEX-RL) agent which treats explainability as a first-class citizen in its design by using a readily interpretable knowledge graph state representation coupled with a Hierarchical Graph Attention network. This agent is able to produce step-by-step commentary-like immediate explanations and also a condensed temporal trajectory level explanation via a post-hoc analysis. We show that with careful design, it is possible to create inherently explainable RL agents that do not lose performance when compared to contemporary state-of-the-art agents and simultaneously are able to generate significantly higher quality explanations of actions.
+
+# References
+
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+Prithviraj Ammanabrolu and Matthew Hausknecht. 2020. Graph constrained reinforcement learning for natural language action spaces. In International Conference on Learning Representations.  
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+Marc-Alexandre Côté, Akos Kádár, Xingdi Yuan, Ben Kybartas, Tavian Barnes, Emery Fine, James Moore, Matthew Hausknecht, Layla El Asri, Mahmoud Adada, et al. 2018. Textworld: A learning environment for text-based games. In Workshop on Computer Games, pages 41-75. Springer.  
+Upol Ehsan, Pradyumna Tambwekar, Larry Chan, Brent Harrison, and Mark O Riedl. 2019. Automated rationale generation: a technique for explainable ai and its effects on human perceptions. In Proceedings of the 24th International Conference on Intelligent User Interfaces, pages 263-274.  
+Wenbo Guo, Xian Wu, Usmann Khan, and Xinyu Xing. 2021. Edge: Explaining deep reinforcement learning policies. In Thirty-Fifth Conference on Neural Information Processing Systems.  
+Matthew Hausknecht, Prithviraj Ammanabrolu, Marc-Alexandre Côté, and Xingdi Yuan. 2020. Interactive fiction games: A colossal adventure. In Proceedings of the AAAI Conference on Artificial Intelligence, volume 34, pages 7903-7910.  
+Bradley Hayes and Julie A Shah. 2017. Improving robot controller transparency through autonomous policy explanation. In 2017 12th ACM/IEEE International Conference on Human-Robot Interaction (HRI, pages 303-312. IEEE.  
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+Zoe Juozapaitis, Anurag Koul, Alan Fern, Martin Erwig, and Finale Doshi-Velez. 2019. Explainable reinforcement learning via reward decomposition. In *IJCAI/ECAI Workshop on Explainable Artificial Intelligence*.  
+Graham Klyne. 2004. Resource description framework (pdf): Concepts and abstract syntax. http://www.w3.org/TR/2004/REC-rdf-concepts-20040210/.  
+Zhenzhong Lan, Mingda Chen, Sebastian Goodman, Kevin Gimpel, Piyush Sharma, and Radu Soricut. 2019. Albert: A lite bert for self-supervised learning of language representations. arXiv preprint arXiv:1909.11942.  
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+Prashan Madumal, Tim Miller, Liz Sonenberg, and Frank Vetere. 2020. Explainable reinforcement learning through a causal lens. In Proceedings of the AAAI Conference on Artificial Intelligence, volume 34, pages 2493-2500.  
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+
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+
+# Checklist
+
+1. For all authors...
+
+(a) Do the main claims made in the abstract and introduction accurately reflect the paper's contributions and scope? [Yes]  
+(b) Did you describe the limitations of your work? [Yes]  
+(c) Did you discuss any potential negative societal impacts of your work? [Yes]  
+(d) Have you read the ethics review guidelines and ensured that your paper conforms to them? [Yes]
+
+2. If you are including theoretical results...
+
+(a) Did you state the full set of assumptions of all theoretical results? [N/A]  
+(b) Did you include complete proofs of all theoretical results? [N/A]
+
+3. If you ran experiments...
+
+(a) Did you include the code, data, and instructions needed to reproduce the main experimental results (either in the supplemental material or as a URL)? [Yes]  
+(b) Did you specify all the training details (e.g., data splits, hyperparameters, how they were chosen)? [Yes]  
+(c) Did you report error bars (e.g., with respect to the random seed after running experiments multiple times)? [Yes]  
+(d) Did you include the total amount of compute and the type of resources used (e.g., type of GPUs, internal cluster, or cloud provider)? [Yes]
+
+4. If you are using existing assets (e.g., code, data, models) or curating/releasing new assets...
+
+(a) If your work uses existing assets, did you cite the creators? [Yes]  
+(b) Did you mention the license of the assets? [Yes]  
+(c) Did you include any new assets either in the supplemental material or as a URL? [Yes]  
+(d) Did you discuss whether and how consent was obtained from people whose data you're using/curating? [Yes]  
+(e) Did you discuss whether the data you are using/curating contains personally identifiable information or offensive content? [Yes]
+
+5. If you used crowdsourcing or conducted research with human subjects...
+
+(a) Did you include the full text of instructions given to participants and screenshots, if applicable? [Yes]  
+(b) Did you describe any potential participant risks, with links to Institutional Review Board (IRB) approvals, if applicable? [Yes]  
+(c) Did you include the estimated hourly wage paid to participants and the total amount spent on participant compensation? [Yes]

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+# Injecting Domain Knowledge from Empirical Interatomic Potentials to Neural Networks for Predicting Material Properties
+
+Zeren Shui
+
+University of Minnesota shuix007@umn.edu
+
+Daniel S. Karls*
+
+University of Minnesota karl0100@umn.edu
+
+Mingjian Wen*
+
+University of Houston  
+mjwen@uh.edu
+
+IIia A. Nikiforov
+
+University of Minnesota nikif002@umn.edu
+
+Ellad B. Tadmor
+
+University of Minnesota tadmor@umn.edu
+
+George Karypis
+
+University of Minnesota / AWS karypis@umn.edu
+
+# Abstract
+
+For decades, atomistic modeling has played a crucial role in predicting the behavior of materials in numerous fields ranging from nanotechnology to drug discovery. The most accurate methods in this domain are rooted in first-principles quantum mechanical calculations such as density functional theory (DFT). Because these methods have remained computationally prohibitive, practitioners have traditionally focused on defining physically motivated closed-form expressions known as empirical interatomic potentials (EIPs) that approximately model the interactions between atoms in materials. In recent years, neural network (NN)-based potentials trained on quantum mechanical (DFT-labeled) data have emerged as a more accurate alternative to conventional EIPs. However, the generalizability of these models relies heavily on the amount of labeled training data, which is often still insufficient to generate models suitable for general-purpose applications. In this paper, we propose two generic strategies that take advantage of unlabeled training instances to inject domain knowledge from conventional EIPs to NNs in order to increase their generalizability. The first strategy, based on weakly supervised learning, trains an auxiliary classifier on EIPs and selects the best-performing EIP to generate energies to supplement the ground-truth DFT energies in training the NN. The second strategy, based on transfer learning, first pretrains the NN on a large set of easily obtainable EIP energies, and then fine-tunes it on ground-truth DFT energies. Experimental results on three benchmark datasets demonstrate that the first strategy improves baseline NN performance by  $5\%$  to  $51\%$  while the second improves baseline performance by up to  $55\%$ . Combining them further boosts performance.
+
+# 1 Introduction
+
+Predictive modeling of materials is a field with manifold applications that has been the subject of many cross-disciplinary studies. While modeling is conducted at different length scales, all material behavior ultimately has its origins at the nano scale, where the interactions between individual atoms must be understood. The most accurate methods at this scale are based on quantum mechanics theory, requiring explicit consideration of the electronic degrees of freedom described by the Schrödinger equation. However, these methods are presently limited to systems containing at most several
+
+thousand atoms, precluding their use in investigation of important microstructural phenomena such as crack propagation. To overcome this limitation, practitioners have long relied upon heuristic models known as empirical interatomic potentials (EIPs), which consist of physically motivated analytical functional forms that strive to model the complex electronic interactions between atoms using only the nuclear coordinates of the atoms and their elemental species as input [34]. In the past several years, there has been a surge of interest in the development of machine learning EIPs, particularly those based on neural networks (NNs), as a more accurate alternative to traditional EIPs [4]. In contrast to traditional EIPs, NN potentials contain little inductive bias and, accordingly, require large volumes of training data labeled using first-principles quantum mechanical methods. The first-principles method most commonly used is density functional theory (DFT), which scales as  $\mathcal{O}(n_{\mathrm{e}}^{3})$  with the number of valence electrons  $n_{\mathrm{e}}$  in a given configuration of atoms [23]. As a result of this high computational cost, it is difficult to acquire a sufficient number of labeled training instances to create an NN potential that performs accurately over a wide range of applications.
+
+One potential solution to this problem is to seek additional supervision signals. Traditional EIPs are attractive sources of such additional supervision for two reasons. First, their functional forms incorporate prior physical information that allows them to correlate with DFT, and, in regions of relevance to common applications, are often quite accurate. That is, they contain domain knowledge that could benefit the training of NN potentials. Second, EIPs scale linearly with the number of atoms and are thus orders of magnitude faster than DFT, permitting the labeling of massive datasets. Despite the advantages, no research has focused on using EIP supervision signals in training NN potentials.
+
+In this paper, we leverage physically motivated EIPs and unlabeled configurations to tackle the label scarcity challenge for training NN potentials. We propose two generic strategies, weakly supervised learning and transfer learning, for exploiting this additional source of information. In the first strategy, we expand the DFT-labeled training set with unlabeled configurations and their EIP energies. To achieve this goal, we train an auxiliary classifier on the original DFT-labeled training configurations that predicts which one of a selected set of EIPs is likely to produce the most accurate estimate of the DFT energy for each of a large set of unlabeled configurations. The unlabeled configurations are then labeled by their corresponding predicted best-performing EIPs and appended to the training set. We train NN potentials on the expanded training set by optimizing a robust regression loss to mitigate the influence of noise and outliers introduced by the EIP energies. In the second strategy, we adopt a transfer learning approach by way of multi-task pretraining. We first pretrain the representation module of an NN potential to reproduce the energies predicted by the EIPs. During the subsequent fine-tuning stage, the representation module of the NN is paired with a prediction head and trained on the DFT-labeled configurations. These two strategies can be flexibly used and coupled to train any NN potential.
+
+The contributions of this work are three-fold: 1) We demonstrate that EIPs are capable of providing high quality supervision signals for training NN potentials, which opens a new direction for future development of NN potentials; 2) We propose two effective and generic strategies that take advantage of EIPs and unlabeled configurations to tackle the label scarcity challenge for training NN potentials; 3) We conduct comprehensive experiments on three benchmark datasets and four representative NN potentials that cover most of the NN potential forms currently in use. Experimental results show that the proposed strategies successfully inject domain knowledge from EIPs to NN potentials and improve the performance of the NN potentials by up to  $55\%$ .
+
+# 2 Preliminaries
+
+# 2.1 Atomic Configurations
+
+The fundamental input in atomistic modeling is an atomic configuration. An atomic configuration is a spatial arrangement of atoms  $C = \{(Z_i, \mathbf{r}_i)\}_{i=1}^N$  where  $Z_i$  and  $\mathbf{r}_i$  are the atomic number and the three-dimensional Euclidean coordinates of atom  $i$ , respectively. In the simplest scenario, an atomic configuration corresponds to an isolated cluster of atoms that comprise a molecule. However, it may more generally describe a bulk system such as a crystal, which contains an infinite number of atoms distributed over space. These systems are modeled using a small collection of atoms in a finite simulation cell that is effectively repeated across all of space with the aid of periodic boundary conditions (PBCs) [34] (see Appendix A.1).
+
+# 2.2 Physics-based Potentials
+
+Mathematically, an EIP is a function,  $E = \mathcal{V}(C;\theta)$ , that takes an atomic configuration  $C$  as input and returns its total potential energy  $E$ ; here,  $\theta$  denotes a set of fitting parameters to be determined. The functional form  $\mathcal{V}$  of a physics-based EIP is made up of carefully designed analytic expressions that strive to capture the underlying physics in the material it models [43, 44]. Because the functional form itself is intended to capture most of the relevant physics, such models need relatively few parameters (typically on the order of ten) and are usually fitted to a set of material properties deemed most relevant to real-world applications. It is expected that physics-based EIPs will approximate the first-principles energy surface well in the vicinity of atomic configurations corresponding to the material properties to which they were fit. However, their generalizability can be inconsistent, as shown in Fig. 1.
+
+# 2.3 Machine Learning Potentials
+
+In contrast to physics-based EIPs, machine learning EIPs employ general-purpose regression algorithms as the functional form  $\mathcal{V}$  that do not encode any knowledge of the material it models. Therefore, machine learning EIPs are almost exclusively trained on large sets of DFT data so as to include as much physical knowledge as possible.
+
+One important class of machine learning EIPs are neural network (NN)-based potentials. In this paper, we define an NN potential  $\mathcal{M}:\mathcal{C}\mapsto \mathcal{E}$  as
+
+$$
+\mathcal {M} (C) = f _ {\text {p r e d}} \circ f _ {\text {r e p}} (C),
+$$
+
+where  $f_{\mathrm{rep}}: \mathcal{C} \mapsto \mathcal{R}^{n \times d}$  is the representation learning module that maps each of the  $n$  atoms of  $C$  to a feature vector of length  $d$  based on its local environment, and  $f_{\mathrm{pred}}: \mathcal{R}^{n \times d} \mapsto \mathcal{E}$  is the prediction module that maps the feature vectors of the atoms to the total potential energy.
+
+A concrete instance of this type of model are those that use a graph neural network
+
+(GNN) [24, 16, 40, 31, 21] as the representation learning module and a multilayer perceptron (MLP) as the prediction module. Before being passed to the representation learning module  $f_{\mathrm{rep}}$ , an atomic configuration  $C$  is first converted to a graph  $\mathcal{G} = (V,E)$  with nodes  $V$  and edges  $E$ . One node is defined for each atom in the simulation cell, as well as for additional padding atoms representing PBCs if present. An edge is created between any two nodes with a distance smaller than a prescribed cutoff radius. Next, for each atom  $i$ , its atomic number  $Z_{i}$  is one-hot encoded into an initial feature vector  $h_{i}^{(0)}$  of the node corresponding to the atom. The GNN representation learning module  $f_{\mathrm{rep}}$  then updates the feature vectors using a message passing paradigm [15], i.e., the nodes iteratively aggregate information from their neighbors. Formally, the feature vector of node  $i$  at the  $l + 1$ -th layer  $\mathbf{h}_{i}^{(l + 1)}$  is updated as a function of the feature vectors of its neighbors  $\mathcal{N}(i)$  and itself at the previous layer,
+
+$$
+\mathbf {m} _ {i} ^ {(l)} = \operatorname {A g g} ^ {(l)} \left(\left\{f ^ {(l)} \left(\mathbf {h} _ {i} ^ {(l)}, \mathbf {h} _ {j} ^ {(l)}, \mathbf {e} _ {i j} ^ {(l)}\right) \mid j \in \mathcal {N} (i) \right\}\right), \tag {1}
+$$
+
+$$
+\mathbf {h} _ {i} ^ {(l + 1)} = \operatorname {U p d a t e} ^ {(l)} \left(\mathbf {h} _ {i} ^ {(l)}, \mathbf {m} _ {i} ^ {(l)}\right), \tag {2}
+$$
+
+where  $f^{(l)}$  and  $\mathrm{Update}^{(l)}$  are learnable functions,  $\mathrm{Agg}^{(l)}$  is a permutation-invariant function that operates on sets of feature vectors, and  $\mathbf{e}_{ij}^{(l)}$  denotes the feature vector associated with the edge connecting node  $i$  and node  $j$  at the  $l$ -th layer. Finally, the prediction module  $f_{\mathrm{pred}}$  maps the feature vector of each atom at the last layer  $\mathbf{h}_i$  to a corresponding energy contribution and sums them to arrive at the total energy of the configuration, i.e.,  $E = \sum_{i}\mathrm{MLP}(\mathbf{h}_{i})$ .
+
+![](images/0e57f64ff96a6bd5292f458c73c9f51e0f0e2164285faab87d11ec0925f909fb.jpg)  
+Figure 1: Schematic illustration of the energy landscape defined over atomic configuration space by two physics-based EIPs and by DFT. Both EIPs are fitted to reproduce properties of the DFT energy landscape near atomic configuration A. Away from configuration A, the relative accuracy of the EIPs compared to DFT varies: at point B, EIP 1 is a fair approximation while EIP 2 is less accurate; at point C, neither EIP is accurate.
+
+$$
+\left. \left. ^ {l)} \mathbf {h} _ {j} ^ {(l)}, \mathbf {e} _ {i j} ^ {(l)}\right) | j \in \mathcal {N} (i) \right\}, \tag {1}
+$$
+
+# 3 Related Works
+
+# 3.1 Neural Network Potentials
+
+The first modern NN potential was proposed by Behler and Parrinello [3]. In their formulation, an atomic descriptor (i.e., basis functions that transform an atomic configuration into a fixed-length fingerprint vector) based on the bond lengths and bond angles is passed to an MLP. On top of this formulation, a Monte Carlo dropout technique can be applied to the MLP to equip the potential with the ability to quantify its predictive uncertainty [42]. The DeePMD method is a similar approach to that of Behler and Parrinello, except that a novel atomic environment descriptor is used [48]. More recently, researchers have developed GNN potentials based on a message passing paradigm [30, 46]. Another class of GNN potentials such as NequIP [2] and GemNet [14] pass equivariant messages rather than invariant ones based on the formulation of tensor field networks [37] and achieve state-of-the-art performance. All of these models are trained with supervised learning, without exploring the possibilities of leveraging weakly supervised learning or transfer learning to take advantage of unlabeled data.
+
+# 3.2 Weakly Supervised Learning
+
+Weakly supervised learning refers to techniques that attempt to train machine learning models from incomplete (only a portion of training instances are labeled) or inaccurate (noisy labels) supervision signals [50]. Solutions for incomplete supervision usually fall into the category of semi-supervised learning, which assumes that nearby instances have similar labels [25, 33, 29]. For inaccurate supervision, a model either learns directly from noisy labels with noise-robust algorithms [12, 49, 13] or resorts to a small portion of clean labeled data to reduce the noise [39, 45].
+
+# 3.3 Transfer Learning
+
+Transfer learning [51] refers to a machine learning paradigm that transfers knowledge to a model learns from one or more relevant tasks to benefit a target task. Transfer learning has enjoyed great success, especially in the low-data regime, as demonstrated by the rise of pretrained neural networks. This recent trend began with natural language processing when BERT [9] and successive large pretrained language models [26, 7] were released and quickly gained popularity in other domains such as computer vision [17, 8, 10] and graph learning [47, 19, 28]. These methods pretrain large neural networks on self-supervised tasks in order to encode common contextual knowledge in the structured input. Another approach imparts domain-specific knowledge by pretraining models on tasks related to the target task but for which abundant labeled data is available [18, 27]. The pretrained model is then fine-tuned on the limited training data of the target task.
+
+# 4 Methods
+
+# 4.1 Problem Definition
+
+Let  $\mathcal{P}$  be a set of physics-based EIPs,  $\mathcal{C}$  be the space of all possible atomic configurations,  $\mathcal{C}_{\mathrm{DFT}} = \{C_i,\{E_i^p\}_{p\in \mathcal{P}},E_i^{\mathrm{DFT}}\}_{i = 1}^m$  be a set of  $m$  configurations with corresponding DFT and EIP energies, and  $\mathcal{C}_{\mathrm{EIP}} = \{C_i,\{E_i^p\}_{p\in \mathcal{P}}\}_{i = m + 1}^{m + n}$  be a set of  $n$  configurations with only physics-based EIP energies. Here,  $E_{i}^{\mathrm{DFT}}$  and  $E_{i}^{p}$  denote the energies predicted for configuration  $C_i$  by DFT and the  $p$ -th physics-based EIP, respectively. In practice, physics-based EIPs are much less expensive than DFT, and so the size of  $\mathcal{C}_{\mathrm{EIP}}$  is much larger than  $\mathcal{C}_{\mathrm{DFT}}$ , i.e.,  $n\gg m$ . Our goal is to use this data from physics-based EIPs to train an NN-based EIP  $\mathcal{M}$  to closely approximate the DFT energy surface over the space of all atomic configurations, i.e.,
+
+$$
+\mathcal {M} (C) \approx E ^ {\mathrm {D F T}}, \forall C \in \mathcal {C}.
+$$
+
+# 4.2 Label Augmentation
+
+Because physics-based EIPs are developed as approximations to DFT, it is intuitive to use their predictions as surrogate labels for configurations without DFT energies in order to resolve the label scarcity issue of training neural networks. However, there are two fundamental challenges. First, as
+
+discussed in Sec. 2.2, while physics-based EIPs are designed to be accurate in specific regions of the configuration space and generalize better than those based on machine learning, they may still be inaccurate in other regions. Given an arbitrary configuration for which no DFT energy is available, it is unknown which EIP from a given set will yield the most accurate prediction and how large its error will be. Second, surrogating DFT energies with EIP energies inevitably introduces noise, and potentially outliers, into the training set that may have pathological effects.
+
+EIP prediction using an auxiliary classification model. In order to augment training sets with unlabeled configurations and their corresponding EIP-approximated energies, we use an auxiliary classification model to predict the best-performing EIP for a given configuration. For a configuration  $C$ , the classification model predicts a discrete distribution over the EIP set  $\mathcal{P}' = \mathcal{P} \bigcup \{p_{\emptyset}\}$  that indicates their probability of being the most accurate EIP for  $C$ , i.e.,  $\mathbb{P}(\mathcal{P}' \mid C)$ . We introduce a dummy EIP,  $p_{\emptyset}$ , to represent the case where none of the physics-based EIPs in  $\mathcal{P}$  is predicted to approximate DFT to an accuracy level  $c$ , i.e.,  $\frac{1}{N_i} ||E_i^p - E_i^{\mathrm{DFT}}||_1 > c, \forall p \in \mathcal{P}$ , where  $N_i$  is the number of atoms in configuration  $i$ ; throughout this work,  $c$  is set to 0.1. By excluding configurations that are labeled with the dummy class from the training set,  $c$  acts as a confidence threshold to control the noise and outliers introduced by using the surrogate EIP energies.
+
+The classification model consists of a representation learning module that converts an atomic configuration to fixed-length feature vectors (one for each atom in the configuration), a permutation-invariant readout function that aggregates them to form a feature vector describing the entire configuration, and a prediction module that maps the configuration representation to a set of probabilities. We train the classification model on  $\mathcal{C}_{\mathrm{DFT}}$  by optimizing a cross-entropy loss and apply it to  $\mathcal{C}_{\mathrm{EIP}}$ . Configurations with a predicted EIP other than  $p_{\emptyset}$  are assigned the corresponding EIP energy and are merged with the configurations that have DFT energies to arrive at the final training set. We denote the set of EIP-labeled configurations as  $\hat{\mathcal{C}}_{\mathrm{EIP}} = \{C_i, \{E_i^p\}_{p \in \mathcal{P}}, E_i^{\hat{p}_i}\}_{i=m+1}^{m+s}$  where  $1 \leq s \leq n$  is the number of selected configurations not labelled by DFT, and  $\hat{p}_i$  and  $E_i^{\hat{p}_i}$  are the predicted best-performing EIP and its prediction on  $C_i$ .
+
+Regression with robust loss functions. We train the NN potential using configurations with ground-truth DFT energies and configurations with EIP energies selected by the classification model. In regression problems, models are usually trained by optimizing the mean square error (MSE) loss. The MSE loss is sensitive to outliers, as the magnitude of its gradient is linearly proportional to the difference between the predicted value and the ground truth value. To lessen the impact of outliers, we optimize the MSE loss on DFT-labeled configurations, while on EIP labeled-configurations, we optimize the Tukey biweight (bisquare) loss [6, 11, 5], i.e.,
+
+$$
+\mathcal {L} = \frac {1}{m} \sum_ {i = 1} ^ {m} \left(E _ {i} ^ {\mathrm {D F T}} - \hat {E} _ {i}\right) ^ {2} + \frac {\alpha}{s} \sum_ {i = m + 1} ^ {m + s} l _ {\text {T u k e y}} \left(E _ {i} ^ {\hat {p} _ {i}} - \hat {E} _ {i}\right) \tag {3}
+$$
+
+where  $\hat{E}_i$  is the model prediction of the energy for configuration  $C_i$ , and  $\alpha > 0$  is a hyper-parameter that controls the contribution of EIP-labeled configurations to the loss and its gradient.
+
+The Tukey biweight loss falls under the M-estimation method [20] and is intended to screen outliers using robust statistics of the regression residuals such as median absolute residuals (MAR) and suppress their influence on the gradient. The Tukey loss function is computed as
+
+$$
+l _ {\text {T u k e y}} \left(r _ {i}\right) = \left\{ \begin{array}{l l} \frac {k ^ {2}}{6} \left[ 1 - \left(1 - \left(\frac {r _ {i}}{k}\right) ^ {2}\right) ^ {3} \right], & \left| r _ {i} \right| \leq k \\ \frac {k ^ {2}}{6}, & \left| r _ {i} \right| > k \end{array} \right. \tag {4}
+$$
+
+where  $r_i = E_i^{\hat{p}i} - \hat{E}_i$  is the residual and  $k$  is a tuning constant that is commonly set to  $4.685\sigma$  to produce  $95\%$  efficiency when the errors are normally distributed with standard deviation  $\sigma$ . To set the value of  $k$ , we estimate the standard deviation as  $\hat{\sigma} = \mathrm{MAR} / 0.6745$ . From Eq. 4, residuals with absolute values greater than  $k$  are considered outliers and are rejected for gradient computation. In each gradient step, we sample from both DFT- and EIP-labeled configurations to form a batch for gradient back-propagation, and the MAR is estimated on residuals of both DFT- and EIP-labeled configurations in the batch. Since the MAR and  $\hat{\sigma}$  are dynamically estimated during training, the Tukey loss does not introduce additional hyperparameters.
+
+# 4.3 Multi-task Pretraining
+
+We propose a multi-task pretraining strategy to encode the domain knowledge in physics-based EIPs to the parameters of the representation learning module by jointly predicting the set of EIP calculations for configurations with only EIP predictions,  $C_{\mathrm{EIP}}$ , and optimizing the following multi-task regression loss:
+
+$$
+\mathcal {L} = \frac {1}{n | \mathcal {P} |} \sum_ {p \in \mathcal {P}} \sum_ {i = m + 1} ^ {m + n} (\hat {E} _ {i} ^ {p} - E _ {i} ^ {p}) ^ {2}.
+$$
+
+During pretraining, we couple the representation learning module with  $|\mathcal{P}|$  prediction modules (MLPs) to generate predictions corresponding to different physics-based EIPs, i.e.,  $\hat{E}_i^p = f_{\mathrm{pred}}^p \circ f_{\mathrm{rep}}(C_i)$ . The pretrained representation module is then fine-tuned with a randomly initialized prediction module for the downstream DFT prediction task. Note that the representation module could either be naively fine-tuned on configurations with DFT energies by optimizing an MSE loss, or on the training set generated by our proposed label augmentation method by optimizing Eq. 3.
+
+Although transfer learning has been successful in various application domains, it could easily hinder model performance on the target task if the pretraining tasks are unrelated to the target task (negative transfer). We argue that predicting the output of physics-based EIPs is relevant and beneficial to the target task of predicting DFT energies. Although physics-based EIPs are not perfectly accurate across the space of all possible spatial arrangements of atoms, their functional forms incorporate prior physical information that allow them to correlate with DFT over this space. In Sec. 5, we empirically demonstrate that the multi-task pretraining strategy successfully encodes domain knowledge into configuration representations and creates a smoother DFT energy surface.
+
+# 4.4 Combining Label Augmentation and Multi-task Pretraining
+
+The label augmentation and multi-task pretraining methods outlined above can be combined with relative ease. The procedure is similar to the ordinary label augmentation strategy, but rather than using a randomly initialized representation module  $f_{\mathrm{rep}}$  for the final NN training, the representation module produced by the multi-task pretraining method is used during fine-tuning. Fig. 2 provides a schematic overview of both strategies, how they relate to one another, and how they can be combined.
+
+# 5 Experiments
+
+To test our methodology, we experiment with three datasets: the ANI-A1 [32] dataset and the KIM-Si [22] dataset each with a single species, as well as a multispecies AgAu dataset [41]. Detailed descriptions of the three datasets are provided in Appendix A.2. For each dataset, we generate three splits by randomly assigning  $20\%$  of the DFT-labeled configurations as test sets and the other  $80\%$  as training sets. During training, we use  $20\%$  of the training set as a validation set for model selection. All of the reported experimental results are averaged over three different splits to avoid over-fitting to a specific split. We release our code  ${}^{2}$  and the KIM-Si dataset  ${}^{3}$  for reproducing our experimental results and continuous works.
+
+# 5.1 Experimental Setting
+
+# 5.1.1 Neural Network-based Potentials
+
+We evaluate our proposed strategies on two classes of neural network potentials that reflect the majority of machine learning potentials currently in use. The first represents atomic environments using pre-computed descriptors and learns non-linear transformations (MLPs) to map the descriptors to atomic embeddings, while the second uses GNNs to learn atomic representations from configuration graphs. We select one representative potential from each class. For our MLP-based potential, we use the Smooth Overlap of Atomic Positions (SOAP) [1] atomic environment descriptor together with a representation module and prediction module consisting of MLPs; we term this potential SOAPNet in later discussions. For our GNN-based potential, we select SchNet [30], CGCNN [46],
+
+![](images/fc19d84238e9d11ee42804f370b64aed03347238e18c6a06a3dba6c1a45e9363.jpg)  
+Figure 2: Illustration of the Label Augmentation (LA) and Multi-task Pretraining (MP) strategies and their usage in the training of NN-based potentials for the case of two physics-based EIPs,  $p_1$  and  $p_2$ . In LA, a classifier is trained to predict the most accurate EIP energies  $\hat{\mathcal{C}}_{\mathrm{EIP}}$  for the unlabeled training instances, which are combined with the DFT energies from  $\mathcal{C}_{\mathrm{DFT}}$  in the loss function when training the final NN (blue arrow + green arrow). In MP, the representation module  $f_{\mathrm{rep}}$  is pretrained by simultaneously fitting the energies of each physics-based EIP before it is inherited as the initial state for the representation module in the final NN training, where the DFT-labeled instances are again used (purple arrow + green arrow). The two strategies can be combined by incorporating  $\hat{\mathcal{C}}_{\mathrm{EIP}}$  into the loss while also using multi-task pretraining to initialize  $f_{\mathrm{rep}}$  (blue arrow + purple arrow + green arrow).
+
+and GemNet [14]. In our label augmentation experiments, we set the representation module of the auxiliary classification model to be the same kind as the corresponding NN potential, e.g., the classification model used for training the SchNet potential has a SchNet GNN as its representation module. Details about the model and training hyper-parameters can be found in Appendix A.4.
+
+# 5.1.2 Selection of Physics-based EIPs
+
+The physics-based EIPs used in our experiments were selected to encompass differing levels of functional complexity. Because physics-based EIPs are designed for specific elemental species (in this case, aluminum, silicon, and gold and silver systems), a different set of physics-based EIPs had to be chosen for each dataset. A total of ten physics-based EIPs were used for aluminum, eight for silicon, and two for the gold-silver system (see Tab. 2 in Appendix). They were taken mainly from the Open Knowledgebase of Interatomic Models (OpenKIM) repository. [35, 36]. For detailed information, see Appendix A.3.
+
+# 5.2 Experimental Results
+
+# 5.2.1 Performance of Label Augmentation and Multi-task Pretraining
+
+As shown in Tab. 1, our proposed strategies improve the performance of the four baseline NNs on the three benchmark datasets. In particular, the label augmentation strategy improves the baseline NNs by  $5\%$  to  $51\%$ , while the multi-task pretraining strategy improves the baselines by  $2\%$  to  $55\%$ . Combining the two strategies gives further improvement.
+
+Table 1: Performance of the two proposed strategies on DFT energy prediction tasks. We report the configuration-level and atom-level mean absolute error (MAE, lower is better) in eV and eV/atom, respectively. We denote the label augmentation strategy by LA and the multi-task pretraining strategy by MP. Best performance is shown in bold. Cases where the training procedure failed due to running out of memory are marked OOM.  
+
+<table><tr><td></td><td>Config</td><td>KIM-Si Atom</td><td>Improv.</td><td>Config</td><td>ANI-Al Atom</td><td>Improv.</td><td>Config</td><td>AgAu Atom</td><td>Improv.</td></tr><tr><td>Best EIP</td><td>1.6326</td><td>0.2524</td><td>-</td><td>46.4869</td><td>0.3561</td><td>-</td><td>4.4587</td><td>0.2063</td><td>-</td></tr><tr><td>SOAPNet</td><td>0.7706</td><td>0.0975</td><td>-</td><td>0.2153</td><td>0.0017</td><td>-</td><td>0.5422</td><td>0.0226</td><td>-</td></tr><tr><td>+LA</td><td>0.5595</td><td>0.0704</td><td>27.58%</td><td>0.1786</td><td>0.0014</td><td>18.14%</td><td>0.5067</td><td>0.0205</td><td>07.92%</td></tr><tr><td>+MP</td><td>0.5717</td><td>0.0733</td><td>25.28%</td><td>0.1744</td><td>0.0014</td><td>19.12%</td><td>0.3962</td><td>0.0154</td><td>29.34%</td></tr><tr><td>+MP+LA</td><td>0.5307</td><td>0.0657</td><td>31.88%</td><td>0.1697</td><td>0.0013</td><td>22.13%</td><td>0.3858</td><td>0.0154</td><td>30.23%</td></tr><tr><td>SchNet</td><td>0.4805</td><td>0.0718</td><td>-</td><td>0.1693</td><td>0.0014</td><td>-</td><td>0.7290</td><td>0.0290</td><td>-</td></tr><tr><td>+LA</td><td>0.4015</td><td>0.0549</td><td>19.99%</td><td>0.0845</td><td>0.0007</td><td>51.24%</td><td>0.6815</td><td>0.0266</td><td>07.33%</td></tr><tr><td>+MP</td><td>0.4034</td><td>0.0569</td><td>18.40%</td><td>0.1296</td><td>0.0010</td><td>26.00%</td><td>0.3353</td><td>0.0130</td><td>54.65%</td></tr><tr><td>+MP+LA</td><td>0.3719</td><td>0.0490</td><td>27.17%</td><td>0.0816</td><td>0.0006</td><td>53.27%</td><td>0.3496</td><td>0.0135</td><td>52.80%</td></tr><tr><td>CGCNN</td><td>0.9314</td><td>0.1410</td><td>-</td><td>0.2410</td><td>0.0019</td><td>-</td><td>1.6683</td><td>0.0625</td><td>-</td></tr><tr><td>+LA</td><td>0.7476</td><td>0.1050</td><td>22.61%</td><td>0.1786</td><td>0.0014</td><td>25.44%</td><td>1.6065</td><td>0.0589</td><td>04.71%</td></tr><tr><td>+MP</td><td>0.8457</td><td>0.1253</td><td>10.16%</td><td>0.2206</td><td>0.0017</td><td>07.80%</td><td>1.4377</td><td>0.0532</td><td>14.38%</td></tr><tr><td>+MP+LA</td><td>0.7435</td><td>0.1005</td><td>24.44%</td><td>0.1392</td><td>0.0011</td><td>41.65%</td><td>1.3857</td><td>0.0499</td><td>18.55%</td></tr><tr><td>GemNet</td><td>0.5138</td><td>0.0546</td><td>-</td><td>OOM</td><td>OOM</td><td>OOM</td><td>0.9257</td><td>0.0342</td><td>-</td></tr><tr><td>+LA</td><td>0.4691</td><td>0.0511</td><td>07.55%</td><td>OOM</td><td>OOM</td><td>OOM</td><td>0.8381</td><td>0.0300</td><td>10.87%</td></tr><tr><td>+MP</td><td>0.5024</td><td>0.0531</td><td>02.48%</td><td>OOM</td><td>OOM</td><td>OOM</td><td>0.5057</td><td>0.0185</td><td>45.71%</td></tr><tr><td>+MP+LA</td><td>0.4651</td><td>0.0476</td><td>11.12%</td><td>OOM</td><td>OOM</td><td>OOM</td><td>0.6074</td><td>0.0218</td><td>35.30%</td></tr></table>
+
+# 5.2.2 EIP Energies as High-Quality Supervision Signals for Training NN Potentials
+
+Recall that in the label augmentation strategy, an auxiliary classification model selects unlabeled configurations and predicts their corresponding best-performing physics-based EIPs, which are subsequently used to label them for the training of the NN potential. These configurations are then labeled by the predicted best-performing physics-based EIPs for NN potential training. We investigate the quality of EIP labels and the auxiliary classification model by training NN potentials on three augmented training sets in which the selected unlabeled configurations are labeled by three different sources: DFT-based energies, ground-truth best-performing-EIP energies, and predicted best-performing EIP energies. The DFT-labeled configurations (0.8K) are the same for the three training sets. We only conduct this experiment on the ANI-Al dataset, as all of its configurations have DFT energies available. The number of selected unlabeled configurations is shown in Tab. 6 (in Appendix).
+
+Fig. 3a shows the performance of the NN potentials trained on the three augmented training sets and the original training set (where only DFT-labeled configurations are used). As shown in the figure, expanding the training set with ground truth DFT calculations (blue) greatly improves the baseline MAE (yellow bar, model trained on the original training set). Labeling configurations with the ground-truth best-performing physics-based EIPs (red bar) performs slightly worse than with DFT energies (blue bar) but still much better than the baseline (yellow bar). This demonstrates that physics-based EIPs are valuable sources of supervision signals for training NN-based potentials. Using the predicted best-performing physics-based EIPs for labeling (green bar) performs on par with the ground-truth best-performing physics-based EIP labeling (red), revealing the utility of the auxiliary classification model.
+
+# 5.2.3 Importance of the Robust Tukey Loss
+
+We next conduct experiments to investigate the importance of the Tukey loss and its ability to reject outliers during training. As before, we only conduct this experiment on the ANI-Al dataset, using DFT energies for unlabeled configurations to determine noise and outliers introduced by the predicted best-performing physics-based EIPs. We define an unlabeled configuration with a predicted best-performing physics-based EIP to be an outlier if the absolute difference between its physics-based
+
+![](images/371fcb6f866f1cf90d0ed6af5495a16199ce5eab03ea74bf560b365ed0a7e95c.jpg)  
+(a)
+
+![](images/eaed78c0aa3dd169f4cd9961f089aab4f8639f5c0d3190c565ad44517b18e280.jpg)  
+(b)  
+Figure 3: (a) Performance of NN potentials trained on the original training sets (yellow bar, DFT-labeled configurations only) and three augmented training sets whose unlabeled configurations are labeled by DFT energies (blue bar), ground-truth best-performing physics-based EIP energies (red bar), and predicted best-performing physics-based EIP energies (green bar). (b) Percentage of outliers used for computing gradient during training. (c) Standard deviation of residuals during training.
+
+![](images/134cd42335c5444eb607bf5c1d3123418a935aa302ae945fb7c1dd9ebfcb4b1f.jpg)  
+(c)
+
+EIP energy and its DFT energy is larger than a threshold, i.e.,  $|r_i| = \frac{1}{N_i} |E_i^{\mathrm{DFT}} - E_i^{\hat{p}_i}| > c$  where  $N_{i}$  is the number of atoms in configuration  $i$ . We categorize the outliers as mild, normal, and severe by setting  $c$  to  $\{0.1, 0.2, 0.3\}$ . The initial number of outliers introduced by the unlabeled configurations can be found in Tab. 6 (in Appendix). Figs. 3b and 3c show that the number of outliers of all kinds included for computing gradients decreases as the training proceeds, demonstrating that as the NN potential gets progressively more accurate, the Tukey loss can effectively eliminate outliers from the training set. We also conduct an ablation study by replacing the Tukey loss in Eq. 3 with the MSE loss. The results in Tab. 4 (in Appendix) show that the models' performance degrades without the Tukey loss.
+
+# 5.2.4 Visualization of Pretrained Configuration Representations
+
+Fig. 4 plots the t-SNE [38] 2D projections of the training silicon configuration representations colored by their per-atom DFT energies. The left-hand figure plots representations generated by a SchNet with random weights and the right-hand figure plots representations generated by a SchNet pretrained by the multi-task strategy. The representations generated by the randomly initialized SchNet do not exhibit any clear patterns and the energy surface is rough. In the right-hand figure, representations of the atomic cluster configurations (i.e., isolated groups of atoms) and the bulk configurations (crystals) are clearly separated and form clusters in the t-SNE 2D space. The per-atom DFT energy surface of the right-hand figure is much smoother, i.e., configurations with similar energies are close to one another after pretraining. This verifies our previous statement that, although physics-based EIPs lack complete generalizability, they nonetheless correlate reasonably well with DFT over atomic configuration space, and demonstrates that our proposed multi-task pretraining strategy successfully encodes domain knowledge into the NN-based potentials. T-SNE plots for SOAPNet on both datasets and SchNet on the ANI-Al dataset show similar patterns. These figures may be found in Appendix.
+
+# 6 Conclusion
+
+In this paper, we show that physics-based EIPs can be used to incorporate domain knowledge into machine learning EIPs by providing additional supervision signals on a large body of unlabeled training instances. Two generic strategies are formulated. The first, label augmentation, is based on weakly supervised learning and uses a classifier to select unlabeled training instances to supplement the original labeled training set. The second, multi-task pretraining, is based on transfer learning and uses the predictions of a set of physics-based EIPs on unlabeled training instances to pretrain a machine learning EIP that is subsequently fine-tuned on the original training set. Our methodology is proven using several experiments, including a cross-validation study that demonstrates significant performance gains for either strategy individually and further improvement when they are combined, a label augmentation study that shows that using EIP energies as surrogate labels provides nearly as much robustness as using additional DFT-labeled training instances, and a statistical experiment that illustrates the utility of the Tukey loss in diminishing noise and excluding outliers. Finally, we present
+
+![](images/1e73640a06016d2e833da74642d7e490ca2a1826b0d8ecfc5d43a75df1efb65f.jpg)  
+Figure 4: T-SNE plots of silicon configuration representations generated by a randomly initialized SchNet (left) and a SchNet pretrained with our proposed multi-task pretraining strategy (right). Configurations are colored by their per-atom DFT energies. Representations generated by the pretrained SchNet naturally form two clusters that correspond to the atomic cluster configurations (i.e., isolated groups of atoms) and the bulk configurations (i.e., crystals).
+
+reduced-dimension visualizations that indicate the multi-task pretraining strategy yields training set feature vectors over which the energy varies smoothly.
+
+One direction for future work is to replace the heuristic EIPs by more accurate but still computationally cheap methods, such as the semi-empirical tight binding method. These methods better approximate DFT and thus can lead to further improvement when combined with our proposed approaches.
+
+# 7 Acknowledgement
+
+This work was supported in part by NSF (1447788, 1704074, 1757916, 1834251, 1834332), Army Research Office (W911NF1810344), the startup funds from the Presidential Frontier Faculty Program at the University of Houston, Intel Corp, and Amazon Web Services. Access to research and computing facilities was provided by the Minnesota Supercomputing Institute. We thank the anonymous reviewers for their feedback during NeurIPS 2022 review process. We are grateful to Sijie He and Yingxue Zhou for their insightful discussion and inspiration.
+
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+# Insights into Pre-training via Simpler Synthetic Tasks
+
+Yuhuai  $\mathbf{W}\mathbf{u}^{12*}$
+
+yuhuai@cs.stanford.edu
+
+Felix Li\*
+
+fzli@berkeley.edu
+
+Percy Liang
+
+pliang@cs.stanford.edu
+
+$^{1}$ Stanford University
+
+2Google Research
+
+$^{3}$ UC Berkeley
+
+# Abstract
+
+Pre-training produces representations that are effective for a wide range of downstream tasks, but it is still unclear what properties of pre-training are necessary for effective gains. Notably, recent work shows that even pre-training on synthetic tasks can achieve significant gains in downstream tasks. In this work, we perform three experiments that iteratively simplify pre-training and show that the simplifications still retain much of its gains. First, building on prior work, we perform a systematic evaluation of three existing synthetic pre-training methods on six downstream tasks. We find the best synthetic pre-training method, LIME, attains an average of  $67\%$  of the benefits of natural pre-training. Second, to our surprise, we find that pre-training on a simple and generic synthetic task defined by the Set function achieves  $65\%$  of the benefits, almost matching LIME. Third, we find that  $39\%$  of the benefits can be attained by using merely the parameter statistics of synthetic pre-training. We release the source code at https://github.com/felixzli/synthetic_pretraining.
+
+# 1 Introduction
+
+Pre-training on a large amount of data from sources such as text from the web—natural pre-training—is effective for a wide range of downstream tasks (Devlin et al., 2019; Brown et al., 2020; Chen et al., 2021; Lu et al., 2021). More surprisingly, recent works (Wu et al., 2021; Krishna et al., 2021; Ri & Tsuruoka, 2022; Anderson & Farrell, 2022) show that pre-training on data that is fully synthetically generated—synthetic pre-training—can also provide substantial gains, partially closing the gap between training from a randomly initialized model and a naturally pre-trained model.
+
+What properties of pre-training are necessary for effective gains? In this paper, we conduct a careful empirical study to help understand this question. We perform three experiments that iteratively simplify natural pre-training, while still retaining much of the benefits of natural pre-training. See Figure 1 for an overview of the results obtained in our three experiments.
+
+First, we perform a systematic evaluation of three previously proposed synthetic pre-training methods over six downstream tasks. Prior works only evaluated each synthetic pre-training method on a single downstream domain: Wu et al. (2021) on mathematical reasoning benchmarks, Chiang & Lee (2021) on GLUE, Krishna et al. (2021) on summarization, and Ri & Tsuruoka (2022) on language modeling and dependency parsing. We instead show how general these methods are as well as how different synthetic pre-training methods compare against each other. We find that the best performing synthetic task, LIME (Wu et al., 2021), closes a significant fraction of the gap between random initialization and natural pre-training for summarization (85%), semantic parsing (83%), reading comprehension (55%), code translation (43%), and retrosynthesis (49%).
+
+![](images/5a226e7e5df3cb03350b5b3626ceb1653c29f2538c11a0cf1c2c9a879a280e60.jpg)  
+Figure 1: We compare downstream task performance between natural language pre-training (Pre-trained_T5), synthetic pre-training (LIME), simpler synthetic pre-training (Set), initializing from the statistics of Set pre-training (Set Statistics), and initializing just the pre-attention layer norms to a lower value (Pre-attention Layer Norm Init). The average percent improvement is computed by taking the average over our six downstream tasks of  $\frac{\text{Performance(Initialization)} - \text{Performance(Random_Initial)}}{\text{Performance(Pre-trained_T5)} - \text{Performance(Random_Initial)}}$ .
+
+Second, we discover a surprisingly simple and generic synthetic task that captured nearly the same effectiveness provided by the existing best synthetic task LIME, closing an average of  $65\%$  of the gap to natural pre-training (compared to  $67\%$  for LIME). This simple task, which we call Set, consists of outputting the input sequence without duplicate tokens in its original order, for example:
+
+Input: a a c b a b  $\Rightarrow$  Output: a c b.
+
+Third, to simplify the initialization even more, we ask whether only the statistics of the pre-trained weights can explain the effectiveness of synthetic pre-training. We find that initializing parameters with statistics of a synthetic pre-trained model provides large benefits for some downstream tasks, closing  $39\%$  of the gap to natural pre-training. Simplifying further allows us to see that by initializing all pre-attention layer norm parameters to a lower value (just one number!), we close more than  $40\%$  of the gap to natural pre-training performance for summarization and semantic parsing, and an average of  $21\%$  over all six tasks.
+
+# 2 Experimental Setup
+
+Architecture We trained a T5-Small model, which is a 60 million parameter encoder-decoder transformer model with 6 encoder layers, 6 decoder layers, 8 heads, head dimension 64, and MLP dimension 2048 (Raffel et al., 2020).
+
+Training Details For synthetic pre-training, we use the same hyperparameters that the off-the-shelf language pre-trained T5-small was trained with: AdaFactor optimizer, batch size 128, sequence length 512, and inverse square root learning rate  $1 / \sqrt{\max(n,10000)}$  where  $n$  is the current training step. We evaluate token validation accuracy every 5000 training steps. For all synthetic tasks besides Dyck, we fine-tuned with the first checkpoint that the model reaches above  $99\%$  token validation accuracy. For Dyck, the model's validation accuracy plateaued without reaching above  $99\%$ , so we chose to fine-tune using the checkpoint with max validation accuracy of  $77.7\%$ . We provide further pre-training details in Appendix A and fine-tuning details in Appendix B.
+
+Downstream Tasks We fine-tuned synthetically pre-trained models on a diverse suite of downstream tasks: 1) Java to C# code translation (10K training examples) (Lu et al., 2021); 2) two semantic parsing benchmarks, MTOP (17K training examples) (Li et al., 2021) and WebQSP (2.7K training
+
+Table 1: Evaluation of three previously proposed synthetic pre-training tasks and two simpler synthetic pre-training tasks. For tasks where we report both Exact Match (EM) and F1, the Average column is computed using EM.  
+
+<table><tr><td></td><td>CNNDM-10K ROUGE1</td><td>MTOP EM/F1</td><td>WebQSP EM/F1</td><td>SQuAD EM/F1</td><td>Code Trans. EM</td><td>Retrosyn. EM</td><td colspan="2">Average</td></tr><tr><td colspan="9">Baselines</td></tr><tr><td>Pre-trained_T5</td><td>35.81</td><td>81.0/95.2</td><td>83.1/91.9</td><td>77.5/86.0</td><td>61.6</td><td>43.1</td><td>63.7</td><td>100%</td></tr><tr><td>Random_Init</td><td>18.9</td><td>38.6/83.0</td><td>28.2/72.9</td><td>17.2/25.2</td><td>57.0</td><td>39.2</td><td>33.2</td><td>0%</td></tr><tr><td colspan="9">Section 3: Benchmarking Existing Synthetic Pre-training</td></tr><tr><td>LIME</td><td>33.2</td><td>73.7/94.0</td><td>75.2/88.7</td><td>50.4/62.1</td><td>59.0</td><td>41.1</td><td>55.4</td><td>67%</td></tr><tr><td>Dyck</td><td>27.1</td><td>65.9/91.9</td><td>58.5/83.5</td><td>50.3/62.4</td><td>58.8</td><td>40.4</td><td>50.2</td><td>49%</td></tr><tr><td>Nons_Summary2</td><td>32.0</td><td>68.0/92.7</td><td>65.2/85.2</td><td>48.4/60.1</td><td>57.3</td><td>39.6</td><td>51.8</td><td>47%</td></tr><tr><td colspan="9">Section 4: Simpler Synthetic Pre-Training</td></tr><tr><td>Set</td><td>32.8</td><td>71.7/93.7</td><td>72.2/88.2</td><td>48.3/60.0</td><td>59.4</td><td>40.9</td><td>54.2</td><td>65%</td></tr><tr><td>Identity</td><td>30.2</td><td>68.6/93.0</td><td>69.8/86.8</td><td>26.1/35.4</td><td>57.8</td><td>40.5</td><td>48.8</td><td>46%</td></tr></table>
+
+examples) (Yih et al., 2016), consisting of converting a natural language query to a logical form; 3) USPTO-50K retrosynthesis (40K training examples) (Liu et al., 2017), a task that consists of predicting possible reactants when given a product as input; 4) the reading comprehension benchmark SQuAD 1.1 (87K training examples) (Rajpurkar et al., 2016); and 5) the summarization benchmark CNNDM-10K<sup>3</sup> which is 10K training examples from the CNNDM (Krishna et al., 2021). See full descriptions and input-output examples for every downstream task in Appendix D.
+
+Baselines We compared against randomly initialized (Random_Init) and off-the-shelf natural language pre-trained T5-Small (Pre-trained_T5). Pre-trained_T5 was trained for 524K steps on the C4 dataset with the span-corruption objective (Raffel et al., 2020). We also compared against T5-Small trained on Wikipedia (Guo et al., 2020) for 10K steps (Wiki_10K) with the same pre-training objective as Pre-trained_T5.
+
+Source code We release the source code to reproduce the experiments at https://github.com/felixzli/synthetic_pretraining.
+
+# 3 Benchmarking Existing Synthetic Pre-training
+
+We benchmark three previously proposed synthetic pre-training tasks across six downstream tasks. We compute how much of the benefits using natural pre-training are gained by each synthetic pre-training method by taking the average over six downstream tasks of Performance(Initialization) - Performance(Random_Init) / Performance(Pre-trained_T5) - Performance(Random_Init). We show these numbers in the right-most column of Table 1. We find that LIME achieves the best performance, gaining  $67\%$  of the benefits using natural pre-training.
+
+# 3.1 Synthetic Tasks
+
+For each synthetic task, we generated one million examples. To generate synthetic task data, we sampled tokens a vocabulary of size 32K, matching the vocabulary size of Pre-trained_T5. Details about data generation can be found in the original papers. We provide an example of each task in Figure 2.
+
+![](images/1f63b7acc4dbcce341dbe553ea6e411967dc376c07ed7ba00b06df818a7c1af3.jpg)
+
+![](images/347a676b013a3239148e3cd043448b9b85f254efe2c21e15f27ccb7299c6b989.jpg)
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+![](images/02479d1c57cd093bf62020bf5e767f42203eda9e5d5ed22992c9906725abbd20.jpg)
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+![](images/f02faf98ae620f1581fa9bf99c0502eb7a503f6506099918e85fc4731de09a52.jpg)  
+Figure 2: We show examples of Section 3 previously proposed synthetic tasks in the top row and Section 4 simpler synthetic tasks in the bottom row. Top Left: LIME deduct task, whose input consists of a "rule string" and "substitution dictionary" and output is a "result string." The "rule symbols" and "math symbols" are written as actual letters and math operations for reader clarity. In real generated data, each example's rule and math symbols are randomly sampled tokens. Top Middle: T5 style masked language modelling with Dyck language. In the input, the grey boxes indicate masked out spans. The model is trained to predict the masked spans, each separated by a special token  $\langle s \rangle$ . In this example, the "noise_density" is 0.33 (so 5 out of 15 input brackets are corrupted) and the "span-corruption_length" is 1 (so each masked span is of length 1). In actual generated data, the "span-corruption_length" is 3 and the "noise_density" is 0.15, which are the same parameter values used for natural language Pre-trained_T5. Top Right: Nonsense_Summary task. In this example, two operations to create the summary are "copy sentence containing a keyword" and "identify keyword sentiment." Bottom Left: Set task. Bottom Right: Identity task.
+
+![](images/030dd1034cee02880c0770645a57fda89d89582f7f5589c029b4c93a0ad46cbf.jpg)
+
+LIME LIME is a set of three tasks—Deduct, Induct, and Abduct—inspired by Charles Peirce's three reasoning primitives (Wu et al., 2021). Each task consists of three elements: a rule string, a dictionary that represents substitutions, and a result string that is the result of applying those substitutions to the rule string. The three tasks are then constructed by using two of the three elements as inputs to predict the remaining element.
+
+Dyck Artificial Language We consider a Dyck language with multiple bracket types (taken from Ri & Tsuruoka (2022)), which we refer as Dyck. We turn generated text into a sequence-to-sequence task by using the T5 span-corruption pre-training objective (Raffel et al., 2020). The motivation behind the design of Dyck is how "sentences of natural language often have dependency relations where the existence of a certain word can be predictive of another word" (Ri & Tsuruoka, 2022).
+
+Nonsense Summary Krishna et al. (2021) developed the nonsense summarization task, which we refer to as Nonsense_Summary, by conducting a qualitative analysis of summarization data and identifying 21 elementary operations used to create summaries such as "CopyFirstSentence" and "CopyQuoted." Each synthetic task example consists of an input that is a nonsense document made of randomly sampled tokens and an output that is the corresponding nonsense summary consisting of applying a random subset of the 21 operations to the document.
+
+# 3.2 Results
+
+Table 1 shows results of evaluating the synthetic tasks across six downstream tasks, and Appendix Table 4 shows results of ablating the dataset size for retrosynthesis, CNNDM, and SQuAD.
+
+Synthetic pre-training provides large gains over random initialization. We observed large gains from all three synthetic tasks we evaluated. Out of these, LIME provided the most benefits across all six downstream tasks. We computed how much of the natural pre-training benefit is captured by LIME as  $\frac{\text{Performance(LIME) - Performance(Random_ Init)}}{\text{Performance(Pre-trained_T5) - Performance(Random_ Init)}}$ . LIME closed  $84.6\%$ ,  $82.8\%$ ,  $85.6\%$ ,  $55.1\%$ ,  $43.5\%$ , and  $48.7\%$  of the gap for CNNDM-10K, MTOP, WebQSP, SQuAD, Code Translation, and Retrosynthesis respectively.
+
+Task or domain prior is not required for pre-training benefits. For example, LIME benefits tasks including summarization (CNNDM-10K) and reading comprehension (SQuAD) despite LIME pre-training data involving no natural language like Pre-trained_T5, representing no structure of natural language like Dyck, and not being explicitly designed to reflect the operations used in summarization like Nonsense_Summary.
+
+Varying degrees of benefits. Comparing Nonsense_Summary and Dyck, we see that the two synthetic tasks benefit MTOP, WebQSP, and SQuAD similarly, yet their benefits for the three other tasks are very different. The difference between Nonsense_Summary and Dyck as well as the superiority of LIME are examples of how different synthetic pre-training can have significantly different effectiveness.
+
+Synthetic pre-training is not more efficient. One of the advantages argued for synthetic pretraining is its computational efficiency (Wu et al., 2021; Krishna et al., 2021). To explore this potential advantage, we pre-trained our model using the T5 span-corruption pre-training objective on Wikipedia (Guo et al., 2020) for 10K steps, and the result is shown in Table 1 labelled Wiki_10K. Wikipedia pre-training for 10K steps provided more significant gains than LIME pre-training for 30K steps and Set pre-training for 10K steps. Based on this result, we see no evidence that the evaluated synthetic pre-training tasks provide computational efficiency benefits over natural language pre-training, unlike what Wu et al. (2021) suggested.
+
+Natural language pre-training benefits non-natural language tasks. Natural language pretraining, including Wiki_10K, outperforms synthetic pre-training on the non-natural language tasks of Code Translation and Retrosynthesis.
+
+# 4 Simpler Synthetic Pre-training
+
+Encouraged by the results in the previous section, we want to explore what is necessary for synthetic pre-training benefits. We evaluate pre-training with simpler and more generic synthetic tasks similar to basic Python API functions. Surprisingly, a simple and generic task defined by the Set function provides significant benefits (65%), almost matching the previous best synthetic task LIME (67%). Furthermore, another simple task defined by the Identity function also provided significant benefits (46%).
+
+# 4.1 Simpler Synthetic Tasks
+
+We constructed 18 simpler synthetic tasks similar to basic Python API functions. As we did for the three existing synthetic tasks, for each simpler synthetic task we generated one million data points with a vocabulary of 32K tokens, matching the vocabulary size of Pre-trained_T5. In Appendix G.1, we define each task, explain how examples are randomly generated, and provide input-output examples. We first evaluated all tasks on CNNDM-10K. We then evaluated Set, one of the best performing tasks on CNNDM-10K, and Identity, one of the simplest tasks, on all six downstream tasks.
+
+![](images/5871bdad095882eaa0c965111c116d0b2f36585156dd39bfe2dc35624c2f51dd.jpg)  
+Figure 3: For each simple synthetic task, we pre-train on it and then fine-tune on CNNDM-10K. Set performed the best and Identity, one of the simplest tasks, provided significant benefit. The Set and Identity tasks are defined in 4.1, and the other tasks are defined in G.
+
+Set The Set task consists of outputting the input sequence in its original order except without tokens appearing more than once. An example input is [a b b a a c d c], and the corresponding output is [a b c d]. The Set task data generation is explained in Appendix G.
+
+Identity The Identity task consists of copying the input tokens to the output. An example input is [a d e a a], and the corresponding output is [a d e a a]. To generate one example, we uniformly sample a sequence of tokens between length 10 and length 220 to be the input and output.
+
+# 4.2 Results: simpler synthetic tasks provide similar benefits as existing synthetic tasks
+
+Figure 3 shows results on CNNDM-10K for models pre-trained on each of 18 simpler synthetic tasks. Table 1 shows results on all six downstream tasks for Set and Identity pre-trained models. Averaged across the six downstream tasks, Set and Identity gain  $65\%$  and  $46\%$  of the natural pretraining benefits compared to  $67\%$  for LIME, the best previously proposed synthetic task evaluated in Section 3. Identity provides at least as much benefit as Nested Language and Nonsense Summary for all tasks besides Code Translation, for which Nested Language benefits more.
+
+The significant benefits of Set and Identity pre-training suggest that the complexities of the three previously proposed synthetic tasks we evaluated—representing three fundamental logical operations, having a structural property that mimics natural language, and reflecting operations used for summarization—may not be the necessary factors for why pre-training on those tasks provides benefits. Also, due to their simplicity, Set and Identity may be useful for future work towards understanding pre-training and how to design better synthetic tasks.
+
+# 5 Even Simpler Synthetic Pre-training with Statistics
+
+Inspired by the previous section, we tried to find even simpler initializations that achieve the benefits of pre-training. We explored how benefits of synthetic pre-training can be attained by initialization based only on statistics of pre-trained weights. We found: (1) Initialization using the mean and standard deviation (SD) of synthetic pre-trained parameters can gain more than  $70\%$  of the benefit from synthetic pre-training for CNNDM-10K and MTOP (or an average of  $39\%$  of natural pre-training benefits). (2) Initialization using simpler schemes involving less statistics can still provide benefits. (3) Specifically the pre-attention layer norm plays an important role for initialization benefits.
+
+![](images/5d5b94cd2a56de50947c849d9e0237f8b04929136698658a1f57d9a041e57988.jpg)  
+Figure 4: Illustration of Section 5.1 Per_Param_Mean_SD init scheme. We use this init scheme with LIME, Set, or Identity pre-trained models before fine-tuning on the downstream tasks. As an example, if we were to use this init scheme with an Identity pre-trained model, we would initialize the encoder layer 2 query matrix weights by sampling from a Gaussian with the mean and SD of the encoder layer 2 query matrix from an Identity pre-trained model, and we would initialize every other parameter in the same manner. The Per_Param_Mean_SD init scheme recovered more than  $70\%$  of the benefit of normally fine-tuning a synthetic pre-trained model (or an average of  $39\%$  of natural pre-training benefits).
+
+# 5.1 How much of synthetic pre-training benefit can be attributed to better statistics for initialization?
+
+We compare normal fine-tuning, which we call Full1_Ini t, with fine-tuning a model that is initialized with the mean and standard deviation (SD) of each synthetic pre-trained model's parameter. We call this initialization scheme Per_Param_Mean_SD and illustrate it in Figure 4. Specifically, for each parameter  $p$  (e.g. an attention query matrix) in the synthetic pre-trained model, we compute the the mean  $\mu_p$  and the standard deviation  $\sigma_p$  of all entries in  $p$ . We then randomly initialize the model with the same  $\mu_p$  and  $\sigma_p$  for each parameter  $p$ . For example, if a pre-trained model had a  $2 \times 2$  parameter matrix  $p = [[-1,2],[2,5]]$ , then  $\mu_p = 2$  and  $\sigma_p = 2.4$ , and we would initialize  $p$  by sampling the each entry of this  $2 \times 2$  matrix from a normal distribution  $\mathcal{N}(2,2.4^2)$ . We list all the parameters we initialize for T5-Small in this manner in Appendix K.
+
+As shown in Table 2, using the Per_Param_Mean_SD initialization scheme achieves a large proportion of the benefit from normally fine-tuning a synthetic pre-trained model. For example, for LIME Identity and Set, the Per_Param_Mean_SD initialization recovered  $78\%$ ,  $85\%$ , and  $97\%^2$  of MTOP Full_Init performance (or  $65\%$ ,  $66\%$ ,  $69\%$  of the benefits by natural pre-training). The remaining gap between Per_Param_Mean_SD and Full_Init supports the possibility that other factors besides better initialization statistics may cause the benefits of synthetic pre-training. We note that it does not preclude the possibility that better initialization statistics is the sole cause of benefits: there may exist other initialization schemes such that the statistics of the pre-trained model are a subset of the statistics from the initialization scheme.
+
+# 5.2 Can simpler initialization schemes still produce benefits?
+
+Next we explore if simpler initialization schemes that use a smaller number of statistics than Per_Param_Mean_SD can still provide benefits. We present three initialization schemes: Per_Param_Scale, Whole_model_Scale, and Subset_Per_Param_Scale, with results shown in Table 2. We also tried two other initialization schemes, Across_Layers_Scale and Per Layers_Scale, discussed in Appendix H.
+
+Table 2: Evaluating different initialization schemes with statistics of LIME, Set, and Identity pre-trained models. Results for Per_Param_Mean_SD for the four other tasks as well as five seeds for CNNDM-10K, MTOP, and SQuAD are in Appendix Table 7. Results for two other initialization ablations and three other Subset_Per_Param_Scale subset choices we tried are in Appendix Table 5.  
+
+<table><tr><td></td><td colspan="3">CNNDM-10K</td><td colspan="3">MTOP</td></tr><tr><td></td><td>LIME</td><td>Set</td><td>Identity</td><td>LIME</td><td>Set</td><td>Identity</td></tr><tr><td colspan="7">Section 5.1</td></tr><tr><td>Full_Init</td><td>33.2</td><td>32.8</td><td>30.2</td><td>73.7/94.0</td><td>71.7/93.7</td><td>68.6/93.0</td></tr><tr><td>Per_Param_Mean_SD</td><td>29.3</td><td>29.8</td><td>28.8</td><td>66.1/92.3</td><td>66.6/92.3</td><td>67.8/92.6</td></tr><tr><td colspan="7">Section 5.2</td></tr><tr><td>Per_Param_Scale</td><td>29.0</td><td>29.5</td><td>29.0</td><td>67.3/92.5</td><td>66.5/92.4</td><td>66.3/92.3</td></tr><tr><td>Whole_Mode_Scale</td><td>15.5</td><td>26.1</td><td>20.5</td><td>34.3/81.4</td><td>38.7/83.1</td><td>59.8/91.0</td></tr><tr><td>Pre-attn_LN_Per_Param_Scale</td><td>27.8</td><td>25.1</td><td>27.3</td><td>59.9/90.3</td><td>58.9/89.0</td><td>46.7/86.3</td></tr><tr><td>Baseline</td><td colspan="3">CNNDM-10K</td><td colspan="3">MTOP</td></tr><tr><td>Random_Init</td><td colspan="3">18.9</td><td colspan="3">38.6/83.0</td></tr><tr><td>Pre-trained_T5</td><td colspan="3">35.8</td><td colspan="3">81.0/95.2</td></tr></table>
+
+Per Param Scale We use the term "scale statistic" to denote the mean of layer norm parameters or the SD of non layer norm parameters. Unlike the Per_Param_Mean_SD init scheme in the previous subsection, where we initialize each parameter with two statistics (mean and SD), Per_Param_Scale initializes each parameter with only the scale statistic: initialize layer norm parameters by setting their value to the mean of the corresponding pre-trained parameter weights, and initialize non layer norm parameters by sampling from a Gaussian with mean 0 and the SD of the corresponding pre-trained parameter weights.
+
+Whole Model Scale We initialize the whole model with only two statistics: the SD over all non layer norm weights  $\sigma_{m}$  and the mean over all layer norm weights  $\mu_{m}$ . Initialize each non layer norm weight by sampling from  $\mathcal{N}(0, \sigma_m^2)$  and Initialize each layer norm weight as the value  $\mu_{m}$ .
+
+Per Param Scale: Subset The motivation for this init scheme is to find a smaller set of parameters whose initialization can provide significant benefit. We initialize a subset of the parameters (e.g., attention parameters) with the Per_Param_Scale initialization and initialize the rest of the parameters with the default Random_Init. With this initialization scheme, we tried four different subsets: attention parameters, MLP parameters, pre-attention layer norms, and pre-MLP layer norms.
+
+Results Per_Param_Scale provided almost identical benefits as the Per_Param_Mean_SD initialization, suggesting that just the scale statistics, rather than both the mean and SD, are sufficient to provide benefits of initialization.
+
+Whole_Connection_Scale initialization was much worse than Per_Param_Mean_SD, yielding almost no improvements. A finer scale initialization rather than simply increasing the scale of weights across the whole model as Whole_Connection_Scale does may be necessary for benefits.
+
+For the Subset_Per_Param_Scale init scheme, pre-attention layer norms was the only subset we tried that provided significant initialization benefits, covering  $46\%$  and  $39\%$  of the gap between From Scratch and Pre-trained_T5 for CNNDM-10K and MTOP. See results in Table 2 (labeled Pre-attn_LN_Per_Param_Scale) and results for all four subsets tried in Appendix Table 5.
+
+# 5.3 A Better Pre-attention Layer Norm Initialization
+
+Synthetic pre-training results in lower pre-attention layer norm means. Motivated by benefits from the Subset_Per_Param_Scale initialization scheme—initializing pre-attention layer norms using their synthetic pre-trained means and all other parameters with the default initialization—we wanted to see if the pre-attention layer norm mean values in synthetic pre-trained models have a
+
+Table 3: Results from initializing pre-attention layer norm parameters to a particular value. They are initialized to the value of 1.0 in the Random_Init baseline which is the default T5 initialization.  
+
+<table><tr><td>Pre-attn LN Init Value</td><td>CNNDM-10K</td><td>MTOP</td><td>WebQSP</td><td>SQuAD</td><td>Code Trans.</td><td>Retrosyn.</td></tr><tr><td>0.05</td><td>28.2</td><td>25.7/73.3</td><td>30.5/71.2</td><td>22.8/31.2</td><td>57.9</td><td>39.3</td></tr><tr><td>0.1</td><td>28.3</td><td>32.5/77.5</td><td>28.9/70.6</td><td>23.5/31.7</td><td>57.6</td><td>39.9</td></tr><tr><td>0.2</td><td>28.2</td><td>56.4/89.3</td><td>30.6/72.3</td><td>28.2/37.3</td><td>57.0</td><td>39.4</td></tr><tr><td>0.4</td><td>24.4</td><td>56.9/89.9</td><td>30.0/72.8</td><td>36.3/46.9</td><td>57.3</td><td>39.6</td></tr><tr><td>0.8</td><td>18.6</td><td>50.7/88.5</td><td>28.0/71.6</td><td>34.2/44.6</td><td>57.2</td><td>39.9</td></tr><tr><td>Baseline</td><td>CNNDM-10K</td><td>MTOP</td><td>WebQSP</td><td>SQuAD</td><td>Code Trans.</td><td>Retrosyn.</td></tr><tr><td>Random_Init</td><td>18.9 (0.5)</td><td>38.6/83.0 (2.1/1.2)</td><td>25.8/71.5 (0.7/0.5)</td><td>16.0/24.2 (2.1/2.0)</td><td>57.6 (0.3)</td><td>39.0 (0.5)</td></tr><tr><td>Pre-trained_T5</td><td>35.8 (0.1)</td><td>81.0/95.2 (0.3/2.2)</td><td>82.3/91.7 (0.8/0.4)</td><td>77.4/86.1 (0.2/0.2)</td><td>61.2 (0.7)</td><td>43.5 (0.3)</td></tr></table>
+
+pattern. In Appendix Figure 6, we plotted pre-attention layer norm parameter means of each layer for LIME, Set, and Identity pre-trained models. We noticed that the pre-attention layer norm means from the synthetic pre-trained models are much smaller values (below 0.4 for most layers) compared to the default initialization of 1.0.
+
+Lower value pre-attention layer norm initialization provides benefits. Inspired by the above observation, we tried an initialization scheme where all the pre-attention layer norm parameters are initialized to a specific value, and all other parameters are initialized with the default initialization. We tried five different values ranging from 0.05 to 0.8. Results in Table 3 show that initializing the pre-attention layer norm parameters to a lower value than the default initialization of 1.0 provided significant benefits: initializing them to 0.2 improved CNNDM-10K from 18.9 to 28.2 ROUGE1, MTOP from  $38.6\%$  to  $56.4\%$  EM, and SQuAD from 16.0 to 22.8 EM.
+
+# 6 Related Work
+
+Synthetic Pre-training Past work has shown that synthetic pre-training can benefit a wide range of downstream tasks. For example, it has been shown that pre-training on artificial languages that intuitively mimic properties of natural language can benefit language modeling, dependency parsing, and GLUE (Ri & Tsuruoka, 2022; Papadimitriou & Jurafsky, 2020; Chiang & Lee, 2021). Pretraining on LIME, a mixture of three synthetic tasks motivated from reasoning primitives, benefits mathematical reasoning benchmarks (Wu et al., 2021). For summarization, Krishna et al. (2021) designed a suite of synthetic summarization tasks on a corpus of random tokens, and they show such pre-training can almost match up with natural pre-training. Synthetic pre-training also has been applied outside of text-based domains: Anderson & Farrell (2022) shows pre-training on synthetically generated fractal images nearly matches the benefits of pre-training with ImageNet.
+
+Cross-domain Transfer Learning Synthetic pre-training benefiting downstream tasks across a wide range of domains is an instance of cross-domain transfer learning, which past work has demonstrated the efficacy of. Pre-training language models with many languages benefits downstream tasks in a new unseen language (Liu et al., 2020). More surprisingly, pre-training on music, code, and amino acid sequences benefits natural language downstream tasks (Papadimitriou & Jurafsky, 2020; Chiang & Lee, 2020). It was also shown recently that pre-training on natural language benefits offline RL (Reid et al., 2022).
+
+Understanding Pre-training Past work has explored what properties of the pre-training data are important for benefits. Ri & Tsuruoka (2022) and Papadimitriou & Jurafsky (2020) use synthetic pretraining to identify general properties of pre-training data that correlate with improved downstream language task performance by leveraging how synthetic data is easy to modify in a controlled manner. Papadimitriou & Jurafsky (2020) found that changing the distribution of tokens in an artificial language can affect downstream performance, and Ri & Tsuruoka (2022) found that for an artificial brackets language using a nesting structure as well as different tokens for bracket pairs results in better performance than using a flat structure and the same token for bracket pairs. K et al. (2020), Conneau et al. (2020), and Dufter & Schütze (2020) explore data properties related to cross-language transfer learning. Sinha et al. (2021) shows that word order within sentences is not essential for natural language pre-training benefits.
+
+Other past work has used probing methods to explore what potentially transferable knowledge is gained from pre-training. Ri & Tsuruoka (2022) explored how much knowledge of "position-aware context dependence" was gained from artificial language and natural language pre-training. Chi et al. (2020) and Papadimitriou et al. (2021) explored potentially transferable knowledge gained from multi-lingual language pre-training.
+
+Initialization Our work tried to understand pre-training through the lens of initialization statistics that benefit optimization. There has been past work on improving transformer initialization to achieve more stable training and benefits with deeper models (Zhang et al., 2019; Huang et al., 2020; Zhang et al., 2020; Wang et al., 2022). These papers do not explore benefits from initializing with pre-trained model statistics.
+
+# 7 Discussion
+
+Limitations of current synthetic tasks In this work, we show that synthetic pre-training can provide significant benefits across many different tasks. However, synthetic pre-training still lags behind natural pre-training by an average of  $33\%$  for the six downstream tasks we fine-tuned on. This is expected for natural language downstream tasks, but it is important to note that even on non-natural language tasks, natural language pre-training still outperforms synthetic pre-training. This result suggests that natural language pre-training produces effects that are missing from the evaluated synthetic pre-training. Exploring the differences between different pre-training may be an interesting direction for future work.
+
+We emphasize that this paper does not support any claims regarding the practical efficacy of synthetic pre-training. The metric of "percentage of natural pre-training gap closed" measures relative performance along a meaningless linear scale, and for downstream tasks retrosynthesis and code translation, the magnitude of the improvement over from scratch for natural pre-training is low to begin with. This paper's goal is not to communicate that synthetic pre-training is practically effective, but rather to communicate that synthetic pre-training produces nontrivial differences in model behavior compared to a randomly initialized model, and also that there is nontrivial variation in model behavior from differing synthetic pre-training and initialization schemes involving pre-trained weight statistics.
+
+Synthetic pre-training requires deeper understanding Our results show that initialization using the statistics from synthetic pre-trained models can recover more than  $70\%$  of the benefit from synthetic pre-training. The remaining gap leaves open the possibility that the benefits of synthetic pretraining are due to other factors besides better statistics for initialization. Due to their simplicity, Set and Identity may be useful for future work towards deeper understanding of synthetic pre-training.
+
+In the long term, as we learn more about synthetic pre-training and pre-training in general, we believe that for some downstream tasks it may be possible to develop synthetic pre-training that outperforms natural pre-training: the complexity of existing natural data is fixed, while in some sense the complexity of fully synthetically generated data is infinite.
+
+Privacy and Ethics This work focuses on the understanding of synthetic pre-training. But synthetic pre-training can also have practical implications for privacy and ethics. Standard pre-training data, even when its public, can contain private user information. Large pre-trained models are capable of memorizing training examples, which makes them vulnerable to privacy attacks (Carlini et al., 2020). Synthetic data on the other hand is divorced from the real world, which can help mitigate privacy issues. Large pre-trained corpora can contain harmful text (e.g., hate speech). With synthetic pre-training, since we have complete control over what goes into the data, we can potentially mitigate some of the harmful behavior of existing models. Unfortunately, synthetic pre-training is currently not as performant as natural pre-training, so real-world data would still have to be used to close the performance gap.
+
+# Acknowledgements
+
+We thank Google TPU Research Cloud for the experimental support. We thank Christian Szegedy, Eric Zelikman, Tri Dao, Dan Fu, Sidd Karamcheti and Rishi Bommasani, for their useful feedbacks on the draft.
+
+# References
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+
+# Checklist
+
+1. For all authors...
+
+(a) Do the main claims made in the abstract and introduction accurately reflect the paper's contributions and scope? [Yes]  
+(b) Did you describe the limitations of your work? [Yes]  
+(c) Did you discuss any potential negative societal impacts of your work? [N/A]  
+(d) Have you read the ethics review guidelines and ensured that your paper conforms to them? [Yes]
+
+2. If you are including theoretical results...
+
+(a) Did you state the full set of assumptions of all theoretical results? [N/A]  
+(b) Did you include complete proofs of all theoretical results? [N/A]
+
+3. If you ran experiments...
+
+(a) Did you include the code, data, and instructions needed to reproduce the main experimental results (either in the supplemental material or as a URL)? [Yes]  
+(b) Did you specify all the training details (e.g., data splits, hyperparameters, how they were chosen)? [Yes] Sections 2,3.1,4.1,G,D  
+(c) Did you report error bars (e.g., with respect to the random seed after running experiments multiple times)? [N/A]  
+(d) Did you include the total amount of compute and the type of resources used (e.g., type of GPUs, internal cluster, or cloud provider)? [Yes] Appendix Section I
+
+4. If you are using existing assets (e.g., code, data, models) or curating/releasing new assets...
+
+(a) If your work uses existing assets, did you cite the creators? [Yes]  
+(b) Did you mention the license of the assets? [Yes]  
+(c) Did you include any new assets either in the supplemental material or as a URL? [Yes] We release the source code at https://github.com/felixzli/synthetic_pretraining.  
+(d) Did you discuss whether and how consent was obtained from people whose data you're using/curating? [N/A]  
+(e) Did you discuss whether the data you are using/curating contains personally identifiable information or offensive content? [N/A]
+
+5. If you used crowdsourcing or conducted research with human subjects...
+
+(a) Did you include the full text of instructions given to participants and screenshots, if applicable? [N/A]  
+(b) Did you describe any potential participant risks, with links to Institutional Review Board (IRB) approvals, if applicable? [N/A]  
+(c) Did you include the estimated hourly wage paid to participants and the total amount spent on participant compensation? [N/A]

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+# INSNET: An Efficient, Flexible, and Performant Insertion-based Text Generation Model
+
+Sidi Lu, Tao Meng, Nanyun Peng
+
+University of California, Los Angeles
+
+{sidilu, tmeng, violetpeng}@cs.ucla.edu
+
+# Abstract
+
+We propose INSNET, an expressive insertion-based text generator with efficient training and flexible decoding (parallel or sequential). Unlike most existing insertion-based text generation works that require re-encoding of the context after each insertion operation and thus are inefficient to train, INSNET only requires one pass of context encoding for the entire sequence during training by introducing a novel insertion-oriented position encoding and a light-weighted slot representation strategy to enable computation sharing. Furthermore, we propose an algorithm INSNET-Dinic to better determine the parallelization of insertion operations that provides a controllable switch between parallel and sequential decoding, making it flexible to handle more parallelizable tasks such as machine translation with efficient decoding, or less parallelizable tasks such as open-domain text generation to guarantee high-quality outputs. Experiments on two lexically constrained text generation datasets and three machine translation datasets demonstrate INSNET's advantages over previous insertion-based methods in terms of training speed, inference efficiency, and generation quality.
+
+# 1 Introduction
+
+Insertion-based text generation that formulates the generation process as a sequence of token insertion operations has received increasing attention in recent years. There are two major advantages of insertion-based generation over the prevalent left-to-right auto-regressive paradigm: 1) It reduces the decoding cost to sub-linear w.r.t. the sequence length with parallel decoding (Stern et al., 2019; Gu et al., 2019b), and 2) the flexible insertion orders may better recover/utilize the underlying linguistic structures of languages (Welleck et al., 2019; Gu et al., 2019a).
+
+However, this new paradigm of text generation brings unique challenges, mostly in the training efficiency. Unlike left-to-right auto-regressive decoders which monotonically expand the context, the insertion operations complicate the position information of each token as the context expands. Concretely, as is shown in Figure 1, the absolute position of a token in a sequence constantly changes along with the insertion operations. As a result, a naive implementation of insertion-based models (e.g., Stern et al. (2019); Gu et al. (2019b)) needs to re-encode the context with updated positional information for each token as the insertions proceed, yielding inefficient training with  $O(n)$  times of context re-encoding (with  $n$  indicating the sequence length).
+
+To tackle this problem, previous insertion-based generation models such as Insertion Transformer (InsT) (Stern et al., 2019) and Levenshtein Transformer (LevT) (Gu et al., 2019b) propose parallel token insertion to reduce the insertion/re-encoding steps from  $O(n)$  to  $\Theta (\log n)$  for both training and inference. However, while it works well for machine translation, such parallel insertion falls short on high-entropy generation tasks such as open-domain dialogue systems(Li et al., 2017a), creative generation such as stories (Yao et al., 2019; Goldfarb-Tarrant et al., 2020; Han et al., 2022), poetry (Manurung et al., 2000; Tian and Peng, 2022), and humor generation (Hempelmann, 2008; He et al.,
+
+![](images/15d18f2094236a26374044b43eb101df65b2dc76d9a12d96a78c613ac5185e2a.jpg)  
+Step t
+
+![](images/bfb56c2da2b2487208892994e6c6c2334539cbd1d533f0801e0e6d4db91cc7a2.jpg)
+
+![](images/4105da85378e2a17c9a4c9c8bd45feb3dfd054d4df4874a96c76acb599a1be21.jpg)
+
+![](images/633065abef626b0fed7da33b927b033eb1445217b1d6699769f86bdbbf54e6ad.jpg)  
+Step t+1
+
+![](images/b1af9547bdc6f47c9a3ec4a31590d5625ebc4f8210b5a4dfa41fbc81604949a2.jpg)
+
+![](images/1884e1536ea37a9d5b814d00d3c64cbcd6d7e3ceab13bc932c143140d155e3e6.jpg)  
+Figure 1: The absolute position information of each token is volatile for insertion-based models. Thus, naive models with absolute position encoding have to re-encode the sequence after each insertion operation during training.
+
+![](images/e5a6160988017b5460868f7882468a1b50eff7037f8b158cefba88ac435dc7eb.jpg)
+
+![](images/2eafabe618263c46606c1f7a886848d8ab132865c989f25c776105c1a7f965c7.jpg)
+
+![](images/6b8099ac6611d2edd236c8d3f05b19d5d64b0befb1b4344a5ea3488154ab07d3.jpg)
+
+<table><tr><td>Model</td><td>#Re-Enc. w/ ParaDec</td><td>#Re-Enc. w/ SeqDec</td><td>PosInfo</td></tr><tr><td>Ins. Trans.</td><td>Θ(log n)</td><td>O(n)</td><td>Absolute</td></tr><tr><td>Lev. Trans.</td><td>Θ(log n)</td><td>Ω(n)</td><td>Absolute</td></tr><tr><td>NMSTG</td><td>O(n)</td><td>O(n)</td><td>Markovian/Absolute</td></tr><tr><td>InDIGO</td><td>N/A</td><td>O(1)</td><td>Direction-only</td></tr><tr><td>INSNET (Ours)</td><td>O(1)</td><td>O(1)</td><td>Relative</td></tr></table>
+
+Table 1: Comparisons between INSNET and existing insertion-based models regarding training-time re-encoding steps (#Re-Enc. columns) and how the models encode positional information (the PosInfo column).
+
+2019; Mittal et al., 2022; Tian et al., 2022; Sun et al., 2022), where the output is open-ended and the tokens are usually highly dependent on each other, thus cannot be inserted in parallel. Another thread of work seeks to maintain sequential insertions but simplify the position encoding for the tokens so that they do not change as the insertion operations proceed (Gu et al., 2019a) to save computation. However, this sacrifices model capability and limits its applications to complex sequences.
+
+In this paper, we propose INSNET, which addresses the training efficiency issue of insertion-based text generators by enabling computation sharing similar as that in vanilla decoder transformers. Our first contribution is an insertion-oriented, relative positional encoding coined offset that allows INSNET to achieve computation sharing without compromising model capacity. At each insertion step, previously computed position encodings of the existing tokens remain unchanged, while the position encodings of newly inserted tokens accurately recording the updated pairwise token spatial relations of all the inserted tokens. A corresponding offset can be efficiently computed for any given insertion order with a novel process named offset compression. In this way, we avoid expensive re-encoding of the context, and based on the encoded context, our second contribution is the design of an effective aggregation strategy that allows us to parallelly generate expressive slot representations for every slot in the partially generated sequence, flexibly support both sequential and parallel decoding. Finally, inspired by the layerization idea of Dinic's algorithm(Dinitz, 2006), we propose an algorithm for INSNET to better determine the parallelization of insertion operations in order to reduce the likelihood discrepancy before and after the parallelization. With all the components, INSNET is a novel framework that enables efficient training, flexible decoding, and expressive positional encoding. Table 1 shows a comparison between INSNET and all prior insertion-based models.
+
+# 2 Related Works and Background
+
+Auto-regressive Language Models minimize the negative-log-likelihood of a sequence of n tokens  $s_{<n} = [x_0, x_1, \dots, x_{n-1}]$  with a left-to-right factorization. With the transformer architecture (Vaswani et al., 2017), each step of likelihood estimation can be calculated in parallel while sharing the prefix context encoding calculations. This makes it possible to build powerful and efficient text generation models, like the GPT family (Radford et al., 2018, 2019; Brown et al., 2020). A lot of successful applications are based on this paradigm of models, such as automatic story generation (Yao et al., 2019; Tan et al., 2020), image captioning (Vinyals et al., 2015; Xu et al., 2015), machine translation (Bahdanau et al., 2015; Liu et al., 2020), and dialogue system (Li et al., 2017b,a).
+
+Insertion-based Models with Parallel Decoding Insertion transformer (Stern et al., 2019) (InsT) proposes a design for insertion-based text generation. In each step, a bi-directional encoder transformer is performed on the partial sequence to compute the representation for each candidate slot between every two consecutive positions. Then, a model for the joint distribution of position-token is built to insert one or more token(s). Variants of InsT share the common atomic objective that models the step log-likelihood. On step  $t$  where a token  $x_{i\downarrow i + 1}$  is inserted in slot  $l_{i\downarrow i + 1}$  between position  $i$  and  $i + 1$  of context  $s_t^o$ , the step log-likelihood can be written as:
+
+$$
+\log p \left(x _ {i \downarrow i + 1}, \boldsymbol {l} _ {i \downarrow i + 1} \mid \boldsymbol {s} _ {t} ^ {o}\right) = \log p _ {\text {p o s i t i o n}} (i + 1 \mid \boldsymbol {s} _ {t} ^ {o}) + \log p _ {\text {t o k e n}} \left(x _ {i \downarrow i + 1} \mid e \left(\boldsymbol {s} _ {t} ^ {o}\right) _ {i} \oplus e \left(\boldsymbol {s} _ {t} ^ {o}\right) _ {i + 1}\right),
+$$
+
+where  $e(\cdot)_i$  stands for the i-th position of bi-directional encoding of the sequence and  $\oplus$  stands for vector concatenation. InS T adopts the original absolute positional encoding of transformers, the
+
+representation of the generated sequence has to be completely re-encoded after each step of context expansion to match the position changes of tokens. The expectation of the negative log step likelihood over all permitted context-insertion pairs at each step is computed as the step loss. The step losses from the first step to the last one are summed up as the sequence loss. Benefiting from the partially parallelized prediction of tokens, InsT can reduce the number of re-encoding steps to  $\Theta (\log n)$ .
+
+Levenshtein Transformer (Gu et al., 2019b) (LevT) contains two phases during generation: 1) Insertion phase: it first uses a similar strategy as InsT, but only inserts placeholders; an MLM is applied to fulfill the placeholders. 2) Deletion phase: the model is trained as a token-wise discriminator to determine where to delete, under the evaluation of the Levenshtein distance by dynamic programming. In practice, this would result in a slower process compared to InsT, but better generation quality as it alleviates the incoherence caused by parallel insertions.
+
+Sequential Insertion-based Model Previous exploration in insertion-based models mostly assumed the usage of parallel decoding in each decoding step, resulting in a partially auto-regressive procedure. For those are trained to only generate one token per decoding step, a vanilla implementation would result in an  $O(n)$  factor in training time complexity. NMSTG (Non-monotonic Sequential Text Generation, Welleck et al. (2019)) is one of the first attempts at modeling a non-monotonic sequential insertion-based generation process. It constrains the dependencies of each inserted token to pseudo-Markovian on an expansion tree. InDIGO(Gu et al., 2019a) is proposed as a sequentially-decoded insertion-based model with the transformer architecture. It supports the efficient likelihood estimation by working around the aforementioned volatility problem (see Figure 1) at a cost of omitting the distance information in its position encoding, and is thus able to adopt the conventional computation sharing trick to boost the multiplicative factor in training time complexity for each sequence to as fast as  $O(1)$ . However, InDIGO uses an encoding of all previously inserted tokens to predict the next token regardless of their tentative position. It is only after the next token is predicted, the inserted position for it is predicted. Thus, there's no trivial solution to use InDIGO as a (partially) non-autoregressive insertion-based generator.
+
+Absolute vs. Relative Position Embedding The original transformer (Vaswani et al., 2017) uses sinusoidal, absolute position embeddings. Relative position embedding in transformers (Shaw et al., 2018; Dai et al., 2019) was originally proposed to make the modeling of spatial relation invariant to position translation, and to improve the long-term dependency performance of the model. In replacement of absolute positions, which are tied to each token in the sequence, relative positions try to encode the spatial layout of the tokens with a matrix that records a directed distance from the column token to the row token. We further develop the idea of relative position embedding as the key component to overcome the issue of volatile position information of absolute positions in an insertion-based generation process. A noticeable fact is that InDIGO's implementation of position encoding can also be regarded as a direction-only relative position embedding system.
+
+# 3 The INSNet Model
+
+There are three major components of INsNET: 1) An context encoder based on the transformer architecture (Vaswani et al., 2017) that uses a novel way to compute insertion-oriented relative position encoding to better suit the insertion-based nature of the generation process and enable computation sharing (Section 3.1); 2) a module to compose expressive slot representation for predicting tokens to insert in different slots simultaneously (Section 3.2); and 3) an algorithm that adaptively determines the parallelization of the insertion operations to minimize conflicts (Section 3.3). Figure 2 illustrate the components of the full model.
+
+# 3.1 Context Encoder: A Transformer with Insertion-Oriented Relative Position
+
+In the introduction, we discussed the challenges in (efficient) computation sharing caused by the volatile position information of the context tokens as shown in Figure 1. We address this problem by empowering the transformer-based context encoder with a distance-aware, insertion-oriented pairwise relative position encoding. The proposed insertion-oriented relative position encoding shares important designs with previous relative position embeddings(Dai et al., 2019; Yang et al., 2019; Shih et al., 2019), but differs in how it accurately depicts the spatial layout (i.e., the actual sequential order) of the inserted tokens in an insertion-based generation process.
+
+![](images/b5b45131c3586827f7a90eb46fa704a156bbf73802fec57fd43e4dab9afa871a.jpg)  
+Figure 2: Illustration of the full InsNet model. Given an input sequence with a specified insertion order as illustrated in the [Inputs] panel (the upper right panel in blue), a context encoder (Section 3.1) illustrated in the [Encoder] panel (the upper left corner in purple) with insertion-oriented position encoding (Section 3.1.1) is applied to the sequence of insertion operations to obtain step-contextualized token embeddings as the transformer outputs. Then, the slot representation module (Section 3.2) illustrated in the [Slot Representation] panel (the bottom panel in yellow) compose slot representations for each step from InsNet transformer outputs using its left, right token representation and a global, step-wise token representation. The slot representations are then used to compute the token/position likelihood and also to determine the auto-parallelization in InsNet-Dinic (Section 3.3) illustrated in the [InsNet-Dinic] panel (the middle right panel in green).
+
+# 3.1.1 Offset: Insertion-Oriented Relative Position Encoding
+
+To illustrate our relative position encoding, considering the partial context (to be completed by further insertions) of "I have pen" with an insertion order of "have pen I". To insert an "a" in between "have" and "pen", the distance vector for token "a" against the rest of the context inserted so far should be [(“have”, -1), ("pen", +1), ("I", -2)], simplified as  $[-1, +1, -2]$ . This relative position encoding clearly defines where the insertion happens by only describing the pairwise spatial relationship between the incoming token and the existing context tokens. We can pack the relative position vectors for each insertion step to get a matrix that reflects the relative spatial relation along the trajectory of insertions, with each row corresponding to an insertion step. We name it the offset matrix.
+
+![](images/d6993ddda25597b97db9361a4c8dc86b6f81a173651617cb5201db743d782c0d.jpg)  
+(a) Absolute Positions
+
+![](images/d551c465817b5f70cfe2353c2d71f946eff84a51a6cf5fe5639bf541c4dc3bb1.jpg)  
+Figure 3: Comparison of different position encodings for an insertion-based generation process. From left to right, we illustrate 3(a) the volatile absolute positions, 3(b) the traditional non-insertion-oriented relative positions, 3(c) the proposed offset matrix presented in the insertion order, and 3(d) the same offset matrix permuted to show how it looks like if we restore the actual (partial-)sequence.
+
+![](images/5dd20f411b61a96aea62f43879715e83ce7485bcbe9df04c8d3e936922fb8cdb.jpg)  
+(b) Vanilla Relative Positions  
+(c) Our Relative Positions
+
+![](images/60b51bbf6ef880c2dc758d40ff7140ac73e7acc45b6d8bc3a7eec4f0791e7607.jpg)  
+(d) Ours, Order-Restored
+
+![](images/b93b9a4a5b5bc23420072156ce932ca6933ee791a54a685089c47ebb3e384067.jpg)  
+Figure 4: Illustration of the offset compression algorithm. It efficiently transforms the insertion order/permutation indices to an offset matrix.
+
+As a concrete example, to generate the sentence "I have a pen." with an insertion order  $\langle \mathrm{BOS} \rangle \langle \mathrm{EOS} \rangle$  have pen I a .", the complete offset matrix is shown in Figure 3(c). The effective elements of this matrix are all in the lower triangular region, making the application of the computation sharing trick for the decoder transformers trivial. This reduces the number of context (re-)encoding steps from  $n$  in naive insertion-based models (Stern et al., 2019) to 1 during training.
+
+Discussion: Comparisons With Other Positional Encoding Strategies To illustrate how the proposed insertion-oriented relative position encoding is different from existing position encoding strategies, Figure 3 shows the volatile, un-reusable absolute positions in 3(a), the traditional non-insertion-oriented relative positions as used in models such as XLNet Yang et al. (2019) in 3(b), the proposed offset matrix in the insertion order in 3(c), and the same offset matrix permuted to show the restored actual (partial-)sequence in 3(d). We can see that the absolute and the traditional relative positional encoding strategies are not applicable for efficient insertion-based generators. The former requires constant updates of the positional encoding of the existing tokens as the insertion progresses, disabling computation sharing. The latter requires knowledge about the final sequence length at the beginning of the insertion process, which is unrealistic and destroys the flexibility of insertion-based models. Our relative position encoding, on the other hand, reflects the order of the original sequence. Tokens that are on the left of the current one in the original sequence have a negative relative position to the current one while preserving the insertion order. As is illustrated in Figure 3(d), all the "later inserted tokens" are masked out and excluded from the previously inserted token's computation.
+
+# 3.1.2 Efficient Computation of Offset Matrix
+
+During training, given an insertion order to generate a sequence, we can pre-compute the offset matrix. However, naively constructing the offset matrix with element-wise operations takes  $O(n^{2})$  complexity for each sequence, limiting the training efficiency. Therefore we design an efficient algorithm to construct the offset matrix. Assume the insertion order is described in the form of absolute position permutation indices (i.e., in the previous example, [0, 6, 2, 4, 1, 3, 5]), we design a matrix algorithm called offset compression for our purpose. Figure 4 illustrates the algorithm with our exemplar sentence. Specifically, we first convert the absolute position vector into a matrix by duplication. Then, the upper triangular elements are masked by "infinity" to remove their impact in relative position computation because the inserting token should not attend to future to-be-inserted tokens. In the third step, each element is replaced by its in-row ranking i.e. its absolute position skipping the masked positions. In the final step, each row is baselined by the diagonal element to reflect that the relative position is between the last inserted token (the diagonal element) and previously inserted ones. The algorithm can be efficiently executed as a series of fast matrix operations.
+
+# 3.1.3 An INSNET Layer
+
+We hereby describe how to incorporate the offset matrix into each transformer layer in INSNET for efficient context encoding. In general, we inherit most designs from previous ones with relative position encoding (Dai et al., 2019; Yang et al., 2019). The encoder panel in Figure 2 (the upper left panel in purple) illustrate the process. Specifically, for layer number  $i = 1 \dots N$ , given the output embedding  $\mathbf{E}^{i-1}$  from the last layer and the offset matrix  $\mathbf{R}$ , the formulation of a transformer layer in an  $N$ -layer INSNET can be written as:
+
+$$
+\mathbf {Q} ^ {i}, \mathbf {K} ^ {i}, \mathbf {V} ^ {i}, \mathbf {P} ^ {i} = \mathbf {W} _ {q} ^ {i} \mathbf {E} ^ {i - 1}, \mathbf {W} _ {k, E} ^ {i} \mathbf {E} ^ {i - 1}, \mathbf {W} _ {v} ^ {i} \mathbf {E} ^ {i - 1}, \mathbf {W} _ {k, R} ^ {i} \mathbf {R}
+$$
+
+$$
+\mathbf {A} ^ {i} = \mathbf {Q} ^ {i \top} \mathbf {K} ^ {i} + \mathbf {Q} ^ {i \top} \mathbf {P} ^ {i} + \mathbf {u} ^ {i \top} \mathbf {K} ^ {i} + \mathbf {v} ^ {i \top} \mathbf {P} ^ {i}
+$$
+
+$$
+\mathbf {V} _ {\text {r e d u c e d}} ^ {i} = \text {M a s k e d - S o f t m a x} (\mathbf {A} ^ {i}) \mathbf {V} ^ {i}
+$$
+
+$$
+\mathbf {V} _ {\text {s k i p c o n n}} ^ {i} = \mathbf {V} _ {\text {r e d u c e d}} ^ {i} + \mathbf {E} ^ {i - 1}
+$$
+
+$$
+\mathbf {E} ^ {i} = \text {F e e d - F o r w a r d} \left(\mathbf {V} _ {\text {s k i p c o n n}} ^ {i}; \theta^ {i}\right),
+$$
+
+where  $\mathbf{Q}^i, \mathbf{K}^i, \mathbf{V}^i, \mathbf{P}^i$  are query, key, value, position matrices, respectively, for layer  $i$ .  $\mathbf{u}^i, \mathbf{v}^i, \mathbf{W}_q^i, \mathbf{W}_{k,E}^i, \mathbf{W}_v^i, \mathbf{W}_{k,R}^i$  and  $\theta^i$  are learnable model parameters of each INSNET layer. We denote the input word embeddings as  $\mathbf{E}^0$  for notation consistency. The last layer of transformer in INSNET outputs the step-wise context-aware token embeddings. They will be used to compute slot representations for the insertion steps.
+
+# 3.2 Slot Representation and Insertion Prediction
+
+With the context efficiently encoded, the subsequent step of an insertion-based generator is to predict the next position and token to insert. Thus, a representation for each potential insertion slot should be computed. We hereby show how slot representations can be aggregated from INSNET outputs and the potential challenges during this aggregation process.
+
+A naive design of the slot representation is to simply concatenate the representation vectors from the left-neighbor and right-neighbor (in the natural observation order) of the slot as is done in prior works (Stern et al., 2019). This slot representation can efficiently compute the slot representation for all possible slots in parallel for each time step. However, the context encoding we obtained is insertion-order sensitive and unaware of the tokens inserted later in INsNET. This naive slot representation thus falls short of capturing the global sequence information. As a remedy, we propose to also include the last insertion token's representation to compose the slot representation  $e_{i\downarrow j}^{(t)}$ . Specifically, given the representations of the left-side neighbor  $e_{i-}$ , the right-side neighbor  $e_{+j}$  and the last inserted token  $e_t$ , the slot representation can be computed as:
+
+$$
+\boldsymbol {e} _ {i \downarrow j} ^ {(t)} = \operatorname {L a y e r N o r m} \left(\left(f _ {l} \left(\boldsymbol {e} _ {i -}\right) \oplus f _ {r} \left(\boldsymbol {e} _ {+ j}\right)\right) + \boldsymbol {e} _ {t}\right)
+$$
+
+where  $f_{l}$  and  $f_{r}$  are linear projections for left-point and right-point representation vectors and  $\oplus$  stands for vector concatenation. The slot representation panel in Figure 2 (the bottom panel in yellow) illustrates the process. Note that comparing step 3 and step 4, both compute the slot representation for  $\mathrm{B}\downarrow < / \mathrm{s} >$ , but the resulting slot representation changes in response to a new step due to the introduction of the global vector.
+
+The slot representation can then be converted into a probabilistic distribution over the vocabulary to predict the log-likelihood of inserted tokens using a log-linear transformation i.e.  $\log p(x_{i\downarrow j}^{(t)}|e_{i\downarrow j}^{(t)}) =$  log-softmax  $(\mathbf{W}_p\pmb {e}_{i\downarrow j}^{(t)} + \pmb {b})_{x_{i\downarrow j}^{(t)}}$  . To decide which slot to insert next, a position logit  $\alpha_{k}^{(t)} = w_{o}^{\top}e_{i_{k}\downarrow j_{k}}^{(t)}$  is computed for each slot  $k$  with a linear layer. Then a (log-)soft-max operation is applied on top of the position logits to obtain the log-probability to insert to each candidate slot. i.e.  $\log o(x_{i_k\downarrow j_k}^{(t)}|\{e_\downarrow^{(t)}\}) =$  log-softmax  $([\alpha_0^{(t)},\alpha_1^{(t)},\dots ])_k$
+
+We apply a binary classification  $q(0/1|\cdot)$  on the last inserted token's representation  $e_t$  to decide the termination of the generation at each step. During training, only the final step  $q(0/1|e_{n-1})$  is trained to predict 1; all intermediate steps are trained to predict 0. In the following section, we will continue to discuss how to formulate the final objective function of INsNET using these step-wise distributions.
+
+# 3.3 Adaptive Parallelization of Insertions
+
+In addition to our efforts for better training efficiency, we also propose an algorithm to adaptively parallelize the generation process of INSNET to speed up the decoding while preserving the generation quality. The idea mimics the graph layerization process in the Dinic's algorithm (Dinitz, 2006). Specifically, assume that we partition the tokens into different layers, such that the tokens within the
+
+![](images/25be12e658b764e776147923266066ff22e42c6af2e41fbd469937abdad1ae24.jpg)  
+(a) Initial Layerization
+
+![](images/8d6469eed0e9f26fe6bca3bd4b7987c0e86672607d798af4e2c49d2e67114e81.jpg)  
+(b) Merged Layerization  
+Figure 5: (a) Initialized layerization of tokens after the pre-training stage. It assumes a fully sequential decoding process for now. (b) As the parallelization fine-tuning proceeds, the algorithm gradually merges different layers. Tokens within the same layer are safely parallelizable in decoding without sacrificing log-likelihood over the tolerance threshold  $\tau$ .
+
+same layer are safe to insert in parallel without affecting the coherence of context. The algorithm aims to determine the partition such that the log-likelihood estimation of the resulting parallel insertion-based model is as close to the sequential insertion-based models as possible. The intuition is that the sequential insertion process maximally captures the inter-dependencies between the output tokens. By staying close to the sequential insertion-based model regarding likelihood prediction ability, we speed up the inference without severely sacrificing the likelihood estimation quality.
+
+We initialize the layerization with a sequential insertion order, which is equivalent to single-node layerization, as is illustrated in Figure 5(a). Starting from this initialization, we perform INsNET-Dinic to gradually evolve this fully-sequential layerization into a non-autoregressive, parallelized layerization, as shown in Figure 5(b). We promote a token  $x \gets$  from its respective slot  $(i^l, j^l)$  in layer  $l$  to  $(i^{l - 1}, j^{l - 1})$  in the previous layer  $l - 1$  if two conditions are satisfied: 1) In layer  $l - 1$ , slot  $(i^{l - 1}, j^{l - 1})$  does not yet have an assigned insertion; 2) log  $p(x \gets |e_{i^l \downarrow j^l}^{(l)}) - \log p(x \gets |e_{i^{l - 1} \downarrow j^{l - 1}}^{(l - 1)}) \leq \tau$ .
+
+Here  $\tau$  is a hyper-parameter that controls how much to parallel. Intuitively, larger  $\tau$  usually indicates more tolerance for token likelihood loss caused by parallel decoding, i.e. more parallelization with tradeoff on likelihood estimation accuracy. The choice of  $\tau$  is highly task/data-dependent as it is a value associated with the likelihood. Note that the promotion of tokens only affects the slot representation computation and the insertion likelihood calculation of the respective insertion. When encoding the context (as is discussed in Section 3.1), we always treat it as if these insertion operations are performed sequentially.
+
+For a target sequence  $s_{<n} = [x_0, x_1, \dots, x_{n-1}]$ , after the layerization algorithm, if token  $x_{i\downarrow j}$  is layered in the  $l$ -th layer and to be inserted in the slot  $(i,j)$ , we denote it as  $x_{i\downarrow j}^{l}$ . Suppose we have  $m$  effective layers in total. Denote the corresponding representation for slot  $(i,j)$  given available context in layer  $l$  as  $e_{i\downarrow j}^{(l)}$  and the transformer's encoding of the partial context until the end of layer  $l$  as  $e_{-1}^{(l)}$ . We denote the token likelihood, position likelihood and termination likelihood of the token  $x_{i\downarrow j}^{l}$  to be  $p(x_{i\downarrow j}^{l}|e_{i\downarrow j}^{(l)})$ ,  $o(x_{i\downarrow j}^{l}|\{e_{\downarrow}^{(l)}\})$  and  $q(0/1|e_{-1}^{(l)})$ , respectively. The general objective of INsNET parameterized by  $\phi$  can be formulated as:
+
+$$
+\mathcal {L} (\boldsymbol {s}, \phi) = - \sum_ {l = 0} ^ {m - 1} \sum_ {x _ {i \downarrow j} ^ {l}} [ \log p _ {\phi} (x _ {i \downarrow j} ^ {l} | \boldsymbol {e} _ {i \downarrow j} ^ {(l)}) + \log o _ {\phi} (x _ {i \downarrow j} ^ {l} | \{\boldsymbol {e} _ {\downarrow} ^ {(l)} \}) ] - \sum_ {l = 0} ^ {m} \log q _ {\phi} (\boldsymbol {1} (l = m) | \boldsymbol {e} _ {- 1} ^ {(l)})
+$$
+
+# 4 Experiments
+
+We demonstrate the efficiency, flexibility, and model capability of INSNET with two sets of experiments. 1) To show INSNET's performance and the flexibility to switch between sequential and parallel decoding on datasets with high inter-token dependency (i.e., less suitable for parallel decoding). We follow the setup in Zhang et al. (2020) and address the unsupervised lexically constrained text generation problem on two datasets Yelp Review and News. 2) To further verify the effectiveness of INSNET-Dinic, we evaluate INSNET as a (partially) non-autoregressive machine translation model
+
+Table 2: Performance comparison on Yelp Review and News datasets. For Levenshtein Transformer, insertion and deletion stages are both counted in # of decoding-time iterations. It's non-trivial to do unsupervised lexically constrained text generation with auto-regressive models. To work around this, we implemented a Plan-And-Write(Yao et al., 2019) style auto-regressive transformer-based model for better reference. Models with a star mark * are re-implemented by us. Other baseline results are directly taken from the original papers.  
+
+<table><tr><td rowspan="2">Model</td><td rowspan="2">BLEU-2/4</td><td colspan="2">Yelp Review</td><td rowspan="2">BLEU-2/4</td><td colspan="2">News</td></tr><tr><td>NIST-2/4</td><td># Dec. Steps</td><td>NIST-2/4</td><td># Dec. Steps</td></tr><tr><td>Auto-regressive Transformer (Plan-And-Write-static, Yao et al. (2019))</td><td>16.68/5.46</td><td>2.79/2.86</td><td>39.24</td><td>8.79/2.40</td><td>1.65/1.67</td><td>36.74</td></tr><tr><td>NMSTG (Welleck et al., 2019)</td><td>10.06/1.92</td><td>1.11/1.12</td><td>27.92</td><td>10.67/1.58</td><td>2.70/2.70</td><td>27.85</td></tr><tr><td>InDIGO* (Gu et al., 2019a) (w/ Searched Adaptive Order)</td><td>16.14/4.63</td><td>3.08/3.10</td><td>45.63</td><td>13.89/3.62</td><td>3.08/3.10</td><td>26.78</td></tr><tr><td>Levenshtein Transformer (Parallel Decoding, (Gu et al., 2019b))</td><td>14.84/3.96</td><td>2.84/2.89</td><td>14.28</td><td>11.76/1.89</td><td>2.64/2.71</td><td>16.13</td></tr><tr><td>InsT-POINTER-Base (BERT init)</td><td>11.48/2.16</td><td>2.15/2.15</td><td>6.00</td><td>12.13/1.63</td><td>2.90/2.80</td><td>6.00</td></tr><tr><td>InsT-POINTER-Base (BERT init+Wiki)</td><td>15.63/3.32</td><td>3.27/3.30</td><td>6.00</td><td>13.01/2.51</td><td>3.04/3.06</td><td>6.00</td></tr><tr><td>InsT-POINTER-Large (BERT init+Wiki)</td><td>16.78/3.79</td><td>3.49/3.53</td><td>6.00</td><td>14.04/3.04</td><td>3.28/3.30</td><td>6.00</td></tr><tr><td>INSNET (Ours, Fully-Sequential)</td><td>19.36/5.78</td><td>3.51/3.54</td><td>48.73</td><td>16.31/4.96</td><td>3.10/3.13</td><td>32.69</td></tr><tr><td>INSNET-uniform (Ours)</td><td>12.31/2.30</td><td>2.19/2.17</td><td>7.00</td><td>12.89/2.01</td><td>2.99/2.90</td><td>7.00</td></tr><tr><td>INSNET-Dinic (Ours, τ = 10.0)</td><td>16.73/4.35</td><td>3.19/3.20</td><td>11.83</td><td>14.13/3.75</td><td>2.97/3.00</td><td>8.13</td></tr></table>
+
+with parallel decoding on three classical datasets: WMT Ro-En, WMT En-De and WAT En-Ja. The general setup about how hyper-parameters are determined can be found in appendix??
+
+# 4.1 Lexically Constrained Generation
+
+Experimental Setup Yelp Review dataset consists of 160K training sequences, 10K sequences for validation and 1k test sequences. News dataset consists of 268586 sentences in total, of which 10k are randomly selected as validation set, 1k for testing. YAKE<sup>1</sup> is performed to the test split of each dataset to extract lexical constraints.
+
+We vary the hyperparameter  $\tau$  in the range of  $\{10.0, 3.0, 1.0, 0.3, -\infty$  (fully-sequential) and collect the results on both datasets. For position prediction, we are inserting into slots with positions lying in the top  $70\%$  of the position distribution mass. For token prediction, we are doing top- $\{1, 1, 3, 3, 5\}$  sampling over the vocabulary distribution. The results are shown in Table 2, Figure 6(a) and 6(b).
+
+![](images/e36cb66c09e84a719621294fb43ca7786307e37c91e327107e1ee9a7eaac3b8c.jpg)  
+(a) Yelp Review
+
+![](images/56d64823aad6d10cdcbd056bb592c3358a16dc49ff8b7aeebb8af7e8c0effd31.jpg)  
+(b) News  
+Figure 6: Illustration of the spectral study results in terms of BLEU-4/Sequential Ratio in decoding on Yelp Review and News dataset. Sequential Ratio is computed by  $\frac{Avg.\#Dec.\text{Steps}}{Avg.\text{Length}}$   
+Discussion 1) INSNET is able to significantly outperform most previous works on the quality-latency trade-off spectrum. With more sequential flavor (i.e. smaller  $\tau$ ), it is generally to achieve the new state-of-the-art in generation quality. Compared to a Plan-And-Write style autoregressive baseline,
+
+Table 3: Generation examples with insertion steps. Every row shows 3 consecutive steps with orange, green, blue represents the first, second, and third insertion, respectively.  
+
+<table><tr><td>Input</td><td colspan="5">day decided started focus on</td><td colspan="5">local group hurt rule out</td></tr><tr><td>Step 3</td><td colspan="5">the day decided to started focus on .</td><td colspan="5">the local group hurt rule out of .</td></tr><tr><td>Step 6</td><td colspan="5">the day , he decided to get started focus on .</td><td colspan="5">the local group hurt the rule out of the of .</td></tr><tr><td>Step 9</td><td colspan="5">on the day , he decided to get started focus on the court .</td><td colspan="5">the local group hurt the government rule out of the of the year .</td></tr><tr><td>Step 12</td><td colspan="5">but on the next day , he decided to get started to focus on the court .</td><td colspan="5">the local group has hurt the government to rule out of the of the last year .</td></tr><tr><td>Step 15</td><td colspan="5">but , on the next day , he decided to get started to focus on the court for the .</td><td colspan="5">the local group has been hurt the government to rule out of for the rest of the last year .</td></tr><tr><td>Step 17</td><td colspan="5">but , on the next day , he decided to get started to focus on the court for the first time .</td><td colspan="5">the local group has been hurt by the government to rule out of support for the rest of the last year .</td></tr></table>
+
+INSNET guarantees the perfect incorporation of keywords due to its insertion-based nature. Also, its improved performance is consistent on both datasets. 2) When setting  $\tau$  larger to encourage more parallelization of decoding, INSNET is mostly able to parallelize the insertions with good preservation of the generative quality. Compared to the vanilla uniformly-parallel setup (INSNET-uniform), INSNET-Dinic significantly improved the effectiveness of parallel decoding.
+
+Generation sample We also show two samples of the generation process in Table 3. It is interesting to observe that the model tends to insert punctuation and function words (e.g., Articles and Prepositions) first, and then more concrete content words.
+
+# 4.2 Non-autoregressive Machine Translation
+
+We train INsNET-Dinic models with  $\tau = 2.00$  for extended investigation of the performance on machine learning problems. The results in Table 4 show that, INsNET-Dinic is able to achieve comparable or even better performance compared to previous state-of-the-art non-autoregressive (insertion-based) machine translation models in terms of generation quality and latency.
+
+# 4.3 Training Time Analysis
+
+To show the training efficiency of the proposed model, we hereby do an empirical and theoretical analysis on the candidate models' training procedure. See Table 5. For empirical results, the statistics are collected on the Yelp Review unsupervised lexically constrained text generation problem. All results are collected on a single NVIDIA RTX3090 GPU. The transformer Seq2seq baseline is trained
+
+Table 4: Machine translation evaluation. Each generation iteration of Levenshtein Transformer requires at least two full executions of the transformer model. Results with a star mark * are collected from our re-implementation. Other baseline results are directly taken from the original papers. The results for a vanilla transformer is taken from the LevT paper (Gu et al., 2019b).  
+
+<table><tr><td>Model</td><td>Ro-En</td><td>En-De</td><td>En-Ja</td><td>#Dec Step.</td><td>Latency</td></tr><tr><td>InDIGO-SAO (w/o KD) (Gu et al., 2019a)</td><td>32.47</td><td>26.14*</td><td>40.87*</td><td>n</td><td>516ms</td></tr><tr><td>InDIGO-random (w/o KD) (Gu et al., 2019a)</td><td>20.20</td><td>17.48*</td><td>23.91*</td><td>n</td><td>502ms</td></tr><tr><td>InsT-uniform (+KD) (Stern et al., 2019)</td><td>28.52</td><td>26.72</td><td>41.89*</td><td>Θ(log n) ≤ 10</td><td>107ms</td></tr><tr><td>InsT-binary (τ = 0.5, +KD) (Stern et al., 2019)</td><td>30.66</td><td>27.41</td><td>42.17*</td><td>Θ(log n) ≤ 10</td><td>92ms</td></tr><tr><td>LevT (+KD)(Gu et al., 2019b)</td><td>33.26</td><td>27.27</td><td>42.36</td><td>Θ(log n) ≤ 2 × 10</td><td>116ms</td></tr><tr><td>Vanilla Transformer (w/o KD)</td><td>32.30</td><td>27.17</td><td>43.68</td><td>n</td><td>389ms</td></tr><tr><td>INSNET-uniform</td><td>29.13</td><td>26.45</td><td>41.67</td><td>Θ(log n) ≤ 10</td><td>92ms</td></tr><tr><td>INSNET-Dinic (Ours, τ = 2.00)</td><td>33.41</td><td>27.36</td><td>43.71</td><td>Θ(log n) ≈ 15.8</td><td>105ms</td></tr><tr><td>+ KD</td><td>33.91</td><td>28.05</td><td>44.10</td><td>Θ(log n) ≈ 16.1</td><td>103ms</td></tr></table>
+
+Table 5: Empirical & theoretical comparative study of different algorithms' training efficiency. # of Steps/Seq means during training how many time the transformer needs to be executed per sequence.  
+
+<table><tr><td>Model</td><td>Ttrain/Epoch</td><td># of Steps/Seq</td></tr><tr><td>Auto-regressive Decoder Transformer</td><td>35min48s</td><td>1</td></tr><tr><td>InDIGO-SAO</td><td>58min36s + 2h12min(SAO)</td><td>1 + O(n)</td></tr><tr><td>Insertion Transformer-Sequential</td><td>26h31min</td><td>O(n)</td></tr><tr><td>Levenshtein Transformer-Sequential</td><td>37h24min</td><td>O(n)</td></tr><tr><td>Insertion Transformer</td><td>8h24min</td><td>Θ(log n)</td></tr><tr><td>Levenshtein Transformer</td><td>16h33min</td><td>Θ(log n)</td></tr><tr><td>InsT- POINTER</td><td>9h36min</td><td>Θ(log n)</td></tr><tr><td>INSNET/INSNET-Dinic</td><td>1h12min</td><td>1</td></tr></table>
+
+as a Plan-And-Write model. POINTER's training time per epoch is calculated with the official implementation with mixed precision supported by the NVIDIA APEX library  $^{2}$ .
+
+Discussion The results in Table 5 confirm the significant improvement in training efficiency of INSNET over most previous baselines. In addition, this concretely shows that it is not practically affordable to obtain a sequential insertion-based generator directly with InsT/LevT. Note that SAO of InDIGO is an offline, inference-only algorithm that does not require any computation of gradients. Thus its constant factor is significantly smaller than other  $O(n)$  gradient-propagating procedures. According to its original paper and our previous experiments, InDIGO practically needs this process to achieve reasonable performance.
+
+# 4.4 Ablation Study on the Effectiveness of Components
+
+Since the model components of InsNet are closely entangled, it is nontrivial to do ablation study. We design experiments to show that 1). The proposed offset matrix is a powerful insertion-oriented position encoding with significantly better capacity. We show that this particularly helps the model learn to terminate the sequence in correct timings. To study this, we truncate the offset matrix to range  $[-1.0, 1.0]$  (InsNet-truncated) so that it is similar to InDIGO's position encoding, and 2) the global representation is necessary for computing the slot representation in INSNET. It in general helps to improve the sequence termination performance, and leads to convergence to better local optimum. We report the results of an ablated version of InsNet without global representation (InsNet-noglobal).
+
+The termination NLL (see Sec 3.3 for its definition) and BLEU scores for each model can be found in appendix Table ??. We also show some random samples from the ablated variants and the failure cases can be find in appendix Table ??
+
+# 5 Conclusion & Future Work
+
+We propose INSNET, an efficient and performant insertion-based generator that supports sequential and parallel decoding. Experiments on two unsupervised lexically constrained text generation datasets and three machine translation datasets show the advantages of INSNET over previous methods.
+
+Future work can explore obtaining a large-scale pre-trained version of INsNET for further fine-tuning under different downstream scenarios. We anticipate such insertion-based models to have better compositional generalizability and controllability.
+
+# Acknowledgement
+
+We thank I-Hung Hsu, Dr. Rujun "RJ" Han, Te-Lin Wu, Kai-Wei Chang, Sarik Ghazarian, Alexander Spangher, Yining Hong, Mingyu Derek Ma and all other members from PlusLabNLP/UCLANLP group for their participation in initial discussions and comments on paper writing. We would like to thank Huggingface for their great work of the Transformers project. The work is partially supported by a Meta SRA and an Amazon research gift.
+
+# References
+
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+Tom B Brown, Benjamin Mann, Nick Ryder, Melanie Subbiah, Jared Kaplan, Prafulla Dhariwal, Arvind Neelakantan, Pranav Shyam, Girish Sastry, Amanda Askell, et al. 2020. Language models are few-shot learners. arXiv preprint arXiv:2005.14165.  
+Zihang Dai, Zhilin Yang, Yiming Yang, Jaime Carbonell, Quoc Le, and Ruslan Salakhutdinov. 2019. Transformer-XL: Attentive language models beyond a fixed-length context. In Proceedings of the 57th Annual Meeting of the Association for Computational Linguistics, pages 2978-2988, Florence, Italy. Association for Computational Linguistics.  
+Yefim Dinitz. 2006. Dinitz'algorithm: The original version and even's version. In Theoretical computer science, pages 218-240. Springer.  
+Seraphina Goldfarb-Tarrant, Tuhin Chakrabarty, Ralph Weischedel, and Nanyun Peng. 2020. Content planning for neural story generation with aristotelian rescoring. In the 2020 Conference on Empirical Methods in Natural Language Processing (EMNLP), pages 4319-4338.  
+Jiatao Gu, Qi Liu, and Kyunghyun Cho. 2019a. Insertion-based decoding with automatically inferred generation order. Transactions of the Association for Computational Linguistics, 7:661-676.  
+Jiatao Gu, Changhan Wang, and Jake Zhao. 2019b. Levenshtein transformer. arXiv preprint arXiv:1905.11006.  
+Rujun Han, Hong Chen, Yufei Tian, and Nanyun Peng. 2022. Go back in time: Generating flashbacks in stories with event temporal prompts. In 2022 Annual Conference of the North American Chapter of the Association for Computational Linguistics (NAACL).  
+He He, Nanyun Peng, and Percy Liang. 2019. Pun generation with surprise. In 2019 Annual Conference of the North American Chapter of the Association for Computational Linguistics (NAACL-HLT 2019), volume 1.  
+Christian F Hempelmann. 2008. Computational humor: Beyond the pun? The Primer of Humor Research. Humor Research, 8:333-360.  
+Jiwei Li, Will Monroe, Tianlin Shi, Sebastien Jean, Alan Ritter, and Dan Jurafsky. 2017a. Adversarial learning for neural dialogue generation. arXiv preprint arXiv:1701.06547.  
+Xiujun Li, Yun-Nung Chen, Lihong Li, Jianfeng Gao, and Asli Celikyilmaz. 2017b. End-to-end task-completion neural dialogue systems. arXiv preprint arXiv:1703.01008.  
+Yinhan Liu, Jiatao Gu, Naman Goyal, Xian Li, Sergey Edunov, Marjan Ghazvininejad, Mike Lewis, and Luke Zettlemoyer. 2020. Multilingual denoising pre-training for neural machine translation. Transactions of the Association for Computational Linguistics, 8:726-742.  
+Hisar Manurung, Graeme Ritchie, and Henry Thompson. 2000. Towards a computational model of poetry generation. Technical report, The University of Edinburgh.  
+Anirudh Mittal, Yufei Tian, and Nanyun Peng. 2022. Ambipun: Generating humorous puns with ambiguous context. In 2022 Annual Conference of the North American Chapter of the Association for Computational Linguistics (NAACL), short.  
+Alec Radford, Karthik Narasimhan, Tim Salimans, and Ilya Sutskever. 2018. Improving language understanding by generative pre-training.  
+Alec Radford, Jeffrey Wu, Rewon Child, David Luan, Dario Amodei, and Ilya Sutskever. 2019. Language models are unsupervised multitask learners. OpenAI blog, 1(8):9.  
+Peter Shaw, Jakob Uszkoreit, and Ashish Vaswani. 2018. Self-attention with relative position representations. arXiv preprint arXiv:1803.02155.
+
+Yong-Siang Shih, Wei-Cheng Chang, and Yiming Yang. 2019. Xl-editor: Post-editing sentences with xlnet. arXiv preprint arXiv:1910.10479.  
+Mitchell Stern, William Chan, Jamie Kiros, and Jakob Uszkoreit. 2019. Insertion transformer: Flexible sequence generation via insertion operations. In International Conference on Machine Learning, pages 5976-5985. PMLR.  
+Jiao Sun, Anjali Narayan-Chen, Shereen Oraby, Shuyang Gao, Tagyoung Chung, Jing Huang, Yang Liu, and Nanyun Peng. 2022. Context-situated pun generation. In Proceedings of the 2022 Conference on Empirical Methods in Natural Language Processing (EMNLP).  
+Bowen Tan, Zichao Yang, Maruan Al-Shedivat, Eric P Xing, and Zhiting Hu. 2020. Progressive generation of long text. arXiv preprint arXiv:2006.15720.  
+Yufei Tian, Divyanshu Arun Sheth, and Nanyun Peng. 2022. A unified framework for pun generation with humor principles. In Proceedings of the 2022 Conference on Empirical Methods in Natural Language Processing (EMNLP).  
+Yufei Tian and Nanyun Peng. 2022. Zero-shot sonnet generation with discourse-level planning and aesthetics features. In 2022 Annual Conference of the North American Chapter of the Association for Computational Linguistics (NAACL).  
+Ashish Vaswani, Noam Shazeer, Niki Parmar, Jakob Uszkoreit, Llion Jones, Aidan N Gomez, Lukasz Kaiser, and Illia Polosukhin. 2017. Attention is all you need. In NIPS.  
+Oriol Vinyals, Alexander Toshev, Samy Bengio, and Dumitru Erhan. 2015. Show and tell: A neural image caption generator. In Proceedings of the IEEE conference on computer vision and pattern recognition, pages 3156-3164.  
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+
+# Checklist
+
+1. For all authors...
+
+(a) Do the main claims made in the abstract and introduction accurately reflect the paper's contributions and scope? [Yes]  
+(b) Did you describe the limitations of your work? [Yes]  
+(c) Did you discuss any potential negative societal impacts of your work? [No]  
+(d) Have you read the ethics review guidelines and ensured that your paper conforms to them? [Yes]
+
+2. If you are including theoretical results...
+
+(a) Did you state the full set of assumptions of all theoretical results? [Yes]
+
+(b) Did you include complete proofs of all theoretical results? [Yes] Most theoretical results are self-explanatory.
+
+3. If you ran experiments...
+
+(a) Did you include the code, data, and instructions needed to reproduce the main experimental results (either in the supplemental material or as a URL)? [No] They will be released upon camera ready.  
+(b) Did you specify all the training details (e.g., data splits, hyperparameters, how they were chosen)? [Yes]  
+(c) Did you report error bars (e.g., with respect to the random seed after running experiments multiple times)? [No]  
+(d) Did you include the total amount of compute and the type of resources used (e.g., type of GPUs, internal cluster, or cloud provider)? [Yes]
+
+4. If you are using existing assets (e.g., code, data, models) or curating/releasing new assets...
+
+(a) If your work uses existing assets, did you cite the creators? [Yes]  
+(b) Did you mention the license of the assets? [Yes]  
+(c) Did you include any new assets either in the supplemental material or as a URL? [Yes]  
+(d) Did you discuss whether and how consent was obtained from people whose data you're using/curating? [Yes]  
+(e) Did you discuss whether the data you are using/curating contains personally identifiable information or offensive content? [No]
+
+5. If you used crowdsourcing or conducted research with human subjects...
+
+(a) Did you include the full text of instructions given to participants and screenshots, if applicable? [N/A]  
+(b) Did you describe any potential participant risks, with links to Institutional Review Board (IRB) approvals, if applicable? [N/A]  
+(c) Did you include the estimated hourly wage paid to participants and the total amount spent on participant compensation? [N/A]

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+# InsPro: Propagating Instance Query and Proposal for Online Video Instance Segmentation
+
+Fei He $^{1,2}$ , Haoyang Zhang $^{4}$ , Naiyu Gao $^{4}$ , Jian Jia $^{1,2}$ , Yanhu Shan $^{4}$ , Xin Zhao $^{1,2*}$ , Kaiqi Huang $^{1,2,3}$
+
+$^{1}$  CRISE, Institute of Automation, Chinese Academy of Sciences
+
+$^{2}$  School of Artificial Intelligence, University of Chinese Academy of Sciences
+
+3 CAS Center for Excellence in Brain Science and Intelligence Technology 4 Horizon Robotics
+
+{hefei2018, jiajian2018}@ia.ac.cn, {haoyang.zhang, naiyu01.gao, yanhu.shan}@horizon.ai{xzhao, kaiqi.huang}@nlpr.ia.ac.cn
+
+# Abstract
+
+Video instance segmentation (VIS) aims at segmenting and tracking objects in videos. Prior methods typically generate frame-level or clip-level object instances first and then associate them by either additional tracking heads or complex instance matching algorithms. This explicit instance association approach increases system complexity and fails to fully exploit temporal cues in videos. In this paper, we design a simple, fast and yet effective query-based framework for online VIS. Relying on an instance query and proposal propagation mechanism with several specially developed components, this framework can perform accurate instance association implicitly. Specifically, we generate frame-level object instances based on a set of instance query-proposal pairs propagated from previous frames. This instance query-proposal pair is learned to bind with one specific object across frames through conscientiously developed strategies. When using such a pair to predict an object instance on the current frame, not only the generated instance is automatically associated with its precursors on previous frames, but the model gets a good prior for predicting the same object. In this way, we naturally achieve implicit instance association in parallel with segmentation and elegantly take advantage of temporal clues in videos. To show the effectiveness of our method InsPro, we evaluate it on two popular VIS benchmarks, i.e., YouTube-VIS 2019 and YouTube-VIS 2021. Without bells-and-whistles, our InsPro with ResNet-50 backbone achieves 43.2 AP and 37.6 AP on these two benchmarks respectively, outperforming all other online VIS methods.
+
+# 1 Introduction
+
+Video instance segmentation (VIS) [1] is a challenging but important computer vision task. It requires not only segmenting object instances on each video frame but also associating them across all frames. Due to its fine-grained object representation form, it has got a wide range of applications in various areas such as autonomous driving and video editing.
+
+Existing VIS methods can be categorized into two groups: frame-level methods and clip-level methods. Frame-level methods [1, 2, 3, 4] generally follow a 'tracking-by-detection' paradigm, which first generate per-frame object instances by existing instance segmentation models [5, 6], and then associate them across frames via additional tracking heads (as shown in Figure 1 (a)). In comparison, clip-level methods [7, 8, 9, 10] take a 'clip-matching' paradigm, which divide a video
+
+![](images/268cb16941d2bb54403386cb983994bdd816d729754b51097e8aaf8d9f197b20.jpg)  
+(a) Previous VIS paradigm
+
+![](images/34520a48fc27e9fe333f2adc847a0ed5371f6c9ce15a08660a69b452a5c9bd10.jpg)  
+(b) Our InsPro  
+Figure 1: (a) Previous methods take a two-step approach to VIS. They first generate object instances and then perform explicit instance association to link them across frames. (b) Our InsPro implements implicit instance association through a temporal propagation mechanism, achieving object instance segmentation and tracking in one shot. It generates frame-level object instances based on a set of instance query and proposal pairs propagated from previous frames. Since the instance query-proposal pair is learned to represent one specific object across frames, the instance association is naturally achieved in parallel with segmentation and video temporal clues are elegantly exploited meanwhile.
+
+into multiple overlapped clips, generate instance predictions for each clip, and then associate these clip-level predictions by some hand-crafted instance matching algorithms. Whether frame-level or clip-level methods, both of them inevitably need an explicit instance association step to fulfill object tracking. This generally requires to design a complicated association strategy to achieve good tracking performance, which is not trivial. More importantly, the explicit association step increases model complexity and slows inference speed. Furthermore, this extra step also indicates that the temporal clues intrinsic in videos are not well utilized, as the instance prediction is performed separately on each frame or each clip.
+
+In this work, inspired by the recent success of query-based object detectors [11, 12], we propose a simple, fast and yet effective query-based framework for online VIS. Our system, dubbed as InsPro, segments and tracks objects in one shot through an instance query and proposal propagation strategy with carefully designed modules (Figure 1 (b)), which eliminates the explicit instance association step. Specifically, our approach generates frame-level object instances based on a set of instance query-proposal pairs propagated from previous frames. In the learning process, we develop several techniques to make sure that the generated instance query-proposal pair corresponds to one specific object across frames. Thus, when an object instance is generated using such a query-proposal pair on the current frame, it is automatically associated with its precursors on all previous frames. In this way, we achieve implicit object association without a linking step. Meanwhile, this instance query-proposal propagation mechanism also enables our VIS system to achieve a better prediction accuracy (see Table 1). This benefits from the instance query-proposal pair's encoding one object's temporal and spatial information across all previous frames, which provides a very good prior for the model to infer the same object on the current frame. In this sense, our query-based VIS method actually implements an efficient way to exploit the intrinsic temporal clues in videos.
+
+To fulfill the advantages of our VIS system, learning exclusive and expressive instance query-proposal pairs is the key. In this work, we develop several strategies to ensure the learning effectiveness. First, we design a temporally consistent matching mechanism to enforce the one-to-one correspondence between the instance query-proposal pair and a specific ground truth object across frames during training. Second, we propose a box dedduplication loss to enlarge the distance between instance proposals. This helps suppress duplicate proposals on the same object and increase the exclusivity of the generated instance query-proposal pair. At the same time, the sparsely distributed unoccupied query-proposal pairs can serve as candidates in the next frame to detect new objects, allowing our system to achieve new object detection and tracking effortlessly. Third, we propose an intra-query attention module that enhances instance query with its predecessors encoding the same object. This explicitly aggregates long-range object information into the query, augmenting its representation capacity, which helps handle occlusion and motion blur.
+
+To validate the effectiveness and efficiency of our InsPro, we conduct extensive experiments on two popular VIS benchmarks [1], i.e., YouTube-VIS 2019 and YouTube-VIS 2021. Without bells-and-whistles, our InsPro with ResNet-50 [13] backbone achieves 43.2 AP on YouTube-VIS 2019 and 37.6 AP on YouTube-VIS 2021 respectively, outperforming all other online VIS models. Moreover, our
+
+lite variant, InsPro-lite, reaches 38.7 AP on YouTube-VIS 2019 at impressive 45.7 FPS on a Nvidia RTX2080Ti GPU.
+
+In summary, we make the following contributions in this paper. 1) We propose a simple, fast and yet effective query-based framework for online VIS. 2) We develop several techniques to make the query-proposal pair propagation mechanism work smoothly. These techniques distinguish our work from other query propagation-based object association methods [14, 15, 16], and make our work simpler, more elegant and more effective than them. 3) Our VIS system achieves the state-of-the-art performances on two popular VIS benchmarks.
+
+# 2 Related Work
+
+Frame-level VIS Methods mainly adopt a 'tracking-by-detection' paradigm and can run in an online fashion. They first generate instance predictions frame by frame and then perform explicit instance association. MaskTrack R-CNN [1] first proposes the VIS task and simply adds an additional tracking head to Mask R-CNN [5] for instance association. Follow-up works [2, 3, 4, 17] improve either the segmentation or the tracking algorithm to achieve better performance. On the other hand, some works [8, 18, 19, 20, 21, 22] attempt to perform temporal feature fusion to improve instance segmentation and association. For example, PCAN [22] proposes frame- and instance-level prototypical cross-attention modules to leverage rich spatio-temporal information to facilitate better segmentation. All these methods require additional modules to achieve explicit instance association, which expands model complexity and reduces inference speed. By contrast, our method performs instance association implicitly through an instance query and proposal propagation mechanism, which is simpler and naturally exploits the temporal and spatial consistency in videos.
+
+Clip-level VIS Methods take a 'clip-matching' paradigm, which process multiple frames within a clip simultaneously and then perform instance matching between clips to complete VIS. While some methods [7, 23, 9] propagate instance information within a clip with well-designed propagation modules to model temporal context, recent works [8, 10] utilize transformer [24] to model temporal context in an end-to-end manner. These methods normally need hand-crafted matching algorithms to complete instance association between clips. Although they usually achieve high performance, they can only run in an offline mode, which restricts their application to limited areas. In contrast, our method achieves comparable performance but can run online.
+
+Query-based Methods have attracted increasing attention in recent years due to their flexibility and simplicity. DETR [11] first uses a set of learned queries interacting with image features to encode objects, and then directly outputs detections by decoding the transformed queries. Following works [25, 26, 27, 28, 29] improve DETR in terms of either training efficiency or detection performance. Sparse R-CNN [12] builds a query-based detector on top of R-CNN architecture [30, 31]. Its follow-up works [3, 32] extend it to instance segmentation and video object detection. Besides, Max-DeepLab [33] proposes a box-free panoptic segmentation method with external query. DAFL [34] uses a set of queries to encode pedestrian information for pedestrian attribute recognition.
+
+The success of DETR has also inspired query-based VIS methods. VisTR [8] adapts DETR to the VIS task. It takes a video clip as input and directly outputs the sequence of masks for each instance orderly. IFC [10] proposes inter-frame communication transformers to reduce the heavy computation and memory usage of VisTR-like VIS methods. Similar to VisTR, Mask2Former [35] applies masked attention to a video clip and directly predicts a 3D instance volume. To learn a powerful video-level instance query, SeqFormer [36] aggregates temporal information from each frame to the instance query. These methods work on clips rather than frames, and achieve object association through sharing of the queries within a clip rather than query propagation. Thus, they still need instance matching between clips. Instead, our method applies to frames, and can propagate query-proposal pairs through the entire video and thereby can associate object instances over any frame length.
+
+Query Propagation-based Object Association Methods have been recently explored in several works, such as TransTrack [14], TrackFormer [15], MOTR [16] and EfficientVIS [37], which are also inspired by query-based methods [11, 12]. This shows the effectiveness and potential of such a new object linking approach. The differences between our InsPro and them are as follows.
+
+![](images/aaa4b3f75fe54aa6955135d3e5722d77eb9b619056439fb3cae9ac2bdf08d23f.jpg)  
+(a) Overview of InsPro
+
+![](images/54846e9511fda037d5f7b25ef9558159a3b3bf866b1a26c4799a430da0ec0372.jpg)  
+(b) Details of SegHead  
+Figure 2: (a) Overview of our InsPro. It performs VIS by propagating instance query-proposal pairs across frames.  $\pmb{q}_{init} \in \mathbb{R}^{N \times C}$  and  $\pmb{p}_{init} \in \mathbb{R}^{N \times 4}$  are initial instance queries and proposals on the first video frame, respectively. They are used in SegHead to predict instance results  $\pmb{r}_0$  on frame  $I_0$ , and to produce updated  $\pmb{q}_0$  and  $\pmb{p}_0$  which are propagated to the next frame. By repeating this process, we complete the VIS task. (b) Details of SegHead. It is a multi-stage network, consisting of a dynamic instance interaction module (DIIM) and an instance segmentation module. The former transforms instance queries with RoI features of corresponding proposals and produces object features, while the latter predicts object instances based on the object features and conditional convolution [39].
+
+First, our InsPro is different from them in the way of either tracking seen objects or detecting new objects. TransTrack is basically a 'tracking-by-detection' method, because it still needs to explicitly match detection boxes to tracked boxes in each frame, while our InsPro performs implicit association. More importantly, TransTrack, TrackFormer and MOTR adopt a track query subset to track seen objects and an extra object query subset to detect new objects. This requires additional heuristic rules to combine two type queries, and may miss occluded or blurred objects with low scores, which can result in object trajectory break [38]. Our InsPro simply propagates all object queries produced in the previous frame to the current frame, and keeps using this set to track seen objects and detect new objects, which is much simpler and more elegant. As for EfficientVIS, our concurrent work, it does not consider this new object detection problem, and its performance will probably be impacted greatly if there are new objects in the next clip.
+
+Furthermore, we design a more intelligent strategy to suppress duplicates. TransTrack and TrackFormer employ score filtering or NMS to reduce duplicate predictions. MOTR builds a temporal aggregation network to learn more discriminative features to address this problem, while EfficientVIS does not discuss this problem. By contrast, we design a Box Dedduplication Loss to suppress duplicates and an Inter-query attention module to enhance queries with their predecessors. Our solution avoids heuristic rules and post-processing steps, and is more effective according to the experimental results (see Table 2 (e)).
+
+# 3 InsPro
+
+We aim to design a simple and fast VIS system that performs instance association implicitly and exploits video temporal clues elegantly. To this end, we take a query-based VIS approach that predicts object instances on each frame based on a set of instance query-proposal pairs propagated from previous frames. In this section, we introduce our VIS system, InsPro, including an instance query and proposal propagation mechanism and an instance segmentation head. Meanwhile, we also describe those proposed techniques that make our propagation mechanism work well.
+
+# 3.1 Instance Query and Proposal Propagation
+
+The instance query and proposal propagation mechanism enables our VIS system to perform object instance association implicitly in parallel with instance segmentation. Since it is inspired by the recent query-based object detector Sparse R-CNN [12], we first briefly review Sparse R-CNN.
+
+Sparse R-CNN [12] formulates object detection as a set prediction problem and achieves state-of-the-art performance. It simplifies the detection pipeline and removes heuristic components like NMS. Specifically, it first initializes a fixed set of learnable instance queries  $(N\times C,N$  denotes the number of queries and  $C$  the query dimension) paired with learnable instance proposals  $(N\times 4)$  to describe objects in an image. As illustrated in Figure 2 (b), each instance query is convolved with the RoI feature of the corresponding proposal to output a more discriminate feature  $o_{t}$  [12]. After multi-stage iterative updating, the instance query encodes more accurate object appearance information while the proposal captures more precise location. Finally, decoding the object feature  $o_{t}$  produced using the instance query-proposal pairs, we get the detection results.
+
+Inspired by this instance query-proposal representation of an object, we design a query-proposal temporal propagation mechanism (as shown in Figure 2 (a)) to achieve implicit object instance association and temporal cue utilization in VIS. Our key insight is that there is a one-to-one correspondence between the learned instance query-proposal pair and a specific object. If we manage to preserve this correspondence from the first frame to the one where the object finally disappears, then we realize object tracking and object information propagation spontaneously.
+
+To this end, we first initialize a set of instance queries  $\pmb{q}_{init} \in \mathbb{R}^{N \times C}$  and proposals  $\pmb{p}_{init} \in \mathbb{R}^{N \times 4}$  on the first video frame  $I_0$ , where  $\pmb{q}_{init}$  and  $\pmb{p}_{init}$  are learnable parameters and arranged in pairs. After learning, they are able to encode objects on the first frame. Decoding them with the first frame image feature inside the SegHead, we obtain instance results  $\pmb{r}_0$  as well as a new set of updated pairs  $(\pmb{q}_0, \pmb{p}_0)$ . Then we propagate this pair set  $(\pmb{q}_0, \pmb{p}_0)$  to the next frame as input to the SegHead. Similarly, we get the instance results  $r_1$  and another new set of  $(\pmb{q}_1, \pmb{p}_1)$  on this frame. Among them, the object instance produced on this frame shares the same ID with the one on the previous frame if they are both decoded by the same slice of the instance queries. In this way, we automatically link object instances belonging to an identical object across frames and elegantly make use of object priors from the past. Repeating the above process until the last video frame, we then accomplish the VIS task on this video.
+
+Please note that our InsPro simply propagates all object queries produced in the previous frame to the current frame, and keeps using this set to track seen objects and detect new objects. Instead, recent works [14, 15, 16] that take a similar query-propagation mechanism use a track query set to track seen objects and a new object query set to detect new objects respectively, which requires additional heuristic rules to combine these two type queries. Moreover, they rely on hand-crafted rules like a score threshold to select a subset of track queries, and occluded objects with low prediction scores are probably filtered out, which results in non-negligible true object missing and fragmented trajectories [38]. In comparison, our method is obviously simpler, more elegant and more effective (see Table 2 (e)).
+
+Intra-query Attention Since frame-by-frame temporal propagation encodes only short-range temporal cues, the instance query from just the last frame shows limitations in dealing with tough scenarios, e.g., occlusion and motion blur. To boost the representation capacity of instance query, we augment it in practice with instance features from previous  $T$  frames [40, 41]. Specifically, we build a feature bank that caches instance features from previous  $T$  frames and perform intra-query attention inside this bank to aggregate long-range temporal cues into the current instance query, as shown in the upper part of Figure 2 (b). Formally, at frame  $I_{t}$ , instance features  $\mathbf{o}$  from previous  $T$  frames are put together to form a feature bank  $fb = \{o_{t - T + 1},\dots,o_t\}$ . Then, the enhanced instance query is computed as:
+
+$$
+\boldsymbol {q} _ {t} ^ {i} = \frac {\sum_ {n = 0} ^ {T - 1} \boldsymbol {o} _ {t - n} ^ {i} \exp (\varepsilon \left(\boldsymbol {o} _ {t - n} ^ {i}\right))}{\sum_ {m = 0} ^ {T - 1} \exp (\varepsilon \left(\boldsymbol {o} _ {t - m} ^ {i}\right))} + \boldsymbol {o} _ {t} ^ {i}, \tag {1}
+$$
+
+where  $i$  denotes the  $i$ -th query and  $\varepsilon(\cdot)$  is a linear transformation function. The enhanced  $\mathbf{q}_t$  is basically a weighted sum of instance features inside the feature bank, and the weights are learned upon the quality of the queries. Experiments (Table 6 (c)) show that this augmentation improves the query representation capacity greatly.
+
+# 3.2 Segmentation Head
+
+The segmentation head transforms the instance query and performs instance segmentation on each frame. As illustrated in Figure 2 (b), it is a multi-stage network and has two main parts: a dynamic instance interaction module and an instance segmentation module.
+
+Dynamic Instance Interaction Module transforms the instance query with the proposal RoI feature and yields object features. It has  $M$  stages and forms an iterative structure. At the first stage, given a pair of instance queries  $\pmb{q}_{t-1} \in \mathbb{R}^{N \times C}$  and proposals  $\pmb{p}_{t-1} \in \mathbb{R}^{N \times 4}$  that propagated from the previous frame  $I_{t-1}$ , it first augments the instance queries by a self-attention module [24], which models the inter-query relations. At the same time, it extracts the RoI feature of each proposal on the feature map by RoIAAlign [5]. Then, each enhanced instance query convolves with its corresponding RoI feature through dynamic convolution [42] to get the object feature  $\pmb{o}_t \in \mathbb{R}^{N \times C}$ . Since the object feature absorbs temporal cues encoded in instance queries, it has better representation ability than the single-frame RoI feature (see Table 2 (d)). The object feature is used to predict object instances in the following instance segmentation module, and the newly generated object boxes together with the object features proceed as input to the next stage in the iterative process. At the final stage, the object feature is processed together with its precursors from previous frames through the aforementioned intra-query attention module, to produce instance queries for the next frame.
+
+Instance Segmentation Module decodes the object feature  $\pmb{o}_t$  and produces VIS predictions. It has three main heads. While the classification head predicts object classes, the regression one generates object boxes. Another mask head is responsible for producing instance masks through a conditional convolution [39] approach. Specifically, it first uses  $\pmb{o}_t$  to generate conditional convolution weights in weight generator. As shown in Figure 2 (b), it inputs  $\pmb{o}_t^i$  that represents the  $i$ -th object instance feature to the weight generator and outputs a set of convolution parameters  $\omega_i$ . Meanwhile, it produces mask feature maps by transforming FPN [43] feature maps through a mask branch. Note that the output mask feature maps have 8 channels and a  $\frac{1}{8}$  resolution of the input image, and are boosted by concatenating a 2-channel relative coordinates map to it. This relative coordinates map is computed using the center of predicted object boxes and provides strong location cues for predicting instance masks. Finally, we feed the combined feature maps  $f_t^{mask} \in \mathbb{R}^{10 \times \frac{H}{8} \times \frac{W}{8}}$  and convolution parameters  $\omega_i$  to a mask FCN head, predicting the  $i$ -th object instance mask via a conditional convolution:
+
+$$
+\boldsymbol {m} _ {t} ^ {i} = \operatorname {C o n d C o n v} \left(\boldsymbol {f} _ {t} ^ {\text {m a s k}}, \boldsymbol {\omega} _ {i}\right). \tag {2}
+$$
+
+For more details about the instance segmentation module, please refer to [39].
+
+# 3.3 Temporally Consistent Matching
+
+The key to the success of our InsPro is to make sure that the evolving instance query-proposal pair corresponds to the same object across frames in a video. To ensure this, one technique we propose is temporally consistent matching. This technique matches predictions and ground truth during training, assigns each ground truth object a proper prediction, and propagates the matching made on previous frames to subsequent frames.
+
+Specifically, given a training batch consisting of multiple consecutive frames, we first compute the matching cost  $L_{match}$  between predictions and ground truth objects on the first frame:
+
+$$
+\mathcal {L} _ {\text {m a t c h}} = \lambda_ {c l s} \cdot \mathcal {L} _ {c l s} + \lambda_ {L 1} \cdot \mathcal {L} _ {L 1} + \lambda_ {\text {g i o u}} \cdot \mathcal {L} _ {\text {g i o u}}, \tag {3}
+$$
+
+where  $\mathcal{L}_{cls}$  is the focal loss [44] between predicted classifications and ground-truth labels,  $\mathcal{L}_{L1}$  and  $\mathcal{L}_{\mathrm{giou}}$  are L1 loss and the generalized IoU loss [45] between predicted boxes and ground-truth boxes, respectively.  $\lambda_{cls}$ ,  $\lambda_{L1}$  and  $\lambda_{\mathrm{giou}}$  are loss weights and set as 2, 5, and 2, respectively.
+
+We search for the best bipartite matching that minimizes the matching cost  $L_{match}$  with the Hungarian algorithm [46]. After finding the best matching on the first frame, we propagate this matching to other frames. Concretely, if one ground truth object still exists on subsequent frames, it will be matched to the prediction that is generated by the same instance query on the first frame. If there are new objects emerging, new matching will be made between the new objects and yet unmatched predictions. If a ground truth object disappears, its corresponding predictions will not participate in a new matching process. Through this temporally consistent matching mechanism, we bind one ground truth object to a single instance query during training.
+
+# 3.4 Loss Function
+
+Box Dedduplication Loss Although the self-attention mechanism between queries has driven the model to generate fewer duplication predictions [11], we still observe multiple overlapped proposal
+
+![](images/d766ed51bf8c31808a23f63d8b28a6839b3f16986c5d10fd33c8193c31896c0f.jpg)  
+Figure 3: (a) Multiple duplicate boxes exist on the same object across frames. (b) After applying the proposed box dedduplication loss (BDL) in training, the duplicate predictions are significantly suppressed along with temporal propagation.
+
+boxes on the same object across many frames, as displayed in Figure 3 (a). We conjecture this is caused by those unmatched queries which cannot be pushed away from those matched queries due to lack of supervision. To address this problem, we propose a box dedduplication loss to push away prediction boxes in terms of the center-to-center distance between them. As a result, not only the duplication problem is alleviated, but the sparsely distributed unmatched query-proposal pairs can serve as candidates in the next frame to detect and track new objects (see Figure 7 in Appendix). The loss is defined as:
+
+$$
+\mathcal {L} _ {\text {d e d u p}} = \frac {1}{k} \sum_ {i = 1} ^ {k} \max  (\beta - \frac {\mathcal {C} ^ {2} (\boldsymbol {b} , \hat {\boldsymbol {b}} _ {\text {n e g}} ^ {i})}{\mathcal {D} ^ {2} (\boldsymbol {b})}, 0), \tag {4}
+$$
+
+where  $\pmb{b}$  is a ground truth box,  $\hat{\pmb{b}}_{\text{neg}}^i$  is a negative box that has the  $i$ -th largest IoU with  $\pmb{b}$  among those unmatched predicted boxes,  $\mathcal{C}(\cdot)$  is the center distance calculation function, and  $\mathcal{D}(\cdot)$  is the diagonal length calculation function.  $\beta$  is set as 0.1 and  $k$  as 5. This loss penalizes the short distance between  $\pmb{b}$  and  $\hat{\pmb{b}}_{\text{neg}}^i$ , and drags all other duplicate boxes away from  $\pmb{b}$  [47]. With this new loss, our final box loss function is formed as:
+
+$$
+\mathcal {L} _ {b o x} = \lambda_ {L 1} \cdot \mathcal {L} _ {L 1} + \lambda_ {\text {g i o u}} \cdot \mathcal {L} _ {\text {g i o u}} + \lambda_ {d e d u p} \cdot \mathcal {L} _ {d e d u p}, \tag {5}
+$$
+
+where  $\lambda_{L1}$  and  $\lambda_{\mathrm{giou}}$  have the same values as in Equation 3, and  $\lambda_{dedup}$  is set as 1.
+
+Overall Loss Given the one-to-one matching results, the final loss on each training frame is a sum of classification, box and mask losses:
+
+$$
+\mathcal {L} = \lambda_ {c l s} \cdot \mathcal {L} _ {c l s} + \lambda_ {b o x} \cdot \mathcal {L} _ {b o x} + \lambda_ {d i c e} \cdot \mathcal {L} _ {d i c e} + \lambda_ {f o c a l} \cdot \mathcal {L} _ {f o c a l}, \tag {6}
+$$
+
+where  $\mathcal{L}_{\text{dice}}$  and  $\mathcal{L}_{\text{focal}}$  are dice loss [48] and focal loss [44] for foreground mask prediction, respectively. We set  $\lambda_{box} = 1$ ,  $\lambda_{dice} = 5$  and  $\lambda_{focal} = 5$ . Finally, the losses of all training frames inside a batch are summed together and normalized by the number of frames.
+
+# 4 Experiments
+
+# 4.1 Datasets and Evaluation Metrics
+
+We evaluate our method on YouTube-VIS 2019 and 2021 benchmarks [1]. YouTube-VIS 2019 consists of 2,238 training videos and 302 validation videos, and labels 40 object categories. YouTube-VIS 2021 is an extended version, which comprises 2,985 training videos and 421 validation videos, and labels improved 40 categories. All videos in these two datasets are annotated every 5 frames with object bounding box, object category, instance mask and instance ID. Following [1], we report the video-level average precision (AP) and average recall (AR) on the validation sets as the evaluation metrics, where both accurate instance segmentation and instance association are necessary to achieve high performance.
+
+Table 1: Comparison of our InsPro to state-of-the-art methods. All methods use ResNet-50 as backbone. C: additionally using COCO train2017 images that contain YouTube-VIS categories for training. The inference speed is tested on a Nvidia RTX2080Ti GPU. * indicates using deformable convolution [52] in backbone. ‡ indicates that the FPS is measured by parallel processing of images in one clip rather than sequential processing.  
+
+<table><tr><td colspan="2"></td><td colspan="5">YouTube-VIS 2019 Val.</td><td colspan="5">YouTube-VIS 2021 Val.</td></tr><tr><td>Method</td><td>Online</td><td>AP</td><td>\( AP_{50} \)</td><td>\( AP_{75} \)</td><td>\( AR_1 \)</td><td>\( AR_{10} \)</td><td>AP</td><td>\( AP_{50} \)</td><td>\( AP_{75} \)</td><td>\( AR_1 \)</td><td>\( AR_{10} \)</td></tr><tr><td>STEm-Seg [7] (C)</td><td>✘</td><td>30.6</td><td>50.7</td><td>33.5</td><td>31.6</td><td>37.1</td><td>-</td><td>-</td><td>-</td><td>-</td><td>4.4</td></tr><tr><td>VisTR [8]</td><td>✘</td><td>35.6</td><td>56.8</td><td>37.0</td><td>35.2</td><td>40.2</td><td>-</td><td>-</td><td>-</td><td>-</td><td>\( 30.0^‡ \)</td></tr><tr><td>Propose-Reduce [9] (C)</td><td>✘</td><td>40.4</td><td>63.0</td><td>43.8</td><td>41.1</td><td>49.7</td><td>-</td><td>-</td><td>-</td><td>-</td><td>&lt; 20</td></tr><tr><td>MaskProp* [23]</td><td>✘</td><td>40.0</td><td>-</td><td>42.9</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>&lt; 10</td></tr><tr><td>IFC [10]</td><td>✘</td><td>39.0</td><td>60.4</td><td>42.7</td><td>41.7</td><td>51.6</td><td>35.2</td><td>57.2</td><td>37.5</td><td>-</td><td>\( 46.5^‡ \)</td></tr><tr><td>EfficientVIS [37]</td><td>✘</td><td>37.9</td><td>59.7</td><td>43.0</td><td>40.3</td><td>46.6</td><td>34.0</td><td>57.5</td><td>37.3</td><td>33.8</td><td>\( 42.5 \)</td></tr><tr><td>MaskTrack R-CNN [1]</td><td>✓</td><td>30.3</td><td>51.1</td><td>32.6</td><td>31.0</td><td>35.5</td><td>28.6</td><td>48.9</td><td>29.6</td><td>26.5</td><td>33.8</td></tr><tr><td>SipMask [4]</td><td>✓</td><td>33.7</td><td>54.1</td><td>35.8</td><td>35.4</td><td>40.1</td><td>31.7</td><td>52.5</td><td>34.0</td><td>30.8</td><td>37.8</td></tr><tr><td>STMask* [20]</td><td>✓</td><td>33.5</td><td>52.1</td><td>36.9</td><td>31.1</td><td>39.2</td><td>-</td><td>-</td><td>-</td><td>-</td><td>28.6</td></tr><tr><td>SG-Net [2]</td><td>✓</td><td>34.8</td><td>56.1</td><td>36.8</td><td>35.8</td><td>40.8</td><td>-</td><td>-</td><td>-</td><td>-</td><td>23.0</td></tr><tr><td>PCAN [22]</td><td>✓</td><td>36.1</td><td>54.9</td><td>39.4</td><td>36.3</td><td>41.6</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td></tr><tr><td>CrossVIS [18]</td><td>✓</td><td>36.3</td><td>56.8</td><td>38.9</td><td>35.6</td><td>40.7</td><td>34.2</td><td>54.4</td><td>37.9</td><td>30.4</td><td>38.2</td></tr><tr><td>HybridVIS [21] (C)</td><td>✓</td><td>41.3</td><td>61.5</td><td>43.5</td><td>42.7</td><td>47.8</td><td>35.8</td><td>56.3</td><td>39.1</td><td>33.6</td><td>40.3</td></tr><tr><td>InsPro-lite</td><td>✓</td><td>38.7</td><td>60.9</td><td>41.7</td><td>36.9</td><td>43.6</td><td>-</td><td>-</td><td>-</td><td>-</td><td>45.7</td></tr><tr><td>InsPro</td><td>✓</td><td>40.2</td><td>62.9</td><td>43.1</td><td>37.6</td><td>44.5</td><td>36.1</td><td>57.6</td><td>39.6</td><td>30.9</td><td>40.4</td></tr><tr><td>InsPro (C)</td><td>✓</td><td>43.2</td><td>65.3</td><td>48.0</td><td>38.8</td><td>49.0</td><td>37.6</td><td>58.7</td><td>40.9</td><td>32.7</td><td>41.4</td></tr></table>
+
+# 4.2 Implementation Details
+
+We implement our InsPro with Detector2 [49], and most hyperparameters are set following Sparse R-CNN [12] and CondInst [39] unless otherwise specified. More implementation details can be found in Appendix A.1.
+
+Training Details We employ AdamW [50] with an initial learning rate of  $2.5 \times 10^{-5}$  and weight decay 0.0001 as our model optimizer. We initialize our model with parameters pre-trained on COCO [51], and train it for 32k iterations where the learning rate is divided by 10 at iterations 24k and 28k, respectively. The training is performed end-to-end on 8 Nvidia RTX2080Ti GPUs and each GPU holds one mini-batch which contains three frame images randomly sampled from the same video. Data augmentation includes only random horizontal flip and multi-scale training where the training image is resized so that the length of its shortest side is at least 288 and at most 512. Unless otherwise noted, our InsPro adopts ResNet-50 [13] as backbone and uses 100 instance queries in our experiments.
+
+Inference Details In inference, we resize the frame image size to  $640 \times 360$ , which follows MaskTrack R-CNN [1]. The size of the feature bank is set to 18 by default. If the generated proposal box exceeds the frame's boundaries, it will be clipped to corresponding boundaries. No multi-scale testing is adopted in our experiments.
+
+InsPro-lite We also build a lite version of our method, named InsPro-lite. In this variant, inspired by [32], we divide video frames into key frames and non-key frames, i.e., we select one key frame per  $K$  frames in a video and treat other frames as non-key ones.  $K$  is 10 by default. On key frames, we conduct the dynamic instance interaction 6 times while only once on non-key frames. This takes advantage of the redundancy of videos and helps reduce inference computation time. Our InsPro-lite reaches a high inference speed of 45.7 FPS at a small accuracy loss (Table 1).
+
+# 4.3 Main Results
+
+We perform a thorough comparison of our InsPro to state-of-the-art VIS methods on YouTube-VIS 2019 and 2021. Existing VIS methods can be divided into two categories according to whether they run online or offline [53]. Since some methods [7, 9] use  $80\mathrm{k}$  transformed COCO images [51] as extra training data to prevent overfitting to YouTube-VIS, for a fair comparison, we also report our
+
+(a) Effectiveness of instance query and proposal propagation, and temporally consistent matching (TCM).
+
+Table 2: Ablation studies on YouTube-VIS 2019.  
+
+<table><tr><td></td><td>query</td><td>proposal</td><td>TCM</td><td>AP</td><td>\( AP_{50} \)</td><td>\( AP_{75} \)</td></tr><tr><td>(A)</td><td></td><td></td><td></td><td>24.0</td><td>41.3</td><td>24.2</td></tr><tr><td>(B)</td><td>✓</td><td>✓</td><td></td><td>36.3</td><td>56.3</td><td>38.9</td></tr><tr><td>(C)</td><td>✓</td><td>✓</td><td>✓</td><td>37.4</td><td>57.6</td><td>41.1</td></tr><tr><td>(D)</td><td>✓</td><td></td><td>✓</td><td>36.7</td><td>57.3</td><td>39.9</td></tr><tr><td>(E)</td><td></td><td>✓</td><td>✓</td><td>36.6</td><td>55.5</td><td>40.3</td></tr></table>
+
+(b) Effectiveness of the proposed box dedduplication loss (BDL).
+
+<table><tr><td></td><td>AP</td><td>\( AP_{50} \)</td><td>\( AP_{75} \)</td><td>FPS</td></tr><tr><td>w/o BDL</td><td>37.4</td><td>57.6</td><td>41.1</td><td>26.3</td></tr><tr><td>w/ BDL</td><td>38.4</td><td>57.7</td><td>41.6</td><td>26.3</td></tr></table>
+
+(c) Intra-query attention.  $T$  is the length of the feature bank.
+
+<table><tr><td></td><td>AP</td><td>AP50</td><td>AP75</td><td>FPS</td></tr><tr><td>T=1</td><td>38.4</td><td>57.7</td><td>41.6</td><td>26.3</td></tr><tr><td>T=9</td><td>39.7</td><td>61.6</td><td>42.1</td><td>26.3</td></tr><tr><td>T=18</td><td>40.2</td><td>62.9</td><td>43.1</td><td>26.3</td></tr><tr><td>T=27</td><td>40.1</td><td>62.6</td><td>42.2</td><td>26.3</td></tr><tr><td>T=36</td><td>40.1</td><td>62.5</td><td>42.2</td><td>26.3</td></tr></table>
+
+(d) Comparison between our temporal propagation paradigm and 'tracking-by-detection' paradigm.
+
+<table><tr><td></td><td>AP</td><td>\( AP_{50} \)</td><td>\( AP_{75} \)</td><td>Param (M)</td><td>FLOPs (G)</td><td>FPS</td></tr><tr><td>Tracking-by-detection</td><td>31.5</td><td>49.3</td><td>34.1</td><td>119.9</td><td>48.3</td><td>25.4</td></tr><tr><td>Ours</td><td>37.4</td><td>57.6</td><td>41.1</td><td>106.1</td><td>45.5</td><td>26.3</td></tr></table>
+
+(e) Comparison between our united query and 'track-and-detect query'.
+
+<table><tr><td></td><td>AP</td><td>\( AP_{50} \)</td><td>\( AP_{75} \)</td></tr><tr><td>Track-and-detect query</td><td>37.4</td><td>56.9</td><td>40.3</td></tr><tr><td>Ours</td><td>38.4</td><td>57.7</td><td>41.6</td></tr></table>
+
+results with and without extra COCO training data. Table 1 presents all the results obtained with a ResNet-50 backbone on a Nvidia RTX2080Ti GPU.
+
+YouTube-VIS 2019 Table 1 (left) shows the comparison between our InsPro and other state-of-the-art methods on YouTube-VIS 2019 validation set. We can see that, in the online group, our InsPro outperforms all other popular methods under the same data setting. Specifically, our InsPro achieves 40.2 AP without COCO data and 43.2 AP with COCO data respectively, surpassing other online VIS methods by a large margin. Even our lite version, InsPro-lite, performs better than all other online methods trained without COCO data, reaching 38.7 AP at an impressive speed of 45.7 FPS.
+
+YouTube-VIS 2021 Table 1 (right) displays results on YouTube-VIS 2021 validation set. It shows a similar comparison pattern to YouTube-VIS 2019 and our InsPro achieves the state-of-the-art performance once again.
+
+# 4.4 Ablation Study
+
+We conduct extensive experiments on YouTube-VIS 2019 to study the effectiveness and individual performance contribution of our proposed modules.
+
+Temporal Propagation and Matching Our InsPro is built on the proposed instance query and proposal temporal propagation mechanism. Table 2 (a) shows how this mechanism contributes to our high performance. In this table, method A represents the video instance segmentation baseline, where each frame is processed individually without any temporal propagation, and object instances generated on each frame are linked if they are produced from the same instance query slice. Since this method lacks the mechanism to ensure the instance query-proposal pair corresponds to the same object across frames, it only achieves 24.0 AP due to inaccurate instance association. By contrast, when we add the temporal propagation (method B), we can easily improve the performance significantly to 36.3 AP. This evidences the importance and effectiveness of the proposed temporal propagation technique in a query-based VIS framework. If we further adopt the temporally consistent matching strategy during training (method C), we achieve an even better performance of 37.4 AP.
+
+We also analyze the separate performance of propagating only instance query (method D) or instance proposal (method E). The results show that these two settings achieve a similar performance boost (36.7 AP vs 36.6 AP). Applying them together yields 37.4 AP, bringing further performance gain.
+
+Box Dedduplication Loss We propose a box dedduplication loss to suppress the duplicate proposal boxes on the same object across frames. With the qualitative results shown in Figure 3, we show the quantitative comparison in Table 2 (b). We can see that supervising the learning with this loss during training can lead 1.0 AP improvement (38.4 AP vs 37.4 AP). This performance gain is brought by fewer duplicate boxes and fewer missed detections.
+
+Intra-query Attention We perform intra-query attention inside a feature bank to augment the instance query so that it can capture long-range temporal cues. As we can see in Table 2 (c), this simple method works well and improves the performance considerably. In particular,  $T = 1$  indicates no intra-query attention is used and 38.4 AP is achieved. When we increase the volume  $T$  of the feature bank, the performance rises and saturates at 40.2 AP with  $T = 18$ . It is worthwhile to note that this lightweight yet effective intra-query attention module brings almost no speed drop.
+
+Temporal Propagation vs. Tracking-by-Detection Despite the fact that our InsPro does not perform explicit instance association, it still outperforms all other online methods implementing explicit tracking or matching. To verify that our superior performance comes from the temporal propagation mechanism rather than the instance segmentation model design, we compare our temporal propagation VIS approach to the typical 'tracking-by-detection' paradigm with the same instance segmentation baseline. We implement a 'tracking-by-detection' VIS system by replacing the Mask R-CNN part in MaskTrack R-CNN [1] with our instance segmentation model. In this case, the only independent variable is the object tracking method.
+
+As shown in Table 2 (d), our InsPro surpasses the 'tracking-by-detection' model by a large margin even if our design is simpler and faster, which soundly proves the effectiveness of our method. We argue again that this is because the evolving instance query-proposal pair in propagation encodes object temporal and spatial cues intrinsic in videos, whereas 'tracking-by-detection' methods are generally incapable of exploiting this advantage.
+
+Our United Query vs. Track-and-Detect Query We further compare our method to those MOT methods that adopt a similar query-propagation method for object tracking. These methods rely on two different query sets, i.e., a track query set and an object query set, to track seen objects and detect new objects respectively, while we only maintain one united query set. They need heuristic rules to combine these two type queries. Meanwhile, they manually select track queries with high scores from the previous frame to build the track query set. This makes them complex and less effective in tracking since occluded objects with low scores probably have broken trajectories because of the filtering.
+
+To show the superiority of our method, we compare the 'track-and-detect query' paradigm adopted in the most recent MOTR [16] to ours using the same instance segmentation baseline. We follow MOTR [16] exactly to set up the model and experiment settings. To exclude the influence of other factors, we do not use temporal feature aggregation in both methods. Table 2 (e) shows the comparisons on YouTube-VIS 2019. It can be seen that our InsPro achieves a higher performance even using a simpler query design. We attribute this advantage to our conscientiously designed modules described in Sec 3.
+
+# 5 Conclusion
+
+In this paper, we propose a simple, fast and yet effective query-based framework for online VIS. In this framework, we rely on a novel instance query and proposal propagation mechanism to undertake VIS, where we generate object instances based on a set of evolving instance query-proposal pairs propagated from previous frames. This mechanism enables our model not only to associate object instances implicitly, but to utilize video temporal cues elegantly. To make this propagation mechanism work well, we develop several modules to ensure that the learned instance query-proposal pair keeps being bound to one object. These modules include an intra-query attention unit, a temporally consistent matching mechanism and a box dedduplication loss. Extensive experiments on YouTube-VIS 2019 and 2021 verify the effectiveness of our designs, and show that our InsPro achieves superior VIS performance, outperforming all other online VIS methods.
+
+# Acknowledgments
+
+This work is supported in part by the National Natural Science Foundation of China (Grant No. 61721004), the Projects of Chinese Academy of Science (Grant No. QYZDB-SSW-JSC006), the Strategic Priority Research Program of Chinese Academy of Sciences (Grant No. XDA27000000), and the Youth Innovation Promotion Association CAS.
+
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+
+# Checklist
+
+1. For all authors...
+
+(a) Do the main claims made in the abstract and introduction accurately reflect the paper's contributions and scope? [Yes]  
+(b) Did you describe the limitations of your work? [Yes] We discuss limitations and future work in Appendix.  
+(c) Did you discuss any potential negative societal impacts of your work? [Yes] We discuss the potential negative societal impacts in Appendix.  
+(d) Have you read the ethics review guidelines and ensured that your paper conforms to them? [Yes]
+
+2. If you are including theoretical results...
+
+(a) Did you state the full set of assumptions of all theoretical results? [N/A]  
+(b) Did you include complete proofs of all theoretical results? [N/A]
+
+3. If you ran experiments...
+
+(a) Did you include the code, data, and instructions needed to reproduce the main experimental results (either in the supplemental material or as a URL)? [Yes] We list the data and instructions in Sec. 4 and Appendix, and the code will be released upon publication.  
+(b) Did you specify all the training details (e.g., data splits, hyperparameters, how they were chosen)? [Yes] All training details are listed in Sec. 4 and Appendix.  
+(c) Did you report error bars (e.g., with respect to the random seed after running experiments multiple times)? [No]  
+(d) Did you include the total amount of compute and the type of resources used (e.g., type of GPUs, internal cluster, or cloud provider)? [Yes]
+
+4. If you are using existing assets (e.g., code, data, models) or curating/releasing new assets...
+
+(a) If your work uses existing assets, did you cite the creators? [Yes]  
+(b) Did you mention the license of the assets? [No]  
+(c) Did you include any new assets either in the supplemental material or as a URL? [No]  
+(d) Did you discuss whether and how consent was obtained from people whose data you're using/curating? [Yes]  
+(e) Did you discuss whether the data you are using/curating contains personally identifiable information or offensive content? [Yes]
+
+5. If you used crowdsourcing or conducted research with human subjects...
+
+(a) Did you include the full text of instructions given to participants and screenshots, if applicable? [N/A]  
+(b) Did you describe any potential participant risks, with links to Institutional Review Board (IRB) approvals, if applicable? [N/A]  
+(c) Did you include the estimated hourly wage paid to participants and the total amount spent on participant compensation? [N/A]

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+# Instability and Local Minima in GAN Training with Kernel Discriminators
+
+Evan Becker
+
+Dept. CS
+
+UCLA
+
+evbecker@cs.ucla.edu
+
+Parthe Pandit
+
+HDSI
+
+UC, San Diego
+
+parthepandit@ucsd.edu
+
+Sundep Rangan
+
+Dept. ECE
+
+NYU
+
+srangan@nyu.edu
+
+Alyson K. Fletcher
+
+Dept. Statistics
+
+UCLA
+
+akfletcher@ucla.edu
+
+# Abstract
+
+Generative Adversarial Networks (GANs) are a widely-used tool for generative modeling of complex data. Despite their empirical success, the training of GANs is not fully understood due to the min-max optimization of the generator and discriminator. This paper analyzes these joint dynamics when the true samples as well as the generated samples are discrete, finite sets, and the discriminator is kernel-based. A simple yet expressive framework for analyzing training called the Isolated Points Model is introduced. In the proposed model, the distance between true samples greatly exceeds the kernel width, so each generated point is influenced by at most one true point. Our model enables precise characterization of the conditions for convergence, both to good and bad minima. In particular, the analysis explains two common failure modes: (i) an approximate mode collapse and (ii) divergence. Numerical simulations are provided that predictably replicate these behaviors.
+
+# 1 Introduction
+
+Generative Adversarial Networks (GANs) are the most widely-used method for learning generative models of complex and structured data in an unsupervised manner. Indeed, GANs have seen incredible empirical success in a wide variety of domains ranging from image generation, speech generation, text generation, and many more. Models trained in this manner have also become critical in downstream applications. See [1, 29] for an overview.
+
+In the GAN methodology, a generator model is trained to output samples that emulate a target dataset, which we call true samples. A critic model, called the discriminator, is trained to tell apart (discriminate) the true and generated samples. The generator is trained in parallel to fool the discriminator.
+
+Correctly tuning the joint training of the discriminator and generator is one of the key challenges in GANs and is the source of several empirically observed problematic phenomena. For example, it is well-known that the resulting distributions can suffer from mode collapse and catastrophic forgetting. The optimization can also lead to divergence or slow convergence of min-max optimization algorithms. See [10, 19] for more details. Practical GANs methods overcome these issues with a combination of careful hyper-parameter optimization and heuristics. Significant effort has strived to develop theoretical frameworks that can better analyze and optimize GAN training.
+
+![](images/190ffa361715efd7af721bb6d8d9cf6a86ee27106129b75155a59f62ab3a57bb.jpg)  
+Figure 1: (Isolated Points Model) An illustration of our results. Each isolated region (blue circle) has a single true point (green disk) with point mass  $p_i$ . There are eight generated points with fixed point mass  $\widetilde{p}_j = 1/8$  starting at the locations shown in the orange squares. In  $V_1$  and  $V_3$ , the excess point mass is non-negative  $(\Delta_i \geq 0)$  and the generated points converge to the true point. In  $V_2$ , the excess point mass  $\Delta_4 < 0$  and the generated points converge to a stable equilibrium around the true point analogous of mode collapse. Finally, a point outside all four isolated regions may diverge to  $\infty$  in a linear velocity.
+
+In this work, we propose a simple theoretical model, called the Isolated Points Model, that is analytically tractable and allows us to rigorously study the stability and convergence properties of training a GAN. In the proposed model, the true and generated points are discrete distributions over finite sets, and the discriminator is kernel-based meaning that it is linearly parametrized. We make an additional critical assumption that the true points are sufficiently separated such that the kernel interaction between points near two distinct true points is negligible. For distance-based kernels, this assumption essentially requires that the true points are separated much greater than the kernel width. A simple example of this model with four true points is illustrated in Fig. 1.
+
+Main Contributions We show that this simple isolated points model provides sufficient richness to exhibit several interesting phenomena and insights:
+
+(1) Local stability and instability: We provide necessary and sufficient conditions for local convergence of generated points to a true point within each isolated region (Theorem 1). The results show that the stability is determined by the excess point mass, meaning the difference between the true and generated mass in the region. A consequence of these results, Corollary 1, is that exact mode collapse where an excess of generated points concentrate in a single true point, is provably not possible.  
+(2) Approximate mode collapse: Although an excess of generated point mass cannot exactly concentrate on a single true point, we prove (see Theorem 2) the existence of locally stable equilibria where an arbitrarily large excess point mass concentrates near the true point. We call this phenomena approximate mode collapse.  
+(3) Divergence: We also demonstrate that given a initial perturbation, an isolated generated point can lock into an trajectory moving away from any true point in an arbitrary direction. Interestingly, this trajectory is driven solely by the generated point's own kernel history.  
+(4) Role of the kernel width: The results provide insights into the role of the kernel width in training - See Section 6. In particular, wider kernel widths reduce the likelihood of different modes of the distribution being isolated, which is the critical for the approximate mode collapse and divergence that we observe. At the same time, discriminators with wide kernels make are unable to distinguish points that are close, resulting in slow convergence near the true distribution.
+
+Prior work Convergence problems for GANs have been widely-recognized and studied since their inception [1, 29]. Indeed, many of the developments in GANs, notably the popular Wasserstein GAN and its variants, were motivated to overcome these issues [2, 3, 13, 17].
+
+For analytic tractability, we focus on a relatively simple GAN with a kernel discriminator and maximum mean discrepancy loss. This methodology has been applied in other works such as [6,9,12,27]. Our focus is on joint optimization of a kernel discriminator and a generator with multiple discrete points. An important avenue of future work would be to extend our results to more complex losses such as the Wasserstein loss [2,3,13].
+
+The early Dirac-GAN [19], and its extension [26], considered a linear discriminator and a single Dirac-distribution for the generator and true data. Our model extends this work by considering multiple points and a general kernel discriminator. Importantly our model allows for a new parameter called the excess point mass which enables modeling complex behaviours in the isolated regions.
+
+Our results also heavily rely on linearization methods derived from control theory, which have been studied in the GAN context in [21, 22, 30]. See, also general functional descriptions in [9, 16, 20, 23]. These prior works demonstrated the local stability of the generated distribution close to the true distribution. Stability results with a single-layer NN generator and linear discriminator has been studied in [4]. Due to the isolated points model, our results also prove the existence of locally stable bad local minima.
+
+Metrics for understanding the performance of generative models remains an open challenge. Some efforts in this direction were presented in [8, 25], and we take advantage of this progress to report our numerical simulations.
+
+# 2 Isolated Points Model
+
+We propose the following model for studying the training of GANs.
+
+Discrete true and generated distributions: We assume that the true and generated distributions,  $\mathbb{P}_r$  and  $\mathbb{P}_g$ , are over discrete sets in  $\mathbb{R}^d$ :
+
+$$
+\mathbb {P} _ {r} (\boldsymbol {x}) = \sum_ {i = 1} ^ {N _ {r}} p _ {i} \delta (\boldsymbol {x} - \boldsymbol {x} _ {i}), \quad \mathbb {P} _ {g} (\boldsymbol {x}) = \sum_ {j = 1} ^ {N _ {g}} \widetilde {p} _ {j} \delta (\boldsymbol {x} - \widetilde {\boldsymbol {x}} _ {j}), \tag {1}
+$$
+
+where  $N_{r}$  and  $N_{g}$  are the number of true and generated points,  $\pmb{X} = \{\pmb{x}_i\}_{i=1}^{N_r}$  and  $\widetilde{\pmb{X}} = \{\widetilde{\pmb{x}}_j\}_{j=1}^{N_g}$  are the true and generated points, respectively, and  $\{p_i\}$  and  $\{\widetilde{p_j}\}$  are their probabilities. For the generator, we assume that the probabilities  $\widetilde{p_j}$  are fixed. The problem is to learn the point mass locations  $\widetilde{\pmb{X}}$  so that the generated and true distributions match.
+
+Kernel discriminator: We consider a GAN where the discriminator has a linear parametrization of the form:
+
+$$
+f (\boldsymbol {x}, \boldsymbol {\theta}) = a (\boldsymbol {x}) ^ {\top} \boldsymbol {\theta}, \tag {2}
+$$
+
+for some vector of basis functions  $a(\pmb{x})$  and parameter vector  $\theta$ . The discriminator is trained to maximize a maximum mean discrepancy (MMD) metric [6,9,12,27]
+
+$$
+\mathcal {L} _ {d} \left(\boldsymbol {\theta}, \left\{\widetilde {\boldsymbol {x}} _ {j} \right\} _ {j = 1} ^ {N _ {g}}\right) := \sum_ {i} p _ {i} f \left(\boldsymbol {x} _ {i}, \boldsymbol {\theta}\right) - \sum_ {j} \widetilde {p} _ {j} f \left(\widetilde {\boldsymbol {x}} _ {j}, \boldsymbol {\theta}\right) - \frac {\lambda}{2} \| \boldsymbol {\theta} \| ^ {2}, \tag {3}
+$$
+
+for some regularization parameter  $\lambda > 0$ .
+
+Discriminator updates: We assume simple gradient ascent of the MMD metric (3):
+
+$$
+\boldsymbol {\theta} ^ {k + 1} = \boldsymbol {\theta} ^ {k} + \eta_ {d} \left(\sum_ {i} p _ {i} a \left(\boldsymbol {x} _ {i}\right) - \sum_ {j} \widetilde {p} _ {j} a \left(\widetilde {\boldsymbol {x}} _ {j}\right) - \lambda \boldsymbol {\theta} ^ {k}\right), \tag {4}
+$$
+
+where  $\eta_d > 0$  is the discriminator step-size. If we let  $f^k(\boldsymbol{x}) \coloneqq f(\boldsymbol{x}, \theta^k)$ , then for any fixed  $\boldsymbol{x}$ , we have
+
+$$
+f ^ {k + 1} (\boldsymbol {x}) = f ^ {k} (\boldsymbol {x}) + \eta_ {d} \left(\sum_ {i = 1} ^ {N _ {r}} p _ {i} K (\boldsymbol {x}, \boldsymbol {x} _ {i}) - \sum_ {j = 1} ^ {N _ {g}} \widetilde {p} _ {j} K (\boldsymbol {x}, \widetilde {\boldsymbol {x}} _ {j}) - \lambda f ^ {k} (\boldsymbol {x})\right), \tag {5}
+$$
+
+where  $K(\pmb {x},\pmb{x}^{\prime})$  is the kernel:
+
+$$
+K \left(\boldsymbol {x}, \boldsymbol {x} ^ {\prime}\right) := a (\boldsymbol {x}) ^ {\intercal} a \left(\boldsymbol {x} ^ {\prime}\right). \tag {6}
+$$
+
+To make the analysis concrete, some of our results will apply to the radial basis function (RBF) kernel
+
+$$
+K \left(\boldsymbol {x}, \boldsymbol {x} ^ {\prime}\right) = e ^ {- \frac {1}{2 \sigma^ {2}} \left\| \boldsymbol {x} - \boldsymbol {x} ^ {\prime} \right\| ^ {2}}, \tag {7}
+$$
+
+where  $\sigma > 0$  is the kernel width.
+
+Generator updates: We will assume the generator distribution  $\mathbb{P}_g$  in (1) is directly parameterized by the point mass locations,  $\widetilde{\pmb{X}} = \{\widetilde{\pmb{x}}_j\}_{j=1}^{N_g}$ , that are updated to minimize the loss:
+
+$$
+\mathcal {L} _ {g} \left(\boldsymbol {\theta}, \left\{\widetilde {\boldsymbol {x}} _ {j} \right\} _ {j = 1} ^ {N _ {g}}\right) := - \sum_ {j} \widetilde {p} _ {j} f \left(\widetilde {\boldsymbol {x}} _ {j}, \boldsymbol {\theta}\right). \tag {8}
+$$
+
+We consider a simple gradient descent update for this loss
+
+$$
+\widetilde {\boldsymbol {x}} _ {j} ^ {k + 1} = \widetilde {\boldsymbol {x}} _ {j} ^ {k} + \eta_ {g} \widetilde {p} _ {j} \nabla f ^ {k} \left(\widetilde {\boldsymbol {x}} _ {j} ^ {k}\right). \tag {9}
+$$
+
+Together, (5) and (9) define the joint dynamics of the GAN.
+
+Isolated Points: Our final key assumption is that the true samples are separated far enough so that there exists a non-empty isolated neighborhood  $V_{i}$  around each sample  $\pmb{x}_i$  such that,
+
+$$
+K \left(\boldsymbol {x}, \boldsymbol {x} ^ {\prime}\right) = 0 \text {f o r a l l} \boldsymbol {x} \in V _ {i} \text {a n d} \boldsymbol {x} ^ {\prime} \in V _ {j} \text {f o r a l l} i \neq j \tag {10}
+$$
+
+In other words, the samples are separated sufficiently far apart such that they are outside the width of the kernel evaluated at another sample. See Figure 1 for an illustration.
+
+The assumption (10) is obviously strict and is an idealization of what occurs in practice. The supplementary material discusses modifications of the results to the case where  $|K(\pmb{x},\pmb{x}^{\prime})| \leq \epsilon$  for some small  $\epsilon$  and all  $\pmb{x} \in V_{i}$  and  $\pmb{x}^{\prime} \in V_{j}$ .
+
+Now consider an isolated region  $V_{i}$  around a true point  $\pmb{x}_i$ . At each training step  $k$ , let  $N_i^k$  be the set of indices  $j$  such that the generated points  $\widetilde{\pmb{x}}_j^k \in V_i$ . We further suppose that  $N_i^k$  is constant over time, so the points  $\widetilde{\pmb{x}}_j^k$  do not enter or exit this region. Then, their dynamics within the region are given by:
+
+$$
+f ^ {k + 1} (\boldsymbol {x}) = f ^ {k} (\boldsymbol {x}) + \eta_ {d} \left(p _ {i} K (\boldsymbol {x}, \boldsymbol {x} _ {i}) - \sum_ {j \in N _ {i}} \widetilde {p} _ {j} K (\boldsymbol {x}, \widetilde {\boldsymbol {x}} _ {j} ^ {k}) - \lambda f ^ {k} (\boldsymbol {x})\right) \quad \forall \boldsymbol {x} \in V _ {i}, \tag {11a}
+$$
+
+$$
+\widetilde {\boldsymbol {x}} _ {j} ^ {k + 1} = \widetilde {\boldsymbol {x}} _ {j} ^ {k} + \eta_ {g} \widetilde {p} _ {j} \nabla f ^ {k + 1} \left(\widetilde {\boldsymbol {x}} _ {j} ^ {k}\right) \quad \forall j \in N _ {i} \tag {11b}
+$$
+
+We will call the updates (11) the dynamical system in the region  $V_{i}$ .
+
+We may also choose to write the updates (11) in terms of the components of  $\pmb{\theta}$ . Let  $\pmb{\theta}_i \coloneqq \pmb{P}_i\pmb{\theta}$  where  $\pmb{P}_i$  is the projection onto the range space of  $a(\pmb{x})$  for  $\pmb{x} \in V_i$ . Then, it can be verified that the updates in (11) can be written as:
+
+$$
+\boldsymbol {\theta} _ {i} ^ {k + 1} = \boldsymbol {\theta} _ {i} ^ {k} + \eta_ {d} \left(p _ {i} a (\boldsymbol {x} _ {i}) - \sum_ {j \in N _ {i}} \widetilde {p} _ {j} a \left(\widetilde {\boldsymbol {x}} _ {j} ^ {k}\right) - \lambda \boldsymbol {\theta} _ {i} ^ {k} (\boldsymbol {x})\right) \tag {12a}
+$$
+
+$$
+\widetilde {\boldsymbol {x}} _ {j} ^ {k + 1} = \widetilde {\boldsymbol {x}} _ {j} ^ {k} + \eta_ {g} \widetilde {p} _ {j} \nabla f \left(\widetilde {\boldsymbol {x}} _ {j} ^ {k}, \boldsymbol {\theta} _ {i}\right), \quad \forall j \in N _ {i}. \tag {12b}
+$$
+
+We will use the notation
+
+$$
+\widetilde {\boldsymbol {X}} _ {i} = \left\{\widetilde {\boldsymbol {x}} _ {j}, j \in N _ {i} \right\}, \tag {13}
+$$
+
+to denote the set of generated points  $\widetilde{\boldsymbol{x}}_j$  in the isolated region. The state variables for the dynamics (12) can be represented by the pair  $(\theta_i, \widetilde{\boldsymbol{X}}_i)$ . Note that we allow separate learning rates for the two update equations. Our convergence result in Theorem 1 relies on the ratio of these learning rates.
+
+# 3 Behavior Near the True Point
+
+Consider an isolated region  $V_{i}$  around some true point  $\pmb{x}_i$ . We first analyze the dynamics where
+
+$$
+\widetilde {\boldsymbol {x}} _ {j} ^ {k} \approx \boldsymbol {x} _ {i} \forall j \in N _ {i}. \tag {14}
+$$
+
+That is, all the generated points in  $V_{i}$  are close to the true point. We follow a standard linearization analysis used in several other GAN works such as [6, 12, 18]. However, a critical difference in our model is that the generated mass may not equal the true mass in a particular isolated region. We thus define the probability mass difference
+
+$$
+\Delta_ {i} := p _ {i} - \sum_ {j \in N _ {i}} \widetilde {p} _ {j}, \tag {15}
+$$
+
+which represents the difference the true probability mass in the region,  $p_i$ , and total probability mass of the generated points in that region. Note that this parameter is missing from the analysis of Dirac-GAN [20] since  $\Delta_i = 0$  trivially when only one true point exists. Several numerical behaviors of GANs emerge due to  $\Delta_i \neq 0$ .
+
+Assumption 1. The kernel  $K(\pmb{x}, \pmb{x}^{\prime})$  is smooth and, at each true point  $\pmb{x}_i$ :
+
+$$
+\left. \frac {\partial K (\boldsymbol {x} , \boldsymbol {x} _ {i})}{\partial \boldsymbol {x}} \right| _ {\boldsymbol {x} = \boldsymbol {x} _ {i}} = \mathbf {0}, \tag {16}
+$$
+
+and
+
+$$
+\left. - \frac {\partial^ {2} K (\boldsymbol {x} , \boldsymbol {x} _ {i})}{\partial \boldsymbol {x} ^ {2}} \right| _ {\boldsymbol {x} = \boldsymbol {x} _ {i}} \in [ k _ {1}, k _ {2} ] \boldsymbol {I}, \quad \left. \frac {\partial^ {2} K (\boldsymbol {x} , \boldsymbol {x} ^ {\prime})}{\partial \boldsymbol {x} \partial \boldsymbol {x} ^ {\prime}} \right| _ {\boldsymbol {x} = \boldsymbol {x} ^ {\prime} = \boldsymbol {x} _ {i}} \in [ k _ {3}, k _ {4} ] \boldsymbol {I}, \tag {17}
+$$
+
+for some  $k_{1}, k_{2}, k_{3}, k_{4} > 0$  where we use the notation that  $\pmb{Q} \in [q_{1}, q_{2}] \pmb{I}$  to mean  $q_{1} \pmb{I} \preceq \pmb{Q} \preceq q_{2} \pmb{I}$ .
+
+The condition is mild and simply requires that the kernel  $K(\pmb{x}, \pmb{x}_i)$  has a local maxima at  $\pmb{x} = \pmb{x}_i$  with negative curvature, and also satisfies a strict positivity condition. The assumption, for example, is satisfied by the RBF kernel (7) with  $k_1 = k_2 = k_3 = k_4 = 1 / \sigma^2$ .
+
+Theorem 1. Fix any isolated region  $V_{i}$ , suppose the kernel satisfies Assumption 1 and let
+
+$$
+\widetilde {\boldsymbol {X}} _ {i} ^ {*} = \left\{\widetilde {\boldsymbol {x}} _ {j} ^ {*}, j \in N _ {i} \right\}, \quad \widetilde {\boldsymbol {x}} _ {j} ^ {*} = \boldsymbol {x} _ {i}. \tag {18}
+$$
+
+Then, there is a unique  $\pmb{\theta}_i^*$  such that  $(\pmb{\theta}_i^*, \overline{\pmb{X}}_i^*)$  is an equilibrium point of the GAN dynamics (12) in the region  $V_i$ . In addition, if the parameters  $k_i$ ,  $\lambda$  and  $\mu = \eta_g / \eta_d$  are fixed, then, for sufficiently small step size  $\eta_d$ :
+
+(a) If  $\Delta_i > 0$ , the equilibrium point is locally stable.  
+(b) If  $\Delta_{i} < 0$  and  $|N_{i}| \geq 1$ , the equilibrium point is locally unstable.  
+(c) If  $|N_{i}| = 1$  and
+
+$$
+\mu \Delta_ {i} k _ {1} \widetilde {p} _ {1} + \min  \left\{\lambda^ {2}, \mu \widetilde {p} _ {1} ^ {2} k _ {3} \right\} > 0, \tag {19}
+$$
+
+the equilibrium point is locally stable.
+
+(d) If  $|N_{i}| = 1$  and
+
+$$
+\mu \Delta_ {i} k _ {2} \widetilde {p} _ {1} + \min  \left\{\lambda^ {2}, \mu \widetilde {p} _ {1} ^ {2} k _ {4} \right\} <   0, \tag {20}
+$$
+
+the equilibrium point is locally unstable.
+
+In cases (c) and (d),  $\widetilde{p}_1$  denotes the point mass of the single generated point in region.
+
+The theorem provides simple necessary and sufficient conditions on the stability of equilibrium points at the true point. In case (a), we see that when the probability mass difference,  $\Delta_i > 0$ , the generated points will converge locally to the true point. That is, the generated points will converge to the true point as long as the true point mass exceeds the total generated mass. Also, (19) will always be satisfied when  $\Delta_i = 0$ . So, when  $|N_i| = 1$  (i.e., there is a single generated point), the generated point will converge to the true point when the generated and true point have the same mass. Examples of these convergence situations are shown in Figure 1, in  $V_1$  (where  $\Delta_1 = 0$ ) and  $V_3$  (where  $\Delta_3 > 0$ ).
+
+Conversely, case (b) shows that when  $\Delta_{i} < 0$  (i.e. the generated probability exceeds the true probability), the generated mass can no longer stably settle in the true point. This case is shown in  $V_{2}$  in Fig. 1 where the generated points are repelled from the true points.
+
+Single vs. multiple points: Note there is a slight difference between the cases when  $|N_i| = 1$  (i.e., there is a single generated point in the region), and  $|N_i| > 1$ , when there is more than one generated point. As the proof of the theorem shows, when  $|N_i| > 1$ , the mean of the generated points may converge to the true point, but the system may have other modes that are unstable in other directions.
+
+Boundary cases: Our theorem does not discuss certain boundary cases. For example, when  $\Delta_i = 0$  and  $|N_i| > 1$ , the theorem does not state whether the equilibrium point is stable or unstable. These boundary cases are standard in linearization analyses when the eigenvalues of the linearized system are only critically stable. Critically stable linear systems will have limit cycles [28]. The stability of general nonlinear systems will be determined by the higher order dynamics, which, in our case, would be determined by the higher order terms of the kernel. However, standard non-linear systems theory results such as [28] show that any such points cannot be exponentially stable. Hence, if the points locally converge, the convergence rate may be slow.
+
+Relation to Dirac-GAN: Note that our theorem recovers the stability result [20, Theorem 4.1] as a special case: When there is a single true and generated point (whereby  $|N_i| = 1$  and  $\Delta_i = 0$ ), and regularization is used  $(\lambda > 0)$ , the theorem shows the true point is locally stable. When there is no regularization  $(\lambda = 0)$ , the criteria in both (19) and (20) are both exactly zero, meaning the equilibrium point is not exponentially stable or unstable. In this case, [20] shows that the system has a limit cycle.
+
+Non-existence of exact mode collapse: An important corollary of these results is provided by the following result.
+
+Corollary 1. Suppose the kernel satisfies Assumption 1, and the number of true and generated points are equal so that  $N_r = N_g = N$  for some  $N$ . Also, suppose that  $p_j = \widetilde{p}_j = 1 / N$  for all  $j$ . Then, the only stable equilibrium of the generated distribution  $\mathbb{P}_g$  with
+
+$$
+\operatorname {s u p p} \left(\mathbb {P} _ {g}\right) \subseteq \operatorname {s u p p} \left(\mathbb {P} _ {r}\right) \tag {21}
+$$
+
+$$
+i s \mathbb {P} _ {g} = \mathbb {P} _ {r}.
+$$
+
+In Corollary 1,  $\mathrm{supp}(\cdot)$  denotes the support of the distributions. So,  $\mathrm{supp}(\mathbb{P}_r)$  denotes the set of non-unique values of true points  $\boldsymbol{x}_i$  and  $\mathrm{supp}(\mathbb{P}_g)$  denotes the set of non-unique values of generated points  $\widetilde{\boldsymbol{x}}_j$ . Thus, the result rules out the possibility that the generated distribution can find a locally stable equilibrium where multiple generated points converge to a single true point.
+
+# 4 Bad Local Minima and Approximate Mode Collapse
+
+Mode collapse is one of the most widely-known failure mechanisms of GANs, especially for the simple GANs considered in this work [2, 11, 19, 26]. Corollary 1 appears to rule out mode collapse in the sense that there are no locally stable equilibria where multiple generated point arrive in a single true point. Indeed, Theorem 1 suggests that the generated points will be "repelled" from the true point when the generated mass exceeds the true mass, i.e.,  $\Delta_{i} < 0$ . This repulsive force offers the possibility that the excess generated mass can move towards other true points.
+
+However, in this section, we will show that even when  $\Delta_{i} < 0$ , the generated points may get stuck close to the true point. This phenomena is what we call approximate mode collapse.
+
+Theorem 2. Fix a region  $V_{i}$  and consider the dynamical system (12) with the RBF kernel (7). Assume  $|N_i| > 1$  so there is more than one generated point in the region. Then, there exists a constant  $c > 0$ , independent of the kernel width  $\sigma$ , and  $N_{\mathrm{max}}$ , which is only a function of  $d$ , such that, if  $|N_{i}| \leq N_{\mathrm{max}}$ , the system has a locally stable equilibrium with  $\widetilde{\mathbf{X}}_i^* = \{\widetilde{\mathbf{x}}_j^*, j \in N_i\}$  and
+
+$$
+\left\| \widetilde {\boldsymbol {x}} _ {j} ^ {*} - \boldsymbol {x} _ {i} \right\| \leq c \sigma , \tag {22}
+$$
+
+for all  $j\in N_{i}$
+
+The theorem states that, under certain conditions, the generated points can get stuck in a local minima, even when the generated mass exceeds the true mass. An example of this situation is illustrated in Figure 1 in the region  $V_{2}$ . The true mass (the green disk at the center of  $V_{2}$ ) is itself an unstable location for the generated points since  $\Delta_{i} < 0$ . But, the points may find a stable local minima around
+
+the true point. Moreover, the distance of the local minima to the  $x_{i}$  scales with the kernel width  $\sigma$ . Hence, for very small  $\sigma$ , the generated points with arbitrarily high mass may accumulate near a single true point. This phenomena can thus be seen as a type of approximate mode collapse.
+
+The proof of Theorem 2 shows that the bad minima holds for kernels other than the RBF as well.
+
+Prior work in mode collapse have identified at least two failure mechanisms: The first is that the discriminator may have zero gradient on the generated data when it comes from a low dimensional manifold [2]. The above result provides a constructive proof for the existence of such modes, and additionally shows that the local minima is stable, meaning the gradients push the generated data locally to the bad minima.
+
+A second failure mechanism is catastrophic forgetting [26], that past generated values are not remembered by the discriminator. Since we use a regularized discriminator  $(\lambda > 0)$ , our discriminator also "forgets" past values, and thus, the existence of the bad minima would be consistent with catastrophic forgetting.
+
+# 5 Divergence
+
+What happens to generated points isolated from all the true points  $\{\pmb{x}_i\}$ ? Consider a single generated point  $\widetilde{\pmb{x}}_0$  whose trajectory  $\{\widetilde{\pmb{x}}_0^k\}_{k\geq 0}$  satisfies
+
+$$
+K \left(\tilde {\boldsymbol {x}} _ {0} ^ {k}, \tilde {\boldsymbol {x}} _ {j} ^ {k}\right) = 0 \quad \forall j \neq 0, \quad K \left(\tilde {\boldsymbol {x}} _ {0} ^ {k}, \boldsymbol {x} _ {i}\right) = 0 \quad \forall i \quad \text {f o r a l l} k. \tag {23}
+$$
+
+That is,  $\widetilde{x}_0^k$  is sufficiently far from all the other generated points and true points so that the kernel can be treated as zero. In this case, the dynamics (5) and (9) reduce to
+
+$$
+f ^ {k + 1} (\boldsymbol {x}) = (1 - \eta_ {d} \lambda) f ^ {k} (\boldsymbol {x}) - K (\boldsymbol {x}, \widetilde {\boldsymbol {x}} _ {0} ^ {k}) \tag {24a}
+$$
+
+$$
+\widetilde {\boldsymbol {x}} _ {0} ^ {k + 1} = \widetilde {\boldsymbol {x}} _ {0} ^ {k} + \eta_ {g} \nabla f ^ {k} \left(\widetilde {\boldsymbol {x}} _ {0} ^ {k + 1}\right). \tag {24b}
+$$
+
+Theorem 3. Consider the dynamical system (24) with a translation-invariant kernel of the form  $K(\pmb{x},\pmb{x}^{\prime}) = \phi (\| \pmb {x} - \pmb{x}^{\prime}\|)$  for some smooth, integrable function  $\phi (\cdot)$  with  $\phi (0) > 0$ . Then, for any initial condition,  $\widetilde{\pmb{x}}_0^0$ , and unit vector  $\pmb {u}\in \mathbb{R}^{d}$ , and  $\lambda$  sufficiently small, there exists an initial discriminator function  $f^0 (\pmb {x})$  and velocity  $v_{0} > 0$  such that the solution to the system in equation (24) is
+
+$$
+\widetilde {\boldsymbol {x}} _ {0} ^ {k} = \widetilde {\boldsymbol {x}} _ {0} ^ {0} + k v _ {0} \boldsymbol {u}. \tag {25}
+$$
+
+Remarkably, the theorem shows that an isolated generated point can enter a trajectory where it continues to move linearly in a direction simply propelled by its own kernel history  $K(\pmb{x}, \widetilde{\pmb{x}}_0^k)$ . This situation is illustrated in Figure 1, where the generated point in the top right moves to the right in a straight line. We will see several examples of divergence in the numerical results as well.
+
+# 6 Role of the Kernel Width
+
+A key parameter of any distance based kernels, such as the RBF (7), is its kernel width, meaning the approximate distance at which the kernel begins to decay significantly. Our results, combined with previous analyses, provide useful insights into the role of the kernel width in GAN training.
+
+For example, consider the simple case of the Dirac-GAN where there is a single true point  $x_0$ , and single generated point  $\widetilde{x}_0$ . If we fix the generated point,  $\widetilde{x}_0$ , and let an RBF discriminator run to the convergence, it is known from results in [6, 12, 18] that the discriminator parameters  $\theta^k$  will converge to some values  $\theta^k \rightarrow \theta^*$ , and the resulting generator loss in (8) is given by
+
+$$
+\mathcal {L} _ {g} \left(\boldsymbol {\theta} ^ {*}, \widetilde {\boldsymbol {x}} _ {0}\right) = \frac {1}{\lambda} \left(1 - e ^ {- \left\| \boldsymbol {x} _ {0} - \widetilde {\boldsymbol {x}} _ {0} \right\| ^ {2} / 2 \sigma^ {2}}\right). \tag {26}
+$$
+
+The supplementary material reviews these results in a more general setting.
+
+Now, if the kernel width  $\sigma$  is very large, the loss will have a small gradient when  $\widetilde{x}_0$  is close to  $x_0$ , i.e., the generated point is close to the true value. On the other hand, if  $\sigma$  is very small, the loss will have a small gradient when  $\widetilde{x}_0$  is far from  $x_0$ . The selection of the appropriate  $\sigma$  must balance the convergence rates in these two regimes. In addition, when  $\sigma$  is large, the discriminator is approximately linear and therefore cannot distinguish nonlinear regions.
+
+![](images/98d102f5963531f2664f246f556f8d43f14f934b7c12530427dd64fa0229a97a.jpg)
+
+![](images/a89855921d5598fb01bf514597fce92cd5171bf2f9b6931f205cbfe21c4c0bb4.jpg)
+
+![](images/cfd9799ca62eb10f9bf2e737de1b75f7842091c85d03bd8c02bb28d139026748.jpg)
+
+![](images/64e8bc02fbf3a558edacce5b3bb3cd71e467ad660c9771d7bddaab7f8bdaa0be.jpg)
+
+![](images/7fef5016cec12ecb0f533b638fd25ae3f36b78c903393a5c58b1dfa952e0cd57.jpg)
+
+![](images/e13279885a2248393892d798a732f4eaaa2eb1a7c90c262c8f17f4284508d8fb.jpg)
+
+![](images/acc9cc51648bee86b184cbc6eb98e688a67e14dfb479ad6f2a0d0c259156f4c4.jpg)
+
+![](images/bdc899a2e8898fd19854d6ceae866591bd66752ef14a57d9f62345b11e89f7cf.jpg)  
+Figure 2: Behavior of joint GAN training with kernel width: Example trajectories of generated points over the course of training (red lines with final point marked as a cross), true distribution (green), final discriminator (blue and yellow colormap).
+
+![](images/e0cb7b741f28675c2b0bce8f763f51cc3432d6228a0f4a7976f1a0d2d3e783e2.jpg)
+
+![](images/e03f04fc593cf5558c3df3872c1513dc3c7a5c1336b773b87a7bc68a47dd6b3a.jpg)
+
+The results for the isolated points model provide additional insight. The isolated points assumption arises when the true points become separated at a distance much greater than  $\sigma$ . In such cases, we have seen that at least two failure modes become possible: (1) approximate mode collapse, as described in Theorem 2; and (2) divergence, as described in Theorem 3, where generated points isolated from all true points diverge in arbitrary directions. These two failure mechanisms may be prevented with a wider kernel width to preclude the isolated points assumption. However, wide kernel widths come at the price of poor discriminator power and slow convergence near the true point.
+
+# 7 Numerical Examples
+
+We conduct a simple experiment on low-dimensional, synthetic data to illustrate the behavior that the theory predicts. We also look at the frequency of GAN failure modes as a function of kernel width of the discriminator. Our main observation is that failure most often occurs when the model is in an isolated points regime at small kernel width.
+
+Two simple datasets for the true data are used: A set of  $N_r = 4$  points arranged on uniformly on the unit circle in dimension  $d = 2$ ; and a set of  $N_r = 10$  points randomly distributed on the unit sphere in dimension  $d = 10$ . In both cases, we initialize  $N_g = N_r$  generated point as Gaussians with zero mean and  $\mathbb{E} \| \widetilde{\boldsymbol{x}}_j \|^2 = 1$ . We approximate the RBF discriminator using a random Fourier feature map as in [24]. We set  $\lambda = 0.01$  and  $\eta_d = \eta_g = 10^{-3}$  and use 40000 training steps. Other details are in the Supplementary material.
+
+Figure 2 show example trajectories for the  $d = 2$  case for different kernel widths  $\sigma$ . For the two dimensional setting we visualize the trajectories of generated points over the course of training in Figure 2. We see a diverging behavior at very small kernel widths, as these generated points are not influenced by the kernels of the true points. Instead they "wander" off. Although Theorem 3 shows linear diverging trajectories, other diverging trajectories may be possible due to the randomness
+
+![](images/9a8a33c484c0e3a86d15a6b0d189879719612cdfbe43cdaeaa22e90437479810.jpg)  
+(a) Normalized Wasserstein distance
+
+![](images/bf8f6de80f856f8c1fa61d23f8f2bc936d1f81dc39ad15d998f903ac1a334a44.jpg)  
+(b) Frequency of divergence  
+Figure 3: Frequency of different GAN failure modes in a four point, two dimensional setting and a 10 point, 10 dimensional setting. (a) The median change in Wasserstein distance between true and generated distributions after 40k iterations. Different kernel widths of an RBF kernel discriminator are tested. (b) The average percentage of diverging generated points after training a GAN with RBF kernel discriminator. A generated point is considered diverging if  $\| \widetilde{\boldsymbol{x}}\| _2 > 2$
+
+in Fourier feature map. Also, in the example at  $\sigma = 0.6$ , we see an approximate mode collapse where two generated points converge to a single true point. Finally, at very large kernel widths the discriminator is not able to properly distinguish between distributions, and as a result we see large oscillations. The discriminator gradients are much smoother, meaning they are closer to the linear discriminator regime from the original Dirac-GAN analysis [19, 26].
+
+To measure how these failure mechanisms affect the overall performance, for each kernel width  $\sigma$ , we run 100 trials of the true and generated points. Figure 3 plots the median performance of the GAN along two metrics. In Figure 3(a), we measure the Wasserstein-2 distance [14] between true and generated distributions before and after training period for a fixed number of iterations. The Wasserstein-2 distance is estimated by [7] and provides a error due to diverging points and mode collapse. We see that at very low kernel widths, the normalized error can be even greater than one, meaning that the generated points are further from the true points than their initial value. This behavior is a result of the divergence. At very high kernel widths, there is also significant error due to the slow convergence and lack of power of the discriminator.
+
+As a simple measure of frequency of divergence, Figure 3b measures the average fraction of generated points,  $\widetilde{\boldsymbol{x}}_j$ , where  $\| \widetilde{\boldsymbol{x}} \|_2 > 2$  at the end of the iterations. Since the true points are on the unit sphere  $(\| \boldsymbol{x}_i \| = 1)$ , this condition can be considered as a case where the generated points have move significantly from all the true points. As expected, we see that the frequency of divergence increases with small kernel width. However, for dimension  $d = 10$ , the frequency of divergence is small for very low kernel widths. This phenomena is a result of the fact that we are approximating the RBF with random features. As shown in [24], the number of features required for a good approximation to the RBF grows with  $d^2$ .
+
+# 8 Potential Solutions via Multi-Scale Kernels
+
+The above discussion and simulations show that there is a fundamental trade-off with respect to the kernel width. On the one hand, very wide kernels tend to provide slow rates of convergence. In addition, they are not able to accurately discriminate between true points that are close. On the other hand, very narrow width kernels can result in true points being isolated from one another. The results in this paper show that isolated points can lead to both approximate mode collapse and divergence of generated points far away from the true distribution. Fortunately, the analysis in the paper suggests a way to avoid both of these conditions. Specifically, suppose we consider a kernel where the discriminator is of the form:
+
+$$
+f (x, \theta_ {1}, \theta_ {2}) := a _ {1} (x) ^ {\mathsf {T}} \theta_ {1} + a _ {2} (x) ^ {\mathsf {T}} \theta_ {2}, \tag {27}
+$$
+
+where  $a_1(x)$  and  $a_2(x)$  are two basis functions and  $\theta = (\theta_1, \theta_2)$  are the parameters. The kernel for this discriminator is:
+
+$$
+K (x, x ^ {\prime}) = K _ {1} \left(x, x ^ {\prime}\right) + K _ {2} \left(x, x ^ {\prime}\right), \tag {28}
+$$
+
+![](images/b7ddc58c590622f5b7fc2e5d3f15b98fc9fea158ed91b9ec9154c859835e0416.jpg)  
+Figure 4: GAN training dynamics using fixed-width kernel discriminators (blue, yellow, and green dashed lines) compared to the GAN training dynamics using a feature map constructed by concatenation of fixed-width feature maps (illustrated by solid line). A concatenated feature map produces a kernel that is a linear combination of its component kernels
+
+where  $K_{i}(x,x^{\prime}) = a_{i}(x)^{\top}a_{i}(x^{\prime})$  are the kernels for each of the bases.
+
+Now, suppose that  $a_1(x)$  and  $a_2(x)$  are selected so that  $K_{1}(x)$  has a wide width and  $K_{2}(x)$  has a small width. We call such a kernel multi-scale. The overall kernel (28) will have a heavy tail component due to the "wide" kernel  $K_{1}(x)$ . Hence, it can avoid the isolated points problem. On the other hand, since  $K_{2}(x)$  has a small width, the kernel will have a "sharp" component. This may enable fast convergence near the true distribution.
+
+To illustrate the potential use of such a multi-scale kernel, Fig. 4 compares the performance of kernels with three fixed widths of  $\sigma \in \{0.1, 1, 10\}$  with a single kernel concatenated with all three widths. The true and generated data are single point masses with an initial distance of 10. As expected, the very low width kernel ( $\sigma = 0.1$ ) fails to converge while the wide width kernels ( $\sigma = 1, 10$ ) converge slowly. In contrast, the concatenated kernel is able to get fast convergence in both the initial and later stages. Of course, further work will be needed to find out the best architectures for networks with multiple widths for complex practical data. In the context of neural network discriminators, such multi-scale kernel behavior may be achieved using parallel networks with varying depths, or the use of skip connections. However, designing such a discriminator requires further investigation.
+
+# 9 Conclusion
+
+In this paper we propose a new framework for understanding behaviors exhibited by training GANs. By assuming isolated neighborhoods around each true point, we can decouple the kernel GAN training dynamics that occur when jointly updating a generator and discriminator. Using this model we theoretically derive conditions that explain well-known training behavior in GANs. While a stable local equilibrium exists when true and generated distributions match exactly, bad local minima exist as well. Specifically a true point can "greedily" hold onto multiple generated points in a region determined by the discriminator kernel width, leading to the phenomenon of approximate mode collapse. We provide theoretical and empirical evidence of a diverging failure mode as well, where one or more generated points completely escape the influence of the true distribution and travel along arbitrary trajectories. Through this analysis, it becomes clear that kernel width in particular plays an important role in this behavior.
+
+There are several possible lines of future work. Most importantly, we have studied a simple GAN without gradient penalties, as is commonly used in methods such as [2, 3, 13, 17]. Future work can also consider the convergence. Our linearized analysis provides the eigenvalues which can be used for rates close to the equilibrium as performed in [21, 22, 30]. Finally, there is a large body of literature connecting kernels to the neural networks via the so-called neural tangent kernel (NTK) [15] and may be useful to study here as well.
+
+# Acknowledgments and Disclosure of Funding
+
+S. Rangan was supported in part by NSF grants 1952180, 1925079, 1564142, 1547332, the SRC, and the industrial affiliates of NYU WIRELESS.
+
+# References
+
+[1] Hamed Alqahtani, Manolya Kavakli-Thorne, and Gulshan Kumar. Applications of Generative Adversarial Networks (GANs): An Updated Review. Archives of Computational Methods in Engineering, 28(2):525-552, March 2021.  
+[2] Martin Arjovsky and Léon Bottou. Towards principled methods for training generative adversarial networks, 2017.  
+[3] Martin Arjovsky, Soumith Chintala, and Léon Bottou. Wasserstein generative adversarial networks. In International conference on machine learning, pages 214-223. PMLR, 2017.  
+[4] Yogesh Balaji, Mohammadmahdi Sajedi, Neha Mukund Kalibhat, Mucong Ding, Dominik Stöger, Mahdi Soltanolkotabi, and Soheil Feizi. Understanding overparameterization in generative adversarial networks, 2021.  
+[5] Alberto Bietti and Julien Mairal. On the Inductive Bias of Neural Tangent Kernels. 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.  
+[6] Mikołaj Binkowski, Danica J Sutherland, Michael Arbel, and Arthur Gretton. Demystifying MMD GANs. arXiv preprint arXiv:1801.01401, 2018.  
+[7] Nicolas Bonneel, Michiel Van De Panne, Sylvain Paris, and Wolfgang Heidrich. Displacement interpolation using lagrangian mass transport. In Proceedings of the 2011 SIGGRAPH Asia conference, pages 1-12, 2011.  
+[8] Ali Borji. Pros and cons of GAN evaluation measures: New developments. CoRR, abs/2103.09396, 2021.  
+[9] Jean-Yves Franceschi, Emmanuel de Bezenac, Ibrahim Ayed, Mickaël Chen, Sylvain Lamprier, and Patrick Gallinari. A neural tangent kernel perspective of gans, 2021.  
+[10] Ian Goodfellow. Nips 2016 tutorial: Generative adversarial networks. arXiv preprint arXiv:1701.00160, 2016.  
+[11] Ian J. Goodfellow, Jean Pouget-Abadie, Mehdi Mirza, Bing Xu, David Warde-Farley, Sherjil Ozair, Aaron Courville, and Yoshua Bengio. Generative adversarial networks, 2014.  
+[12] Arthur Gretton, Karsten M Borgwardt, Malte J Rasch, Bernhard Schölkopf, and Alexander Smola. A kernel two-sample test. The Journal of Machine Learning Research, 13(1):723-773, 2012.  
+[13] Ishaan Gulrajani, Faruk Ahmed, Martin Arjovsky, Vincent Dumoulin, and Aaron Courville. Improved training of wasserstein gans. arXiv preprint arXiv:1704.00028, 2017.  
+[14] Martin Heusel, Hubert Ramsauer, Thomas Unterthiner, Bernhard Nessler, and Sepp Hochreiter. Gans trained by a two time-scale update rule converge to a local nash equilibrium. Advances in neural information processing systems, 30, 2017.  
+[15] Arthur Jacot, Franck Gabriel, and Clément Hongler. Neural tangent kernel: Convergence and generalization in neural networks. In Advances in neural information processing systems, pages 8571-8580, 2018.  
+[16] Valentin Khrulkov, Artem Babenko, and Ivan Oseledets. Functional space analysis of local gan convergence. In International Conference on Machine Learning, pages 5432-5442. PMLR, 2021.  
+[17] Naveen Kodali, Jacob D. Abernethy, James Hays, and Zsolt Kira. How to train your DRAGAN. CoRR, abs/1705.07215, 2017.  
+[18] Chun-Liang Li, Wei-Cheng Chang, Yu Cheng, Yiming Yang, and Barnabás Póczos. MMD GAN: Towards deeper understanding of moment matching network. Advances in neural information processing systems, 30, 2017.  
+[19] Lars Mescheder, Andreas Geiger, and Sebastian Nowozin. Which training methods for gans do actually converge?, 2018.  
+[20] Lars Mescheder, Sebastian Nowozin, and Andreas Geiger. The numerics of gans. Advances in neural information processing systems, 30, 2017.
+
+[21] Youssef Mroueh and Truyen Nguyen. On the convergence of gradient descent in gans: Mmd gan as a gradient flow. In International Conference on Artificial Intelligence and Statistics, pages 1720-1728. PMLR, 2021.  
+[22] Vaishnavh Nagarajan and J. Zico Kolter. Gradient descent gan optimization is locally stable, 2018.  
+[23] Weili Nie and Ankit B Patel. Towards a better understanding and regularization of gan training dynamics. In Uncertainty in Artificial Intelligence, pages 281-291. PMLR, 2020.  
+[24] Ali Rahimi and Benjamin Recht. Random features for large-scale kernel machines. In J. Platt, D. Koller, Y. Singer, and S. Roweis, editors, Advances in Neural Information Processing Systems, volume 20. Curran Associates, Inc., 2007.  
+[25] Eitan Richardson and Yair Weiss. On gans and gmms. CoRR, abs/1805.12462, 2018.  
+[26] Hoang Thanh-Tung, Truyen Tran, and Svetha Venkatesh. On catastrophic forgetting and mode collapse in generative adversarial networks. CoRR, abs/1807.04015, 2018.  
+[27] Thomas Unterthiner, Bernhard Nessler, Calvin Seward, Günter Klambauer, Martin Heusel, Hubert Ramsauer, and Sepp Hochreiter. Coulomb GANs: Provably optimal Nash equilibria via potential fields. arXiv preprint arXiv:1708.08819, 2017.  
+[28] Mathukumalli Vidyasagar. Nonlinear systems analysis, volume 42. Siam, 2002.  
+[29] Zhengwei Wang, Qi She, and Tomás Ward. Generative adversarial networks in computer vision: A survey and taxonomy. ACM Computing Surveys, 54:1-38, 02 2021.  
+[30] Kun Xu, Chongxuan Li, Jun Zhu, and Bo Zhang. Understanding and stabilizing gans' training dynamics with control theory. arXiv preprint arXiv:1909.13188, 2019.
+
+# Checklist
+
+1. For all authors...
+
+(a) Do the main claims made in the abstract and introduction accurately reflect the paper's contributions and scope? [Yes] The Introduction has a section on main contributions to highlight the key items.  
+(b) Did you describe the limitations of your work? [Yes] The introduction and model clearly define the specific model we are considering. The conclusions describes avenues of future work not considered here.  
+(c) Did you discuss any potential negative societal impacts of your work? [No] As a theoretical result, we did not see any negative societal impacts.  
+(d) Have you read the ethics review guidelines and ensured that your paper conforms to them? [Yes] The guidelines are reviewed.
+
+2. If you are including theoretical results...
+
+(a) Did you state the full set of assumptions of all theoretical results? [Yes] All assumptions are stated in the Model section and theorem statements  
+(b) Did you include complete proofs of all theoretical results? [Yes] They are included in the supplementary material.
+
+3. If you ran experiments...
+
+(a) Did you include the code, data, and instructions needed to reproduce the main experimental results (either in the supplemental material or as a URL)? [Yes] Code is added as supplementary material  
+(b) Did you specify all the training details (e.g., data splits, hyperparameters, how they were chosen)? [Yes] Main hyperparameters are discussed in the main results, details included in the supplementary  
+(c) Did you report error bars (e.g., with respect to the random seed after running experiments multiple times)? [No] We considered a large number of trials and took the median.  
+(d) Did you include the total amount of compute and the type of resources used (e.g., type of GPUs, internal cluster, or cloud provider)? [No] Due to the simplicity of the experiments, this was not necessary
+
+4. If you are using existing assets (e.g., code, data, models) or curating/releasing new assets...
+
+(a) If your work uses existing assets, did you cite the creators? [N/A] We only used standard existing code packages such as pytorch.  
+(b) Did you mention the license of the assets? [N/A]  
+(c) Did you include any new assets either in the supplemental material or as a URL? [No] No new assets were needed.  
+(d) Did you discuss whether and how consent was obtained from people whose data you're using/curating? [N/A] No subjects were tested.  
+(e) Did you discuss whether the data you are using/curating contains personally identifiable information or offensive content? [N/A] No personally identifiable was used.
+
+5. If you used crowdsourcing or conducted research with human subjects...
+
+(a) Did you include the full text of instructions given to participants and screenshots, if applicable? [N/A]  
+(b) Did you describe any potential participant risks, with links to Institutional Review Board (IRB) approvals, if applicable? [N/A]  
+(c) Did you include the estimated hourly wage paid to participants and the total amount spent on participant compensation? [N/A]

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+# Instance-based Learning for Knowledge Base Completion
+
+Wanyun Cui $^{1}$ , Xingran Chen $^{2}$
+
+Shanghai University of Finance and Economics<sup>1</sup>
+
+University of Michigan
+
+cui.wanyun@sufe.edu.cn, chenxran@umich.edu
+
+# Abstract
+
+In this paper, we propose a new method for knowledge base completion (KBC): instance-based learning (IBL). For example, to answer (Jill Biden, lived city,?) instead of going directly to Washington D.C., our goal is to find Joe Biden, who has the same lived city as Jill Biden. Through prototype entities, IBL provides interpretability. We develop theories for modeling prototypes and combining IBL with translational models. Experiments on various tasks confirmed the IBL model's effectiveness and interpretability.
+
+In addition, IBL shed light on the mechanism of rule-based KBC models. Previous research has generally agreed that rule-based models provide rules with semantically compatible premises and hypotheses. We challenge this view. We begin by demonstrating that some logical rules represent instance-based equivalence (i.e. prototypes) rather than semantic compatibility. These are denoted as IBL rules. Surprisingly, despite occupying only a small portion of the rule space, IBL rules outperform non-IBL rules in all four benchmarks. We use a variety of experiments to demonstrate that rule-based models work because they have the ability to represent instance-based equivalence via IBL rules. The findings provide new insights of how rule-based models work and how to interpret their rules.
+
+# 1 Introduction
+
+In knowledge base completion (KBC), the learner attempts to infer new facts about the world from given training facts (head, relation, tail). This problem has intrigued many researchers' interest as it's both a fundamental task of relational data representation learning [16, 33] and a benefit for downstream KB-based applications [34, 39].
+
+One typical way of KBC is to answer (head, relation, ?) or (?, relation, tail). Knowledge graph embedding [4, 10] and rule-based reasoning [17, 38] are two typical KBC methods. Knowledge graph embedding learns low-dimensional vectors for entities and relations. One disadvantage of knowledge graph embedding is its lack of interpretability. Rule-based reasoning provides interpretability by transforming queries into human-readable logical rules. However, the predictions are constrained by the rule length  $k$ . As  $k$  increases, the rule-based search space expands exponentially, increasing the difficulty of mining high-quality rules.
+
+In this paper, we explore an alternative method: instance-based learning (IBL) [15, 26]. Instead of performing explicit generalization, IBL generates predictions by comparing query instances with instances seen in training. Despite its rich history in machine learning, IBL has not seen applied in KBC. In terms of KBC, instead of directly finding the target entity to the query, we ask which alternative entities share the same value with the query entity and relation? For example, for the query (Jill Biden, lived city, ?), instead of directly finding Washington D.C. from the vector space or by rules, we first identify the entity Joe Biden, who has the same lived city as Jill Biden. In this
+
+paper, Joe Biden is referred to the prototype for that query. To answer the query, we will use the prototype's known fact (i.e., (Joe Biden, lived city, Washington D.C.)).
+
+The main motivation behind IBL is the low-rank assumption, which holds that real-world knowledge is interdependent. Such assumption has been popularized with the Netflix challenge [9] and now frequently employed in knowledge graph embedding [10, 30, 37]. In the instance-based setting, we found that the low-rank assumption is significant and clear. For example, Joe Biden and Jill Biden share the same lived city because they have the same marriage event and the same children.
+
+IBL has several advantages over traditional KBC methods: first, it enables interpretability through prototypes. For example, inferring Jill Biden's lived city from Joe Biden rather than Barack Obama makes more sense. Second, in contrast to rule-based reasoning, IBL is not constrained by the  $k$ -step neighborhood. In addition, the search space is always  $O(n\_ entity)$ . Third, it helps KB curation by ensuring high-quality reasoning as the KB grows dynamically [35]. For example, we can still use Joe Biden to infer Jill Biden's lived city if she moves to another city in the future.
+
+Due to the complexity of the knowledge graph, it is still challenging to find correct prototypes among the massive number of entities. Fortunately, although IBL is a new KBC method, we theoretically show that prototypes have closed-form expressions with the well-studied translational models [4, 12, 27] in Sec 3. The derived IBL model has competitive performance with cutting-edge KBC methods. In addition, we establish a joint optimization theory of the IBL model and the translational model in Sec 4, revealing the concordance in terms of their optimization objectives. The combined model inspired by this theory outperforms existing methods in two out of four benchmarks.
+
+Although IBL appears to be irrelevant to rule-based KBC models, we found IBL shed light on its mechanism (Sec 5). Previous literature has commonly agreed that logical rules provide semantically compatible premises and hypotheses. We show that, some rules need to be interpreted through instance-based equivalence, rather than the semantic compatibility. We call them IBL rules. Surprisingly, although IBL rules only occupy a small fraction of the entire rules space, they play a significant role in rule-based reasoning. In Sec 7, we show that IBL rules are more critical than all other rules in all four benchmarks, and can even replace all other rules completely with no effect degradation in two of four benchmarks. This phenomenon challenges the previously common understanding of how logical rules work and provide interpretability. Rule-based models rely much on their ability to represent instance-based equivalence to infer new knowledge. The interpretability of logical rules should be revisited from the perspective of instance-based equivalence. And the IBL method proposed in this paper is a more effective alternative to rule-based methods when modeling the equivalency.
+
+Our contributions are outlined as follows: (1) we explore instance-based learning for KBC. IBL provides interpretability while not being constrained by local neighborhoods. (2) Using translational models, we prove that prototypes have closed-form expressions. (3) We developed a joint optimization theory for IBL and translational models as a foundation for combing different KBC models. Experimental results show that the combined model outperforms existing methods on two out of four benchmarks. (4) We found that rule-based reasoning relies heavily on IBL rules to represent the instance-based equivalence. The results suggest that the mechanism and interpretability of rule-based reasoning should be reassessed in terms of instance-based equivalence.
+
+# 2 Related Work
+
+For a comprehensive survey of knowledge graph completion, we refer readers to [33].
+
+Learning representations for knowledge via prototypes Another theory that provides support to our work is the prototype theory. The term prototypes, as initially defined in psychologist Eleanor Rosch's study "Natural Categories" [20], has been widely studied in psychology [19] and cognitive linguistics [28]. Bobrow and Winograd [3] used the prototype theory to the general knowledge representation. The general consensus is that knowledge reasoning is dominated by a recognition process in which new objects and events are compared to stored sets of expected prototypes. Although the prototype theory is widely studied in a variety of domains, our study is the first to apply the prototype theory for KBC to our knowledge. We concretize the theory as the equivalence of query entities to prototypes on specific relations, thus allowing knowledge reasoning.
+
+Instance-based learning There is a large body of literature on IBL. The k-nearest neighbors algorithm (kNN), which uses the k-nearest examples as prototypes, is the most representative work. Besides, [18] proposed to use model-based methods to calibrate the prototypes in the instance-based method. In recent years, IBL has also shown its great potentials to few-shot learning [1, 25] and self-supervised learning [32]. One trend of IBL is to leverage metric learning for prototype selection [8, 36], in particular, using neural networks for instance embedding [13, 22]. In this paper, similar to [13, 22], we use trainable entity embeddings to represent prototypes.
+
+# 3 Models
+
+In this section, we first formalize the KBC problem (Sec 3.1). Then we demonstrate how translational models allow prototype modeling with closed-form expressions (Sec 3.2 3.3). We also investigate the relation-awareness property for specific translational representations (Sec 3.4).
+
+# 3.1 Problem: Link Prediction for Knowledge Base Completion
+
+KBs are collections of triplet facts that represent world knowledge (head, relation, tail). We denote the fact in the KB as  $\mathcal{D} = \{(h_1,r_1,t_1)\dots (h_n,r_n,t_n)\}$ . As it is impossible to collect all facts, a fundamental problem is to predict the missing facts. In this paper, we formalize the completion task as link prediction. The query format in the test set is  $(h,r,\ref{eq:2})$  or  $(?r,t)$ .
+
+# 3.2 Translational Models
+
+Translational models are one of the mainstream methods of KBC. The main idea behind translational models is to model relations between entities as a translation from head to tail over the vector space. For a new fact, its plausibility can be calculated by the distance between the translated head entity and the tail entity. We formulate the plausibility of  $(h,r,t)$  as below:
+
+$$
+\mathcal {T} (\mathrm {h}, \mathrm {r}, \mathrm {t}) = \| \operatorname {t r a n s} _ {\mathrm {r}} (\operatorname {e m b} (\mathrm {h})) - \operatorname {e m b} (\mathrm {t}) \| \tag {1}
+$$
+
+where  $emb(\cdot)$  denotes the entity embedding, and  $trans_{r}(\cdot)$  denotes the translation w.r.t.  $r$ .
+
+There are numerous variants under the translational framework. Here we present some typical models. The feasibility of exploiting these models for IBL are investigated in Sec 3.4.
+
+TransE [4] stores the embeddings  $emb(\cdot)$  in an embedding matrix. It uses the simplest vector addition as the translation  $trans_r(\cdot)$ . Its scoring function is:
+
+$$
+\operatorname {T r a n s E} (\mathrm {h}, \mathrm {r}, \mathrm {t}) = \| \mathbf {e} _ {\mathrm {h}} + \mathrm {r} - \mathbf {e} _ {\mathrm {t}} \| \tag {2}
+$$
+
+TransR [12] extends TransE by adding a relationship matrix  $\mathbf{W}_{\mathrm{r}}$ , which maps embeddings of entities from the entity space to the relational space:
+
+$$
+\operatorname {T r a n s R} (\mathrm {h}, \mathrm {r}, \mathrm {t}) = \| \mathbf {W} _ {\mathrm {r}} \mathbf {e} _ {\mathrm {h}} + \mathrm {r} - \mathbf {W} _ {\mathrm {r}} \mathbf {e} _ {\mathrm {t}} \| \tag {3}
+$$
+
+RotatE [27] is a cutting-edge translational model. It represents the translation via rotations in the complex space (denoted as  $\circ$ ). It has the following scoring function:
+
+$$
+\operatorname {R o t a t E} (\mathrm {h}, \mathrm {r}, \mathrm {t}) = \| \mathbf {e} _ {\mathrm {h}} \circ \mathrm {r} - \mathbf {e} _ {\mathrm {t}} \| \tag {4}
+$$
+
+# 3.3 Modeling Prototypes in IBL
+
+In IBL, given the query entity and relation, our primary goal is to locate the prototype whose value is the same as the query and is present in the training data. We say  $p$  is the prototype of  $(h,r,?)$  if  $(p,r,?) = (h,r,?)$ . For example, Joe Biden is a prototype of the query (Jill Biden, lived city,?), since (Jill Biden, lived city,?) = (Joe Biden, lived city,?) = Washington D.C. The prototype of  $(? , r , t )$  is defined similarly.
+
+We show that the prototype computation is tractable using translational models. We take advantage of a translational model property:  $\text{trans}_r(\cdot)$  is a mapping from the head and relation to the tail in the vector space. Thus, given the query, the prototype can be determined by Lemma 1.
+
+Lemma 1. For an optimal translational model  $\mathcal{T}$ , the prototypes  $P$  of  $(h,r,?)$  are:
+
+$$
+P = \left\{p \mid \operatorname {t r a n s} _ {r} (\mathrm {e m b} (h)) = \operatorname {t r a n s} _ {r} (\mathrm {e m b} (p)) \right\} \tag {5}
+$$
+
+Lemma 1 employs the absolute equality of vectors for the ideal case. In practice, we relax this restriction by using the differentiable vector distance instead. In particular, we model the plausibility of a candidate prototype  $p$  by:
+
+$$
+\mathrm {f} _ {\mathrm {h r}} (\mathrm {p}) = \max  (\gamma - \| \operatorname {t r a n s} _ {\mathrm {r}} (\mathrm {e m b} (\mathrm {h})) - \operatorname {t r a n s} _ {\mathrm {r}} (\mathrm {e m b} (\mathrm {p})) \|, 0) \tag {6}
+$$
+
+where we use marginal distance because closer distance means higher plausibility. Symmetrically, we model the prototype for the query  $(?,r,t)$  by:
+
+$$
+\mathrm {f} _ {\mathrm {r t}} (\mathrm {p}) = \max  (\gamma - \| \operatorname {t r a n s} _ {\mathrm {r}} ^ {- 1} (\mathrm {e m b} (\mathrm {t})) - \operatorname {t r a n s} _ {\mathrm {r}} ^ {- 1} (\mathrm {e m b} (\mathrm {p})) \|, 0) \tag {7}
+$$
+
+where  $trans_{r}^{-1}$  is the inverse of  $trans_{r}$ . For example, TransE [4] uses vector subtraction as  $trans_{r}^{-1}$ .
+
+For a given query  $(h,r,?)$ , we consider all candidate prototype entities whose relation  $r$  is known in the training data. We aggregate these prototypes based on their scores in Eq. (6). The score of a tail entity  $t$  is the sum of scores of its corresponding prototypes:
+
+$$
+\mathcal {I} _ {\mathrm {h r}} (\mathrm {t}) = 1 / \left(\gamma \mid \{\mathrm {p} \mid (\mathrm {p}, \mathrm {r}, \mathrm {t}) \in \mathcal {D} \} \mid\right) \sum_ {(\mathrm {p}, \mathrm {r}, \mathrm {t}) \in \mathcal {D}} \mathrm {f} _ {\mathrm {h r}} (\mathrm {p}) \tag {8}
+$$
+
+where  $1 / (\gamma |\{p|(p,r,t)\in \mathcal{D}\})$  is used to normalize the score. The score of a head entity  $h$  for  $(?,r,t)$  is computed symmetrically. We denote the above model  $\mathcal{I}$  as IBLE (instance-based learning). We use the cross-entropy between  $\mathcal{I}_{hr}(t)$  and the ground-truth entity as the training objective.
+
+We highlight that IBLE's aggregating strategy in Eq. (8) differs from GNNs [23]. For query  $(\mathrm{h},\mathrm{r},?)$ , regardless of whether the instance is a neighbor of  $\mathrm{h}$ , we aggregate the instance  $\mathfrak{p}$  throughout the full instance space whose relation  $\mathbf{r}$  is known (i.e.  $\{\mathfrak{p}|\left(\mathfrak{p},\mathbf{r},\mathfrak{t}\right)\in \mathcal{D}\}$ ).
+
+# 3.4 Relation-awareness for Prototype Modeling
+
+Next, using Eq. (6), we analyze the property of relation-awareness for prototype modeling. It is obvious that whether an entity is the prototype depends on the query relation. Joe Biden, for example, is a prototype of Jill Biden for lived_city, but not for profession. We summarize the relation-awareness of TransE, TransR and RotatE by substituting them into Eq. (6) in Table 1.
+
+Table 1: Prototype modeling by translational models  
+
+<table><tr><td>Model</td><td>Prototype modeling</td><td>Relation-aware</td></tr><tr><td>TransE</td><td>||eh - ep|| (10)</td><td>No</td></tr><tr><td>TransR</td><td>||Wr e_h - Wr e_p|| (11)</td><td>Yes</td></tr><tr><td>RotatE</td><td>||eh - e_p|| (12)</td><td>No</td></tr><tr><td>R-RotatE</td><td>||Wr e_h - Wr e_p|| (13)</td><td>Yes</td></tr></table>
+
+Prototypes by TransE and RotatE are not relation-aware because Eq. (9) and Eq. (10) consider only entity embeddings and is agnostic to the relation  $r$ . The relational matrix  $\mathbf{W}_r$ , on the other hand, allows TransR to distinguish different relations.
+
+Since RotatE is the cutting-edge translational model, we propose to optimize its representations. We integrate the relational matrix from TransR into RotatE, denoted as relational-RotatE (R-RotatE):
+
+$$
+\operatorname {R - R o t a t E} (\mathrm {h}, \mathrm {r}, \mathrm {t}) = \| \mathbf {W} _ {\mathrm {r}} \mathbf {e} _ {\mathrm {h}} \circ \mathrm {r} - \mathbf {W} _ {\mathrm {r}} \mathbf {e} _ {\mathrm {t}} \| \tag {14}
+$$
+
+We show the prototype modeling of R-RotatE in Table 1.
+
+Remark 1. TransR and R-RotatE are relation-aware for prototype modeling, whereas TransE and RotatE are not.
+
+An interesting phenomenon is, TransE and RotatE actually have the same prototype model, as do TransR and R-RotatE. In this paper, we use the prototype modeling of TransR and R-RotatE by default.
+
+# 4 Combine IBL and Translational Models
+
+Embeddings of entities and relations of IBL can be directly reused in the translational model. In this section, we develop the theory of the joint optimization for IBL and the translational model. We show that their objectives are compatible. The theory motivates us to combine IBL with the translational model.
+
+We denote the IBL model and the translational model with shared entity and relation embeddings  $\theta$  as  $\mathcal{I}_{\theta}$  and  $\mathcal{T}_{\theta}$ , respectively. We show that the optimization objectives of  $\mathcal{I}^{\theta}$  and  $\mathcal{T}^{\theta}$  are compatible in Lemma 2.
+
+Lemma 2. If we use margin loss for the translational model, and cross-entropy loss for the IBL model, and negative sampling to train both models, then the global minimum of the training criterion of  $\mathcal{T}^{\theta}$  can be obtained only if  $\mathcal{T}^{\theta}$  also reaches its optimum.
+
+This inspires us to propose CIBLE (combined instance-based learning) model, which combines  $I_{\theta}$  and  $T_{\theta}$  with the following scoring function:
+
+$$
+\mathcal {C} _ {\mathrm {h r}} ^ {\theta} (\mathrm {t}) = (1 - \alpha) \mathcal {I} _ {\mathrm {h r}} ^ {\theta} (\mathrm {t}) + \frac {\alpha}{\gamma} \max  (\gamma - \mathcal {T} ^ {\theta} (\mathrm {h}, \mathrm {r}, \mathrm {t}), 0) \tag {15}
+$$
+
+where  $\alpha \in (0,1)$  is a hyper-parameter. We use cross-entropy as the training objective for CIBLE. The optimization objectives of  $\mathcal{T}^{\theta},\mathcal{I}^{\theta}$  and  $\mathcal{C}^{\theta}$  are compatible. We show this in Theorem 1
+
+Theorem 1. If we use cross-entropy loss for the combined model and the IBL model, the margin loss for the translational model, and train all three models with negative sampling, then the  $\theta$  values for the global minimum training criteria of  $C_{\theta}$  and  $\mathcal{T}_{\theta}$  are consistent. At that  $\theta$ , the training criterion of  $\mathcal{I}_{\theta}$  is also minimized, and:
+
+$$
+\mathcal {C} _ {\mathrm {h r}} ^ {\theta} (t) = \left\{ \begin{array}{l l} 1 & (h, r, t) \text {i s a v a l i d f a c t}, \\ 0 & \text {o t h e r w i s e}. \end{array} \right. \tag {16}
+$$
+
+# 5 Logical Rules as Prototype-based Inference
+
+In this section, we show that, under the low-rank assumption, the instance-based equivalence and prototypes in IBL can be represented by a particular class of logical rules. We discovered that the interpretability of these rules differs from typical rules in previous papers. More numerical analysis will be presented in Sec 7.
+
+Rule-based KBC models induct rules in the form of Horn clauses:  $\mathrm{rel}_1\wedge \dots \mathrm{rel}_k\Rightarrow \mathrm{rel}_0$  Table 2 shows examples from the body of previous publications [17, 38]. According to these examples, it is commonly believed that the semantic compatibility between the premise and the hypothesis provides the interpretability for humans. For example, the semantics of nationality is equal to the semantics of the path born_in  $\land$  place_in_country, making the rule born_in  $\land$  place_in_country  $\Rightarrow$  nationality human-readable.
+
+However, we found a special class of rules that challenge the understanding. Table 2 shows top-3 rules inducted by RNNLogic [17] for profession in FB15k-237. Clearly, these rules are in one of the following forms:
+
+$$
+\operatorname {r e l} _ {1} \wedge \operatorname {r e l} _ {1} ^ {- 1} \wedge \operatorname {r e l} _ {0} \Rightarrow \operatorname {r e l} _ {0} \tag {17}
+$$
+
+$$
+\mathrm {r e l} _ {0} \wedge \mathrm {r e l} _ {1} \wedge \mathrm {r e l} _ {1} ^ {- 1} \Rightarrow \mathrm {r e l} _ {0}
+$$
+
+The premises of both forms contain the composition of symmetric relations  $\mathrm{rel}_1\wedge \mathrm{rel}_1^{-1}$ , whose meanings are in opposition to each other. The semantics of the premise alone cannot be understood by humans, making the connection between the premise and hypothesis unclear.
+
+![](images/8049353e5472a8309db61e1325d6114083a9a382e1ff355050193fbc791241b6.jpg)  
+Figure 1: An IBL rule indicates the instance-based relational equivalence.
+
+Table 2: Examples of rules.  
+
+<table><tr><td>Top 3 rules for profession by RNNLogic [17]. 
+profession←award-winning-1 ∧ award-winning ∧ profession 
+profession←nationality ∧ nationality-1 ∧ profession 
+profession←webpage_category ∧ webpage_category-1 ∧ profession</td></tr><tr><td>Examples from the RNNLogic paper [17]. 
+nationality←born_in ∧ place_in_country 
+organization_in_state←organization_in_city ∧ city_location_in_state</td></tr><tr><td>Examples from the Neural LP paper [38]. 
+partially.contains←contains ∧ contains 
+marriage_location←nationality ∧ contains</td></tr></table>
+
+However, after transforming such rules into the form of Fig. 1, their meanings become clearer. The semantics of such rules are consistent with IBL's low-rank assumption. The rule in Fig. 1 states that, conditional on the fact that  $x$  and  $x'$  are both the winners of award  $y'$ , they also tend to have the same profession  $y$ . As a result, the rule's true purpose is to find the prototype  $x'$  to deduce the profession of  $x$ .
+
+We employ translational models to provide a theoretical view of IBL rules. We take TransE as an example. In Theorem 2, we show that IBL rules always hold true under the assumption of translational models.
+
+Theorem 2 (The effectiveness of IBL rules). For a KB with an optimal TransE model such that  $|\mathrm{e_h} + \mathrm{r} - \mathrm{e_t}| = 0$  iff  $(\mathrm{h},\mathrm{r},\mathrm{t})\in \mathrm{KB}, \forall \mathrm{r}_0,\mathrm{r}_1,$  the IBL rules  $\mathrm{rel}_1\wedge \mathrm{rel}_1^{-1}\wedge \mathrm{rel}_0\Rightarrow \mathrm{rel}_0$  and  $\mathrm{rel}_0\wedge \mathrm{rel}_1\wedge \mathrm{rel}_1^{-1}\Rightarrow \mathrm{rel}_0$  always hold.
+
+We highlight the difference in interpretability between the claims in previous papers and the rules in Eq. (17). We denote rules in Eq. (17) as IBL rules. IBL rules provide interpretability by establishing instance-based equivalence relations, whereas previous papers claimed that rules typically provide interpretability by semantic relevance between the premise and hypothesis. Experimental results show that the effect and interpretability of instance-based equivalence cannot be ignored. This is elaborated in Sec 7.
+
+# 6 Experiments
+
+# 6.1 Experimental Setup
+
+Datasets: We select four typical KBC datasets for evaluation, including FB15k-237, WN18RR, Kinship, and UMLS  $^{2}$ . For Kinship and UMLS, we use the training/validation/test division in [17]. Their statistics are shown in the Appendix.
+
+Baselines We compared with the following types of KBC methods. Knowledge graph embedding: TransE [4], TransR [12], DistMult [37], ComplEx [30], ConvE [6], TuckER [2], and RotatE [27]. Rule-based: RNNLogic [17], NeuralLP [38], DRUM [21], PathRank [11], MINERVA [5], CTP [14] and M-Walk [24]. Graph neural networks: NBFNet [40], GraIL [29], and RGCN [23].
+
+Evaluation metrics For each test triplet  $(h,r,t)$ , we construct two queries:  $(h,r,?)$  and  $(?,r,t)$  with the answers  $t$  and  $h$ . We choose Mean Rank (MR), Mean Reciprocal Rank (MRR), and hit@k under the filtered setting [27], which is consistent with most existing work.
+
+Implementation details We follow [27] and employ negative sampling for training. All experiments can be run on a single Nvidia Tesla V100 GPU. We illustrate the hyper-parameter search process in the Appendix.
+
+# 6.2 Main Results
+
+We present the main results in Table 3 and Table 4. IBLE is competitive with most existing KBC methods. This verifies the effect and potential of IBL, as the proposed IBLE is still primitive. In
+
+Table 3: Knowledge base completion results on FB15k-237 and WN18RR.  $\ddagger$  means the results are from [17], and  $\dagger$  means the results are from [40].  
+
+<table><tr><td rowspan="2">Category</td><td rowspan="2">Model</td><td colspan="5">FB15k-237</td><td colspan="5">WN18RR</td></tr><tr><td>MR</td><td>MRR</td><td>H@1</td><td>H@3</td><td>H@10</td><td>MR</td><td>MRR</td><td>H@1</td><td>H@3</td><td>H@10</td></tr><tr><td rowspan="6">KG Embedding</td><td>TransE‡</td><td>357</td><td>0.294</td><td>-</td><td>-</td><td>46.5</td><td>3384</td><td>0.226</td><td>-</td><td>-</td><td>50.1</td></tr><tr><td>DistMult‡</td><td>254</td><td>0.241</td><td>15.5</td><td>26.3</td><td>41.9</td><td>5110</td><td>0.430</td><td>39.0</td><td>44.0</td><td>49.0</td></tr><tr><td>ComplEx‡</td><td>339</td><td>0.247</td><td>15.8</td><td>27.5</td><td>42.8</td><td>5261</td><td>0.440</td><td>41.0</td><td>46.0</td><td>51.0</td></tr><tr><td>ConvE‡</td><td>244</td><td>0.325</td><td>23.7</td><td>35.6</td><td>50.1</td><td>4187</td><td>0.430</td><td>40.0</td><td>44.0</td><td>52.0</td></tr><tr><td>TuckER‡</td><td>-</td><td>0.358</td><td>26.6</td><td>39.4</td><td>54.4</td><td>-</td><td>0.470</td><td>44.3</td><td>48.2</td><td>52.6</td></tr><tr><td>RotatE‡</td><td>177</td><td>0.338</td><td>24.1</td><td>37.5</td><td>53.3</td><td>3340</td><td>0.476</td><td>42.8</td><td>49.2</td><td>57.1</td></tr><tr><td rowspan="6">Rule-based Learning</td><td>PathRank‡</td><td>-</td><td>0.087</td><td>7.4</td><td>9.2</td><td>11.2</td><td>-</td><td>0.189</td><td>17.1</td><td>20.0</td><td>22.5</td></tr><tr><td>M-Walk‡</td><td>-</td><td>0.232</td><td>16.5</td><td>24.3</td><td>-</td><td>-</td><td>0.437</td><td>41.4</td><td>44.5</td><td>-</td></tr><tr><td>NeuralLP‡</td><td>-</td><td>0.237</td><td>17.3</td><td>25.9</td><td>36.1</td><td>-</td><td>0.381</td><td>36.8</td><td>38.6</td><td>40.8</td></tr><tr><td>DRUM‡</td><td>-</td><td>0.238</td><td>17.4</td><td>26.1</td><td>36.4</td><td>-</td><td>0.382</td><td>36.9</td><td>38.8</td><td>41.0</td></tr><tr><td>RNNLogic‡</td><td>232</td><td>0.344</td><td>25.2</td><td>38.0</td><td>53.0</td><td>4615</td><td>0.483</td><td>44.6</td><td>49.7</td><td>55.8</td></tr><tr><td>RNNLogic+‡</td><td>178</td><td>0.349</td><td>25.8</td><td>38.5</td><td>53.3</td><td>4624</td><td>0.513</td><td>47.1</td><td>53.2</td><td>59.7</td></tr><tr><td rowspan="3">GNN-based Learning</td><td>RGCN†</td><td>221</td><td>0.273</td><td>18.2</td><td>30.3</td><td>45.6</td><td>2719</td><td>0.402</td><td>34.5</td><td>43.7</td><td>49.4</td></tr><tr><td>GraIL†</td><td>2053</td><td>-</td><td>-</td><td>-</td><td>-</td><td>2539</td><td>-</td><td>-</td><td>-</td><td>-</td></tr><tr><td>NBFNet†</td><td>114</td><td>0.415</td><td>32.1</td><td>45.4</td><td>59.9</td><td>636</td><td>0.551</td><td>49.7</td><td>57.3</td><td>66.6</td></tr><tr><td rowspan="2">IBL-based</td><td>IBLE</td><td>263</td><td>0.284</td><td>20.0</td><td>31.0</td><td>45.2</td><td>7205</td><td>0.394</td><td>37.7</td><td>40.0</td><td>42.7</td></tr><tr><td>CIBLE</td><td>170</td><td>0.341</td><td>24.6</td><td>37.8</td><td>53.2</td><td>3400</td><td>0.490</td><td>44.6</td><td>50.7</td><td>57.5</td></tr></table>
+
+Table 4: Knowledge base completion results on Kinship and UMLS.  
+
+<table><tr><td rowspan="2">Category</td><td rowspan="2">Algorithm</td><td colspan="5">Kinship</td><td colspan="5">UMLS</td></tr><tr><td>MR</td><td>MRR</td><td>H@1</td><td>H@3</td><td>H@10</td><td>MR</td><td>MRR</td><td>H@1</td><td>H@3</td><td>H@10</td></tr><tr><td rowspan="5">KG Embedding</td><td>DistMult‡</td><td>8.5</td><td>0.354</td><td>18.9</td><td>40.0</td><td>75.5</td><td>14.6</td><td>0.391</td><td>25.6</td><td>44.5</td><td>66.9</td></tr><tr><td>ComplEx‡</td><td>7.8</td><td>0.418</td><td>24.2</td><td>49.9</td><td>81.2</td><td>13.6</td><td>0.411</td><td>27.3</td><td>46.8</td><td>70.0</td></tr><tr><td>ComplEx-N3‡</td><td>-</td><td>0.605</td><td>43.7</td><td>71.0</td><td>92.1</td><td>-</td><td>0.791</td><td>68.9</td><td>87.3</td><td>95.7</td></tr><tr><td>TuckER‡</td><td>6.2</td><td>0.603</td><td>46.2</td><td>69.8</td><td>86.3</td><td>5.7</td><td>0.732</td><td>62.5</td><td>81.2</td><td>90.9</td></tr><tr><td>RotatE‡</td><td>3.7</td><td>0.651</td><td>50.4</td><td>75.5</td><td>93.2</td><td>4.0</td><td>0.744</td><td>63.6</td><td>82.2</td><td>93.9</td></tr><tr><td rowspan="8">Rule Learning</td><td>MLN‡</td><td>10.0</td><td>0.351</td><td>18.9</td><td>40.8</td><td>70.7</td><td>7.6</td><td>0.688</td><td>58.7</td><td>75.5</td><td>86.9</td></tr><tr><td>Boosted RDN‡</td><td>25.2</td><td>0.469</td><td>39.5</td><td>52.0</td><td>56.7</td><td>54.8</td><td>0.227</td><td>14.7</td><td>25.6</td><td>37.6</td></tr><tr><td>PathRank†</td><td>-</td><td>0.369</td><td>27.2</td><td>41.6</td><td>67.3</td><td>-</td><td>0.197</td><td>14.8</td><td>21.4</td><td>25.2</td></tr><tr><td>NeuralLP‡</td><td>16.9</td><td>0.302</td><td>16.7</td><td>33.9</td><td>59.6</td><td>10.3</td><td>0.483</td><td>33.2</td><td>56.3</td><td>77.5</td></tr><tr><td>DRUM‡</td><td>11.6</td><td>0.334</td><td>18.3</td><td>37.8</td><td>67.5</td><td>8.4</td><td>0.548</td><td>35.8</td><td>69.9</td><td>85.4</td></tr><tr><td>MINERVA‡</td><td>-</td><td>0.401</td><td>23.5</td><td>46.7</td><td>76.6</td><td>-</td><td>0.564</td><td>42.6</td><td>65.8</td><td>81.4</td></tr><tr><td>CTP‡</td><td>-</td><td>0.335</td><td>17.7</td><td>37.6</td><td>70.3</td><td>-</td><td>0.404</td><td>28.8</td><td>43.0</td><td>67.4</td></tr><tr><td>RNNLogic‡</td><td>3.1</td><td>0.722</td><td>59.8</td><td>81.4</td><td>94.9</td><td>3.1</td><td>0.842</td><td>77.2</td><td>89.1</td><td>96.5</td></tr><tr><td>GNN-based</td><td>NBFNet</td><td>3.7</td><td>0.606</td><td>43.5</td><td>72.5</td><td>93.7</td><td>3.8</td><td>0.778</td><td>68.8</td><td>84.0</td><td>93.8</td></tr><tr><td rowspan="2">IBL-based</td><td>IBLE</td><td>3.7</td><td>0.650</td><td>51.3</td><td>75.5</td><td>93.7</td><td>3.2</td><td>0.816</td><td>71.7</td><td>90.0</td><td>96.1</td></tr><tr><td>CIBLE</td><td>3.0</td><td>0.728</td><td>60.3</td><td>82.0</td><td>95.6</td><td>2.6</td><td>0.856</td><td>78.7</td><td>91.6</td><td>97.0</td></tr></table>
+
+FB15k-237 and WN18RR, CIBLE outperforms previous state-of-the-art methods except the NBFNet. In Kinship and UMLS, CIBLE outperforms all existing methods. This verifies the effectiveness of CIBLE.
+
+# 6.3 Instance-based Interpretations
+
+Understanding model behavior via prototypes Table 5 shows the top 10 prototypes for predicting Taylor Swift and Barack Obama's nationalities in FB15k-237 according to Eq. (6). Despite the fact that they are both Americans, they have different prototypes. Taylor Swift prefers to use American singers and actors as prototypes, while Barack Obama prefers to use American politicians. These selections of prototypes are comprehensible to humans. In addition, we believe that prototypes are more transparent than logical rules, as many logical rules are semantically unclear to humans (see Table 2 for some examples). Sec 7 will delve deeper into this.
+
+Visualization of global model behavior In CIBLE, the distance between entity embeddings (i.e.  $\text{trans}_r(emb)$ ) measures the plausibility of a candidate prototype. Here we visualize
+
+Table 5: Top prototypes. For the same relation (i.e. nationality), Taylor Swift and Barack Obama choose different prototypes. This illustrates the rationality of how the model infers.  
+
+<table><tr><td>Entity &amp; Relation</td><td>Prototypes</td></tr><tr><td>Taylor Swift /nationality</td><td>Carrie Underwood, Miley Cyrus, Selena Gomez, Taylor Lautner, Demi Lovato, Joe Jonas, Randy Travis, Garth Brooks, Brad Paisley, Trisha Yearwood</td></tr><tr><td>Barack Obama /nationality</td><td>Hillary Rodham Clinton, Al Gore, Jimmy Carter, Ossie Davis, James Madison, Martin Luther King, Jr., Paul Rudd, Daniel Inouye, Colin Powell, Herbert Hoover</td></tr></table>
+
+the entity embeddings (i.e.  $\mathbf{W}_r\mathbf{e}$ ) for relation official language in FB15k-237 using t-SNE [31]. The visualization provides global interpretation, since it explains the prediction for all entities, rather than for one prediction.
+
+The visualization provides rich information about how CIBLE makes predictions: (1) entities with the same value exhibit distinct clustering patterns, and (2) the distance between entities reflects knowledge relatedness beyond the target value. For example, Argentina (Spanish) and Brazil (Portuguese) are close to each other, since they share other relations. (3) The location interprets 1-to-N relations. For example, English, Mandarin, and Tamil are all official languages of Singapore. As a result, Singapore is close to China (Mandarin), India (English) and Sri Lanka (Tamil).
+
+![](images/426a56564eebc70d20bef87c5cc7c34d538504093e6a7ac0b0472c4d313793d1.jpg)  
+Figure 2: t-SNE visualization of entities on " /country/official_language" in FB15k-237.
+
+# 6.4 Analysis
+
+![](images/45cfdb35ab8a59989c464a086e5adf3b6b24fab1c74267c7d37751d198c6d70f.jpg)  
+Figure 3: Training loss on UMLS.
+
+Table 6: Effect of relation-awareness.  
+
+<table><tr><td rowspan="2">Model</td><td rowspan="2">Relation -aware</td><td colspan="2">FB15k-237</td><td colspan="2">WN18RR</td></tr><tr><td>MRR</td><td>H@10</td><td>MRR</td><td>H@10</td></tr><tr><td>CIBLE-TransE</td><td>No</td><td>0.286</td><td>44.7</td><td>0.236</td><td>53.8</td></tr><tr><td>CIBLE-TransR</td><td>Yes</td><td>0.341</td><td>52.2</td><td>0.250</td><td>49.7</td></tr><tr><td>CIBLE-RotatE</td><td>No</td><td>0.315</td><td>50.6</td><td>0.430</td><td>54.2</td></tr><tr><td>CIBLE-R-RotatE</td><td>Yes</td><td>0.341</td><td>53.2</td><td>0.490</td><td>57.5</td></tr></table>
+
+Consistency of model optimization In Theorem 1, we show the consistency of the optimization objectives for the three models. To verify this theory, we show their training losses when optimizing CIBLE in Fig. 3. IBLE-baseline and RotatE-baseline denote the convergence of the training loss when we optimize them separately. Overall, as we optimizing CIBLE, the training loss of IBLE and RotatE both decreases. In addition, optimizing CIBLE helps IBLE converge to its minimum training loss. This verifies the consistency of the three optimization objectives.
+
+Effect of relation-awareness We show in Sec 3.4 that prototypes modeled by TransR and R-RotatE are relation-aware while TransE and RotatE are not. To validate the effect of relation-awareness, we compare their effect when adapting to CIBLE. Table 6 shows the results. Models with relation-awareness perform better on most metrics. This is in line with our intuition about prototypes.
+
+# 7 Rule-based Reasoners: Semantic Relevance or Instance-based Equivalence?
+
+In Sec 5, we defined a special class of logical rules, namely IBL rules. In this section, we reassess the mechanism and interpretability of logical rules by validating the effect of IBL rules.
+
+Table 7: Effect of IBL rules for FB15k-237 and WN18RR.  
+
+<table><tr><td rowspan="2">Model</td><td colspan="5">FB15k-237</td><td colspan="5">WN18RR</td></tr><tr><td>MR</td><td>MRR</td><td>H@1</td><td>H@3</td><td>H@10</td><td>MR</td><td>MRR</td><td>H@1</td><td>H@3</td><td>H@10</td></tr><tr><td>all rules</td><td>178</td><td>0.349</td><td>25.8</td><td>38.5</td><td>53.3</td><td>4624</td><td>51.3</td><td>47.1</td><td>53.2</td><td>59.7</td></tr><tr><td>non-IBL rules only</td><td>1241</td><td>0.317</td><td>23.2</td><td>35.1</td><td>48.6</td><td>9849</td><td>44.1</td><td>41.9</td><td>45.3</td><td>48.5</td></tr><tr><td>IBL-rules only</td><td>1011</td><td>0.326</td><td>24.0</td><td>36.1</td><td>49.9</td><td>8946</td><td>44.0</td><td>40.6</td><td>46.3</td><td>49.9</td></tr></table>
+
+Table 8: Effect of IBL rules for Kinship and UMLS.  
+
+<table><tr><td rowspan="2">Model</td><td colspan="5">Kinship</td><td colspan="5">UMLS</td></tr><tr><td>MR</td><td>MRR</td><td>H@1</td><td>H@3</td><td>H@10</td><td>MR</td><td>MRR</td><td>H@1</td><td>H@3</td><td>H@10</td></tr><tr><td>all rules</td><td>3.1</td><td>0.842</td><td>77.2</td><td>89.1</td><td>96.5</td><td>3.1</td><td>0.722</td><td>59.8</td><td>81.4</td><td>94.9</td></tr><tr><td>non-IBL rules only</td><td>9.9</td><td>0.667</td><td>53.7</td><td>76.9</td><td>86.2</td><td>4.0</td><td>0.681</td><td>55.0</td><td>77.3</td><td>93.2</td></tr><tr><td>IBL-rules only</td><td>3.0</td><td>0.840</td><td>75.1</td><td>91.3</td><td>96.6</td><td>3.3</td><td>72.1</td><td>59.5</td><td>81.4</td><td>94.7</td></tr></table>
+
+Earlier rule-based models learn logical rules in a differential way [21, 38]. However, their search space is exponentially large, which results in limited rule length. The recently proposed method, RNNLogic [17], was a breakthrough in its capacity to generate rules with maximum length of 5. With these longer rules, RNNLogic achieves 53.3 hit@10 in FB15k-237, compared to the previous highest hit@10 of only 36.4 for rule-based models [21]. Therefore, we use RNNLogic to investigate the effect of IBL rules.
+
+To demonstrate the effect of IBL rules, we control the candidate rule space of RNNLogic by modifying its rule collection module. We carried out the following ablations: (1) collecting rules from the entire rule space; (2) collecting only non-IBL rules; (3) collecting only IBL rules. Table 7 and Table 8 show the effects of these ablations.
+
+The results are surprising. Despite the fact that IBL rules only cover a small portion of the entire rule space, learning IBL rules outperforms learning non-IBL rules in all four benchmarks. In Kinship and UMLS, learning only IBL rules has almost no performance degradation when compared to learning from the entire rule space. The results indicate that, the small number of IBL rules are even more critical than non-IBL rules.
+
+These findings challenge the widely-held belief that rule-based reasoning use rules whose premises and hypotheses are semantically compatible. The results imply that rule-based KBC models work largely because its capability to represent instance-based equivalence (i.e. prototypes) via IBL rules. The interpretability from instance-based equivalency should not be overlooked in rule-based models.
+
+To understand why IBL rules outperform other semantic non-IBL rules, we investigate the quality of each rule. More concretely, we show the average precision and support [7] of each collected rule for different rule types in Table 9. The results show that both the average precision and support of IBL rules are substantially higher than those of non-IBL rules. The high quality of IBL rules explains why they outperform non-IBL rules despite occupying only a small fraction of the entire rule space.
+
+IBL-based vs rule-based models When representing the instance-based equivalence relationship, CIBLE is more flexible and theoretically sound than rule-based models using IBL rules. In Table 3 4, despite the fact that RNNLogic employs a complex EM algorithm for learning, CIBLE outperforms RNNLogic in three of the four benchmark. Therefore, we consider CIBLE to be a more effective alternative to the rule-based models in instance-based equivalency modeling.
+
+Table 9: Average support and precision of IBL and non-IBLE rules.  
+
+<table><tr><td></td><td>FB15k237</td><td>WN18RR</td><td>UMLS</td><td>Kinships</td></tr><tr><td>IBL Rule</td><td>708.3 / 3.7%</td><td>2374.3 / 12.7%</td><td>3.0 / 11.6%</td><td>8.7 / 11.6%</td></tr><tr><td>Non-IBL Rule</td><td>281.4 / 1.7%</td><td>188.3 / 5.0%</td><td>3.0 / 9.5%</td><td>6.7 / 5.1%</td></tr></table>
+
+# 8 Conclusion
+
+In this paper, we explore a novel IBL-based method for KBC. We validate its effectiveness in two aspects. First, we show that the prototypes can be expressed in a closed-form with the well-studied translational models, and IBL-based models can be cotypecombined with translational models. Second, we found that rule-based reasoning relies heavily on IBL rules. This challenges previous common understanding of how logical rules work and provide interpretability.
+
+Broader impact Our proposed IBL method provides a new paradigm for learning representations for knowledge bases. Based on the results of the experiments, the IBL paradigm can be combined with translational models and rule-based reasoners and enhance the effectiveness. Our theory has provided the foundation for combing IBL models and translational models. We expect to better exploit the theory to combine different KBC models in the future.
+
+# Acknowledgments and Disclosure of Funding
+
+This paper was supported by National Natural Science Foundation of China (No. 61906116), by Shanghai Sailing Program (No. 19YF1414700). We thank Wenting Ba for her valuable plotting assistance.
+
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