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+# WOULDA, COULDA, SHOULDA: COUNTERFACTUALLY-GUIDED POLICY SEARCH
+
+Lars Buesing, Theophane Weber, Yori Zwols, ´   
+Sebastien Racani ´ ere, Arthur Guez, Jean-Baptiste Lespiau, Nicolas Heess \`   
+DeepMind   
+lbuesing@google.com
+
+# ABSTRACT
+
+Learning policies on data synthesized by models can in principle quench the thirst of reinforcement learning algorithms for large amounts of real experience, which is often costly to acquire. However, simulating plausible experience de novo is a hard problem for many complex environments, often resulting in biases for modelbased policy evaluation and search. Instead of de novo synthesis of data, here we assume logged, real experience and model alternative outcomes of this experience under counterfactual actions, i.e. actions that were not actually taken. Based on this, we propose the Counterfactually-Guided Policy Search (CF-GPS) algorithm for learning policies in POMDPs from off-policy experience. It leverages structural causal models for counterfactual evaluation of arbitrary policies on individual off-policy episodes. CF-GPS can improve on vanilla model-based RL algorithms by making use of available logged data to de-bias model predictions. In contrast to off-policy algorithms based on Importance Sampling which re-weight data, CF-GPS leverages a model to explicitly consider alternative outcomes, allowing the algorithm to make better use of experience data. We find empirically that these advantages translate into improved policy evaluation and search results on a non-trivial grid-world task. Finally, we show that CF-GPS generalizes the previously proposed Guided Policy Search and that reparameterization-based algorithms such Stochastic Value Gradient can be interpreted as counterfactual methods.
+
+# 1 INTRODUCTION
+
+Imagine that a month ago Alice had two job offers from companies $a _ { 1 }$ and $a _ { 2 }$ . She decided to join $a _ { 1 }$ because of the larger salary, in spite of an awkward feeling during the job interview. Since then she learned a lot about $a _ { 1 }$ and recently received information about $a _ { 2 }$ from a friend, prodding her now to imagine what would have happened had she joined $a _ { 2 }$ . Re-evaluating her decision in hindsight in this way, she concludes that she made a regrettable decision. She could and should have known that $a _ { 2 }$ was a better choice, had she only interpreted the cues during the interview correctly... This example tries to illustrate the everyday human capacity to reason about alternate, counterfactual outcomes of past experience with the goal of “mining worlds that could have been” (Pearl & Mackenzie, 2018). Social psychologists theorize that such cognitive processes are beneficial for improving future decision making (Roese, 1997). In this paper we aim to leverage possible advantages of counterfactual reasoning for learning decision making in the reinforcement learning (RL) framework.
+
+In spite of recent success, learning policies with standard, model-free RL algorithms can be notoriously data inefficient. This issue can in principle be addressed by learning policies on data synthesized from a model. However, a mismatch between the model and the true environment, often unavoidable in practice, can cause this approach to fail (Talvitie, 2014), resulting in policies that do not generalize to the real environment (Jiang et al., 2015). Motivated by the introductory example, we propose the Counterfactually-Guided Policy Search (CF-GPS) algorithm: Instead of relying on data synthesized from scratch by a model, policies are trained on model predictions of alternate outcomes of past experience from the true environment under counterfactual actions, i.e. actions that had not actually been taken, while everything else remaining the same (Pearl, 2009). At the heart of CF-GPS are structural causal models (SCMs) which model the environment with two ingredients (Wright, 1920): 1) Independent random variables, called scenarios here, summarize all aspects of the environment that cannot be influenced by the agent, e.g. the properties of the companies in Alice’s job search example. 2) Deterministic transition functions (also called causal mechanisms) take these scenarios, together with the agent’s actions, as input and produce the predicted outcome. The central idea of CF-GPS is that, instead of running an agent on scenarios sampled de novo from a model, scenarios are inferred in hindsight from given off-policy data, and then evaluate and improve the agent on these specific scenarios using given or learned causal mechanisms (Balke & Pearl, 1994). CF-GPS can be regarded as a meta-algorithm that extends a given model-base RL algorithm by “grounding” or “anchoring” model-based predictions in inferred scenarios. As a result, this approach explicitly allows to trade-off historical data for model bias. We show empirically in a conceptually simple setting, where unknown initial states are inferred in hindsight and re-used to evalute to counterfactual actions, that this can mitigate model mismatch. CF-GPS differs substantially from standard off-policy RL algorithms based on Importance Sampling (IS), where historical data is re-weighted with respect to the importance weights to evaluate or learn new policies (Precup, 2000). In contrast, CF-GPS explicitly reasons counterfactually about given off-policy data. Our main contributions are:
+
+1. We formulate model-based RL in POMDPs in terms of structural causal models, thereby connecting concepts from reinforcement learning and causal inference. 2. We provide the first results, to the best of our knowledge, showing that counterfactual reasoning in structural causal models on off-policy data can facilitate solving non-trivial RL tasks. 3. We show that two previously proposed classes of RL algorithms, namely Guided Policy Search (Levine & Koltun, 2013) and Stochastic Value Gradient methods (Heess et al., 2015), can be interpreted as counterfactual methods, opening up possible generalizations.
+
+The paper is structured as follows. We first give a self-contained, high-level recapitulation of structural causal models and counterfactual inference, as these are less widely known in the RL and generative model communities. In particular we show how to model POMDPs with SCMs. Based on this exposition, we first consider the task of policy evaluation and discuss how we can leverage counterfactual inference in SCMs to improve over standard model-based methods. We then generalize this approach to the policy search setting resulting in the CF-GPS algorithm. We close by highlighting connections to previously proposed algorithms and by discussing assumptions and limitations of the proposed method.
+
+# 2 PRELIMINARIES
+
+We denote random variables (RVs) with capital letters, e.g. $X$ , and particular values with lower caps, e.g. $x$ . For a distribution $P$ over a vector-valued random variable $X$ , we denote the marginal over $Y \subset X$ by $P _ { Y }$ (and density $p _ { Y }$ ); however we often omit the subscript if it is clear from the context, e.g. as in $Y \sim P$ . We assume the episodic, Partially Observable Markov Decision Process (POMDP) setting with states $S _ { t }$ , actions $A _ { t }$ and observations $O _ { t }$ , for $t = 1 , \dots , T$ . For n, we assume that . We consider sto $O _ { t }$ includes theastic policies $R _ { t }$ . The undiscounted return is denoted byover actions conditioned on observation $\begin{array} { r } { G = \sum _ { t = 1 } ^ { T } R _ { t } } \end{array}$ $\pi ( a _ { t } | h _ { t } )$   
+histories $\hat { H _ { t } } = ( O _ { 1 } , A _ { 1 } , \ldots , A _ { t - 1 } , O _ { t } )$ . We denote the resulting distribution over trajectories $( S _ { 1 } , O _ { 1 } , A _ { 1 } , \ldots , A _ { T - 1 } , S _ { T } , O _ { T } )$ induced by running $\pi$ in the environment with $\mathcal { T } \sim \mathfrak { P } ^ { \pi }$ and the corresponding density by ${ \mathfrak { p } } ^ { \pi } ( \tau )$ .
+
+# 2.1 STRUCTURAL CAUSAL MODELS
+
+Definition 1 (Structural causal model). $A$ structural causal model (SCM) $\mathcal { M }$ over $\begin{array} { r l } { X } & { { } = } \end{array}$ $( X _ { 1 } , \ldots , X _ { N } )$ is given by a $D A G ~ { \mathcal { G } }$ over nodes $X$ , independent noise RVs $U = ( U _ { 1 } , \ldots , U _ { N } )$ with distributions $P _ { U _ { i } }$ and functions $f _ { 1 } , \ldots , f _ { N }$ such that $X _ { i } = f _ { i } ( \mathrm { p a } _ { i } , U _ { i } )$ , where $\operatorname { p a } _ { i } \subset X$ are the parents of $X _ { i }$ in $G$ . An SCM entails a distribution $P$ with density $p$ over $( X , U )$ .
+
+![](images/9592b3f78d13cb29be27d512fb14995ca71647e2853e999782dc1fe5a34e127f.jpg)  
+Figure 1: Structural causal models (SCMs) model environments using random variables $U$ (circles, ‘scenarios’), that summarize immutable aspects, some of which are observed (grey), some not (white). These are fed into deterministic functions $f _ { i }$ (black squares) that approximate causal mechanisms. Left: SCM for a contextual bandit with context $U _ { c }$ , action $A$ , feedback $O$ and scenario $U _ { o }$ . Right: SCM for a POMDP, with initial state $U _ { s 1 } = S _ { 1 }$ , states $S _ { t }$ and histories $H _ { t }$ . The mechanism that generates the actions $A _ { t }$ is the policy $\pi$ .
+
+We also refer to $U$ as scenarios and to $f _ { i }$ as causal mechanisms. We give a (broad) definition of an intervention in an SCM. This also includes what is known as stochastic interventions or mechanism changes (Korb et al., 2004) which generalize atomic interventions (Pearl, 2009).
+
+Definition 2 (Intervention in SCM). An intervention $I$ in an SCM $\mathcal { M }$ consists of replacing some of the original $f _ { i } ( \mathrm { p a } _ { i } , U _ { i } )$ with other functions $f _ { i } ^ { I } ( \mathrm { p a } _ { i } ^ { I } , U _ { i } )$ where $\mathrm { p a } _ { i } ^ { I }$ are the parents in a new $D A G$ $\mathcal { G } ^ { I }$ . We denote the resulting SCM with $\boldsymbol { \mathcal { M } } ^ { \mathrm { d o } ( I ) }$ with distribution $P ^ { \mathrm { d o } ( I ) }$ and density $p ^ { \mathrm { d o } ( I ) }$ .
+
+SCM representation of POMDPs We can represent any given POMDP (under a policy $\pi$ ) by an SCM $\mathcal { M }$ over trajectories $\tau$ in the following way. We express all conditional distributions, e.g. the transition kernel $P _ { S _ { t + 1 } | S _ { t } , A _ { t } }$ , as deterministic functions with independent noise variables $U$ , such as $S _ { t + 1 } = f _ { s t } ( S _ { t } , A _ { t } , \dot { U } _ { s t } )$ . This is always possible using auto-regressive uniformization, see Lemma 2 in the appendix. The DAG $\mathcal { G }$ of the resulting SCM is shown in fig. 1. This procedure is closely related to the ‘reparameterization trick’ for models with location-scale distributions (Kingma & Welling, 2013; Rezende et al., 2014). We denote the distribution over $\tau$ entailed by the SCM with $P ^ { \pi }$ and its density by $p ^ { \pi }$ to highlight the role of $\pi$ ; note the difference to the true environment distribution $\mathfrak { P } ^ { \pi }$ with density ${ \mathfrak { p } } ^ { \pi }$ . Running a different policy $\mu$ instead of $\pi$ in the environment can be expressed as an intervention $I ( \pi \to \mu )$ consisting of replacing ${ A _ { t } } \mathrm { ~ = ~ } f _ { \pi } ( { H _ { t } } , { U _ { a t } } )$ by $A _ { t } \ =$ $f _ { \mu } ( H _ { t } , U _ { a t } )$ . We denote the resulting model distribution over trajectories with $P ^ { \mathrm { d o } ( I ( \pi  \mu ) ) } = P ^ { \mu }$ (analogously $p ^ { \mu }$ ).
+
+Intuition Here, we illustrate the main advantage of SCMs using the example of Alice’s job choice from the introduction. We model it as contextual bandit with feedback shown in fig. 1. Alice has some initial knowledge given by the context $U _ { c }$ that is available to her before taking action $A$ of joining company $A = a _ { 1 }$ or $A = a _ { 2 }$ . We model Alice’s decision as $A = f _ { \pi } ( U _ { c } , \bar { U _ { a } } )$ , where $U _ { a }$ captures potential indeterminacy in Alice’s decision making. The outcome $O = f _ { o } ( A , U _ { c } , U _ { o } )$ also depends on the scenario $U _ { o }$ , capturing all relevant, unobserved and highly complex properties of the two companies such as working conditions etc. Given this model, we can reason about alternate outcomes $f _ { o } ( a _ { 1 } , u _ { c } , u _ { o } )$ and $f _ { o } ( a _ { 2 } , u _ { c } , u _ { o } )$ for same the scenario $u _ { o }$ . This is not possible if we only model the outcome on the level of the conditional distribution $P _ { O | A , U _ { c } }$ .
+
+# 2.2 COUNTERFACTUAL INFERENCE IN SCMS
+
+For an SCM over $X$ , we define a counterfactual query as a triple $( \hat { x } _ { o } , I , X _ { q } )$ of observations $\scriptstyle { \hat { x } } _ { o }$ of some variables $X _ { o } \subset X$ , an intervention $I$ and query variables $X _ { q } \subset X$ . The semantics of the query are that, having observed $\scriptstyle { \hat { x } } _ { o }$ , we want to infer what $X _ { q }$ would have been had we done intervention $I$ , while ‘keeping everything else the same’. Counterfactual inference (CFI) in SCMs answers the query in the following way (Balke & Pearl, 1994):
+
+1. Infer the unobserved noise source $U$ conditioned on the observations $\scriptstyle { \hat { x } } _ { o }$ , i.e. compute $p ( U | \hat { x } _ { o } )$ and replace the prior $p ( U )$ with $p ( U | \hat { x } _ { o } )$ . Denote the resulting SCM by $\mathcal { M } _ { \hat { x } _ { o } }$ .
+
+2. Perform intdistribution $I$ on . R $\mathcal { M } _ { \hat { x } _ { o } }$ . This yieldthe marginal $\mathcal { M } _ { \hat { x } _ { o } } ^ { \mathrm { d o } ( I ) }$ hich entails the counterfactual. $p ^ { \mathrm { d o } ( I ) | \hat { x } _ { o } } ( x )$ $p ^ { \mathrm { d o } ( I ) | \hat { x } _ { o } } ( x _ { q } )$
+
+Note that our definition explicitly allows for partial observations $X _ { o } \subset X$ in accordance with Pearl (2009). A sampled-based version, denoted as CFI, is presented in Algorithm 1. An interesting property of the counterfactual distribution $p ^ { \mathrm { d o } ( I ) | \hat { x } _ { o } }$ is that marginalizing it over observations $\scriptstyle { \hat { x } } _ { o }$ yields an unbiased estimator of the density of $X _ { q }$ under intervention $I$ .
+
+Lemma 1 (CFI for simulation). Let observations $\hat { x } _ { o } \sim p$ come from a SCM $\mathcal { M }$ with density $p$ . Then the counterfactual density $p ^ { \mathrm { d o } ( I ) | \hat { x } _ { o } }$ is an unbiased estimator of $p ^ { \mathrm { d o } ( I ) }$ , i.e.
+
+$$
+\begin{array} { r c l } { \mathbb { E } _ { \hat { x } _ { o } \sim p } [ p ^ { \mathrm { d o } ( I ) | \hat { x } _ { o } } ( x ) ] } & { = } & { p ^ { \mathrm { d o } ( I ) } ( x ) } \end{array}
+$$
+
+The proof is straightforward and outlined in the Appendix A. This lemma and the marginal independence of the $U _ { i }$ leads to the following corollary; the proof is given in the appendix.
+
+Corollary 1 (Mixed counterfactual and prior simulation from an SCM). Assume we have observations $\hat { x } _ { o } \sim p .$ . We can simulate from $\mathcal { M }$ , under any intervention $I$ , i.e. obtain unbiased samples from $\mathcal { M } ^ { \mathrm { d o } ( I ) }$ , by first sampling values $u _ { \mathrm { C F } }$ for an arbitrary subset $U _ { \mathrm { C F } } \subset U$ from the posterior $p ( u _ { \mathrm { C F } } | \hat { x } _ { o } )$ and the remaining $U _ { \mathrm { P r i o r } } : = U \backslash U _ { \mathrm { C F } }$ from the prior $p ( u _ { \mathrm { P r i o r } } )$ , and then computing $X$ with noise $u = u _ { \mathrm { C F } } \cup u _ { \mathrm { P r i o r } }$ .
+
+The corollary essentially states that we can sample from the model $\mathcal { M } ^ { I }$ , by sampling some of the $U _ { i }$ from the prior, and inferring the rest from data $\scriptstyle { \hat { x } } _ { o }$ (as long as the latter was also sampled from $\mathcal { M }$ ). We will make use of this later for running a POMDP model on scenarios $U _ { s t }$ inferred from data while randomizing the action noise $U _ { a t }$ . We note that the noise variables $U _ { \mathrm { C F } }$ from the posterior $P _ { U _ { \mathrm { C F } } | \hat { x } _ { o } }$ are not independent anymore. Nevertheless, SCMs with non-independent noise distributions arising from counterfactual inference, denoted here by $\mathcal { M } _ { \hat { x } _ { o } }$ , are commonly considered in the literature (Peters et al., 2017).
+
+Intuition Returning to Alice’s job example from the introduction, we give some intuition for counterfactual inference in SCMs. Given the concrete outcome $\hat { o }$ , under observed context $\hat { u } _ { c }$ and having joined company $\hat { a } = a _ { 1 }$ , Alice can try to infer the underlying scenario $u _ { o } \sim p ( u _ { o } | a _ { 1 } , \hat { u } _ { c } , \hat { o } )$ that she experiences; this includes factors such as work conditions etc. She can then reason counterfactually about the outcome had she joined the other company, which is given by $f _ { o } ( a _ { 2 } , \hat { u } _ { c } , u _ { o } )$ . This can in principle enable her to make better decisions in the future in similar scenarios by changing her policy $f _ { \pi } ( A , U _ { c } , U _ { a } )$ such that the action with the preferred outcome becomes more likely under $\hat { u } _ { c } , u _ { o }$ . In particular she can do so without having to use her (likely imperfect) prior model over possible companies $p ( U _ { o } )$ . She can use the counterfactual predictions discussed above instead to learn from her experience. We use this insight for counterfactual policy evaluation and search below.
+
+# 3 OFF-POLICY EVALUATION: MODEL-FREE, MODEL-BASED AND COUNTERFACTUAL
+
+To explain how counterfactual reasoning in SCMs can be used for policy search, we first consider the simpler problem of policy evaluation (PE) on off-policy data. The goal of off-policy PE is to determine the value of a policy $\pi$ , i.e. its expected return $\mathbb { E } _ { \mathfrak { p } ^ { \pi } } [ G ]$ , without running the policy itself. We assume that we have data $D ~ = ~ \{ \hat { h } _ { T } ^ { i } \} _ { i = 1 , \dots , N }$ consisting of logged episodes $\hat { h } _ { T } ^ { i } = ( \hat { o } _ { 1 } ^ { i } , \hat { a } _ { 1 } ^ { i } , \dots \hat { a } _ { T - 1 } ^ { i } , \hat { o } _ { T } ^ { i } )$ from running a behavior policy $\mu$ . A standard, model-free approach to PE is to use Importance sampling (IS): We can estimate the policy’s value as $\textstyle \sum _ { i } w ^ { i } { \hat { G } } ^ { i }$ , where $\hat { G } ^ { i }$ is the empirical return of $\hat { h } _ { T } ^ { i }$ and $\begin{array} { r } { w ^ { i } \propto \frac { \mathsf { p } ^ { \pi } ( \hat { h } _ { T } ^ { i } ) } { \mathsf { p } ^ { \mu } ( \hat { h } _ { T } ^ { i } ) } } \end{array}$ are importance weights. However, if the trajectory densities ${ \mathfrak { p } } ^ { \pi }$ and ${ \mathfrak { p } } ^ { \mu }$ are very different, then this estimator has large variance. In the extreme case, IS can be useless if the support of ${ \mathfrak { p } } ^ { \mu }$ does not contain that of ${ \mathfrak { p } } ^ { \pi }$ , irrespective of how much data from ${ \mathfrak { p } } ^ { \mu }$ is available.
+
+If we have access to a model $\mathcal { M }$ , then we can evaluate the policy on synthetic data, i.e. we can estimate $\mathbb { E } _ { p ^ { \pi } } [ G ]$ . This is called model-based policy evaluation (MB-PE). However, any bias in $\mathcal { M }$ propagates from $p ^ { \pi }$ to the estimate $\mathbb { E } _ { p ^ { \pi } } [ G ]$ . In the following, we assume that $\mathcal { M }$ is a SCM and we
+
+# Algorithm 1 Counterfactual policy evaluation and search
+
+<table><tr><td colspan="2">//Counterfactual inference (CFI) procedure CFI(data 𝑥., SCM M,intervention I, query Xq)</td></tr><tr><td colspan="2">1: u~p(ulxo)</td></tr><tr><td>2: 3:</td><td></td></tr><tr><td></td><td>p(u)←δ(u-u)</td></tr><tr><td>4:</td><td>f←f</td></tr><tr><td colspan="2">return xq ~ pdo(1) (xalu) 5: 6: end procedure</td></tr><tr><td colspan="2">/Counterfactual PolicyEvaluation (CF-PE)</td></tr><tr><td>7:</td><td>procedure CF-PE(SCM M, policy π,replay buffer D, number of samples N)</td></tr><tr><td>8: 9:</td><td>fori∈{1,...N}do ~D</td></tr><tr><td>10:</td><td>gi = CFI(h𝑖,M,I(μ→π),G)</td></tr><tr><td>11:</td><td>end for</td></tr><tr><td>12: 13:</td><td></td></tr><tr><td colspan="2">end procedure</td></tr><tr><td></td><td>/Counterfactually-Guided Policy Search (CF-GPS)</td></tr><tr><td>for k =1,... do</td><td>14:procedure CF-GPS(SCM M, initial policy π°, number of trajectory samples N)</td></tr><tr><td>15: 16:</td><td></td></tr><tr><td>17:</td><td>if sometimes then</td></tr><tr><td>18:</td><td>μ←πk end if</td></tr><tr><td>19:</td><td></td></tr><tr><td></td><td>for i=1,...,N do</td></tr><tr><td>20:</td><td>h~pμ</td></tr><tr><td>21:</td><td>T²=CFI(h,M,I(μ→π),T)</td></tr><tr><td></td><td></td></tr><tr><td>22:</td><td></td></tr><tr><td></td><td></td></tr><tr><td></td><td></td></tr><tr><td></td><td>end for</td></tr><tr><td></td><td></td></tr><tr><td></td><td></td></tr><tr><td>23:</td><td>π ←policy improvement on trajectories i=1.,N using eqn. 1</td></tr><tr><td></td><td></td></tr><tr><td></td><td></td></tr><tr><td></td><td></td></tr><tr><td>24:</td><td></td></tr><tr><td></td><td>end for</td></tr><tr><td></td><td></td></tr><tr><td>25: end procedure</td><td></td></tr><tr><td></td><td></td></tr><tr><td></td><td></td></tr><tr><td></td><td></td></tr><tr><td></td><td></td></tr><tr><td></td><td></td></tr><tr><td></td><td></td></tr><tr><td></td><td></td></tr><tr><td></td><td></td></tr><tr><td></td><td></td></tr><tr><td></td><td></td></tr><tr><td></td><td></td></tr><tr><td></td><td></td></tr><tr><td></td><td></td></tr><tr><td></td><td></td></tr><tr><td></td><td></td></tr><tr><td></td><td></td></tr><tr><td></td><td></td></tr><tr><td></td><td></td></tr><tr><td></td><td></td></tr><tr><td></td><td></td></tr><tr><td></td><td></td></tr><tr><td></td><td></td></tr><tr><td></td><td></td></tr><tr><td></td><td></td></tr><tr><td></td><td></td></tr><tr><td></td><td></td></tr><tr><td></td><td></td></tr><tr><td></td><td></td></tr></table>
+
+show that we can use counterfactual reasoning for off-policy evaluation (CF-PE). As the main result for this section, we argue that we expect CF-PE to be less biased than MB-PE, and we illustrate this point with experiments.
+
+# 3.1 COUNTERFACTUAL OFF-POLICY EVALUATION
+
+Naive MB-PE with a SCM $\mathcal { M }$ simply consist of sampling the scenarios $U \sim P _ { U }$ from the prior, and then simulating a trajectory $\tau$ from the functions $f _ { i }$ and computing its return. However, given data $D$ from ${ \mathfrak { p } } ^ { \mu }$ , our discussion of counterfactual inference in SCMs suggests the following alternative strategy: Assuming no model mismatch, i.e. ${ \mathfrak { p } } ^ { \mu } = p ^ { \mu }$ , we can regard the task of off-policy evaluation of $\pi$ as a counterfactual query with data $\hat { h } _ { T } ^ { i }$ , intervention $I ( \mu \to \pi )$ and query variable $G$ . In other words, instead of sampling from the prior as in MB-PE, we are free to the scenarios from the posterior $u ^ { i } \sim p ^ { \mu } ( \cdot | \hat { h } _ { T } ^ { i } )$ . The algorithm is given in Algorithm 1. Lemma 1 guarantees that this results in an unbiased estimate:
+
+Corollary 2 (CF-PE is unbiased). Assuming no model mismatch, CF-PE is unbiased.
+
+Furthermore, Corollary 1 allows us to also sample some of the noise variables from the prior instead of the posterior, we can e.g. randomize the counterfactual actions by re-sampling the action noise Ua.
+
+Motivation When should one prefer CF-PE over the more straightforward MB-PE? Assuming a perfect model, Corollary 2 states that both yield the same answer in expectation for perfect models. For imperfect models however, these algorithms can differ substantially. MB-PE relies on purely synthetic data, sampled from the noise distribution $p ( U )$ . In practice, this is usually approximated by a parametric density model, which can lead to under-fitting in case of complex distributions. This is a well-known effect in generative models with latent variables: In spite of recent research progress, e.g. models of natural images are still unable to accurately model the variability of the true data (Gregor et al., 2016). In contrast, CF-PE samples from the posterior $\begin{array} { r } { N ^ { - 1 } \sum _ { i = 1 } ^ { N } p ^ { \mu } ( U | \hat { h } _ { T } ^ { i } ) } \end{array}$ , which has access to strictly more information than the prior $p ( U )$ by taking into account additional data $\hat { h } _ { T } ^ { i }$ . This semi-nonparametric distribution can help to de-bias the model by effectively winnowing out parts of the domain of $U$ which do not correspond to any real data. We substantiate this intuition with experiments below; a concrete illustration for the difference between the prior and posterior $/$ counterfactual distribution is given in fig. 4 in the appendix and discussed in appendix D. Therefore, we conclude that we expect CF-PE to outperform MB-PE, if the transition and reward kernels $f _ { s t }$ are accurate models of the environment dynamics, but if the marginal distribution over the noise sources $P _ { U }$ is difficult to model.
+
+# 3.2 EXPERIMENTS
+
+Environment As an example, we use a partially-observed variant of the SOKOBAN environment, which we call PO-SOKOBAN. The original SOKOBAN puzzle environment was described in detail by Racaniere et al. (2017); we give a brief summary here. The agent is situated in a \` $1 0 \times 1 0$ grid world and its five actions are to move to one of four adjacent tiles and a NOOP. In our variant, the goal is to push all three boxes onto the three targets. As boxes cannot be pulled, many actions result irreversibly in unsolvable states. Episodes are of length $T = 5 0$ , and pushing a box onto a target yields a reward of 1, removing a box from a target yields $- 1$ , and solving a level results in an additional reward of 10. The state of the environment consists in a $1 0 \times 1 0$ matrix of categorical variables taking values in $\{ 0 , \ldots , 6 \}$ indicating if the corresponding tile is empty, a wall, box, target, agent, or a valid combinations thereof (box+target and agent+target). In order to introduce partial observability, we define the observations as the state corrupted by i.i.d. (for each tile and time step) flipping each categorical variable to the “empty” state with probability 0.9. Therefore, the state of the game is largely unobserved at any given time, and a successful agent has to integrate observations over tens of time steps. Initial states $U _ { s 1 }$ , also called levels, which are the scenarios in this environment, are generated randomly by a generator algorithm which guarantees solvability (i.e. all boxes can be pushed onto targets). The environment is visualized in fig. 3 in the appendix.
+
+Given the full state of PO-SOKOBAN, the transition kernel is deterministic and quite simple as only the agent and potentially an adjacent box moves. Inferring the belief state, i.e. the distribution over states given the history of observations and actions, can however range from trivial to very challenging, depending on the amount of available history. In the limit of a long observed history, every tile is eventually observed and the belief state concentrates on a single state (the true state) that can be easily inferred. With limited observed history however, inferring the posterior distribution over states (belief state) is very complex. Consider e.g. the situation in the beginning of an episode (before pushing the first box). Only the first observation is available, however we know that all POSOKOBAN levels are initially guaranteed to be solvable and therefore satisfy many combinatorial constraints reflecting that the agent is still able to push all boxes onto targets. Learning a compact parametric model of the initial state distribution from empirical data is therefore difficult and likely results in large mismatch between the learned model and the true environment.
+
+Results To illustrate the potential advantages of CF-PE over MB-PE we perform policy evaluation in the PO-SOKOBAN environment. We first generate a policy $\pi$ that we wish to evaluate, by training it using a previously-proposed distributed RL algorithm (Espeholt et al., 2018). The policy is parameterized as a deep, recurrent neural network consisting of a 3-layer deep convolutional LSTM (Xingjian et al., 2015) with 32 channels per layer and kernel size of 3. To further increase computational power, the LSTM ticks twice for each environment step. The output of the agent is a value function and a softmax yielding the probabilities of taking the 5 actions. In order to obtain an SCM of the environment, for the sake of simplicity, we assume that the ground-truth transition, observation and reward kernels are given. Therefore the only part of the model that we need to learn is the distribution $p ( U _ { s 1 } )$ of initial states $S _ { 1 } = U _ { s 1 }$ (for regular MB-PE), and the density $p ( U _ { s 1 } | \hat { h } _ { t } ^ { i } )$ for inferring levels in hindsight for CF-PE. We vary the amount of true data $t$ that we condition this inference on, ranging from $t = 0$ (no real data, equivalent to MB-PE) to $t = T = 5 0$ (a full episode of real data is used to infer the initial state $U _ { s 1 }$ ). We train a separate model for each $t \in \{ 0 , 5 , 1 0 , 2 0 , 3 0 , 4 0 , 5 0 \}$ . To simplify model learning, both models were given access to the unobserved state during training, but not at test time. The models are chosen to be powerful, multilayer, generative DRAW models (Gregor et al., 2015) trained by approximate maximum likelihood learning (Kingma & Welling, 2013; Rezende et al., 2014). The models take as input the (potentially empty) data $\hat { h } _ { t } ^ { i }$ summarized by a backward RNN (a standard convolutional LSTM model with 32 units). The model is shown in fig. 3 in the appendix and additional details are given in appendix C. The data $\hat { h } _ { T } ^ { i }$ was collected under a uniform random policy $\mu$ . For all policy evaluations, we use $\approx > 1 0 ^ { 5 }$ levels $u ^ { i }$ from the inferred model. In order to evaluate policies of different proficiency, we derive from the original (trained) $\pi$ three policies $\pi _ { 0 } , \pi _ { 1 } , \pi _ { 2 }$ ranging from almost perfect to almost random performance by introducing additional stochasticity during action selection.
+
+![](images/b366044c03ad90648e8c61367c9f01c4ca1964cc36322a4e902457ebbae795ad.jpg)  
+Figure 2: Experimental results on PO-SOKOBAN environment. Left: Policy evaluation. Policy evaluation error decreases with amount of off-policy data available (in #transitions per episode) for inferring scenarios (levels) $U _ { s 1 }$ that are used for counterfactual evaluation. No data (data points on the very left) corresponds to standard model-based policy evaluation (MB-PE), yielding large errors, whereas Counterfactual policy evaluation yields more accurate results. This holds for all three policies with different true performances. Right: Policy search. Counterfactually-Guided Policy Search (CF-GPS) outperforms a naive model-based RL (MB-PS) algorithm as well as a version of standard Guided Policy Search (‘GPS-like’) on PO-SOKOBAN.
+
+The policy evaluation results are shown in fig. 2. We found that for $t = 0$ , in spite of extensive hyper-parameter search, the model $p ( U _ { s 1 } )$ was unable to accurately capture the marginal distribution of initial levels in PO-SOKOBAN. As argued above, a solvable level satisfies a large number of complex constraints that span the entire grid world, which are hard for a parametric model to capture. Empirically, we found that the model mismatch manifested itself in samples from $p ( U _ { s 1 } )$ not being well-formed, e.g. not solvable, and hence the performance of the policies $\pi _ { i }$ are very different on these synthetic levels compared to levels sampled form $\mathfrak { p }$ . However, inferring levels from full observed episodes i.e. $p ( U _ { s 1 } | \hat { h } _ { 5 0 } ^ { i } )$ was reliable, and running $\pi$ on these resulted in accurate policy evaluation. The figure also shows the trade-off between policy evaluation accuracy and the amount of off-policy data for intermediate amounts of the data $\hat { h } _ { t } ^ { i }$ . We also want to emphasize that in this setting, model-free policy evaluation by IS fails. The uniform behavior policy $\mu$ was too different from $\pi ^ { i }$ , resulting in a relative error $> 0 . 8$ for all $i = { 1 , 2 , 3 }$ .
+
+# 4 OFF-POLICY IMPROVEMENT: COUNTERFACTUALLY-GUIDED POLICY SEARCH
+
+In the following we show how we can leverage the insights from counterfactual policy evaluation for policy search. We commence by considering a model-based RL algorithm and discuss how we can generalize it into a counterfactual algorithm to increase its robustness to model mismatch. We chose a particular algorithm to start from to make a connection to the previously proposed Guided Policy Search algorithm (Levine & Koltun, 2013; Levine & Abbeel, 2014), but we think a larger class of MBRL algorithms can be generalized in an analogous manner.
+
+4.1 STARTING POINT: VANILLA MODEL-BASED RL WITH RETURN WEIGHTED REGRESSION
+
+We start from the following algorithm. We assume we have a model $\mathcal { M }$ of the environment with trajectory distribution $p ^ { \pi }$ . Our current policy estimate $\pi ^ { k }$ is improved at iteration $k$ using returnweighted regression:
+
+$$
+\begin{array} { r c l } { { \pi ^ { k + 1 } } } & { { = } } & { { \displaystyle \operatorname * { a r g m a x } _ { \pi } \int \exp ( G ( \tau ) ) { p ^ { \pi ^ { k } } ( \tau ) } \log p ^ { \pi } ( \tau ) d \tau , } } \end{array}
+$$
+
+where $G ( \tau )$ is the return of trajectory $\tau$ . This policy improvement step can be motivated by the framework of RL as variational inference (Toussaint, 2009) and is equivalent to minimizing the KL divergence to a trajectory distribution $\propto \exp ( G ) p ^ { \pi ^ { k } }$ which puts additional mass on high-return trajectories. Although not strictly necessary for our exposition, we also allow for a dedicated proposal distribution over trajectories $\bar { p } ^ { \lambda } ( \tau )$ , under a policy $\lambda$ . We refer to $\lambda$ as a planner to highlight that it could consist of a procedure that solves episodes starting from arbitrary, full states $s _ { 1 }$ sampled form the model, by repeatedly calling the model transition kernel, e.g. a search procedure such as MCTS (Browne et al., 2012) or an expert policy. Concretely, we optimize the following finite sample objective:
+
+$$
+\begin{array} { r c l } { \pi ^ { k + 1 } } & { = } & { \underset { \pi } { \mathrm { a r g m a x } } \displaystyle \sum _ { i = 1 } ^ { N } \exp ( G ^ { i } ( \tau ^ { i } ) ) \frac { p ^ { \pi ^ { k } } ( \tau ^ { i } ) } { p ^ { \lambda } ( \tau ^ { i } ) } \log p ^ { \pi } ( \tau ^ { i } ) , \qquad \tau ^ { i } \sim p ^ { \lambda } . } \end{array}
+$$
+
+We refer to this algorithm as model-based policy search (MB-PS). It is based on model rollouts $\tau ^ { i }$ spanning entire episodes. An alternative would be to consider model rollouts starting from states visited in the real environment (if available). Both versions can be augmented by counterfactual methods, but for the sake of simplicity we focus on the simpler MB-PS version detailed above (also we did not find significant performance differences experimentally between both versions).
+
+# 4.2 INCORPORATING OFF-POLICY DATA: COUNTERFACTUALLY-GUIDED POLICY SEARCH
+
+Now, we assume that the model $\mathcal { M }$ is an SCM. Based on our discussion of counterfactual policy evaluation, it is straightforward to generalize the MB-PS described above by anchoring the rollouts $\tau ^ { i }$ under the model $\overline { { p } } ^ { \lambda }$ in off-policy data $D$ : Instead of sampling $\tau ^ { i }$ directly from the prior $p ^ { \lambda }$ , we draw them from counterfactual distribution $p ^ { \lambda | \hat { h } _ { T } ^ { i } }$ with data $\hat { h } _ { T } ^ { i } \sim D$ from the replay buffer, i.e. instead of sampling the scenarios $U$ from the prior we infer them from the given data. Again invoking Lemma 1, this procedure is unbiased under no model mismatch. We term the resulting algorithm Counterfactually-Guided Policy Search (CF-GPS), and it is summarized in Algorithm 1. The motivation for using CF-GPS over MB-PS is analogous to the advantage of CF-PE over MB-PE discussed in sec. 3.1. The policy $\pi$ in CF-GPS is optimized on rollouts $\tau ^ { i }$ that are grounded in data $\hat { h } _ { T } ^ { i }$ by sampling them from the counterfactual distribution $p ^ { \lambda | \hat { h } _ { T } ^ { i } }$ instead of the prior $p ^ { \lambda }$ . If this prior is difficult to model, we expect the counterfactual distribution to be more concentrated in regions where there is actual mass under the true environment ${ \mathfrak { p } } ^ { \lambda }$ .
+
+# 4.3 EXPERIMENTS
+
+We evaluate CF-GPS on the PO-SOKOBAN environment, using a modified distributed actor-learner architecture based on Espeholt et al. (2018): Multiple actors (here 64) collect real data $\hat { h } _ { T }$ by running the behavior policy $\mu$ in the true environment $\mathfrak { p }$ . As in many distributed RL settings, $\mu$ is chosen to be a copy of the policy $\pi$ , often slightly outdated, so the data must be considered to be offpolicy. The distribution $p ( U _ { s 1 } | \hat { h } _ { T } )$ over levels $U _ { s 1 }$ is inferred from the data $\hat { h } _ { T }$ using from the model $\mathcal { M }$ . We sample a scenario $U _ { s 1 }$ for each logged episode, and simulate 10 counterfactual trajectories $\tau ^ { 1 , \ldots , 1 0 }$ under the planner $\lambda$ for each such scenario. Here, for the sake of simplicity, instead of using search, the planner was assumed to be a mixture between $\pi$ and a pre-trained expert policy $\lambda _ { e }$ , i.e. $\lambda = \beta \lambda _ { e } + ( 1 - \beta ) \pi$ . The schedule $\beta$ was set to an exponentially decaying parameter with time constant $1 0 ^ { 5 }$ episodes. The learner performs policy improvement on $\pi$ using $\bar { \tau } ^ { 1 , \ldots , 1 0 }$ according to eqn. 1. $\mathcal { M }$ was trained online, in the same way as in sec. 3.2. $\lambda$ and $\pi$ were parameterized by deep, recurrent neural networks with the same architecture described in sec. 3.2.
+
+We compare CF-GPS with the vanilla MB-PS baseline described in sec. 4.1 (based on the same number of policy updates). MB-PS differs from CF-GPS by just having access to an unconditional model $p ( U _ { s 1 } | \emptyset )$ over initial states. We also consider a method which conditions the scenario model $p ( U _ { s 1 } | o _ { 1 } )$ on the very first observation $o _ { 1 }$ , which is available when taking the first action and therefore does not involve hindsight reasoning. This is more informed compared to MB-PS; however due to the noise on the observations, the state is still mostly unobserved rendering it very challenging to learn a good parametric model of the belief state $p ( U _ { s 1 } | o _ { 1 } )$ . We refer to this algorithm as Guided Policy Search-like (GPS-like), as it roughly corresponds to the algorithm presented by Levine & Abbeel (2014), as discussed in greater detail in sec. 5. Fig. 2 shows that CF-GPS outperforms these two baselines. As expected from the policy evaluation experiments, initial states sampled from the models for GPS and MB-PS are often not solvable, yielding inferior training data for the policy $\pi$ . In CF-GPS, the levels are inferred from hindsight inference $p ( U _ { 1 } | \hat { h } _ { T } )$ , yielding high quality training data. For reference, we also show a policy trained by the model-free method of Espeholt et al. (2018) using the same amount of environment data. Not surprisingly, CF-GPS is able to make better use of the data compared to the model-free baseline as it has access to the true transition and reward kernels (which were not given to the model-free method).
+
+# 5 RELATED WORK
+
+Bottou et al. (2013) provide an in-depth discussion of applying models to off-policy evaluation. However, their and related approaches (Li et al., 2015; Jiang & Li, 2015; Swaminathan & Joachims, 2015; Nedelec et al., 2017; Atan et al., 2016; Thomas & Brunskill, 2016) such as doubly-robust estimators, rely on Importance Sampling (IS), also called Propensity Score method. Although some of these algorithms are also termed counterfactual policy evaluation, they are not counterfactual in the sense used in this paper, where noise variables are inferred from logged data and reused to evaluate counterfactual actions. Hence, they are dogged by high variance in the estimators common to IS, although recent work aims to address this (Munos et al., 2016; Guo et al., 2017). Model-based methods for off-policy evaluation have recently been improved to account for the distribution shift between the data-collecting policy and the policy to be evaluated (Johansson et al., 2016; Liu et al., 2018). Recently (Andrychowicz et al., 2017) proposed the Hindsight Experience Replay (HER) algorithm for learning a family of goal directed policies. In HER one observes an outcome in the true environment, which is kept fixed, and searches for the goal-directed policy that should have achieved this goal in order to positively reinforce it. Therefore, this algorithm is complementary to CF-GPS where we search over alternative outcomes for a given policy. Our CF-GPS algorithm is inspired by and extends work presented by Abbeel et al. (2006) on a method for de-biasing weak models by estimating additive terms in the transition kernel to better match individual, real trajectories. The resulting model, which is a counterfactual distribution in the terminology used in our paper, is then used for model-based policy improvement. Our work generalizes this approach and highlights conceptual connections to causal reasoning. Furthermore, we discuss the connection of CF-GPS to two classes of RL algorithms in greater detail below.
+
+Guided Policy Search (GPS) CF-GPS is closely related to GPS, in particular we focus on GPS as presented by Levine & Abbeel (2014). Consider CF-GPS in the fully-observed MDP setting where $O _ { t } ~ = ~ S _ { t }$ . Furthermore, assume that the SCM $\mathcal { M }$ is structured as follows: Let $S _ { t + 1 } ~ =$ $f _ { s } ( S _ { t } , A _ { t } , U _ { s t } )$ be a linear function in $( S _ { t } , A _ { t } )$ with coefficients given by $U _ { s t }$ . Further, assume an i.i.d. Gaussian mixture model on $U _ { s t }$ for all $t$ . As the states are fully observed, the inference step in the CFI procedure simplifies: we can infer the noise sources $\hat { u } _ { s t }$ (samples or MAP estimates), i.e. the unknown linear dynamics, from pairs of observed, true states $\hat { s } _ { t }$ , $\hat { s } _ { t + 1 }$ . Furthermore assume that the reward is a quadratic function of the state. Then, the counterfactual distribution $p ^ { \lambda } ( \tau | \hat { u } )$ is a linear quadratic regulator (LQR) with time-varying coefficients $\hat { u }$ . An appropriate choice for the planner $\lambda$ is the optimal linear feedback policy for the given LQR, which can be computed exactly by dynamic programming.
+
+Observation 1. In the MDP setting, CF-GPS with a linear SCM and a dynamic programming planner for LQRs λ is equivalent to GPS.
+
+Another perspective is that GPS is the counterfactual version of the MB-PS procedure from sec. 4.1:
+
+Observation 2. In the MDP setting with a linear SCM and a dynamic programming planner for LQRs λ, GPS is the counterfactual variant of the MB-PS procedure outlined above.
+
+The fact that GPS is a successful algorithm in practice shows that the ‘grounding’ of model-based search / rollouts in real, off-policy data afforded by counterfactual reasoning massively improves the naive, ‘prior sample’-based MB-PS algorithm. These considerations also suggest when we expect CF-GPS to be superior compared to regular GPS: If the uncertainty in the environment transition $U _ { s t }$ cannot be reliably identified from subsequent pairs of observations $\hat { o } _ { t } , \hat { o } _ { t + 1 }$ alone, we expect benefits of inferring $U _ { s t }$ from a larger context of observations, in the extreme case from the entire history $\hat { h } _ { T }$ as described above.
+
+Stochastic Value Gradient methods There are multiple interesting connections of CF-GPS to Stochastic Value Gradient (SVG) methods (Heess et al., 2015). In SVG, a policy $\pi$ for a MDP is learned by gradient ascent on the expected return under a model $p$ . Instead of using the scorefunction estimator, SVG relies on a reparameterization of the stochastic model and policy (Kingma & Welling, 2013; Rezende et al., 2014). We note that this reparameterization casts $p$ into an SCM. As in GPS, the noise sources $U _ { s t }$ are inferred from two subsequent observed states $\hat { s } _ { t } , \hat { s } _ { t + 1 }$ from the true environment, and the action noise $U _ { a t }$ is kept frozen. As pointed out in the GPS discussion, this procedure corresponds to the inference step in a counterfactual query. Given inferred values $u$ for $U$ , gradients $\partial _ { \theta } G$ of the return under the model are taken with respect to the policy parameters $\theta$ . We can loosely interpret these gradients as $2 \dim ( \theta )$ counterfactual policy evaluations of policies $\pi ( \theta \pm \Delta \theta _ { i } )$ where a single dimension $i$ of the parameter vector $\theta$ is perturbed.
+
+# 6 DISCUSSION
+
+Simulating plausible synthetic experience de novo is a hard problem for many environments, often resulting in biases for model-based RL algorithms. The main takeaway from this work is that we can improve policy learning by evaluating counterfactual actions in concrete, past scenarios. Compared to only considering synthetic scenarios, this procedure mitigates model bias. However, it relies on some crucial assumptions that we want to briefly discuss here. The first assumption is that off-policy experience is available at all. In cases where this is e.g. too costly to acquire, we cannot use any of the proposed methods and have to exclusively rely on the simulator / model. We also assumed that there are no additional hidden confounders in the environment and that the main challenge in modelling the environment is capturing the distribution of the noise sources $p ( U )$ , whereas we assumed that the transition and reward kernels given the noise is easy to model. This seems a reasonable assumption in some environments, such as the partially observed grid-world considered here, but not all. Probably the most restrictive assumption is that we require the inference over the noise $U$ given data $\hat { h } _ { T } ^ { \phantom { \dagger } }$ to be sufficiently accurate. We showed in our example, that we could learn a parametric model of this distribution from privileged information, i.e. from joint samples $u , h _ { T }$ from the true environment. However, imperfect inference over the scenario $U$ could result e.g. in wrongly attributing a negative outcome to the agent’s actions, instead environment factors. This could in turn result in too optimistic predictions for counterfactual actions. Future research is needed to investigate if learning a sufficiently strong SCM is possible without privileged information for interesting RL domains. If, however, we can trust the transition and reward kernels of the model, we can substantially improve model-based RL methods by counterfactual reasoning on off-policy data, as demonstrated in our experiments and by the success of Guided Policy Search and Stochastic Value Gradient methods.
+
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+
+# A PROOFS
+
+A.1 PROOF OF LEMMA 1
+
+Proof. We start from the fact that the density over noise sources $U$ remains the same for every intervention $I$ as $U$ are root nodes in $\mathcal { G }$ :
+
+$$
+\begin{array} { r c l } { p ^ { \mathrm { d o } ( I ) } ( u ) } & { = } & { p ( u ) . } \end{array}
+$$
+
+This leads to:
+
+$$
+\begin{array} { r l } { p ^ { \mathrm { d o } ( T ) } ( x ) } & { = \ \int p ^ { \mathrm { d o } ( T ) } ( x | u ) p ^ { \mathrm { d o } ( T ) } ( u ) d u } \\ & { = \ \int p ^ { \mathrm { d o } ( T ) } ( x | u ) p ( u ) d u } \\ & { = \ \int p ^ { \mathrm { d o } ( T ) } ( x | u ) \left( \int p ( \hat { x } _ { o } , u ) d \hat { x } _ { o } \right) d u } \\ & { = \ \int \int p ^ { \mathrm { d o } ( T ) } ( x | u ) p \left( u | \hat { x } _ { o } \right) p ( \hat { x } _ { o } ) d u d \hat { x } _ { o } } \\ & { = \ \mathbb { E } _ { \hat { x } _ { o } \sim p \left[ \int p ^ { \mathrm { d o } ( T ) } ( x | u ) p \left( u | \hat { x } _ { o } \right) d u \right] } } \\ & { = \ \mathbb { E } _ { \hat { x } _ { o } \sim p \left[ \int p ^ { \mathrm { d o } ( T ) } ( x | u ) p \left( u | \hat { x } _ { o } \right) d u \right] } } \\ & { = \ \mathbb { E } _ { \hat { x } _ { o } \sim p \left[ p ^ { \mathrm { d o } ( T ) | \hat { x } _ { o } } ( x ) \right] } . } \end{array}
+$$
+
+# A.2 PROOF OF COROLLARY 1
+
+Proof. Given two sets $\mathrm { C F } \subset \{ 1 , \ldots , N \}$ and Prior ${ \textsf { C } } \left\{ 1 , \ldots , N \right\}$ with ${ \mathrm { C F } } \cap { \mathrm { P r i o r } } = \emptyset$ and $\mathrm { C F } \cup \mathrm { P r i o r } = \{ 1 , \ldots , N \}$ , we define $\vec { u _ { \mathrm { C F } } = ( u _ { n } ) _ { n \in \mathrm { C F } } }$ and $u _ { \mathrm { P r i o r } } = \mathsf { \bar { ( } } u _ { n } ) _ { n \in \mathrm { P r i o r } }$ . By construction, the scenarios $U$ are independent under the prior, i.e. $\begin{array} { r } { p ( u ) = \prod _ { n = 1 } ^ { N } p ( u _ { n } ) } \end{array}$ . Therefore $u _ { \mathrm { C F } }$ and $u _ { \mathrm { P r i o r } }$ are independent. We can write:
+
+$$
+\begin{array} { r c l } { p ( u ) } & { = } & { \displaystyle \left( \prod _ { n \in \mathrm { C F } } p ( u _ { n } ) \right) \left( \prod _ { n \in \mathrm { P r i o r } } p ( u _ { n } ) \right) } \\ & { = } & { p ( u _ { \mathrm { C F } } ) p ( u _ { \mathrm { P r i o r } } ) . } \end{array}
+$$
+
+Following the arguments from Lemma 1, the averaged inference distribution is equal to the prior $p ( u ) = \mathbb { \bar { E } } _ { \hat { x } _ { o } \sim p } [ \bar { p ( } u | \hat { x } _ { o } ) ]$ . This also holds for any subset of the variables $u$ , in particular for for $p ( u _ { \mathrm { C F } } )$ . Hence:
+
+$$
+\begin{array} { r c l } { p ( u ) } & { = } & { p ( u _ { \mathrm { C F } } ) p ( u _ { \mathrm { P r i o r } } ) } \\ & { = } & { \mathbb { E } _ { \hat { x } _ { o } \sim p } [ p ( u _ { \mathrm { C F } } | \hat { x } _ { o } ) ] p ( u _ { \mathrm { P r i o r } } ) . } \end{array}
+$$
+
+# B DETAILS ON CASTING A POMDP INTO SCM FORM
+
+Lemma 2 (Auto-regressive uniformization aka Reparametrization). Consider random variables $X _ { 1 } , \ldots , X _ { N }$ with joint distribution $P$ . There exist functions $f _ { n }$ for $n \in \{ 1 , \ldots , N \}$ such that with independent random variables $U _ { n } \sim \mathrm { U n i f o r m } [ 0 , 1 ]$ the random variables $X ^ { \prime }$ equal $X$ in distribution, i.e. $X ^ { \prime } { \overset { d } { = } } X$ , where $X ^ { \prime } = ( X _ { 1 } ^ { \prime } , \ldots , X _ { n } ^ { \prime } )$ are defined as:
+
+$$
+\begin{array} { r } { X _ { n } ^ { \prime } \quad : = \quad f _ { n } ( U _ { n } , X _ { < n } ^ { \prime } ) . } \end{array}
+$$
+
+Proof. We construct the functions $f _ { n }$ by induction on $n$ . Consider the conditional distribution $P _ { X _ { n } | X _ { < n } }$ . For fixed $X _ { < n }$ , denote its CDF with $F _ { n , X < n }$ . We construct a random variable $X ^ { \prime } : = F _ { n , X _ { < n } } ^ { - 1 } ( U _ { n } )$ with $U _ { n } \ \sim \ \mathrm { U n i f o r m [ 0 , 1 ] }$ independent from $X _ { < n }$ and $U _ { < n }$ . By virtue of the inverse-CDF method, we have $X ^ { \prime } | X _ { < n } \overset { d } { = } X | X _ { < n }$ . Therefore, $f _ { n } ( U _ { n } , X _ { < n } ) : = F _ { n , X _ { < n } } ^ { - 1 } ( U _ { n } )$ satisfies the above lemma. □
+
+![](images/1be8fcef4e3a974e21350038df064afa1220d61b230adb3cca5cf03b07073782.jpg)  
+Figure 3: Top: PO-SOKOBAN. Shown on the left is a procedurally generated initial state. The agent is shown in green, boxes in yellow, targets in blue and walls in red. The agent does not observe this state but a sequence of observations, which are masked by iid noise with 0.9 probability, except a $3 \mathrm { x } 3$ window around the agent. Bottom: Inference model. For counterfactual inference in PO-SOKOBAN, we need the (approximate) inference distribution $p ( U _ { s 1 } | \hat { h } _ { T } )$ over the initial state $U _ { s 1 } = S _ { 1 }$ , conditioned on the history of observations $\hat { h } _ { T }$ . We model this distribution using a DRAW generative model with latent variables $Z$ , which are conditioned on the output of a backward RNN summarizing the observation history.
+
+# C MODEL ARCHITECTURE
+
+We assume that we are given the true transition and reward kernels. As the transitions are deterministic in PO-SOKOBAN, the only part of the model that remains to be identified is the initial state distribution $p ( U _ { s 1 } )$ . We learned this model from data using a the DRAW model (Gregor et al., 2015), which is a parametric, multi-layer, latent variable, neural network model for distributions. For our purposes we chose the convolutional DRAW architecture proposed by (Gregor et al., 2016). First, the observation data is summarized by a convolutional LSTM with 32 hidden units and kernel size of 3. The resulting final LSTM state is fed into a conditional Gaussian prior over the latent variables $Z _ { k = 1 , \dots , 8 }$ of the 8-layer conv-DRAW model. Each layer has 32 hidden layers and the canvas had 7 layers, corresponding to the 7 channels of the categorical $U _ { s 1 } \in \{ 0 , 1 \} ^ { 1 0 \times 1 0 \times 7 }$ that we wish to model. The model (together with the backward RNN) was trained with the ADAM optimizer (Kingma & Ba, 2014) on the ELBO loss using the reparametrization trick (Kingma & Welling, 2013; Rezende et al., 2014). The mini-batch size was set to 4 and the learning rate to $3 e - 4$ . We want to emphasize that the DRAW latent variables $Z$ are not directly the noise variables $U$ of the SCM, but integrating out these variables yields this distribution $\begin{array} { r } { p ( U _ { s 1 } | \hat { h } _ { T } ) = \int p ( U _ { s 1 } | z , \hat { h } _ { T } ) p ( z | \hat { h } _ { T } ) d \hat { h } _ { T } } \end{array}$ .
+
+# D MODEL MISMATCH ANALYSIS
+
+Here we provide some analysis of the DRAW model over the initial state $U _ { s 1 }$ , which is the learned part of the SCM $\mathcal { M }$ used for the policy evaluation experiments presented in 3.2. As detailed above, we trained a separate model $p ( U _ { s 1 } | \hat { h } _ { t } )$ for each $t = 1 , \ldots , 5 0$ parameterizing the cardinality of the data the model is conditioned on. We analyze three particular models for $t = 0 , 1$ and 50 which we term the unconditional / filtering $/$ smoothing model, as they are conditioned on no data $/$ on data that is available at test time $/$ all data that is available in hindsight. Directly visualizing the distributions $p ( U _ { s 1 } | \hat { h } _ { t } )$ for an analysis is difficult as the domain $\{ 0 , \ldots , 6 \} ^ { 1 0 \times 1 0 }$ is high-dimensional and discrete. Instead we focus on the latent variables $Z$ which are learned by DRAW to represent this distribution; by construction, these are jointly Normal, facilitating the analysis. In particular, we compare $p ( Z | \hat { h } _ { t } )$ with the inference distribution $q ( Z | \hat { u } _ { s 1 } )$ conditioned on the true state $\hat { U } _ { s 1 }$ . We loosely interpret $q ( Z | \hat { u } _ { s 1 } )$ as the ”true” embedding of the datum $\hat { u } _ { s 1 }$ , whereas $p ( Z | \hat { h } _ { t } )$ is the learned embedding. In a perfect model the prior matches the inference distribution on average:
+
+![](images/194da2f4bafbbb7d129c84352e8f8e98320583d1c3813e1e10719b692d065c74.jpg)  
+Figure 4: Analysis of the model mismatch of the learned inference distributions $p ( U _ { s 1 } | \hat { h } _ { t } )$ over the initial PO-SOKOBAN state $U _ { s 1 }$ , for three different amounts of observations $t = 0 , 1$ and 50. Shown are two dimensions of the learned latent representation $Z$ of $U _ { s 1 }$ . The (whitened) learned prior $p ( Z | \hat { h } _ { t } )$ is indicated by a red contour of one standard deviation. The inferred mean embedding of the true levels are show as crosses, and their aggregated density is shown in blue. With increasing amount data that the model is conditioned on, the learned distributions match the data better.
+
+$$
+\begin{array} { r l r } { \mathbb { E } _ { \hat { h } _ { t } \sim \mathfrak { p } ^ { \mu } } [ p ( Z | \hat { h } _ { t } ) ] } & { \stackrel { ! } { = } } & { \mathbb { E } _ { \hat { u } _ { s 1 } \sim \mathfrak { p } ^ { \mu } } [ q ( Z | \hat { u } _ { s 1 } ) ] , } \end{array}
+$$
+
+i.e. every sample from the prior $p ( Z | \hat { h } _ { t } )$ corresponds to real data and vice versa. We visualize the averaged prior $\mathbb { E } _ { \hat { h } _ { t } \sim \mathfrak { p } ^ { \mu } } [ p ( Z | \hat { h } _ { t } ) ]$ and the averaged posterior $\mathbb { E } _ { \hat { u } _ { s 1 } \sim \mathfrak { p } ^ { \mu } } \left[ q ( Z | \hat { u } _ { s 1 } ) \right]$ in fig. 4. We show the two dimensions of $Z$ where these distributions have the largest KL divergence. Also, the plots were whitened w.r.t. the averaged prior, i.e. the latter is a spherical Gaussian in the plots, represented by an iso-probability contour corresponding to one standard deviation. The inference distribution for each datum $\hat { u } _ { s 1 }$ is visualized by its mean (cross) and a level set corresponding the one standard deviation or less. For the unconditional model $t = 0$ , we find that the distributions are not matched well. In particular, there is a lot of prior mass that sits in regions where there is no or little true data. In the RL setting this results in synthetic data from the model that is unrealistic and training a policy on this data leads to reduced test performance. Also, as apparent from the figure, there is structure in the embedding of the true data, that is not captured at all by the prior. This effect is markedly reduced in the filtering posterior, indicating that the conditional distribution $p ( Z | \hat { h } _ { 1 } )$ already captures the data distribution better. The smoothing model is a very good match to the data. With high probability, all tiles of the game are observed in $\bar { \hat { h } } _ { 5 0 }$ , enabling the model to perfectly learn the belief state, which collapses in this setting to a single state.

parse/train/BJG0voC9YQ/BJG0voC9YQ_content_list.json

@@ -0,0 +1,1490 @@
+[
+    {
+        "type": "text",
+        "text": "WOULDA, COULDA, SHOULDA: COUNTERFACTUALLY-GUIDED POLICY SEARCH ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            99,
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+            146
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Lars Buesing, Theophane Weber, Yori Zwols, ´   \nSebastien Racani ´ ere, Arthur Guez, Jean-Baptiste Lespiau, Nicolas Heess \\`   \nDeepMind   \nlbuesing@google.com ",
+        "bbox": [
+            183,
+            169,
+            691,
+            226
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "ABSTRACT ",
+        "text_level": 1,
+        "bbox": [
+            454,
+            262,
+            544,
+            277
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Learning policies on data synthesized by models can in principle quench the thirst of reinforcement learning algorithms for large amounts of real experience, which is often costly to acquire. However, simulating plausible experience de novo is a hard problem for many complex environments, often resulting in biases for modelbased policy evaluation and search. Instead of de novo synthesis of data, here we assume logged, real experience and model alternative outcomes of this experience under counterfactual actions, i.e. actions that were not actually taken. Based on this, we propose the Counterfactually-Guided Policy Search (CF-GPS) algorithm for learning policies in POMDPs from off-policy experience. It leverages structural causal models for counterfactual evaluation of arbitrary policies on individual off-policy episodes. CF-GPS can improve on vanilla model-based RL algorithms by making use of available logged data to de-bias model predictions. In contrast to off-policy algorithms based on Importance Sampling which re-weight data, CF-GPS leverages a model to explicitly consider alternative outcomes, allowing the algorithm to make better use of experience data. We find empirically that these advantages translate into improved policy evaluation and search results on a non-trivial grid-world task. Finally, we show that CF-GPS generalizes the previously proposed Guided Policy Search and that reparameterization-based algorithms such Stochastic Value Gradient can be interpreted as counterfactual methods. ",
+        "bbox": [
+            233,
+            291,
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "1 INTRODUCTION ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            594,
+            336,
+            609
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Imagine that a month ago Alice had two job offers from companies $a _ { 1 }$ and $a _ { 2 }$ . She decided to join $a _ { 1 }$ because of the larger salary, in spite of an awkward feeling during the job interview. Since then she learned a lot about $a _ { 1 }$ and recently received information about $a _ { 2 }$ from a friend, prodding her now to imagine what would have happened had she joined $a _ { 2 }$ . Re-evaluating her decision in hindsight in this way, she concludes that she made a regrettable decision. She could and should have known that $a _ { 2 }$ was a better choice, had she only interpreted the cues during the interview correctly... This example tries to illustrate the everyday human capacity to reason about alternate, counterfactual outcomes of past experience with the goal of “mining worlds that could have been” (Pearl & Mackenzie, 2018). Social psychologists theorize that such cognitive processes are beneficial for improving future decision making (Roese, 1997). In this paper we aim to leverage possible advantages of counterfactual reasoning for learning decision making in the reinforcement learning (RL) framework. ",
+        "bbox": [
+            174,
+            625,
+            825,
+            791
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "In spite of recent success, learning policies with standard, model-free RL algorithms can be notoriously data inefficient. This issue can in principle be addressed by learning policies on data synthesized from a model. However, a mismatch between the model and the true environment, often unavoidable in practice, can cause this approach to fail (Talvitie, 2014), resulting in policies that do not generalize to the real environment (Jiang et al., 2015). Motivated by the introductory example, we propose the Counterfactually-Guided Policy Search (CF-GPS) algorithm: Instead of relying on data synthesized from scratch by a model, policies are trained on model predictions of alternate outcomes of past experience from the true environment under counterfactual actions, i.e. actions that had not actually been taken, while everything else remaining the same (Pearl, 2009). At the heart of CF-GPS are structural causal models (SCMs) which model the environment with two ingredients (Wright, 1920): 1) Independent random variables, called scenarios here, summarize all aspects of the environment that cannot be influenced by the agent, e.g. the properties of the companies in Alice’s job search example. 2) Deterministic transition functions (also called causal mechanisms) take these scenarios, together with the agent’s actions, as input and produce the predicted outcome. The central idea of CF-GPS is that, instead of running an agent on scenarios sampled de novo from a model, scenarios are inferred in hindsight from given off-policy data, and then evaluate and improve the agent on these specific scenarios using given or learned causal mechanisms (Balke & Pearl, 1994). CF-GPS can be regarded as a meta-algorithm that extends a given model-base RL algorithm by “grounding” or “anchoring” model-based predictions in inferred scenarios. As a result, this approach explicitly allows to trade-off historical data for model bias. We show empirically in a conceptually simple setting, where unknown initial states are inferred in hindsight and re-used to evalute to counterfactual actions, that this can mitigate model mismatch. CF-GPS differs substantially from standard off-policy RL algorithms based on Importance Sampling (IS), where historical data is re-weighted with respect to the importance weights to evaluate or learn new policies (Precup, 2000). In contrast, CF-GPS explicitly reasons counterfactually about given off-policy data. Our main contributions are: ",
+        "bbox": [
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            174,
+            104,
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+    },
+    {
+        "type": "text",
+        "text": "1. We formulate model-based RL in POMDPs in terms of structural causal models, thereby connecting concepts from reinforcement learning and causal inference. 2. We provide the first results, to the best of our knowledge, showing that counterfactual reasoning in structural causal models on off-policy data can facilitate solving non-trivial RL tasks. 3. We show that two previously proposed classes of RL algorithms, namely Guided Policy Search (Levine & Koltun, 2013) and Stochastic Value Gradient methods (Heess et al., 2015), can be interpreted as counterfactual methods, opening up possible generalizations. ",
+        "bbox": [
+            210,
+            353,
+            823,
+            487
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "The paper is structured as follows. We first give a self-contained, high-level recapitulation of structural causal models and counterfactual inference, as these are less widely known in the RL and generative model communities. In particular we show how to model POMDPs with SCMs. Based on this exposition, we first consider the task of policy evaluation and discuss how we can leverage counterfactual inference in SCMs to improve over standard model-based methods. We then generalize this approach to the policy search setting resulting in the CF-GPS algorithm. We close by highlighting connections to previously proposed algorithms and by discussing assumptions and limitations of the proposed method. ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "2 PRELIMINARIES ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            640,
+            338,
+            655
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "We denote random variables (RVs) with capital letters, e.g. $X$ , and particular values with lower caps, e.g. $x$ . For a distribution $P$ over a vector-valued random variable $X$ , we denote the marginal over $Y \\subset X$ by $P _ { Y }$ (and density $p _ { Y }$ ); however we often omit the subscript if it is clear from the context, e.g. as in $Y \\sim P$ . We assume the episodic, Partially Observable Markov Decision Process (POMDP) setting with states $S _ { t }$ , actions $A _ { t }$ and observations $O _ { t }$ , for $t = 1 , \\dots , T$ . For n, we assume that . We consider sto $O _ { t }$ includes theastic policies $R _ { t }$ . The undiscounted return is denoted byover actions conditioned on observation $\\begin{array} { r } { G = \\sum _ { t = 1 } ^ { T } R _ { t } } \\end{array}$ $\\pi ( a _ { t } | h _ { t } )$   \nhistories $\\hat { H _ { t } } = ( O _ { 1 } , A _ { 1 } , \\ldots , A _ { t - 1 } , O _ { t } )$ . We denote the resulting distribution over trajectories $( S _ { 1 } , O _ { 1 } , A _ { 1 } , \\ldots , A _ { T - 1 } , S _ { T } , O _ { T } )$ induced by running $\\pi$ in the environment with $\\mathcal { T } \\sim \\mathfrak { P } ^ { \\pi }$ and the corresponding density by ${ \\mathfrak { p } } ^ { \\pi } ( \\tau )$ . ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "2.1 STRUCTURAL CAUSAL MODELS ",
+        "text_level": 1,
+        "bbox": [
+            176,
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Definition 1 (Structural causal model). $A$ structural causal model (SCM) $\\mathcal { M }$ over $\\begin{array} { r l } { X } & { { } = } \\end{array}$ $( X _ { 1 } , \\ldots , X _ { N } )$ is given by a $D A G ~ { \\mathcal { G } }$ over nodes $X$ , independent noise RVs $U = ( U _ { 1 } , \\ldots , U _ { N } )$ with distributions $P _ { U _ { i } }$ and functions $f _ { 1 } , \\ldots , f _ { N }$ such that $X _ { i } = f _ { i } ( \\mathrm { p a } _ { i } , U _ { i } )$ , where $\\operatorname { p a } _ { i } \\subset X$ are the parents of $X _ { i }$ in $G$ . An SCM entails a distribution $P$ with density $p$ over $( X , U )$ . ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "image",
+        "img_path": "images/9592b3f78d13cb29be27d512fb14995ca71647e2853e999782dc1fe5a34e127f.jpg",
+        "image_caption": [
+            "Figure 1: Structural causal models (SCMs) model environments using random variables $U$ (circles, ‘scenarios’), that summarize immutable aspects, some of which are observed (grey), some not (white). These are fed into deterministic functions $f _ { i }$ (black squares) that approximate causal mechanisms. Left: SCM for a contextual bandit with context $U _ { c }$ , action $A$ , feedback $O$ and scenario $U _ { o }$ . Right: SCM for a POMDP, with initial state $U _ { s 1 } = S _ { 1 }$ , states $S _ { t }$ and histories $H _ { t }$ . The mechanism that generates the actions $A _ { t }$ is the policy $\\pi$ . "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "We also refer to $U$ as scenarios and to $f _ { i }$ as causal mechanisms. We give a (broad) definition of an intervention in an SCM. This also includes what is known as stochastic interventions or mechanism changes (Korb et al., 2004) which generalize atomic interventions (Pearl, 2009). ",
+        "bbox": [
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+            335,
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Definition 2 (Intervention in SCM). An intervention $I$ in an SCM $\\mathcal { M }$ consists of replacing some of the original $f _ { i } ( \\mathrm { p a } _ { i } , U _ { i } )$ with other functions $f _ { i } ^ { I } ( \\mathrm { p a } _ { i } ^ { I } , U _ { i } )$ where $\\mathrm { p a } _ { i } ^ { I }$ are the parents in a new $D A G$ $\\mathcal { G } ^ { I }$ . We denote the resulting SCM with $\\boldsymbol { \\mathcal { M } } ^ { \\mathrm { d o } ( I ) }$ with distribution $P ^ { \\mathrm { d o } ( I ) }$ and density $p ^ { \\mathrm { d o } ( I ) }$ . ",
+        "bbox": [
+            174,
+            383,
+            825,
+            428
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "SCM representation of POMDPs We can represent any given POMDP (under a policy $\\pi$ ) by an SCM $\\mathcal { M }$ over trajectories $\\tau$ in the following way. We express all conditional distributions, e.g. the transition kernel $P _ { S _ { t + 1 } | S _ { t } , A _ { t } }$ , as deterministic functions with independent noise variables $U$ , such as $S _ { t + 1 } = f _ { s t } ( S _ { t } , A _ { t } , \\dot { U } _ { s t } )$ . This is always possible using auto-regressive uniformization, see Lemma 2 in the appendix. The DAG $\\mathcal { G }$ of the resulting SCM is shown in fig. 1. This procedure is closely related to the ‘reparameterization trick’ for models with location-scale distributions (Kingma & Welling, 2013; Rezende et al., 2014). We denote the distribution over $\\tau$ entailed by the SCM with $P ^ { \\pi }$ and its density by $p ^ { \\pi }$ to highlight the role of $\\pi$ ; note the difference to the true environment distribution $\\mathfrak { P } ^ { \\pi }$ with density ${ \\mathfrak { p } } ^ { \\pi }$ . Running a different policy $\\mu$ instead of $\\pi$ in the environment can be expressed as an intervention $I ( \\pi \\to \\mu )$ consisting of replacing ${ A _ { t } } \\mathrm { ~ = ~ } f _ { \\pi } ( { H _ { t } } , { U _ { a t } } )$ by $A _ { t } \\ =$ $f _ { \\mu } ( H _ { t } , U _ { a t } )$ . We denote the resulting model distribution over trajectories with $P ^ { \\mathrm { d o } ( I ( \\pi  \\mu ) ) } = P ^ { \\mu }$ (analogously $p ^ { \\mu }$ ). ",
+        "bbox": [
+            173,
+            452,
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Intuition Here, we illustrate the main advantage of SCMs using the example of Alice’s job choice from the introduction. We model it as contextual bandit with feedback shown in fig. 1. Alice has some initial knowledge given by the context $U _ { c }$ that is available to her before taking action $A$ of joining company $A = a _ { 1 }$ or $A = a _ { 2 }$ . We model Alice’s decision as $A = f _ { \\pi } ( U _ { c } , \\bar { U _ { a } } )$ , where $U _ { a }$ captures potential indeterminacy in Alice’s decision making. The outcome $O = f _ { o } ( A , U _ { c } , U _ { o } )$ also depends on the scenario $U _ { o }$ , capturing all relevant, unobserved and highly complex properties of the two companies such as working conditions etc. Given this model, we can reason about alternate outcomes $f _ { o } ( a _ { 1 } , u _ { c } , u _ { o } )$ and $f _ { o } ( a _ { 2 } , u _ { c } , u _ { o } )$ for same the scenario $u _ { o }$ . This is not possible if we only model the outcome on the level of the conditional distribution $P _ { O | A , U _ { c } }$ . ",
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "2.2 COUNTERFACTUAL INFERENCE IN SCMS ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "For an SCM over $X$ , we define a counterfactual query as a triple $( \\hat { x } _ { o } , I , X _ { q } )$ of observations $\\scriptstyle { \\hat { x } } _ { o }$ of some variables $X _ { o } \\subset X$ , an intervention $I$ and query variables $X _ { q } \\subset X$ . The semantics of the query are that, having observed $\\scriptstyle { \\hat { x } } _ { o }$ , we want to infer what $X _ { q }$ would have been had we done intervention $I$ , while ‘keeping everything else the same’. Counterfactual inference (CFI) in SCMs answers the query in the following way (Balke & Pearl, 1994): ",
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+        "text": "1. Infer the unobserved noise source $U$ conditioned on the observations $\\scriptstyle { \\hat { x } } _ { o }$ , i.e. compute $p ( U | \\hat { x } _ { o } )$ and replace the prior $p ( U )$ with $p ( U | \\hat { x } _ { o } )$ . Denote the resulting SCM by $\\mathcal { M } _ { \\hat { x } _ { o } }$ . ",
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+        "text": "2. Perform intdistribution $I$ on . R $\\mathcal { M } _ { \\hat { x } _ { o } }$ . This yieldthe marginal $\\mathcal { M } _ { \\hat { x } _ { o } } ^ { \\mathrm { d o } ( I ) }$ hich entails the counterfactual. $p ^ { \\mathrm { d o } ( I ) | \\hat { x } _ { o } } ( x )$ $p ^ { \\mathrm { d o } ( I ) | \\hat { x } _ { o } } ( x _ { q } )$ ",
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+        "type": "text",
+        "text": "Note that our definition explicitly allows for partial observations $X _ { o } \\subset X$ in accordance with Pearl (2009). A sampled-based version, denoted as CFI, is presented in Algorithm 1. An interesting property of the counterfactual distribution $p ^ { \\mathrm { d o } ( I ) | \\hat { x } _ { o } }$ is that marginalizing it over observations $\\scriptstyle { \\hat { x } } _ { o }$ yields an unbiased estimator of the density of $X _ { q }$ under intervention $I$ . ",
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+        "text": "Lemma 1 (CFI for simulation). Let observations $\\hat { x } _ { o } \\sim p$ come from a SCM $\\mathcal { M }$ with density $p$ . Then the counterfactual density $p ^ { \\mathrm { d o } ( I ) | \\hat { x } _ { o } }$ is an unbiased estimator of $p ^ { \\mathrm { d o } ( I ) }$ , i.e. ",
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+        "type": "equation",
+        "img_path": "images/af84512d4bc3eb8c1ad1508ce417ddd63580e35f5628de71bfec7313b039129f.jpg",
+        "text": "$$\n\\begin{array} { r c l } { \\mathbb { E } _ { \\hat { x } _ { o } \\sim p } [ p ^ { \\mathrm { d o } ( I ) | \\hat { x } _ { o } } ( x ) ] } & { = } & { p ^ { \\mathrm { d o } ( I ) } ( x ) } \\end{array}\n$$",
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+        "type": "text",
+        "text": "The proof is straightforward and outlined in the Appendix A. This lemma and the marginal independence of the $U _ { i }$ leads to the following corollary; the proof is given in the appendix. ",
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+        "text": "Corollary 1 (Mixed counterfactual and prior simulation from an SCM). Assume we have observations $\\hat { x } _ { o } \\sim p .$ . We can simulate from $\\mathcal { M }$ , under any intervention $I$ , i.e. obtain unbiased samples from $\\mathcal { M } ^ { \\mathrm { d o } ( I ) }$ , by first sampling values $u _ { \\mathrm { C F } }$ for an arbitrary subset $U _ { \\mathrm { C F } } \\subset U$ from the posterior $p ( u _ { \\mathrm { C F } } | \\hat { x } _ { o } )$ and the remaining $U _ { \\mathrm { P r i o r } } : = U \\backslash U _ { \\mathrm { C F } }$ from the prior $p ( u _ { \\mathrm { P r i o r } } )$ , and then computing $X$ with noise $u = u _ { \\mathrm { C F } } \\cup u _ { \\mathrm { P r i o r } }$ . ",
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+        "type": "text",
+        "text": "The corollary essentially states that we can sample from the model $\\mathcal { M } ^ { I }$ , by sampling some of the $U _ { i }$ from the prior, and inferring the rest from data $\\scriptstyle { \\hat { x } } _ { o }$ (as long as the latter was also sampled from $\\mathcal { M }$ ). We will make use of this later for running a POMDP model on scenarios $U _ { s t }$ inferred from data while randomizing the action noise $U _ { a t }$ . We note that the noise variables $U _ { \\mathrm { C F } }$ from the posterior $P _ { U _ { \\mathrm { C F } } | \\hat { x } _ { o } }$ are not independent anymore. Nevertheless, SCMs with non-independent noise distributions arising from counterfactual inference, denoted here by $\\mathcal { M } _ { \\hat { x } _ { o } }$ , are commonly considered in the literature (Peters et al., 2017). ",
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+        "text": "Intuition Returning to Alice’s job example from the introduction, we give some intuition for counterfactual inference in SCMs. Given the concrete outcome $\\hat { o }$ , under observed context $\\hat { u } _ { c }$ and having joined company $\\hat { a } = a _ { 1 }$ , Alice can try to infer the underlying scenario $u _ { o } \\sim p ( u _ { o } | a _ { 1 } , \\hat { u } _ { c } , \\hat { o } )$ that she experiences; this includes factors such as work conditions etc. She can then reason counterfactually about the outcome had she joined the other company, which is given by $f _ { o } ( a _ { 2 } , \\hat { u } _ { c } , u _ { o } )$ . This can in principle enable her to make better decisions in the future in similar scenarios by changing her policy $f _ { \\pi } ( A , U _ { c } , U _ { a } )$ such that the action with the preferred outcome becomes more likely under $\\hat { u } _ { c } , u _ { o }$ . In particular she can do so without having to use her (likely imperfect) prior model over possible companies $p ( U _ { o } )$ . She can use the counterfactual predictions discussed above instead to learn from her experience. We use this insight for counterfactual policy evaluation and search below. ",
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+        "type": "text",
+        "text": "3 OFF-POLICY EVALUATION: MODEL-FREE, MODEL-BASED AND COUNTERFACTUAL ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "To explain how counterfactual reasoning in SCMs can be used for policy search, we first consider the simpler problem of policy evaluation (PE) on off-policy data. The goal of off-policy PE is to determine the value of a policy $\\pi$ , i.e. its expected return $\\mathbb { E } _ { \\mathfrak { p } ^ { \\pi } } [ G ]$ , without running the policy itself. We assume that we have data $D ~ = ~ \\{ \\hat { h } _ { T } ^ { i } \\} _ { i = 1 , \\dots , N }$ consisting of logged episodes $\\hat { h } _ { T } ^ { i } = ( \\hat { o } _ { 1 } ^ { i } , \\hat { a } _ { 1 } ^ { i } , \\dots \\hat { a } _ { T - 1 } ^ { i } , \\hat { o } _ { T } ^ { i } )$ from running a behavior policy $\\mu$ . A standard, model-free approach to PE is to use Importance sampling (IS): We can estimate the policy’s value as $\\textstyle \\sum _ { i } w ^ { i } { \\hat { G } } ^ { i }$ , where $\\hat { G } ^ { i }$ is the empirical return of $\\hat { h } _ { T } ^ { i }$ and $\\begin{array} { r } { w ^ { i } \\propto \\frac { \\mathsf { p } ^ { \\pi } ( \\hat { h } _ { T } ^ { i } ) } { \\mathsf { p } ^ { \\mu } ( \\hat { h } _ { T } ^ { i } ) } } \\end{array}$ are importance weights. However, if the trajectory densities ${ \\mathfrak { p } } ^ { \\pi }$ and ${ \\mathfrak { p } } ^ { \\mu }$ are very different, then this estimator has large variance. In the extreme case, IS can be useless if the support of ${ \\mathfrak { p } } ^ { \\mu }$ does not contain that of ${ \\mathfrak { p } } ^ { \\pi }$ , irrespective of how much data from ${ \\mathfrak { p } } ^ { \\mu }$ is available. ",
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+        "type": "text",
+        "text": "If we have access to a model $\\mathcal { M }$ , then we can evaluate the policy on synthetic data, i.e. we can estimate $\\mathbb { E } _ { p ^ { \\pi } } [ G ]$ . This is called model-based policy evaluation (MB-PE). However, any bias in $\\mathcal { M }$ propagates from $p ^ { \\pi }$ to the estimate $\\mathbb { E } _ { p ^ { \\pi } } [ G ]$ . In the following, we assume that $\\mathcal { M }$ is a SCM and we ",
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+    {
+        "type": "text",
+        "text": "Algorithm 1 Counterfactual policy evaluation and search ",
+        "text_level": 1,
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+    {
+        "type": "table",
+        "img_path": "images/c82a72896e84f3cb46116067e2fd2d35f4d884569a69312570ef163950c089b3.jpg",
+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td colspan=\"2\">//Counterfactual inference (CFI) procedure CFI(data 𝑥., SCM M,intervention I, query Xq)</td></tr><tr><td colspan=\"2\">1: u~p(ulxo)</td></tr><tr><td>2: 3:</td><td></td></tr><tr><td></td><td>p(u)←δ(u-u)</td></tr><tr><td>4:</td><td>f←f</td></tr><tr><td colspan=\"2\">return xq ~ pdo(1) (xalu) 5: 6: end procedure</td></tr><tr><td colspan=\"2\">/Counterfactual PolicyEvaluation (CF-PE)</td></tr><tr><td>7:</td><td>procedure CF-PE(SCM M, policy π,replay buffer D, number of samples N)</td></tr><tr><td>8: 9:</td><td>fori∈{1,...N}do ~D</td></tr><tr><td>10:</td><td>gi = CFI(h𝑖,M,I(μ→π),G)</td></tr><tr><td>11:</td><td>end for</td></tr><tr><td>12: 13:</td><td></td></tr><tr><td colspan=\"2\">end procedure</td></tr><tr><td></td><td>/Counterfactually-Guided Policy Search (CF-GPS)</td></tr><tr><td>for k =1,... do</td><td>14:procedure CF-GPS(SCM M, initial policy π°, number of trajectory samples N)</td></tr><tr><td>15: 16:</td><td></td></tr><tr><td>17:</td><td>if sometimes then</td></tr><tr><td>18:</td><td>μ←πk end if</td></tr><tr><td>19:</td><td></td></tr><tr><td></td><td>for i=1,...,N do</td></tr><tr><td>20:</td><td>h~pμ</td></tr><tr><td>21:</td><td>T²=CFI(h,M,I(μ→π),T)</td></tr><tr><td></td><td></td></tr><tr><td>22:</td><td></td></tr><tr><td></td><td></td></tr><tr><td></td><td></td></tr><tr><td></td><td>end for</td></tr><tr><td></td><td></td></tr><tr><td></td><td></td></tr><tr><td>23:</td><td>π ←policy improvement on trajectories i=1.,N using eqn. 1</td></tr><tr><td></td><td></td></tr><tr><td></td><td></td></tr><tr><td></td><td></td></tr><tr><td>24:</td><td></td></tr><tr><td></td><td>end for</td></tr><tr><td></td><td></td></tr><tr><td>25: end procedure</td><td></td></tr><tr><td></td><td></td></tr><tr><td></td><td></td></tr><tr><td></td><td></td></tr><tr><td></td><td></td></tr><tr><td></td><td></td></tr><tr><td></td><td></td></tr><tr><td></td><td></td></tr><tr><td></td><td></td></tr><tr><td></td><td></td></tr><tr><td></td><td></td></tr><tr><td></td><td></td></tr><tr><td></td><td></td></tr><tr><td></td><td></td></tr><tr><td></td><td></td></tr><tr><td></td><td></td></tr><tr><td></td><td></td></tr><tr><td></td><td></td></tr><tr><td></td><td></td></tr><tr><td></td><td></td></tr><tr><td></td><td></td></tr><tr><td></td><td></td></tr><tr><td></td><td></td></tr><tr><td></td><td></td></tr><tr><td></td><td></td></tr><tr><td></td><td></td></tr><tr><td></td><td></td></tr><tr><td></td><td></td></tr><tr><td></td><td></td></tr></table>",
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+    {
+        "type": "text",
+        "text": "show that we can use counterfactual reasoning for off-policy evaluation (CF-PE). As the main result for this section, we argue that we expect CF-PE to be less biased than MB-PE, and we illustrate this point with experiments. ",
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+        "type": "text",
+        "text": "3.1 COUNTERFACTUAL OFF-POLICY EVALUATION ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "Naive MB-PE with a SCM $\\mathcal { M }$ simply consist of sampling the scenarios $U \\sim P _ { U }$ from the prior, and then simulating a trajectory $\\tau$ from the functions $f _ { i }$ and computing its return. However, given data $D$ from ${ \\mathfrak { p } } ^ { \\mu }$ , our discussion of counterfactual inference in SCMs suggests the following alternative strategy: Assuming no model mismatch, i.e. ${ \\mathfrak { p } } ^ { \\mu } = p ^ { \\mu }$ , we can regard the task of off-policy evaluation of $\\pi$ as a counterfactual query with data $\\hat { h } _ { T } ^ { i }$ , intervention $I ( \\mu \\to \\pi )$ and query variable $G$ . In other words, instead of sampling from the prior as in MB-PE, we are free to the scenarios from the posterior $u ^ { i } \\sim p ^ { \\mu } ( \\cdot | \\hat { h } _ { T } ^ { i } )$ . The algorithm is given in Algorithm 1. Lemma 1 guarantees that this results in an unbiased estimate: ",
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+    {
+        "type": "text",
+        "text": "Corollary 2 (CF-PE is unbiased). Assuming no model mismatch, CF-PE is unbiased. ",
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+    {
+        "type": "text",
+        "text": "Furthermore, Corollary 1 allows us to also sample some of the noise variables from the prior instead of the posterior, we can e.g. randomize the counterfactual actions by re-sampling the action noise Ua. ",
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+        "type": "text",
+        "text": "Motivation When should one prefer CF-PE over the more straightforward MB-PE? Assuming a perfect model, Corollary 2 states that both yield the same answer in expectation for perfect models. For imperfect models however, these algorithms can differ substantially. MB-PE relies on purely synthetic data, sampled from the noise distribution $p ( U )$ . In practice, this is usually approximated by a parametric density model, which can lead to under-fitting in case of complex distributions. This is a well-known effect in generative models with latent variables: In spite of recent research progress, e.g. models of natural images are still unable to accurately model the variability of the true data (Gregor et al., 2016). In contrast, CF-PE samples from the posterior $\\begin{array} { r } { N ^ { - 1 } \\sum _ { i = 1 } ^ { N } p ^ { \\mu } ( U | \\hat { h } _ { T } ^ { i } ) } \\end{array}$ , which has access to strictly more information than the prior $p ( U )$ by taking into account additional data $\\hat { h } _ { T } ^ { i }$ . This semi-nonparametric distribution can help to de-bias the model by effectively winnowing out parts of the domain of $U$ which do not correspond to any real data. We substantiate this intuition with experiments below; a concrete illustration for the difference between the prior and posterior $/$ counterfactual distribution is given in fig. 4 in the appendix and discussed in appendix D. Therefore, we conclude that we expect CF-PE to outperform MB-PE, if the transition and reward kernels $f _ { s t }$ are accurate models of the environment dynamics, but if the marginal distribution over the noise sources $P _ { U }$ is difficult to model. ",
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+    {
+        "type": "text",
+        "text": "3.2 EXPERIMENTS ",
+        "text_level": 1,
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+        "text": "Environment As an example, we use a partially-observed variant of the SOKOBAN environment, which we call PO-SOKOBAN. The original SOKOBAN puzzle environment was described in detail by Racaniere et al. (2017); we give a brief summary here. The agent is situated in a \\` $1 0 \\times 1 0$ grid world and its five actions are to move to one of four adjacent tiles and a NOOP. In our variant, the goal is to push all three boxes onto the three targets. As boxes cannot be pulled, many actions result irreversibly in unsolvable states. Episodes are of length $T = 5 0$ , and pushing a box onto a target yields a reward of 1, removing a box from a target yields $- 1$ , and solving a level results in an additional reward of 10. The state of the environment consists in a $1 0 \\times 1 0$ matrix of categorical variables taking values in $\\{ 0 , \\ldots , 6 \\}$ indicating if the corresponding tile is empty, a wall, box, target, agent, or a valid combinations thereof (box+target and agent+target). In order to introduce partial observability, we define the observations as the state corrupted by i.i.d. (for each tile and time step) flipping each categorical variable to the “empty” state with probability 0.9. Therefore, the state of the game is largely unobserved at any given time, and a successful agent has to integrate observations over tens of time steps. Initial states $U _ { s 1 }$ , also called levels, which are the scenarios in this environment, are generated randomly by a generator algorithm which guarantees solvability (i.e. all boxes can be pushed onto targets). The environment is visualized in fig. 3 in the appendix. ",
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+    {
+        "type": "text",
+        "text": "Given the full state of PO-SOKOBAN, the transition kernel is deterministic and quite simple as only the agent and potentially an adjacent box moves. Inferring the belief state, i.e. the distribution over states given the history of observations and actions, can however range from trivial to very challenging, depending on the amount of available history. In the limit of a long observed history, every tile is eventually observed and the belief state concentrates on a single state (the true state) that can be easily inferred. With limited observed history however, inferring the posterior distribution over states (belief state) is very complex. Consider e.g. the situation in the beginning of an episode (before pushing the first box). Only the first observation is available, however we know that all POSOKOBAN levels are initially guaranteed to be solvable and therefore satisfy many combinatorial constraints reflecting that the agent is still able to push all boxes onto targets. Learning a compact parametric model of the initial state distribution from empirical data is therefore difficult and likely results in large mismatch between the learned model and the true environment. ",
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+        "text": "Results To illustrate the potential advantages of CF-PE over MB-PE we perform policy evaluation in the PO-SOKOBAN environment. We first generate a policy $\\pi$ that we wish to evaluate, by training it using a previously-proposed distributed RL algorithm (Espeholt et al., 2018). The policy is parameterized as a deep, recurrent neural network consisting of a 3-layer deep convolutional LSTM (Xingjian et al., 2015) with 32 channels per layer and kernel size of 3. To further increase computational power, the LSTM ticks twice for each environment step. The output of the agent is a value function and a softmax yielding the probabilities of taking the 5 actions. In order to obtain an SCM of the environment, for the sake of simplicity, we assume that the ground-truth transition, observation and reward kernels are given. Therefore the only part of the model that we need to learn is the distribution $p ( U _ { s 1 } )$ of initial states $S _ { 1 } = U _ { s 1 }$ (for regular MB-PE), and the density $p ( U _ { s 1 } | \\hat { h } _ { t } ^ { i } )$ for inferring levels in hindsight for CF-PE. We vary the amount of true data $t$ that we condition this inference on, ranging from $t = 0$ (no real data, equivalent to MB-PE) to $t = T = 5 0$ (a full episode of real data is used to infer the initial state $U _ { s 1 }$ ). We train a separate model for each $t \\in \\{ 0 , 5 , 1 0 , 2 0 , 3 0 , 4 0 , 5 0 \\}$ . To simplify model learning, both models were given access to the unobserved state during training, but not at test time. The models are chosen to be powerful, multilayer, generative DRAW models (Gregor et al., 2015) trained by approximate maximum likelihood learning (Kingma & Welling, 2013; Rezende et al., 2014). The models take as input the (potentially empty) data $\\hat { h } _ { t } ^ { i }$ summarized by a backward RNN (a standard convolutional LSTM model with 32 units). The model is shown in fig. 3 in the appendix and additional details are given in appendix C. The data $\\hat { h } _ { T } ^ { i }$ was collected under a uniform random policy $\\mu$ . For all policy evaluations, we use $\\approx > 1 0 ^ { 5 }$ levels $u ^ { i }$ from the inferred model. In order to evaluate policies of different proficiency, we derive from the original (trained) $\\pi$ three policies $\\pi _ { 0 } , \\pi _ { 1 } , \\pi _ { 2 }$ ranging from almost perfect to almost random performance by introducing additional stochasticity during action selection. ",
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+    {
+        "type": "image",
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+        "image_caption": [
+            "Figure 2: Experimental results on PO-SOKOBAN environment. Left: Policy evaluation. Policy evaluation error decreases with amount of off-policy data available (in #transitions per episode) for inferring scenarios (levels) $U _ { s 1 }$ that are used for counterfactual evaluation. No data (data points on the very left) corresponds to standard model-based policy evaluation (MB-PE), yielding large errors, whereas Counterfactual policy evaluation yields more accurate results. This holds for all three policies with different true performances. Right: Policy search. Counterfactually-Guided Policy Search (CF-GPS) outperforms a naive model-based RL (MB-PS) algorithm as well as a version of standard Guided Policy Search (‘GPS-like’) on PO-SOKOBAN. "
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+        "type": "text",
+        "text": "The policy evaluation results are shown in fig. 2. We found that for $t = 0$ , in spite of extensive hyper-parameter search, the model $p ( U _ { s 1 } )$ was unable to accurately capture the marginal distribution of initial levels in PO-SOKOBAN. As argued above, a solvable level satisfies a large number of complex constraints that span the entire grid world, which are hard for a parametric model to capture. Empirically, we found that the model mismatch manifested itself in samples from $p ( U _ { s 1 } )$ not being well-formed, e.g. not solvable, and hence the performance of the policies $\\pi _ { i }$ are very different on these synthetic levels compared to levels sampled form $\\mathfrak { p }$ . However, inferring levels from full observed episodes i.e. $p ( U _ { s 1 } | \\hat { h } _ { 5 0 } ^ { i } )$ was reliable, and running $\\pi$ on these resulted in accurate policy evaluation. The figure also shows the trade-off between policy evaluation accuracy and the amount of off-policy data for intermediate amounts of the data $\\hat { h } _ { t } ^ { i }$ . We also want to emphasize that in this setting, model-free policy evaluation by IS fails. The uniform behavior policy $\\mu$ was too different from $\\pi ^ { i }$ , resulting in a relative error $> 0 . 8$ for all $i = { 1 , 2 , 3 }$ . ",
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+        "type": "text",
+        "text": "4 OFF-POLICY IMPROVEMENT: COUNTERFACTUALLY-GUIDED POLICY SEARCH ",
+        "text_level": 1,
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+        "text": "In the following we show how we can leverage the insights from counterfactual policy evaluation for policy search. We commence by considering a model-based RL algorithm and discuss how we can generalize it into a counterfactual algorithm to increase its robustness to model mismatch. We chose a particular algorithm to start from to make a connection to the previously proposed Guided Policy Search algorithm (Levine & Koltun, 2013; Levine & Abbeel, 2014), but we think a larger class of MBRL algorithms can be generalized in an analogous manner. ",
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+    {
+        "type": "text",
+        "text": "4.1 STARTING POINT: VANILLA MODEL-BASED RL WITH RETURN WEIGHTED REGRESSION",
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+        "type": "text",
+        "text": "We start from the following algorithm. We assume we have a model $\\mathcal { M }$ of the environment with trajectory distribution $p ^ { \\pi }$ . Our current policy estimate $\\pi ^ { k }$ is improved at iteration $k$ using returnweighted regression: ",
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+        "type": "equation",
+        "img_path": "images/8383fc1944182b99ef83312b872e7d9753d1d31629f38d295986e54c1771d9b4.jpg",
+        "text": "$$\n\\begin{array} { r c l } { { \\pi ^ { k + 1 } } } & { { = } } & { { \\displaystyle \\operatorname * { a r g m a x } _ { \\pi } \\int \\exp ( G ( \\tau ) ) { p ^ { \\pi ^ { k } } ( \\tau ) } \\log p ^ { \\pi } ( \\tau ) d \\tau , } } \\end{array}\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "where $G ( \\tau )$ is the return of trajectory $\\tau$ . This policy improvement step can be motivated by the framework of RL as variational inference (Toussaint, 2009) and is equivalent to minimizing the KL divergence to a trajectory distribution $\\propto \\exp ( G ) p ^ { \\pi ^ { k } }$ which puts additional mass on high-return trajectories. Although not strictly necessary for our exposition, we also allow for a dedicated proposal distribution over trajectories $\\bar { p } ^ { \\lambda } ( \\tau )$ , under a policy $\\lambda$ . We refer to $\\lambda$ as a planner to highlight that it could consist of a procedure that solves episodes starting from arbitrary, full states $s _ { 1 }$ sampled form the model, by repeatedly calling the model transition kernel, e.g. a search procedure such as MCTS (Browne et al., 2012) or an expert policy. Concretely, we optimize the following finite sample objective: ",
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+    {
+        "type": "equation",
+        "img_path": "images/76c9da3c76ed221c74dd99da1ecaed246e5b9f27ad9b1792f09ad9fe2d912312.jpg",
+        "text": "$$\n\\begin{array} { r c l } { \\pi ^ { k + 1 } } & { = } & { \\underset { \\pi } { \\mathrm { a r g m a x } } \\displaystyle \\sum _ { i = 1 } ^ { N } \\exp ( G ^ { i } ( \\tau ^ { i } ) ) \\frac { p ^ { \\pi ^ { k } } ( \\tau ^ { i } ) } { p ^ { \\lambda } ( \\tau ^ { i } ) } \\log p ^ { \\pi } ( \\tau ^ { i } ) , \\qquad \\tau ^ { i } \\sim p ^ { \\lambda } . } \\end{array}\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "We refer to this algorithm as model-based policy search (MB-PS). It is based on model rollouts $\\tau ^ { i }$ spanning entire episodes. An alternative would be to consider model rollouts starting from states visited in the real environment (if available). Both versions can be augmented by counterfactual methods, but for the sake of simplicity we focus on the simpler MB-PS version detailed above (also we did not find significant performance differences experimentally between both versions). ",
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+    {
+        "type": "text",
+        "text": "4.2 INCORPORATING OFF-POLICY DATA: COUNTERFACTUALLY-GUIDED POLICY SEARCH ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "Now, we assume that the model $\\mathcal { M }$ is an SCM. Based on our discussion of counterfactual policy evaluation, it is straightforward to generalize the MB-PS described above by anchoring the rollouts $\\tau ^ { i }$ under the model $\\overline { { p } } ^ { \\lambda }$ in off-policy data $D$ : Instead of sampling $\\tau ^ { i }$ directly from the prior $p ^ { \\lambda }$ , we draw them from counterfactual distribution $p ^ { \\lambda | \\hat { h } _ { T } ^ { i } }$ with data $\\hat { h } _ { T } ^ { i } \\sim D$ from the replay buffer, i.e. instead of sampling the scenarios $U$ from the prior we infer them from the given data. Again invoking Lemma 1, this procedure is unbiased under no model mismatch. We term the resulting algorithm Counterfactually-Guided Policy Search (CF-GPS), and it is summarized in Algorithm 1. The motivation for using CF-GPS over MB-PS is analogous to the advantage of CF-PE over MB-PE discussed in sec. 3.1. The policy $\\pi$ in CF-GPS is optimized on rollouts $\\tau ^ { i }$ that are grounded in data $\\hat { h } _ { T } ^ { i }$ by sampling them from the counterfactual distribution $p ^ { \\lambda | \\hat { h } _ { T } ^ { i } }$ instead of the prior $p ^ { \\lambda }$ . If this prior is difficult to model, we expect the counterfactual distribution to be more concentrated in regions where there is actual mass under the true environment ${ \\mathfrak { p } } ^ { \\lambda }$ . ",
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+    {
+        "type": "text",
+        "text": "4.3 EXPERIMENTS ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "We evaluate CF-GPS on the PO-SOKOBAN environment, using a modified distributed actor-learner architecture based on Espeholt et al. (2018): Multiple actors (here 64) collect real data $\\hat { h } _ { T }$ by running the behavior policy $\\mu$ in the true environment $\\mathfrak { p }$ . As in many distributed RL settings, $\\mu$ is chosen to be a copy of the policy $\\pi$ , often slightly outdated, so the data must be considered to be offpolicy. The distribution $p ( U _ { s 1 } | \\hat { h } _ { T } )$ over levels $U _ { s 1 }$ is inferred from the data $\\hat { h } _ { T }$ using from the model $\\mathcal { M }$ . We sample a scenario $U _ { s 1 }$ for each logged episode, and simulate 10 counterfactual trajectories $\\tau ^ { 1 , \\ldots , 1 0 }$ under the planner $\\lambda$ for each such scenario. Here, for the sake of simplicity, instead of using search, the planner was assumed to be a mixture between $\\pi$ and a pre-trained expert policy $\\lambda _ { e }$ , i.e. $\\lambda = \\beta \\lambda _ { e } + ( 1 - \\beta ) \\pi$ . The schedule $\\beta$ was set to an exponentially decaying parameter with time constant $1 0 ^ { 5 }$ episodes. The learner performs policy improvement on $\\pi$ using $\\bar { \\tau } ^ { 1 , \\ldots , 1 0 }$ according to eqn. 1. $\\mathcal { M }$ was trained online, in the same way as in sec. 3.2. $\\lambda$ and $\\pi$ were parameterized by deep, recurrent neural networks with the same architecture described in sec. 3.2. ",
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+    {
+        "type": "text",
+        "text": "We compare CF-GPS with the vanilla MB-PS baseline described in sec. 4.1 (based on the same number of policy updates). MB-PS differs from CF-GPS by just having access to an unconditional model $p ( U _ { s 1 } | \\emptyset )$ over initial states. We also consider a method which conditions the scenario model $p ( U _ { s 1 } | o _ { 1 } )$ on the very first observation $o _ { 1 }$ , which is available when taking the first action and therefore does not involve hindsight reasoning. This is more informed compared to MB-PS; however due to the noise on the observations, the state is still mostly unobserved rendering it very challenging to learn a good parametric model of the belief state $p ( U _ { s 1 } | o _ { 1 } )$ . We refer to this algorithm as Guided Policy Search-like (GPS-like), as it roughly corresponds to the algorithm presented by Levine & Abbeel (2014), as discussed in greater detail in sec. 5. Fig. 2 shows that CF-GPS outperforms these two baselines. As expected from the policy evaluation experiments, initial states sampled from the models for GPS and MB-PS are often not solvable, yielding inferior training data for the policy $\\pi$ . In CF-GPS, the levels are inferred from hindsight inference $p ( U _ { 1 } | \\hat { h } _ { T } )$ , yielding high quality training data. For reference, we also show a policy trained by the model-free method of Espeholt et al. (2018) using the same amount of environment data. Not surprisingly, CF-GPS is able to make better use of the data compared to the model-free baseline as it has access to the true transition and reward kernels (which were not given to the model-free method). ",
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+        "type": "text",
+        "text": "",
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+    {
+        "type": "text",
+        "text": "5 RELATED WORK ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Bottou et al. (2013) provide an in-depth discussion of applying models to off-policy evaluation. However, their and related approaches (Li et al., 2015; Jiang & Li, 2015; Swaminathan & Joachims, 2015; Nedelec et al., 2017; Atan et al., 2016; Thomas & Brunskill, 2016) such as doubly-robust estimators, rely on Importance Sampling (IS), also called Propensity Score method. Although some of these algorithms are also termed counterfactual policy evaluation, they are not counterfactual in the sense used in this paper, where noise variables are inferred from logged data and reused to evaluate counterfactual actions. Hence, they are dogged by high variance in the estimators common to IS, although recent work aims to address this (Munos et al., 2016; Guo et al., 2017). Model-based methods for off-policy evaluation have recently been improved to account for the distribution shift between the data-collecting policy and the policy to be evaluated (Johansson et al., 2016; Liu et al., 2018). Recently (Andrychowicz et al., 2017) proposed the Hindsight Experience Replay (HER) algorithm for learning a family of goal directed policies. In HER one observes an outcome in the true environment, which is kept fixed, and searches for the goal-directed policy that should have achieved this goal in order to positively reinforce it. Therefore, this algorithm is complementary to CF-GPS where we search over alternative outcomes for a given policy. Our CF-GPS algorithm is inspired by and extends work presented by Abbeel et al. (2006) on a method for de-biasing weak models by estimating additive terms in the transition kernel to better match individual, real trajectories. The resulting model, which is a counterfactual distribution in the terminology used in our paper, is then used for model-based policy improvement. Our work generalizes this approach and highlights conceptual connections to causal reasoning. Furthermore, we discuss the connection of CF-GPS to two classes of RL algorithms in greater detail below. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Guided Policy Search (GPS) CF-GPS is closely related to GPS, in particular we focus on GPS as presented by Levine & Abbeel (2014). Consider CF-GPS in the fully-observed MDP setting where $O _ { t } ~ = ~ S _ { t }$ . Furthermore, assume that the SCM $\\mathcal { M }$ is structured as follows: Let $S _ { t + 1 } ~ =$ $f _ { s } ( S _ { t } , A _ { t } , U _ { s t } )$ be a linear function in $( S _ { t } , A _ { t } )$ with coefficients given by $U _ { s t }$ . Further, assume an i.i.d. Gaussian mixture model on $U _ { s t }$ for all $t$ . As the states are fully observed, the inference step in the CFI procedure simplifies: we can infer the noise sources $\\hat { u } _ { s t }$ (samples or MAP estimates), i.e. the unknown linear dynamics, from pairs of observed, true states $\\hat { s } _ { t }$ , $\\hat { s } _ { t + 1 }$ . Furthermore assume that the reward is a quadratic function of the state. Then, the counterfactual distribution $p ^ { \\lambda } ( \\tau | \\hat { u } )$ is a linear quadratic regulator (LQR) with time-varying coefficients $\\hat { u }$ . An appropriate choice for the planner $\\lambda$ is the optimal linear feedback policy for the given LQR, which can be computed exactly by dynamic programming. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Observation 1. In the MDP setting, CF-GPS with a linear SCM and a dynamic programming planner for LQRs λ is equivalent to GPS. ",
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+    {
+        "type": "text",
+        "text": "Another perspective is that GPS is the counterfactual version of the MB-PS procedure from sec. 4.1: ",
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+        "page_idx": 8
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+    {
+        "type": "text",
+        "text": "Observation 2. In the MDP setting with a linear SCM and a dynamic programming planner for LQRs λ, GPS is the counterfactual variant of the MB-PS procedure outlined above. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "The fact that GPS is a successful algorithm in practice shows that the ‘grounding’ of model-based search / rollouts in real, off-policy data afforded by counterfactual reasoning massively improves the naive, ‘prior sample’-based MB-PS algorithm. These considerations also suggest when we expect CF-GPS to be superior compared to regular GPS: If the uncertainty in the environment transition $U _ { s t }$ cannot be reliably identified from subsequent pairs of observations $\\hat { o } _ { t } , \\hat { o } _ { t + 1 }$ alone, we expect benefits of inferring $U _ { s t }$ from a larger context of observations, in the extreme case from the entire history $\\hat { h } _ { T }$ as described above. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Stochastic Value Gradient methods There are multiple interesting connections of CF-GPS to Stochastic Value Gradient (SVG) methods (Heess et al., 2015). In SVG, a policy $\\pi$ for a MDP is learned by gradient ascent on the expected return under a model $p$ . Instead of using the scorefunction estimator, SVG relies on a reparameterization of the stochastic model and policy (Kingma & Welling, 2013; Rezende et al., 2014). We note that this reparameterization casts $p$ into an SCM. As in GPS, the noise sources $U _ { s t }$ are inferred from two subsequent observed states $\\hat { s } _ { t } , \\hat { s } _ { t + 1 }$ from the true environment, and the action noise $U _ { a t }$ is kept frozen. As pointed out in the GPS discussion, this procedure corresponds to the inference step in a counterfactual query. Given inferred values $u$ for $U$ , gradients $\\partial _ { \\theta } G$ of the return under the model are taken with respect to the policy parameters $\\theta$ . We can loosely interpret these gradients as $2 \\dim ( \\theta )$ counterfactual policy evaluations of policies $\\pi ( \\theta \\pm \\Delta \\theta _ { i } )$ where a single dimension $i$ of the parameter vector $\\theta$ is perturbed. ",
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+    {
+        "type": "text",
+        "text": "6 DISCUSSION ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Simulating plausible synthetic experience de novo is a hard problem for many environments, often resulting in biases for model-based RL algorithms. The main takeaway from this work is that we can improve policy learning by evaluating counterfactual actions in concrete, past scenarios. Compared to only considering synthetic scenarios, this procedure mitigates model bias. However, it relies on some crucial assumptions that we want to briefly discuss here. The first assumption is that off-policy experience is available at all. In cases where this is e.g. too costly to acquire, we cannot use any of the proposed methods and have to exclusively rely on the simulator / model. We also assumed that there are no additional hidden confounders in the environment and that the main challenge in modelling the environment is capturing the distribution of the noise sources $p ( U )$ , whereas we assumed that the transition and reward kernels given the noise is easy to model. This seems a reasonable assumption in some environments, such as the partially observed grid-world considered here, but not all. Probably the most restrictive assumption is that we require the inference over the noise $U$ given data $\\hat { h } _ { T } ^ { \\phantom { \\dagger } }$ to be sufficiently accurate. We showed in our example, that we could learn a parametric model of this distribution from privileged information, i.e. from joint samples $u , h _ { T }$ from the true environment. However, imperfect inference over the scenario $U$ could result e.g. in wrongly attributing a negative outcome to the agent’s actions, instead environment factors. This could in turn result in too optimistic predictions for counterfactual actions. Future research is needed to investigate if learning a sufficiently strong SCM is possible without privileged information for interesting RL domains. If, however, we can trust the transition and reward kernels of the model, we can substantially improve model-based RL methods by counterfactual reasoning on off-policy data, as demonstrated in our experiments and by the success of Guided Policy Search and Stochastic Value Gradient methods. ",
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+    {
+        "type": "text",
+        "text": "REFERENCES ",
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+        ],
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+    },
+    {
+        "type": "text",
+        "text": "A PROOFS ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            102,
+            277,
+            118
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "A.1 PROOF OF LEMMA 1 ",
+        "bbox": [
+            176,
+            132,
+            359,
+            147
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "Proof. We start from the fact that the density over noise sources $U$ remains the same for every intervention $I$ as $U$ are root nodes in $\\mathcal { G }$ : ",
+        "bbox": [
+            174,
+            157,
+            823,
+            186
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "equation",
+        "img_path": "images/bff68f87835e05c43b2cfe0bfc77ef5e8718e97ed70594e05681ec91719efd8d.jpg",
+        "text": "$$\n\\begin{array} { r c l } { p ^ { \\mathrm { d o } ( I ) } ( u ) } & { = } & { p ( u ) . } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            426,
+            188,
+            571,
+            209
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "This leads to: ",
+        "bbox": [
+            173,
+            210,
+            264,
+            224
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "equation",
+        "img_path": "images/8050ae3ad89b8e44ff4eff2d03a93fa8d11f18cd8ae6a0529ee96aae2d280d29.jpg",
+        "text": "$$\n\\begin{array} { r l } { p ^ { \\mathrm { d o } ( T ) } ( x ) } & { = \\ \\int p ^ { \\mathrm { d o } ( T ) } ( x | u ) p ^ { \\mathrm { d o } ( T ) } ( u ) d u } \\\\ & { = \\ \\int p ^ { \\mathrm { d o } ( T ) } ( x | u ) p ( u ) d u } \\\\ & { = \\ \\int p ^ { \\mathrm { d o } ( T ) } ( x | u ) \\left( \\int p ( \\hat { x } _ { o } , u ) d \\hat { x } _ { o } \\right) d u } \\\\ & { = \\ \\int \\int p ^ { \\mathrm { d o } ( T ) } ( x | u ) p \\left( u | \\hat { x } _ { o } \\right) p ( \\hat { x } _ { o } ) d u d \\hat { x } _ { o } } \\\\ & { = \\ \\mathbb { E } _ { \\hat { x } _ { o } \\sim p \\left[ \\int p ^ { \\mathrm { d o } ( T ) } ( x | u ) p \\left( u | \\hat { x } _ { o } \\right) d u \\right] } } \\\\ & { = \\ \\mathbb { E } _ { \\hat { x } _ { o } \\sim p \\left[ \\int p ^ { \\mathrm { d o } ( T ) } ( x | u ) p \\left( u | \\hat { x } _ { o } \\right) d u \\right] } } \\\\ & { = \\ \\mathbb { E } _ { \\hat { x } _ { o } \\sim p \\left[ p ^ { \\mathrm { d o } ( T ) | \\hat { x } _ { o } } ( x ) \\right] } . } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            316,
+            226,
+            678,
+            415
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "A.2 PROOF OF COROLLARY 1 ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            445,
+            392,
+            460
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "Proof. Given two sets $\\mathrm { C F } \\subset \\{ 1 , \\ldots , N \\}$ and Prior ${ \\textsf { C } } \\left\\{ 1 , \\ldots , N \\right\\}$ with ${ \\mathrm { C F } } \\cap { \\mathrm { P r i o r } } = \\emptyset$ and $\\mathrm { C F } \\cup \\mathrm { P r i o r } = \\{ 1 , \\ldots , N \\}$ , we define $\\vec { u _ { \\mathrm { C F } } = ( u _ { n } ) _ { n \\in \\mathrm { C F } } }$ and $u _ { \\mathrm { P r i o r } } = \\mathsf { \\bar { ( } } u _ { n } ) _ { n \\in \\mathrm { P r i o r } }$ . By construction, the scenarios $U$ are independent under the prior, i.e. $\\begin{array} { r } { p ( u ) = \\prod _ { n = 1 } ^ { N } p ( u _ { n } ) } \\end{array}$ . Therefore $u _ { \\mathrm { C F } }$ and $u _ { \\mathrm { P r i o r } }$ are independent. We can write: ",
+        "bbox": [
+            173,
+            470,
+            826,
+            531
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "equation",
+        "img_path": "images/21022476613c8ba01fde6fed699d37641376fdb10b9797fd461a6e06c44b9539.jpg",
+        "text": "$$\n\\begin{array} { r c l } { p ( u ) } & { = } & { \\displaystyle \\left( \\prod _ { n \\in \\mathrm { C F } } p ( u _ { n } ) \\right) \\left( \\prod _ { n \\in \\mathrm { P r i o r } } p ( u _ { n } ) \\right) } \\\\ & { = } & { p ( u _ { \\mathrm { C F } } ) p ( u _ { \\mathrm { P r i o r } } ) . } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            346,
+            534,
+            647,
+            597
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "Following the arguments from Lemma 1, the averaged inference distribution is equal to the prior $p ( u ) = \\mathbb { \\bar { E } } _ { \\hat { x } _ { o } \\sim p } [ \\bar { p ( } u | \\hat { x } _ { o } ) ]$ . This also holds for any subset of the variables $u$ , in particular for for $p ( u _ { \\mathrm { C F } } )$ . Hence: ",
+        "bbox": [
+            173,
+            598,
+            825,
+            640
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "equation",
+        "img_path": "images/1a3c8116c48c89227775d76aa1ae792dd3c57ec8287b5694e228907cab6acc2b.jpg",
+        "text": "$$\n\\begin{array} { r c l } { p ( u ) } & { = } & { p ( u _ { \\mathrm { C F } } ) p ( u _ { \\mathrm { P r i o r } } ) } \\\\ & { = } & { \\mathbb { E } _ { \\hat { x } _ { o } \\sim p } [ p ( u _ { \\mathrm { C F } } | \\hat { x } _ { o } ) ] p ( u _ { \\mathrm { P r i o r } } ) . } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            364,
+            642,
+            632,
+            680
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "B DETAILS ON CASTING A POMDP INTO SCM FORM ",
+        "text_level": 1,
+        "bbox": [
+            173,
+            714,
+            635,
+            733
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "Lemma 2 (Auto-regressive uniformization aka Reparametrization). Consider random variables $X _ { 1 } , \\ldots , X _ { N }$ with joint distribution $P$ . There exist functions $f _ { n }$ for $n \\in \\{ 1 , \\ldots , N \\}$ such that with independent random variables $U _ { n } \\sim \\mathrm { U n i f o r m } [ 0 , 1 ]$ the random variables $X ^ { \\prime }$ equal $X$ in distribution, i.e. $X ^ { \\prime } { \\overset { d } { = } } X$ , where $X ^ { \\prime } = ( X _ { 1 } ^ { \\prime } , \\ldots , X _ { n } ^ { \\prime } )$ are defined as: ",
+        "bbox": [
+            173,
+            746,
+            825,
+            809
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "equation",
+        "img_path": "images/a6af1e79a33e5d4b2420cfc4cd7d8cfa987a9ef8abe2cc63ba294d6a76424255.jpg",
+        "text": "$$\n\\begin{array} { r } { X _ { n } ^ { \\prime } \\quad : = \\quad f _ { n } ( U _ { n } , X _ { < n } ^ { \\prime } ) . } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            413,
+            810,
+            581,
+            829
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "Proof. We construct the functions $f _ { n }$ by induction on $n$ . Consider the conditional distribution $P _ { X _ { n } | X _ { < n } }$ . For fixed $X _ { < n }$ , denote its CDF with $F _ { n , X < n }$ . We construct a random variable $X ^ { \\prime } : = F _ { n , X _ { < n } } ^ { - 1 } ( U _ { n } )$ with $U _ { n } \\ \\sim \\ \\mathrm { U n i f o r m [ 0 , 1 ] }$ independent from $X _ { < n }$ and $U _ { < n }$ . By virtue of the inverse-CDF method, we have $X ^ { \\prime } | X _ { < n } \\overset { d } { = } X | X _ { < n }$ . Therefore, $f _ { n } ( U _ { n } , X _ { < n } ) : = F _ { n , X _ { < n } } ^ { - 1 } ( U _ { n } )$ satisfies the above lemma. □ ",
+        "bbox": [
+            173,
+            840,
+            826,
+            924
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "image",
+        "img_path": "images/1be8fcef4e3a974e21350038df064afa1220d61b230adb3cca5cf03b07073782.jpg",
+        "image_caption": [
+            "Figure 3: Top: PO-SOKOBAN. Shown on the left is a procedurally generated initial state. The agent is shown in green, boxes in yellow, targets in blue and walls in red. The agent does not observe this state but a sequence of observations, which are masked by iid noise with 0.9 probability, except a $3 \\mathrm { x } 3$ window around the agent. Bottom: Inference model. For counterfactual inference in PO-SOKOBAN, we need the (approximate) inference distribution $p ( U _ { s 1 } | \\hat { h } _ { T } )$ over the initial state $U _ { s 1 } = S _ { 1 }$ , conditioned on the history of observations $\\hat { h } _ { T }$ . We model this distribution using a DRAW generative model with latent variables $Z$ , which are conditioned on the output of a backward RNN summarizing the observation history. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            176,
+            99,
+            826,
+            308
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "C MODEL ARCHITECTURE ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            465,
+            408,
+            482
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "We assume that we are given the true transition and reward kernels. As the transitions are deterministic in PO-SOKOBAN, the only part of the model that remains to be identified is the initial state distribution $p ( U _ { s 1 } )$ . We learned this model from data using a the DRAW model (Gregor et al., 2015), which is a parametric, multi-layer, latent variable, neural network model for distributions. For our purposes we chose the convolutional DRAW architecture proposed by (Gregor et al., 2016). First, the observation data is summarized by a convolutional LSTM with 32 hidden units and kernel size of 3. The resulting final LSTM state is fed into a conditional Gaussian prior over the latent variables $Z _ { k = 1 , \\dots , 8 }$ of the 8-layer conv-DRAW model. Each layer has 32 hidden layers and the canvas had 7 layers, corresponding to the 7 channels of the categorical $U _ { s 1 } \\in \\{ 0 , 1 \\} ^ { 1 0 \\times 1 0 \\times 7 }$ that we wish to model. The model (together with the backward RNN) was trained with the ADAM optimizer (Kingma & Ba, 2014) on the ELBO loss using the reparametrization trick (Kingma & Welling, 2013; Rezende et al., 2014). The mini-batch size was set to 4 and the learning rate to $3 e - 4$ . We want to emphasize that the DRAW latent variables $Z$ are not directly the noise variables $U$ of the SCM, but integrating out these variables yields this distribution $\\begin{array} { r } { p ( U _ { s 1 } | \\hat { h } _ { T } ) = \\int p ( U _ { s 1 } | z , \\hat { h } _ { T } ) p ( z | \\hat { h } _ { T } ) d \\hat { h } _ { T } } \\end{array}$ . ",
+        "bbox": [
+            174,
+            501,
+            825,
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+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "D MODEL MISMATCH ANALYSIS ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            727,
+            457,
+            742
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "Here we provide some analysis of the DRAW model over the initial state $U _ { s 1 }$ , which is the learned part of the SCM $\\mathcal { M }$ used for the policy evaluation experiments presented in 3.2. As detailed above, we trained a separate model $p ( U _ { s 1 } | \\hat { h } _ { t } )$ for each $t = 1 , \\ldots , 5 0$ parameterizing the cardinality of the data the model is conditioned on. We analyze three particular models for $t = 0 , 1$ and 50 which we term the unconditional / filtering $/$ smoothing model, as they are conditioned on no data $/$ on data that is available at test time $/$ all data that is available in hindsight. Directly visualizing the distributions $p ( U _ { s 1 } | \\hat { h } _ { t } )$ for an analysis is difficult as the domain $\\{ 0 , \\ldots , 6 \\} ^ { 1 0 \\times 1 0 }$ is high-dimensional and discrete. Instead we focus on the latent variables $Z$ which are learned by DRAW to represent this distribution; by construction, these are jointly Normal, facilitating the analysis. In particular, we compare $p ( Z | \\hat { h } _ { t } )$ with the inference distribution $q ( Z | \\hat { u } _ { s 1 } )$ conditioned on the true state $\\hat { U } _ { s 1 }$ . We loosely interpret $q ( Z | \\hat { u } _ { s 1 } )$ as the ”true” embedding of the datum $\\hat { u } _ { s 1 }$ , whereas $p ( Z | \\hat { h } _ { t } )$ is the learned embedding. In a perfect model the prior matches the inference distribution on average: ",
+        "bbox": [
+            173,
+            761,
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+            924
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "image",
+        "img_path": "images/194da2f4bafbbb7d129c84352e8f8e98320583d1c3813e1e10719b692d065c74.jpg",
+        "image_caption": [
+            "Figure 4: Analysis of the model mismatch of the learned inference distributions $p ( U _ { s 1 } | \\hat { h } _ { t } )$ over the initial PO-SOKOBAN state $U _ { s 1 }$ , for three different amounts of observations $t = 0 , 1$ and 50. Shown are two dimensions of the learned latent representation $Z$ of $U _ { s 1 }$ . The (whitened) learned prior $p ( Z | \\hat { h } _ { t } )$ is indicated by a red contour of one standard deviation. The inferred mean embedding of the true levels are show as crosses, and their aggregated density is shown in blue. With increasing amount data that the model is conditioned on, the learned distributions match the data better. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            230,
+            102,
+            769,
+            242
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            169,
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+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "equation",
+        "img_path": "images/51ff290fa4e3e9970122e3cc621bec66b4d0b1c8790b1708ed6f1d4413a8e610.jpg",
+        "text": "$$\n\\begin{array} { r l r } { \\mathbb { E } _ { \\hat { h } _ { t } \\sim \\mathfrak { p } ^ { \\mu } } [ p ( Z | \\hat { h } _ { t } ) ] } & { \\stackrel { ! } { = } } & { \\mathbb { E } _ { \\hat { u } _ { s 1 } \\sim \\mathfrak { p } ^ { \\mu } } [ q ( Z | \\hat { u } _ { s 1 } ) ] , } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            352,
+            390,
+            643,
+            415
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "i.e. every sample from the prior $p ( Z | \\hat { h } _ { t } )$ corresponds to real data and vice versa. We visualize the averaged prior $\\mathbb { E } _ { \\hat { h } _ { t } \\sim \\mathfrak { p } ^ { \\mu } } [ p ( Z | \\hat { h } _ { t } ) ]$ and the averaged posterior $\\mathbb { E } _ { \\hat { u } _ { s 1 } \\sim \\mathfrak { p } ^ { \\mu } } \\left[ q ( Z | \\hat { u } _ { s 1 } ) \\right]$ in fig. 4. We show the two dimensions of $Z$ where these distributions have the largest KL divergence. Also, the plots were whitened w.r.t. the averaged prior, i.e. the latter is a spherical Gaussian in the plots, represented by an iso-probability contour corresponding to one standard deviation. The inference distribution for each datum $\\hat { u } _ { s 1 }$ is visualized by its mean (cross) and a level set corresponding the one standard deviation or less. For the unconditional model $t = 0$ , we find that the distributions are not matched well. In particular, there is a lot of prior mass that sits in regions where there is no or little true data. In the RL setting this results in synthetic data from the model that is unrealistic and training a policy on this data leads to reduced test performance. Also, as apparent from the figure, there is structure in the embedding of the true data, that is not captured at all by the prior. This effect is markedly reduced in the filtering posterior, indicating that the conditional distribution $p ( Z | \\hat { h } _ { 1 } )$ already captures the data distribution better. The smoothing model is a very good match to the data. With high probability, all tiles of the game are observed in $\\bar { \\hat { h } } _ { 5 0 }$ , enabling the model to perfectly learn the belief state, which collapses in this setting to a single state. ",
+        "bbox": [
+            173,
+            422,
+            825,
+            642
+        ],
+        "page_idx": 14
+    }
+]

parse/train/H1gax6VtDB/H1gax6VtDB.md

@@ -0,0 +1,485 @@
+# CONTRASTIVE LEARNING OF STRUCTURED WORLD MODELS
+
+Thomas Kipf University of Amsterdam t.n.kipf@uva.nl
+
+Elise van der Pol University of Amsterdam UvA-Bosch Delta Lab e.e.vanderpol@uva.nl
+
+Max Welling   
+University of Amsterdam CIFAR   
+m.welling@uva.nl
+
+# ABSTRACT
+
+A structured understanding of our world in terms of objects, relations, and hierarchies is an important component of human cognition. Learning such a structured world model from raw sensory data remains a challenge. As a step towards this goal, we introduce Contrastively-trained Structured World Models (C-SWMs). CSWMs utilize a contrastive approach for representation learning in environments with compositional structure. We structure each state embedding as a set of object representations and their relations, modeled by a graph neural network. This allows objects to be discovered from raw pixel observations without direct supervision as part of the learning process. We evaluate C-SWMs on compositional environments involving multiple interacting objects that can be manipulated independently by an agent, simple Atari games, and a multi-object physics simulation. Our experiments demonstrate that C-SWMs can overcome limitations of models based on pixel reconstruction and outperform typical representatives of this model class in highly structured environments, while learning interpretable object-based representations.
+
+# 1 INTRODUCTION
+
+Compositional reasoning in terms of objects, relations, and actions is a central ability in human cognition (Spelke & Kinzler, 2007). This ability serves as a core motivation behind a range of recent works that aim at enriching machine learning models with the ability to disentangle scenes into objects, their properties, and relations between them (Chang et al., 2016; Battaglia et al., 2016; Watters et al., 2017; van Steenkiste et al., 2018; Kipf et al., 2018; Sun et al., 2018; 2019b; Xu et al., 2019). These structured neural models greatly facilitate predicting physical dynamics and the consequences of actions, and provide a strong inductive bias for generalization to novel environment situations, allowing models to answer counterfactual questions such as “What would happen if I pushed this block instead of pulling it?”.
+
+Arriving at a structured description of the world in terms of objects and relations in the first place, however, is a challenging problem. While most methods in this area require some form of human annotation for the extraction of objects or relations, several recent works study the problem of object discovery from visual data in a completely unsupervised or self-supervised manner (Eslami et al., 2016; Greff et al., 2017; Nash et al., 2017; van Steenkiste et al., 2018; Kosiorek et al., 2018; Janner et al., 2019; Xu et al., 2019; Burgess et al., 2019; Greff et al., 2019; Engelcke et al., 2019). These methods follow a generative approach, i.e., they learn to discover object-based representations by performing visual predictions or reconstruction and by optimizing an objective in pixel space. Placing a loss in pixel space requires carefully trading off structural constraints on latent variables vs. accuracy of pixel-based reconstruction. Typical failure modes include ignoring visually small, but relevant features for predicting the future, such as a bullet in an Atari game (Kaiser et al., 2019), or wasting model capacity on visually rich, but otherwise potentially irrelevant features, such as static backgrounds.
+
+To avoid such failure modes, we propose to adopt a discriminative approach using contrastive learning, which scores real against fake experiences in the form of state-action-state triples from an experience buffer (Lin, 1992), in a similar fashion as typical graph embedding approaches score true facts in the form of entity-relation-entity triples against corrupted triples or fake facts.
+
+We introduce Contrastively-trained Structured World Models (C-SWMs), a class of models for learning abstract state representations from observations in an environment. C-SWMs learn a set of abstract state variables, one for each object in a particular observation. Environment transitions are modeled using a graph neural network (Scarselli et al., 2009; Li et al., 2015; Kipf & Welling, 2016; Gilmer et al., 2017; Battaglia et al., 2018) that operates on latent abstract representations.
+
+This paper further introduces a novel object-level contrastive loss for unsupervised learning of object-based representations. We arrive at this formulation by adapting methods for learning translational graph embeddings (Bordes et al., 2013; Wang et al., 2014) to our use case. By establishing a connection between contrastive learning of state abstractions (Franc¸ois-Lavet et al., 2018; Thomas et al., 2018) and relational graph embeddings (Nickel et al., 2016a), we hope to provide inspiration and guidance for future model improvements in both fields.
+
+In a set of experiments, where we use a novel ranking-based evaluation strategy, we demonstrate that C-SWMs learn interpretable object-level state abstractions, accurately learn to predict state transitions many steps into the future, demonstrate combinatorial generalization to novel environment configurations and learn to identify objects from scenes without supervision.
+
+# 2 STRUCTURED WORLD MODELS
+
+Our goal is to learn an object-oriented abstraction of a particular observation or environment state. In addition, we would like to learn an action-conditioned transition model of the environment that takes object representations and their relations and interactions into account.
+
+We start by introducing the general framework for contrastive learning of state abstractions and transition models without object factorization in Sections 2.1–2.2, and in the following describe a variant that utilizes object-factorized state representations, which we term a Structured World Model.
+
+# 2.1 STATE ABSTRACTION
+
+We consider an off-policy setting, where we operate solely on a buffer of offline experience, e.g., obtained from an exploration policy. Formally, this experience buffer $\boldsymbol { B } = \{ ( s _ { t } , a _ { t } , \dot { s _ { t + 1 } } ) \} _ { t = 1 } ^ { T }$ contains $T$ tuples of states $s _ { t } \in S$ , actions $a _ { t } \in \mathcal A$ , and follow-up states $s _ { t + 1 } \in S$ , which are reached after taking action $a _ { t }$ . We do not consider rewards as part of our framework for simplicity.
+
+Our goal is to learn abstract or latent representations $z _ { t } \in \mathcal { Z }$ of environment states $s _ { t } ~ \in ~ S$ that discard any information which is not necessary to predict the abstract representation of the followup state $z _ { t + 1 } \in \mathcal { Z }$ after taking action $a _ { t }$ . Formally, we have an encoder $E : { \mathcal { S } }  { \mathcal { Z } }$ which maps observed states to abstract state representations and a transition model $T : \mathcal { Z } \times \mathcal { A }  \mathcal { Z }$ operating solely on abstract state representations.
+
+# 2.2 CONTRASTIVE LEARNING
+
+Our starting point is the graph embedding method TransE (Bordes et al., 2013): TransE embeds facts from a knowledge base $\boldsymbol { K } = \{ ( e _ { t } , \bar { r _ { t } } , o _ { t } ) \} _ { t = 1 } ^ { T }$ , which consists of entity-relation-entity triples $\left( \boldsymbol { e } _ { t } , \boldsymbol { r } _ { t } , \boldsymbol { o } _ { t } \right)$ , where $e _ { t }$ is the subject entity (analogous to the source state $s _ { t }$ in our case), $r _ { t }$ is the relation (analogous to the action $a _ { t }$ in our experience buffer), and $o _ { t }$ is the object entity (analogous to the target state $s _ { t + 1 }$ ).
+
+TransE defines the energy of a triple $\left( \boldsymbol { e } _ { t } , \boldsymbol { r } _ { t } , \boldsymbol { o } _ { t } \right)$ as $H = d ( F ( e _ { t } ) + G ( r _ { t } ) , F ( o _ { t } ) )$ , where $F$ (and $G$ ) are embedding functions that map discrete entities (and relations) to $\dot { \mathbb { R } } ^ { D }$ , where $D$ is the dimensionality of the embedding space, and $d ( \cdot , \cdot )$ denotes the squared Euclidean distance. Training is carried out with an energy-based hinge loss (LeCun et al., 2006), with negative samples obtained by replacing the entities in a fact with random entities from the knowledge base.
+
+We can port TransE to our setting with only minor modifications. As the effect of an action is in general not independent of the source state, we replace $G ( \boldsymbol { r } _ { t } )$ with $T ( z _ { t } , a _ { t } )$ , i.e., with the transition function, conditioned on both the action and the (embedded) source state via $z _ { t } = E ( s _ { t } )$ . The overall energy of a state-action-state triple then can be defined as follows: $H = d ( z _ { t } + T ( z _ { t } , a _ { t } ) , z _ { t + 1 } )$ .
+
+This additive form of the transition model provides a strong inductive bias for modeling effects of actions in the environment as translations in the abstract state space. Alternatively, one could model effects as linear transformations or rotations in the abstract state space, which motivates the use of a graph embedding method such as RESCAL (Nickel et al., 2011), CompleX (Trouillon et al., 2016), or HolE (Nickel et al., 2016b).
+
+With the aforementioned modifications, we arrive at the following energy-based hinge loss:
+
+$$
+\mathcal { L } = d ( z _ { t } + T ( z _ { t } , a _ { t } ) , z _ { t + 1 } ) + \operatorname* { m a x } ( 0 , \gamma - d ( \tilde { z } _ { t } , z _ { t + 1 } ) ) ,
+$$
+
+defined for a single $\left( { { s _ { t } } , { a _ { t } } , { s _ { t + 1 } } } \right)$ with a corrupted abstract state $\tilde { z } _ { t } = E ( \tilde { s } _ { t } )$ . $\tilde { s } _ { t }$ is sampled at random from the experience buffer. The margin $\gamma$ is a hyperparameter for which we found $\gamma = 1$ to be a good choice. Unlike Bordes et al. (2013), we place the hinge only on the negative term instead of on the full loss and we do not constrain the norm of the abstract states $z _ { t }$ , which we found to work better in our context (see Appendix A.3). The overall loss is to be understood as an expectation of the above over samples from the experience buffer $\boldsymbol { B }$ .
+
+# 2.3 OBJECT-ORIENTED STATE FACTORIZATION
+
+Our goal is to take into account the compositional nature of visual scenes, and hence we would like to learn a relational and object-oriented model of the environment that operates on a factored abstract state space $\mathcal { Z } = \mathcal { Z } _ { 1 } \times . . . \times \mathcal { Z } _ { K }$ , where $K$ is the number of available object slots. We further assume an object-factorized action space $\mathcal { A } = \mathcal { A } _ { 1 } \times . . . \times \mathcal { A } _ { K }$ . This factorization ensures that each object is independently represented and it allows for efficient sharing of model parameters across objects in the transition model. This serves as a strong inductive bias for better generalization to novel scenes and facilitates learning and object discovery. The overall C-SWM model architecture using object-factorized representations is shown in Figure 1.
+
+![](images/d744fb95c7de84c97dc3cd7a9915d5bfe5c3d405344450f2571ddcf5113117ac.jpg)  
+Figure 1: The C-SWM model is composed of the following components: 1) a CNN-based object extractor, 2) an MLP-based object encoder, 3) a GNN-based relational transition model, and 4) an object-factorized contrastive loss. Colored blocks denote abstract states for a particular object.
+
+Encoder and Object Extractor We split the encoder into two separate modules: 1) a CNN-based object extractor $E _ { \mathrm { e x t } }$ , and 2) an MLP-based object encoder $E _ { \mathrm { e n c } }$ . The object extractor module is a CNN operating directly on image-based observations from the environment with $K$ feature maps in its last layer. Each feature map $\overset { \vartriangle } { m _ { t } ^ { k } } = [ E _ { \mathrm { e x t } } ( \boldsymbol { s } _ { t } ) ] _ { k }$ can be interpreted as an object mask corresponding to one particular object slot, where $[ \ldots ] _ { k }$ denotes selection of the $k$ -th feature map. For simplicity, we only assign a single feature map per object slot which sufficed for the experiments considered in this work (see Appendix A.4). To allow for encoding of more complex object features (other than, e.g., position/velocity), the object extractor can be adapted to produce multiple feature maps per object slot. After the object extractor module, we flatten each feature map $m _ { t } ^ { k }$ (object mask) and feed it into the object encoder $E _ { \mathrm { e n c } }$ . The object encoder shares weights across objects and returns an abstract state representation: $z _ { t } ^ { k } = E _ { \mathrm { e n c } } ( \bar { m } _ { t } ^ { k } )$ with $z _ { t } ^ { k } \in { \mathcal { Z } } _ { k }$ . We set $\mathcal { Z } _ { k } = \mathbb { R } ^ { D }$ in the following, where $D$ is a hyperparameter.
+
+Relational Transition Model We implement the transition model as a graph neural network (Scarselli et al., 2009; Li et al., 2015; Kipf & Welling, 2016; Battaglia et al., 2016; Gilmer et al., 2017; Battaglia et al., 2018), which allows us to model pairwise interactions between object states while being invariant to the order in which objects are represented. After the encoder stage, we have an abstract state description $z _ { t } ^ { k } \in { \mathcal { Z } } _ { k }$ and an action $a _ { t } ^ { k } \in { \cal { A } } _ { k }$ for every object in the scene. We represent actions as one-hot vectors (or a vector of zeros if no action is applied to a particular object), but note that other choices are possible, e.g., for continuous action spaces. The transition function then takes as input the tuple of object representations $\boldsymbol { z } _ { t } = ( z _ { t } ^ { 1 } , \ldots , \bar { z _ { t } ^ { K } } )$ and actions $a _ { t } = ( a _ { t } ^ { 1 } , \ldots , a _ { t } ^ { K } )$ at a particular time step:
+
+$$
+\Delta z _ { t } = T ( z _ { t } , a _ { t } ) = \mathrm { G N N } ( \{ ( z _ { t } ^ { k } , a _ { t } ^ { k } ) \} _ { k = 1 } ^ { K } ) .
+$$
+
+$T ( z _ { t } , a _ { t } )$ is implemented as a graph neural network (GNN) that takes $z _ { t } ^ { k }$ as input node features. The model predicts updates $\Delta z _ { t } = ( \Delta z _ { t } ^ { 1 } , . . . , \Delta z _ { t } ^ { K } )$ . The object representations for the next time step are obtained via $z _ { t + 1 } = ( z _ { t } ^ { 1 } + \Delta z _ { t } ^ { 1 } , \dots , z _ { t } ^ { K } + \Delta z _ { t } ^ { K } )$ . The GNN consists of node update functions $f _ { \mathrm { n o d e } }$ and edge update functions $f _ { \mathrm { e d g e } }$ with shared parameters across all nodes and edges. These functions are implemented as MLPs and we choose the following form of message passing updates:
+
+$$
+\begin{array} { r l } & { e _ { t } ^ { ( i , j ) } = f _ { \mathrm { e d g e } } ( [ z _ { t } ^ { i } , z _ { t } ^ { j } ] ) } \\ & { \Delta z _ { t } ^ { j } = f _ { \mathrm { n o d e } } ( [ z _ { t } ^ { j } , a _ { t } ^ { j } , \sum _ { i \neq j } e _ { t } ^ { ( i , j ) } ] ) , } \end{array}
+$$
+
+where e(it $e _ { t } ^ { ( i , j ) }$ is an intermediate representation of the edge or interaction between nodes $i$ and $j$ . This corresponds to a single round of node-to-edge and edge-to-node message passing. Alternatively, one could apply multiple rounds of message passing, but we did not find this to be necessary for the experiments considered in this work. Note that this update rule corresponds to message passing on a fully-connected scene graph, which is $\mathcal { O } ( K ^ { 2 } )$ . This can be reduced to linear complexity by reducing connectivity to nearest neighbors in the abstract state space, which we leave for future work. We denote the output of the transition function for the $k$ -th object as $\Delta z _ { t } ^ { k } = T ^ { k } ( z _ { t } , a _ { t } )$ in the following.
+
+Multi-object Contrastive Loss We only need to change the energy function to take the factorization of the abstract state space into account, which yields the following energy $H$ for positive triples and $\tilde { H }$ for negative samples:
+
+$$
+H = \frac { 1 } { K } \sum _ { k = 1 } ^ { K } d ( z _ { t } ^ { k } + T ^ { k } ( z _ { t } , a _ { t } ) , z _ { t + 1 } ^ { k } ) , \quad \tilde { H } = \frac { 1 } { K } \sum _ { k = 1 } ^ { K } d ( \tilde { z } _ { t } ^ { k } , z _ { t + 1 } ^ { k } ) ,
+$$
+
+where $\tilde { z } _ { t } ^ { k }$ is the $k$ -th object representation of the negative state sample $\tilde { z } _ { t } ~ = ~ E ( \tilde { s } _ { t } )$ . The overall contrastive loss for a single state-action-state sample from the experience buffer then takes the form:
+
+$$
+\begin{array} { r } { \mathcal { L } = H + \operatorname* { m a x } ( 0 , \gamma - \tilde { H } ) . } \end{array}
+$$
+
+# 3 RELATED WORK
+
+For coverage of related work in the area of object discovery with autoencoder-based models, we refer the reader to the Introduction section. We further discuss related work on relational graph embeddings in Section 2.2.
+
+Structured Models of Environments Recent work on modeling structured environments such as interacting multi-object or multi-agent systems has made great strides in improving predictive accuracy by explicitly taking into account the structured nature of such systems (Sukhbaatar et al., 2016; Chang et al., 2016; Battaglia et al., 2016; Watters et al., 2017; Hoshen, 2017; Wang et al., 2018; van Steenkiste et al., 2018; Kipf et al., 2018; Sanchez-Gonzalez et al., 2018; Xu et al., 2019). These methods generally make use of some form of graph neural network, where node update functions model the dynamics of individual objects, parts or agents and edge update functions model their interactions and relations. Several recent works succeed in learning such structured models directly from pixels (Watters et al., 2017; van Steenkiste et al., 2018; Xu et al., 2019; Watters et al., 2019), but in contrast to our work rely on pixel-based loss functions. The latest example in this line of research is the COBRA model (Watters et al., 2019), which learns an action-conditioned transition policy on object representations obtained from an unsupervised object discovery model (Burgess et al., 2019). Unlike C-SWM, COBRA does not model interactions between object slots and relies on a pixel-based loss for training. Our object encoder, however, is more limited and utilizing an iterative object encoding process such as in MONet (Burgess et al., 2019) would be interesting for future work.
+
+Contrastive Learning Contrastive learning methods are widely used in the field of graph representation learning (Bordes et al., 2013; Perozzi et al., 2014; Grover & Leskovec, 2016; Bordes et al., 2013; Schlichtkrull et al., 2018; Velickovi ˇ c et al. ´ , 2018), and for learning word representations (Mnih & Teh, 2012; Mikolov et al., 2013). The main idea is to construct pairs of related data examples (positive examples, e.g., connected by an edge in a graph or co-occuring words in a sentence) and pairs of unrelated or corrupted data examples (negative examples), and use a loss function that scores positive and negative pairs in a different way. Most energy-based losses (LeCun et al., 2006) are suitable for this task. Recent works (Oord et al., 2018; Hjelm et al., 2018; Henaff et al. ´ , 2019; Sun et al., 2019a; Anand et al., 2019) connect objectives of this kind to the principle of learning representations by maximizing mutual information between data and learned representations, and successfully apply these methods to image, speech, and video data.
+
+State Representation Learning State representation learning in environments similar to ours is often approached by models based on autoencoders (Corneil et al., 2018; Watter et al., 2015; Ha & Schmidhuber, 2018; Hafner et al., 2019; Laversanne-Finot et al., 2018) or via adversarial learning (Kurutach et al., 2018; Wang et al., 2019). Some recent methods learn state representations without requiring a decoder back into pixel space. Examples include the selectivity objective in Thomas et al. (2018), the contrastive objective in Franc¸ois-Lavet et al. (2018), the mutual information objective in Anand et al. (2019), the distribution matching objective in Gelada et al. (2019) or using causalitybased losses and physical priors in latent space (Jonschkowski & Brock, 2015; Ehrhardt et al., 2018). Most notably, Ehrhardt et al. (2018) propose a method to learn an object detection module and a physics module jointly from raw video data without pixel-based losses. This approach, however, can only track a single object at a time and requires careful balancing of multiple loss functions.
+
+# 4 EXPERIMENTS
+
+Our goal of this experimental section is to verify whether C-SWMs can 1) learn to discover object representations from environment interactions without supervision, 2) learn an accurate transition model in latent space, and 3) generalize to novel, unseen scenes. Our implementation is available under https://github.com/tkipf/c-swm.
+
+# 4.1 ENVIRONMENTS
+
+We evaluate C-SWMs on two novel grid world environments (2D shapes and 3D blocks) involving multiple interacting objects that can be manipulated independently by an agent, two Atari 2600 games (Atari Pong and Space Invaders), and a multi-object physics simulation (3-body physics). See Figure 2 for example observations.
+
+For all environments, we use a random policy to collect experience for both training and evaluation. Observations are provided as $5 0 \times 5 0 \times 3$ color images for the grid world environments and as $5 0 \times$ $5 0 \times 6$ tensors (two concatenated consecutive frames) for the Atari and 3-body physics environments. Additional details on environments and dataset creation can be found in Appendix B.
+
+![](images/571c941d71021101ea83729b42aabe91a9bcf17e7dfc268671925eaeb0f6e126.jpg)  
+Figure 2: Example observations from block pushing environments (a–b), Atari 2600 games (c– d) and a 3-body gravitational physics simulation (e). In the grid worlds (a–b), each block can be independently moved into the four cardinal directions unless the target position is occupied by another block or outside of the scene. Best viewed in color.
+
+# 4.2 EVALUATION METRICS
+
+In order to evaluate model performance directly in latent space, we make use of ranking metrics, which are commonly used for the evaluation of link prediction models, as in, e.g., Bordes et al. (2013). This allows us to assess the quality of learned representations directly without relying on auxiliary metrics such as pixel-based reconstruction losses, or performance in downstream tasks such as planning.
+
+Given an observation encoded by the model and an action, we use the model to predict the representation of the next state, reached after taking the action in the environment. This predicted state representation is then compared to the encoded true observation after taking the action in the environment and a set of reference states (observations encoded by the model) obtained from the experience buffer. We measure and report both Hits at Rank 1 $( \mathrm { H } @ 1 )$ and Mean Reciprocal Rank (MRR). Additional details on these evaluation metrics can be found in Appendix C.
+
+# 4.3 BASELINES
+
+Autoencoder-based World Models The predominant method for state representation learning is based on autoencoders, and often on the VAE (Kingma & Welling, 2013; Rezende et al., 2014) model in particular. This World Model baseline is inspired by Ha & Schmidhuber (2018) and uses either a deterministic autoencoder (AE) or a VAE to learn state representations. Finally, an MLP is used to predict the next state after taking an action.
+
+Physics As Inverse Graphics (PAIG) This model by Jaques et al. (2019) is based on an encoderdecoder architecture and trained with pixel-based reconstruction losses, but uses a differentiable physics engine in the latent space that operates on explicit position and velocity representations for each object. Thus, this model is only applicable to the 3-body physics environment.
+
+# 4.4 TRAINING AND EVALUATION SETTING
+
+We train C-SWMs on an experience buffer obtained by running a random policy on the respective environment. We choose 1000 episodes with 100 environment steps each for the grid world environments, 1000 episodes with 10 steps each for the Atari environments and 5000 episodes with 10 steps each for the 3-body physics environment.
+
+For evaluation, we populate a separate experience buffer with 10 environment steps per episode and a total of 10.000 episodes for the grid world environments, 100 episodes for the Atari environments and 1000 episodes for the physics environment. For the Atari environments, we minimize train/test overlap by ‘warm-starting’ experience collection in these environments with random actions before we start populating the experience buffer (see Appendix B), and we ensure that not a single full test set episode coincides exactly with an episode from the training set. The state spaces of the grid world environments are large (approx. 6.4M unique states) and hence train/test coincidence of a full 10-step episode is unlikely. Overlap is similarly unlikely for the physics environment which has a continuous state space. Hence, performing well on these tasks will require some form of generalization to new environment configurations or an unseen sequence of states and actions.
+
+All models are trained for 100 epochs (200 for Atari games) using the Adam (Kingma & Ba, 2014) optimizer with a learning rate of $5 \cdot 1 0 ^ { - 4 } $ and a batch size of 1024 (512 for baselines with decoders due to higher memory demands, and 100 for PAIG as suggested by the authors). Model architecture details are provided in Appendix D.
+
+# 4.5 QUALITATIVE RESULTS
+
+We present qualitative results for the grid world environments in Figure 3 and for the 3-body physics environment in Figure 4. All results are obtained on hold-out test data. Further qualitative results (incl. on Atari games) can be found in Appendix A.
+
+In the grid world environments, we can observe that C-SWM reliably discovers object-specific filters for a particular scene, without direct supervision. Further, each object is represented by two coordinates which correspond (up to a random linear transformation) to the true object position in
+
+![](images/e5bd435b5c34474359046ca95cc87b8d5cb4938759c939d653b01b18f08679bd.jpg)
+
+(a) Discovered object masks in a scene from the 3D cubes (top) and 2D shapes (bottom) environments.
+
+![](images/9be11a93a604b3c6fa9d6a6e6b3ac82364f7c1a18a6262ba4d6365ec86112f38.jpg)  
+(b) Learned abstract state transition graph of the yellow cube (left) and the green square (right), while keeping all other object positions fixed at test time.
+
+Figure 3: Discovered object masks (left) and direct visualization of the 2D abstract state spaces and transition graphs for a single object (right) in the block pushing environments. Nodes denote state embeddings obtained from a test set experience buffer with random actions and edges are predicted transitions. The learned abstract state graph clearly captures the underlying grid structure of the environment both in terms of object-specific latent states and in terms of predicted transitions, but is randomly rotated and/or mirrored. The model further correctly captures that certain actions do not have an effect if a neighboring position is blocked by another object (shown as colored spheres), even though the transition model does not have access to visual inputs.
+
+![](images/f789ea56fd3413fb571bd253fe7274f9be21ddd44b0048d506b97009835dfc6e.jpg)
+
+(a) Observations from 3-body gravitational physics simulation (bottom) and learned abstract state transition graph for a single object slot (top).
+
+![](images/7db2e6083363c8878c0b0b0108e5dccddeb13309f239e3983b60a886297fbcf0.jpg)  
+(b) Abstract state transition graph from 50 test episodes for single object slot.
+
+Figure 4: Qualitative results for 3-body physics environment for a single representative test set episode (left) and for a dataset of 50 test episodes (right). The model learns to smoothly embed object trajectories, with the circular motion represented in the latent space (projected from four to two dimensions via PCA). In the abstract state transition graph, orange nodes denote starting states for a particular episode, green links correspond to ground truth transitions and violet links correspond to transitions predicted by the model. One trajectory (in the center) strongly deviates from typical trajectories seen during training, and the model struggles to predict the correct transition.
+
+the scene. Although we choose a two-dimensional latent representation per object for easier visualization, we find that results remain unchanged if we increase the dimensionality of the latent representation. The edges in this learned abstract transition graph correspond to the effect of a particular action applied to the object. The structure of the learned latent representation accurately captures the underlying grid structure of the environment. We further find that the transition model, which only has access to latent representations, correctly captures whether an action has an effect or not, e.g., if a neighboring position is blocked by another object.
+
+Similarly, we find that the model can learn object-specific encoders in the 3-body physics environment and can learn object-specific latent representations that track location and velocity of a particular object, while learning an accurate latent transition model that generalizes well to unseen environment instances.
+
+# 4.6 QUANTITATIVE RESULTS
+
+We set up quantitative experiments for evaluating the quality of both object discovery and the quality of the learned transition model. We compare against autoencoder baselines and model variants that do not represent the environment in an object-factorized manner, do not use a GNN, or do not make use of contrastive learning. Performing well under this evaluation setting requires some degree of (combinatorial) generalization to unseen environment instances.
+
+We report ranking scores (in $\%$ ) in latent space, after encoding source and target observations, and taking steps in the latent space using the learned model. Reported results are mean and standard error of scores over 4 runs on hold-out environment instances. Results are summarized in Table 1.
+
+Table 1: Ranking results for multi-step prediction in latent space. Highest (mean) scores in bold.   
+
+<table><tr><td rowspan="2">Model</td><td rowspan="2"></td><td colspan="2">1 Step</td><td colspan="2">5 Steps</td><td colspan="2">10 Steps</td></tr><tr><td>H@1</td><td>MRR</td><td>H@1</td><td>MRR</td><td>H@1</td><td>MRR</td></tr><tr><td rowspan="6">ZAHJSAS IS</td><td>C-SWM</td><td>100±0.0</td><td>100±0.0</td><td>100±0.0</td><td>100±0.0</td><td>99.9±0.0</td><td>100±0.0</td></tr><tr><td>- latent GNN</td><td>99.9±0.0</td><td>100±0.0</td><td>97.4±0.1</td><td>98.4±0.0</td><td>89.7±0.3</td><td>93.1±0.2</td></tr><tr><td>-factored states</td><td>54.5±18.1</td><td>65.0±15.9</td><td>34.4±16.0</td><td>47.4±16.0</td><td>24.1±11.2</td><td>37.0±12.1</td></tr><tr><td>- contrastive loss</td><td>49.9±0.9</td><td>55.2±0.9</td><td>6.5±0.5</td><td>9.3±0.7</td><td>1.4±0.1</td><td>2.6±0.2</td></tr><tr><td>World Model (AE)</td><td>98.7±0.5</td><td>99.2±0.3</td><td>36.1±8.1</td><td>44.1±8.1</td><td>6.5±2.6</td><td>10.5±3.6</td></tr><tr><td>WorldModel (VAE)</td><td>94.2±1.0</td><td>96.4±0.6</td><td>14.1±1.1</td><td>21.4±1.4</td><td>1.4±0.2</td><td>3.5±0.4</td></tr><tr><td></td><td>C-SWM</td><td>99.9±0.0</td><td>100±0.0</td><td>99.9±0.0</td><td>100±0.0</td><td>99.9±0.0</td><td>99.9±0.0</td></tr><tr><td></td><td>-latent GNN</td><td>99.9±0.0</td><td>99.9±0.0</td><td>96.3±0.4</td><td>97.7±0.3</td><td>86.0±1.8</td><td>90.2±1.5</td></tr><tr><td></td><td>-factored states</td><td>74.2±9.3</td><td>82.5±8.3</td><td>48.7±12.9</td><td>62.6±13.0</td><td>65.8±14.0</td><td>49.6±11.0</td></tr><tr><td>BRPEOSSS</td><td>- contrastive loss</td><td>48.9±16.8</td><td>52.5±17.8</td><td>12.2±5.8</td><td>16.3±7.1</td><td>3.1±1.9</td><td>5.3±2.8</td></tr><tr><td></td><td>World Model (AE)</td><td>93.5±0.8</td><td>95.6±0.6</td><td>26.7±0.7</td><td>35.6±0.8</td><td>4.0±0.2</td><td>7.6±0.3</td></tr><tr><td></td><td>World Model (VAE)</td><td>90.9±0.7</td><td>94.2±0.6</td><td>31.3±2.3</td><td>41.8±2.3</td><td>7.2±0.9</td><td>12.9±1.3</td></tr><tr><td rowspan="5">AAAAA DNOd</td><td>C-SWM(K = 5)</td><td>20.5±3.5</td><td>41.8±2.9</td><td>9.5±2.2</td><td>22.2±3.3</td><td>5.3±1.6</td><td>15.8±2.8</td></tr><tr><td>C-SWM(K = 3)</td><td>34.8±5.3</td><td>54.3±5.2</td><td>12.8±3.4</td><td>28.1±4.2</td><td>9.5±1.7</td><td>21.1±2.8</td></tr><tr><td>C-SWM(K=1)</td><td>36.5±5.6</td><td>56.2±6.2</td><td>18.3±1.9</td><td>35.7±2.3</td><td>11.5±1.0</td><td>26.0±1.2</td></tr><tr><td>World Model (AE)</td><td>23.8±3.3</td><td>44.7±2.4</td><td>1.7±0.5</td><td>8.0±0.5</td><td>1.2±0.8</td><td>5.3±0.8</td></tr><tr><td>World Model (VAE)</td><td>1.0±0.0</td><td>5.1±0.1</td><td>1.0±0.0</td><td>5.2±0.0</td><td>1.0±0.0</td><td>5.2±0.0</td></tr><tr><td rowspan="5">LNIAIISS SPAAE</td><td>C-SWM(K = 5)</td><td>48.5±7.0</td><td>66.1±6.6</td><td>16.8±2.7</td><td>35.7±3.7</td><td>11.8±3.0</td><td>26.0±4.1</td></tr><tr><td>C-SWM(K = 3)</td><td>46.2±13.0</td><td>62.3±11.5</td><td>10.8±3.7</td><td>28.5±5.8</td><td>6.0±0.4</td><td>20.9±0.9</td></tr><tr><td>C-SWM(K =1)</td><td>31.5±13.1</td><td>48.6±11.8</td><td>10.0±2.3</td><td>23.9±3.6</td><td>6.0±1.7</td><td>19.8±3.3</td></tr><tr><td>World Model (AE)</td><td>40.2±3.6</td><td>59.6±3.5</td><td>5.2±1.1</td><td>14.1±2.0</td><td>3.8±0.8</td><td>10.4±1.3</td></tr><tr><td>World Model (VAE)</td><td>1.0±0.0</td><td>5.3±0.1</td><td>0.8±0.2</td><td>5.2±0.0</td><td>1.0±0.0</td><td>5.2±0.0</td></tr><tr><td rowspan="4">PPSSSSS BPPD-S</td><td>C-SWM</td><td>100±0.0</td><td>100±0.0</td><td>97.2±0.9</td><td>98.5±0.5</td><td>75.5±4.7</td><td>85.2±3.1</td></tr><tr><td>World Model (AE)</td><td>100±0.0</td><td>100±0.0</td><td>97.7±0.3</td><td>98.8±0.2</td><td>67.9±2.4</td><td>78.4±1.8</td></tr><tr><td>WorldModel (VAE)</td><td>100±0.0</td><td>100±0.0</td><td>83.1±2.5</td><td>90.3±1.6</td><td>23.6±4.2</td><td>37.5±4.8</td></tr><tr><td>Physics WM (PAIG)</td><td>89.2±3.5</td><td>90.7±3.4</td><td>57.7±12.0</td><td>63.1±11.1</td><td>25.1±13.0</td><td>33.1±13.4</td></tr></table>
+
+We find that baselines that make use of reconstruction losses in pixel space (incl. the C-SWM model variant without contrastive loss) typically generalize less well to unseen scenes and learn a latent space configuration that makes it difficult for the transition model to learn the correct transition function. See Appendix A.1 for a visual analysis of such latent state transition graphs. This effect appears to be even stronger when using a VAE-based World Model, where the prior puts further constraints on the latent representations. C-SWM recovers this structure well, see Figure 3.
+
+On the grid-world environments (2D shapes and 3D blocks), C-SWM models latent transitions almost perfectly, which requires taking interactions between latent representations of objects into account. Removing the interaction component, i.e., replacing the latent GNN with an object-wise MLP, makes the model insensitive to pairwise interactions and hence the ability to predict future states deteriorates. Similarly, if we remove the state factorization, the model has difficulties generalizing to unseen environment configurations. We further explore variants of the grid-world environments in Appendix A.5, where 1) certain actions have no effect, and 2) one object moves randomly and independent of agent actions, to test the robustness of our approach.
+
+For the Atari 2600 experiments, we find that results can have a high variance, and that the task is more difficult, as both the World Model baseline and C-SWM struggle to make perfect longterm predictions. While for Space Invaders, a large number of object slots $K = 5$ ) appears to be beneficial, C-SWM achieves best results with only a single object slot in Atari Pong. This suggests that one should determine the optimal value of $K$ based on performance on a validation set if it is not known a-priori. Using an iterative object encoding mechanism, such as in MONet (Burgess et al., 2019), would enable the model to assign ‘empty’ slots which could improve robustness w.r.t. the choice of $K$ , which we leave for future work.
+
+We find that both C-SWMs and the autoencoder-based World Model baseline excel at short-term predictions in the 3-body physics environment, with C-SWM having a slight edge in the 10 step prediction setting. Under our evaluation setting, the PAIG baseline (Jaques et al., 2019) underperforms using the hyperparameter setting recommended by the authors. Note that we do not tune hyperparameters of C-SWM separately for this task and use the same settings as in other environments.
+
+# 4.7 LIMITATIONS
+
+Instance Disambiguation In our experiments, we chose a simple feed-forward CNN architecture for the object extractor module. This type of architecture cannot disambiguate multiple instances of the same object present in one scene and relies on distinct visual features or labels (e.g., the green square) for object extraction. To better handle scenes which contain potentially multiple copies of the same object (e.g., in the Atari Space Invaders game), one would require some form of iterative disambiguation procedure to break symmetries and dynamically bind individual objects to slots or object files (Kahneman & Treisman, 1984; Kahneman et al., 1992), such as in the style of dynamic routing (Sabour et al., 2017), iterative inference (Greff et al., 2019; Engelcke et al., 2019) or sequential masking (Burgess et al., 2019; Kipf et al., 2019).
+
+Stochasticity & Markov Assumption Our formulation of C-SWMs does not take into account stochasticity in environment transitions or observations, and hence is limited to fully deterministic worlds. A probabilistic extension of C-SWMs is an interesting avenue for future work. For simplicity, we make the Markov assumption: state and action contain all the information necessary to predict the next state. This allows us to look at single state-action-state triples in isolation. To go beyond this limitation, one would require some form of memory mechanism, such as an RNN as part of the model architecture, which we leave for future work.
+
+# 5 CONCLUSIONS
+
+Structured world models offer compelling advantages over pure connectionist methods, by enabling stronger inductive biases for generalization, without necessarily constraining the generality of the model: for example, the contrastively trained model on the 3-body physics environment is free to store identical representations in each object slot and ignore pairwise interactions, i.e., an unstructured world model still exists as a special case. Experimentally, we find that C-SWMs make effective use of this additional structure, likely because it allows for a transition model of significantly lower complexity, and learn object-oriented models that generalize better to unseen situations.
+
+We are excited about the prospect of using C-SWMs for model-based planning and reinforcement learning in future work, where object-oriented representations will likely allow for more accurate counterfactual reasoning about effects of actions and novel interactions in the environment. We further hope to inspire future work to think beyond autoencoder-based approaches for object-based, structured representation learning, and to address some of the limitations outlined in this paper.
+
+# ACKNOWLEDGEMENTS
+
+We would like to thank Marco Federici and Adam Kosiorek for helpful discussions. We would further like to thank the anonymous reviewers for valuable feedback. T.K. acknowledges funding by SAP SE.
+
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+
+# A ADDITIONAL RESULTS AND DISCUSSION
+
+# A.1 OBJECT-SPECIFIC REPRESENTATIONS
+
+We visualize abstract state transition graphs separated by object slot for the 3D cubes environment in Figure 5. Discovered object representations in the 2D shapes dataset (not shown) are qualitatively very similar. We apply the same visualization technique to the model variant without contrastive loss, which is instead trained with a decoder model and a loss in pixel space. See Figure 6 for this baseline and note that the regular structure in the latent space is lost, which makes it difficult for the transition model to learn transitions which generalize to unseen environment instances.
+
+Qualitative results for the 3-body physics dataset are summarized in Figures 7 and 8 for two different random seeds.
+
+![](images/4f604aa7d5170e286f0e7898b8781a9dd2ba8c3f13fb68c77e0a32bb0aa645ef.jpg)  
+Figure 5: Abstract state transition graphs per object slot for a trained C-SWM model on the 3D cubes environment (with all objects allowed to be moved, i.e., none are fixed in place). Edge color denotes action type. The abstract state graph is nearly identical for each object, which illustrates that the model successfully represents objects in the same manner despite their visual differences.
+
+![](images/52c8e2a1412a1cb38404c22fdb21728e4c239fc4709a17144f87ca9477aafb0a.jpg)  
+Figure 6: Abstract state transition graphs per object slot for a trained SWM model without contrastive loss, using instead a loss in pixel space, on the 3D cubes environment. Edge color denotes action type.
+
+![](images/275cc87d7443e712211b2964f8ee5061506a111c871a614b7ad8b3a65a4192cf.jpg)
+
+Figure 7: Object filters (left) and abstract state transition graphs per object slot (right) for a trained C-SWM model on unseen test instances of the 3-body physics environment (seed 1).
+
+For the Atari 2600 environments, we generally found latent object representations to be less interpretable. We attribute this to the fact that a) objects have different roles and are in general not exchangeable (in contrast to the block pushing grid world environments and the 3-body physics environment), b) actions affect only one object directly, but many other objects indirectly in two consecutive frames, and c) due to multiple objects in one scene sharing the same visual features. See Figure 9 for an example of learned representations in Atari Pong and Figure 10 for an example in Space Invaders.
+
+![](images/17eeddfabc1418aabfcfc1439537061cc7b09a923a127c3dd5d4376f690df458.jpg)  
+Figure 8: Object filters (left) and abstract state transition graphs per object slot (right) for a trained C-SWM model on unseen test instances of the 3-body physics environment (seed 2).
+
+![](images/fcdc3ce173b867810df6fde8e9af5fcc726464f934e5b76d1034c2bda0871f9f.jpg)  
+Figure 9: Object filters (left) and abstract state transition graphs per object slot (right) for a trained C-SWM model with $K = 3$ object slots on unseen test instances of the Atari Pong environment.
+
+![](images/e6320a5facd8aca13abea7dd3468a08dbcac788137bee56a92327a50f1eb2cd7.jpg)
+
+Figure 10: Object filters (left) and abstract state transition graphs per object slot (right) for a trained C-SWM model with $K = 3$ object slots on unseen test instances of the Space Invaders environment.
+
+![](images/cfc638cb78c5dd911fed1092c4f405c550db5de4f5385fbc3148453eb6dd854a.jpg)  
+Figure 11: Qualitative model comparison in pixel space on a hold-out test instance of the 2D shapes environment. We train a separate decoder model for 100 epochs on both the C-SWM and the World Model baseline using all training environment instances to obtain pixel-based reconstructions for multiple prediction steps into the future.
+
+# A.2 MODEL COMPARISON IN PIXEL SPACE
+
+To supplement our model comparison in latent space using ranking metrics, we here show a direct comparison in pixel space at the example of the 2D shapes environment. This requires training a separate decoder model for C-SWM. For fairness of this comparison, we use the same protocol to train a separate decoder for the World Model (AE) baseline (discarding the one obtained from the original end-to-end auto-encoding training procedure). This decoder has the same architecture as in the other baseline models and is trained for 100 epochs.
+
+For a qualitative comparison, see Figure 11. The C-SWM model, as expected from our ranking analysis in latent space, performs almost perfectly at this task. Although the World Model (AE) baseline makes clear mistakes which compound over time, it nonetheless often gets several object positions correct after many time steps. The ranking loss in latent space captures this behaviour well, and, for example, assigns an almost perfect score for 1-step prediction to the World Model (AE) baseline. The typically used mean-squared error (MSE) in pixel space (see Figure 12), however, differs by several orders of magnitude between the two models and does not capture any of the nuanced differences in qualitative predictive behavior. We hence strongly encourage researchers in this area to consider ranking-based evaluations directly in latent space in addition to, or as a replacement for comparisons in pixel space.
+
+![](images/b2be0057162ac8a6cb6671248d037cc9cfc850f4e7a1e8badd72258876e6a796.jpg)  
+Figure 12: Quantitative model comparison in pixel space on a hold-out test set of the 2D shapes environment. The plot shows mean squared reconstruction error (MSE) in pixel space for multiple transition model prediction steps into the future (lower is better), averaged over 4 runs. Shaded area denotes standard error.
+
+# A.3 HINGE LOSS
+
+In Table 2, we summarize results of a comparison between our multi-object contrastive loss from Eq. 6, denoted by $C$ -SWM (original), and a full-hinge (or triplet) loss that places the hinge over both the positive and the negative energy term, i.e., $\dot { \mathcal { L } } = \operatorname* { m a x } ( 0 , \gamma + H - \overset { \cdot } { H } )$ . The latter loss is similar to the loss employed in Bordes et al. (2013), but without imposing a norm constraint on the embeddings $z _ { t }$ . Reported results are mean and standard error over 4 runs on hold-out environment instances.
+
+Table 2: Comparison of hinge loss variants on 2D shapes environment. Numbers denote ranking scores (in $\%$ ) for multi-step prediction in latent space. Highest (mean) scores in bold.   
+
+<table><tr><td></td><td colspan="2">1 Step</td><td colspan="2">5 Steps</td><td colspan="2">10 Steps</td></tr><tr><td>Model</td><td>H@1</td><td>MRR</td><td>H@1</td><td>MRR</td><td>H@1</td><td>MRR</td></tr><tr><td>C-SWM (original)</td><td>100±0.0</td><td>100±0.0</td><td>100±0.0</td><td>100±0.0</td><td>99.9±0.0</td><td>100±0.0</td></tr><tr><td>with full hinge (y = 1)</td><td>95.9±1.0</td><td>97.8±0.6</td><td>40.4±5.9</td><td>54.4±5.8</td><td>12.6±4.0</td><td>21.8±5.9</td></tr><tr><td>with full hinge (y = 5)</td><td>97.7±0.9</td><td>98.8±0.5</td><td>52.0±2.5</td><td>65.7±2.6</td><td>15.8±2.9</td><td>26.8±3.5</td></tr><tr><td>with full hinge (y = 10)</td><td>97.0±0.4</td><td>98.5±0.2</td><td>49.5±1.5</td><td>63.3±1.4</td><td>18.7±1.2</td><td>30.3±1.4</td></tr></table>
+
+We find that placing the hinge only on the negative energy term significantly improves results, likely because the full hinge loss can be minimized by simply increasing the norms of the embeddings $z _ { t }$ , as outlined in Bordes et al. (2013). This could be addressed by introducing additional constraints, such as by forcing the embeddings $z _ { t }$ to lie on the unit-hypersphere, as done in TransE (Bordes et al., 2013), which we do not consider in this work as most of the environment dynamics in our experiments obey a flat Euclidean geometry (e.g., translation on a grid).
+
+# A.4 MULTIPLE FEATURE MAPS PER OBJECT SLOT
+
+We experimented with assigning multiple feature maps per object slot to investigate whether this can improve performance on the Space Invaders environment (as an example of one of the more complex environments considered in this work). All other model settings are left unchanged. Results are summarized in Table 3. The original model variant with one feature map per object slot (using a total of $K = 5$ slots) is denoted by $C$ -SWM (original). Reported results are mean and standard error over 4 runs on hold-out environment instances.
+
+Table 3: Comparison of model variants with multiple feature maps per object on the Space Invaders environment. Numbers denote ranking scores (in $\%$ ) for multi-step prediction in latent space.   
+
+<table><tr><td></td><td colspan="2">1 Step</td><td colspan="2">5 Steps</td><td colspan="2">10 Steps</td></tr><tr><td>Model</td><td>H@1</td><td>MRR</td><td>H@1</td><td>MRR</td><td>H@1</td><td>MRR</td></tr><tr><td>C-SWM (original)</td><td>48.5±7.0</td><td>66.1±6.6</td><td>16.8±2.7</td><td>35.7±3.7</td><td>11.8±3.0</td><td>26.0±4.1</td></tr><tr><td>with 2 feature maps</td><td>48.2±7.1</td><td>65.7±6.8</td><td>12.8±3.0</td><td>27.7±3.2</td><td>8.2±2.9</td><td>21.2±3.4</td></tr><tr><td>with 5 feature maps</td><td>52.5±3.4</td><td>71.0±2.6</td><td>18.2±4.3</td><td>36.3±4.6</td><td>11.8±2.5</td><td>25.0±3.4</td></tr><tr><td>with 10 feature maps</td><td>50.0±0.9</td><td>68.6±1.0</td><td>15.8±1.4</td><td>32.4±2.3</td><td>9.0±1.4</td><td>22.7±2.3</td></tr></table>
+
+We find that there is no clear advantage in using multiple feature maps per object slot for the Space Invaders environment, but this might nonetheless prove useful for environments of more complex nature.
+
+# A.5 GRID-WORLD ENVIRONMENT VARIANTS
+
+To further evaluate the robustness of our approach, we experimented with two variants of the 2D shapes grid-world environment with five objects. In the first variant, termed no-op action, we include an additional ‘no-op’ action for each object that has no effect. This mirrors the setting present in many Atari 2600 benchmark environments that have an explicit no-op action. In the second variant, termed random object, we assign a special property to one out of the five objects: a random action is applied to this object at every turn, i.e., the agent only has control over the other four objects and in every environment step, a total of two objects receive an action (one by the agent, which only affects one out of the first four objects, and one random action on the last object). This is to test how the model behaves in the presence of other agents that act randomly. As our model is fully deterministic, it is to be expected that adding a source of randomness in the environment will pose a challenge.
+
+We report results in Table 4. Reported numbers are mean and standard error over 4 runs on hold-out environment instances. We find that adding a no-op action has little effect on our model, but adding a randomly moving object reduces predictive performance. Performance in the latter case could potentially be improved by explicitly modeling stochasticity in the environment.
+
+Table 4: Comparison of C-SWM on variants of the 2D shapes environment. Numbers denote ranking scores (in $\%$ ) for multi-step prediction in latent space.   
+
+<table><tr><td></td><td colspan="2">1 Step</td><td colspan="2">5 Steps</td><td colspan="2">10 Steps</td></tr><tr><td>Model</td><td>H@1</td><td>MRR</td><td>H@1</td><td>MRR</td><td>H@1</td><td>MRR</td></tr><tr><td>C-SWM (original env.)</td><td>100±0.0</td><td>100±0.0</td><td>100±0.0</td><td>100±0.0</td><td>99.9±0.0</td><td>100±0.0</td></tr><tr><td>+ no-op action</td><td>100±0.0</td><td>100±0.0</td><td>100±0.0</td><td>100±0.0</td><td>99.9±0.0</td><td>99.9±0.0</td></tr><tr><td>+ random object</td><td>98.4±0.1</td><td>99.2±0.0</td><td>95.9±0.2</td><td>97.6±0.1</td><td>90.4±0.3</td><td>93.7±0.2</td></tr></table>
+
+# A.6 STABILITY
+
+The discovered object representations and identifications can vary between runs. While we found that this process is very stable for the simple grid world environments where actions only affect a single object, we found that results can be initialization-dependent on the Atari environments, where actions can have effects across all objects, and the 3-body physics simulation (see Figures 7 and 8), which does not have any actions. In some cases, the discovered representation can be less suitable for forward prediction in time or for generalization to novel scenes, which explains the variance in some of our results on these datasets.
+
+# A.7 TRAINING TIME
+
+We found that the overall training time of the C-SWM model was comparable to that of the World Model baseline. Both C-SWM and the World Model baseline trained for approx. 1 hour wall-clock time on the 2D shapes dataset, approx. 2 hours on the 3D cubes dataset, and approx. $3 0 \mathrm { { m i n } }$ on the 3-body physics environment using a single NVIDIA GTX1080Ti GPU. The Atari Pong and Space Invaders models trained for typically less than 20 minutes. A notable exception is the PAIG baseline model (Jaques et al., 2019) which trained for approx. 6 hours on a NVIDIA TitanX Pascal GPU using the recommended settings by the authors of the paper.
+
+# B DATASETS
+
+# B.1 GRID WORLDS
+
+To generate an experience buffer for training, we initialize the environment with random object placements and uniformly sample an object and an object-specific action at every time step.
+
+We provide state observations as $5 0 \times 5 0 \times 3$ tensors with RGB color channels, normalized to $[ 0 , 1 ]$ . Actions are provided as a 4-dim one-hot vector (if an action is applied) or a vector of zeros per object slot in the environment. The action one-hot vector encodes the directional movement action applied to a particular object, or is represented as a vector of zeros if no action is applied to a particular object. Note that only a single object receives an action per time step. For the Atari environments, we provide a copy of the one-hot encoded action vector to every object slot, and for the 3-body physics environment, which has no actions, we do not provide an action vector.
+
+2D Shapes This environment is a $5 \times 5$ grid world with 5 different objects placed at random positions. Each location can only be occupied by at maximum one object. Each object is represented by a unique shape/color combination, occupying $1 0 \times 1 0$ pixels on the overall $5 0 \times 5 0$ pixel grid. At each time step, one object can be selected and moved by one position along the four cardinal directions. See Figure 2a for an example. The action has no effect if the target location in a particular direction is occupied by another object or outside of the $5 \times 5$ grid. Thus, a learned transition model needs to take pairwise interactions between object properties (i.e., their locations) into account, as it would otherwise be unable to predict the effect of an action correctly.
+
+3D Blocks To investigate to what degree our model is robust to partial occlusion and perspective changes, we implement a simple block pushing environment using Matplotlib (Hunter, 2007) as a rendering tool. The underlying environment dynamics are the same as in the 2D Shapes dataset, and we only change the rendering component to make for a visually more challenging task that involves a different perspective and partial occlusions. See Figure 2b for an example.
+
+# B.2 ATARI 2600 GAMES
+
+Atari Pong We make use of the Arcade Learning Environment (Bellemare et al., 2013) to create a small environment based on the Atari 2600 game Pong which is restricted to the first interaction between the ball and the player-controlled paddle, i.e., we discard the first couple of frames from the experience buffer where the opponent behaves completely independent of any player action. Specifically, we discard the first 58 (random) environment interactions. We use the PONGDETERMINISTICV4 variant of the environment in OpenAI Gym (Brockman et al., 2016). We use a random policy and populate an experience buffer with 10 environment interactions per episode, i.e., $T = 1 0$ . An observation consists of two consecutive frames, cropped (to exclude the score) and resized to $5 0 \times 5 0$ pixels each. See Figure 2c for an example.
+
+Space Invaders This environment is based on the Atari 2600 game Space Invaders, using the SPACEINVADERSDETERMINISTIC-V4 variant of the environment in OpenAI Gym (Brockman et al., 2016), and processed / restricted in a similar manner as the Pong environment. We discard the first 50 (random) environment interactions for each episode and only begin populating the experience buffer thereafter. See Figure 2d for an example observation.
+
+# B.3 3-BODY PHYSICS
+
+The 3-body physics simulation environment is an interacting system that evolves according to classical gravitational dynamics. Different from the other environments considered here, there are no actions. This environment is adapted from Jaques et al. (2019) using their publicly available implementation1, where we set the step size (dt) to 2.0 and the initial maximum $x$ and $y$ velocities to 0.5. We concatenate two consecutive frames of $5 0 \times 5 0$ pixels each to provide the model with (implicit) velocity information. See Figure 4 for an example observation sequence.
+
+# C EVALUATION METRICS
+
+# C.1 HITS AT RANK K $( \mathrm { H } @ \mathrm { K } )$
+
+This score is 1 for a particular example if the predicted state representation is in the k-nearest neighbor set around the encoded true observation, where we define the neighborhood of a node to include the node itself. Otherwise this score is 0. In other words, this score measures whether the rank of the predicted state representation is smaller than or equal to k, when ranking all reference state representations by distance to the true state representation. We report the average of this score over a particular evaluation dataset.
+
+# C.2 MEAN RECIPROCAL RANK (MRR)
+
+This score is defined as the average inverse rank, i.e., MRR = 1N PNn=1 1rankn , where $\operatorname { r a n k } _ { n }$ is the rank of the $n$ -th sample.
+
+# D ARCHITECTURE AND HYPERPARAMETERS
+
+# D.1 OBJECT EXTRACTOR
+
+For the 3D cubes environment, the object extractor is a 4-layer CNN with $3 \times 3$ filters, zero-padding, and 16 feature maps per layer, with the exception of the last layer, which has $K = 5$ feature maps, i.e., one per object slot. After each layer, we apply BatchNorm (Ioffe & Szegedy, 2015) and a ${ \mathrm { R e L U } } ( x ) = { \mathrm { m a x } } ( 0 , x )$ activation function. For the 2D shapes environment, we choose a simpler CNN architecture with only a single convolutional layer with $1 0 \times 1 0$ filters and a stride of 10, followed by BatchNorm and a ReLU activation or LeakyReLU (Xu et al., 2015) for the Atari 2600 and physics environments. This layer has 16 feature maps and is followed by a channel-wise linear transformation (i.e., a $1 \times 1$ convolution), with 5 feature maps as output. For both models, we choose a sigmoid activation function after the last layer to obtain object masks with values in $( 0 , 1 )$ . We use the same two-layer architecture for the Atari 2600 environments and the 3-body physics environment, but with $9 \times 9$ filters (and zero-padding) in the first layer, and $5 \times 5$ filters with a stride of 5 in the second layer.
+
+# D.2 OBJECT ENCODER
+
+After reshaping/flattening the output of the object extractor, we obtain a vector representation per object (2500-dim for the 3D cubes environment, 25-dim for the 2D shapes environment, and 1000- dim for Atari 2600 and physics environments). The object encoder is an MLP with two hidden layers of 512 units and each, followed by ReLU activation functions. We further use LayerNorm (Ba et al., 2016) before the activation function of the second hidden layer. The output of the final output layer is 2-dimensional (4-dimensional for Atari 2600 and physics environments), reflecting the ground truth object state, i.e., the object coordinates in 2D (although this is not provided to the model), and velocity (if applicable).
+
+# D.3 TRANSITION MODEL
+
+Both the node and the edge model in the GNN-based transition model are MLPs with the same architecture / number of hidden units as the object encoder model, i.e., two hidden layers of 512 units each, LayerNorm and ReLU activations.
+
+# D.4 LOSS FUNCTION
+
+The margin in the hinge loss is chosen as $\gamma = 1$ . We further multiply the squared Euclidean distance $d ( x , y )$ in the loss function with a factor of $0 . 5 / \sigma ^ { 2 }$ with $\sigma \ : = \ : 0 . 5$ to control the spread of the embeddings. We use the same setting in all experiments.
+
+# D.5 BASELINES
+
+World Model Baseline The World Model baseline is trained in two stages: First, we train an autoencoder or a VAE with a 32-dim latent space, where the encoder is a CNN with the same architecture as the object extractor used in the C-SWM model, followed by an MLP with the same architecture as the object encoder module in C-SWM on the flattened representation of the output of the encoder CNN. The decoder exactly mirrors this architecture where we replace convolutional layers with deconvolutional layers. We verified that this architecture can successfully build representations of single frames.
+
+Example reconstructions from the latent code are shown in Figure 13. We experimented both with mean squared error and binary cross entropy (using the continuous channel values in [0, 1] as targets) as reconstruction loss in the (V)AE models, both of which are typical choices in most practical implementations. We generally found binary cross entropy to be more stable to optimize and to produce better results, which is why we opted for this loss in all the baselines using decoders considered in the experimental section.
+
+![](images/9497235a03db3355a0b2eda192eed8be224b1e31f29a7085badbc142dbd1c36b.jpg)  
+Figure 13: Reconstructions from the latent code of a trained VAE-based World Model baseline.
+
+In the second stage, we freeze the model parameters of   
+the auto-encoder and train a transition model with mean-squared error on the latent representations. For the VAE model, we use the predicted mean values of the latent representations. This transition model takes the form of an MLP with the same architecture and number of hidden units as the node model in C-SWM. We experimented both with a translational transition model (i.e., the transition model only predicts the latent state difference, instead of the full next state) and direct prediction of the next state. We generally found that the translational transition model performed better and used it throughout all the reported results.
+
+The World Model baselines are trained with a smaller batch size of 512, which slightly improved performance and simplified memory management.
+
+Ablations We perform the following ablations: 1) we replace the latent GNN with an MLP (per object, i.e., we remove the edge update function) to investigate whether a structured transition model is necessary, 2) we remove the state factorization and embed the full scene into a single latent variable of higher dimensionality (original dimensionality $\times$ number of original object slots) with an MLP as transition model, and 3) we replace the contrastive loss with a pixel-based reconstruction loss on both the current state and the predicted next state (we add a decoder that mirrors the architecture of the encoder).
+
+Physics-as-Inverse-Graphics (PAIG) For this baseline, we train the PAIG model from Jaques et al. (2019) with the code provided by the authors2 on our dataset with the standard settings recommended by the authors for this particular task, namely: mode $\mathbf { l } =$ PhysicsNet, epochs $\mathord { = } 5 0 0$ , batch size $_ { \mathrm { = } 1 0 0 }$ , base lr=1e-3, autoencoder los ${ \mathrm { ; } } { = } 5 . 0$ , anneal lr=true, colo $\mathbf { \bar { \rho } } =$ true, and cell type gravity ode cell. We use the same input size as in C-SWM, i.e., frames are of shape
+
+$5 0 \times 5 0 \times 3$ . We further set input steps $^ { \circ 2 }$ , pred step $_ { \mathrm { = } 1 0 }$ and extrap step ${ } = 0$ to match our setting of predicting for a total of 10 frames while conditioning on a pair of 2 initial frames to obtain initial position and velocity information. We train this model with four different random seeds. For evaluation, we extract learned position representations for all frames of the test set and further run the latent physics prediction module (conditioned on the first two initial frames) to obtain model predictions. We further augment position representations with velocity information by taking the difference between two consecutive position representations and concatenating this representation with the 2-dim position representation, which we found to slightly improve results. In total, we obtain a 12-dim $\left( 4 \mathrm { - } \mathrm { d i m } \times 3 \right.$ objects) representation for each time step, i.e., the same as C-SWM. From this, we obtain ranking metrics. We found that one of the training runs (seed $^ { = 2 }$ ) collapsed all latent representations to a single point. We exclude this run in the results reported in Table 1, i.e., we report average and standard error over the three other runs only.

parse/train/HJ1HFlZAb/HJ1HFlZAb.md

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+# EVALUATION OF GENERATIVE NETWORKS THROUGH THEIR DATA AUGMENTATION CAPACITY
+
+Anonymous authors Paper under double-blind review
+
+# ABSTRACT
+
+Generative networks are known to be difficult to assess. Recent works on generative models, especially on generative adversarial networks, produce nice samples of varied categories of images. But the validation of their quality is highly dependent on the method used. A good generator should generate data which contain meaningful and varied information and that fit the distribution of a dataset. This paper presents a new method to assess a generator. Our approach is based on training a classifier with a mixture of real and generated samples. We train a generative model over a labeled training set, then we use this generative model to sample new data points that we mix with the original training data. This mixture of real and generated data is thus used to train a classifier which is afterwards tested on a given labeled test dataset. We compare this result with the score of the same classifier trained on the real training data mixed with noise. By computing the classifier’s accuracy with different ratios of samples from both distributions (real and generated) we are able to estimate if the generator successfully fits and is able to generalize the distribution of the dataset. Our experiments compare different generators from the VAE and GAN framework on MNIST and fashion MNIST dataset.
+
+# 1 INTRODUCTION AND MOTIVATION
+
+Generative network approaches have been widely used to generate samples in recent years. Methods such as GAN (Goodfellow et al., 2014), WGAN (Arjovsky et al., 2017), CGAN (Mirza & Osindero, 2014), CVAE (Sohn et al., 2015) and VAE (Kingma & Welling, 2014) have produced nice samples on various image datasets such as MNIST, bedrooms (Radford et al., 2015) or imageNet (Nguyen et al., 2017).
+
+One commonly accepted tool to evaluate a generative model trained on images is visual assessment to validate the realistic character of samples. One case of this method is called ’visual Turing tests’, in which samples are visualized by humans who try to guess if the images are generated or not. It has been used to assess generative models of images from ImageNet (Denton et al., 2015) and also on digit images (Lake et al., 2015). Salimans et al. (2016) proposes to automate this method with the inception score, which replaces the human judgment by the use of a pretrained classifier to assess the variability of the samples with an entropy measure of the predicted classes and the confidence in the prediction. Unfortunately, those two methods do not indicate if the generator collapses to a particular mode of the data distribution. Log-likelihood based evaluation metrics were widely used to evaluate generative models but as shown in Lucas Theis & Bethge (2016), those evaluations can be misleading in high dimensional cases.
+
+The solution we propose to estimate both sample quality and global fit of the data distribution is to incorporate generated data into the training phase of a classifier before evaluating it. Using generated samples for training has several advantages over using only the original dataset. First, it can make training more efficient when the amount of data is low. As shown in $\mathrm { N g } \ \&$ Jordan (2002), where the conditional distribution $P ( { Y \vert } X )$ (X represents the samples and Y the classes) learned by a generative model is compared to the same conditional distribution learned by a discriminative model, the generative model performs better in learning this conditional distribution by regularizing the model when the amount of data is low. Secondly, once the generative model is trained, it can sample as much images as needed and can produce interpolations between two images which will induce less risk of overfitting on the data. Other works use generative models for data augmentation (Ratner et al., 2017) or to produce labeled data (Sixt et al., 2016) in order to improve the training of discriminative models, but their intention is not to use it to evaluate or compare generative neural networks.
+
+Our method evaluates the quality of a generative model by assessing its capacity to fit the real distribution of the data. For this purpose, we use the samples generated by a given trained generative model. Our work aims to show how this data augmentation can benefit the training of a classifier and how we can use this benefit as an evaluation tool in order to assess a generative model. This method evaluates whether the information of the original distribution is still present in the generated data and whether the generator is able to produce new samples that are eventually close to unseen data. We compare classifiers trained over mixtures of generated and real data with varying ratios and with varying total amounts of data. This allows us to compare generative models in various data settings (i.e., when there is few or many data points).
+
+The next section will present the related work on generative models, the exploitation of the generated samples and their evaluation. We then present our generative model evaluation framework before presenting experimental results on several generative models with different datasets.
+
+# 2 RELATED WORK
+
+# 2.1 GENERATIVE MODELS
+
+The variational auto-encoder (VAE) framework (Kingma & Welling, 2014), (Rezende et al., 2014) is a particular kind of auto-encoder which has control over its latent space, in which each variable is a sample from a prior distribution, often chosen as an univariate normal distribution $\mathbf { N } ( 0 , I )$ (where $I$ is the identity matrix). The VAE learns to map this low dimensional latent space to the observation space. This characteristic makes the VAE an interesting option for generating new data after training. The particularity of the latent space comes from the minimization of the KL divergence between the distribution of the latent space and the prior $\mathbf { N } ( 0 , I )$ . For the sake of simplicity, in this paper we will speak about the decoder of the VAE as a generator.
+
+Generative adversarial networks (Goodfellow et al., 2014) are a framework of models that learn by a game between two networks: a generator that learns to produce images from a distribution $P$ and a discriminator which learns to discriminate between generated and true images. The generator wants to fool the discriminator and the discriminator wants to beat the generator. This class of generative models can produce visually realistic samples from diverse datasets but they suffer from instabilities in their training. Some recent approaches such as Wasserstein GAN (WGAN) (Arjovsky et al., 2017) try to address those issues by enforcing a Lipschitz constraint on the discriminator.
+
+Conditional neural networks (Sohn et al., 2015) and in particular Conditional Variational Autoencoders (CVAE) or Conditional Generative adversarial networks (CGAN) (Mirza & Osindero, 2014) are a class of generative models that have control over the sample’s class. By imposing a label during training, a conditional generative network can generate from any class and thus produces labeled data. The conditional approach has been used to improve the quality of generative networks and make them more discriminative (Odena et al., 2017). They are particularly adapted for our setup because we need to generate labeled data to train our classifiers.
+
+# 2.2 EXPLOITATION AND EVALUATION OF GENERATED SAMPLES
+
+In Ratner et al. (2017), a generator is used to perform data augmentation. Instead of designing a composition of fine tuned transformations for this objective, the authors use adversarial training to learn a sequence of incremental operations (for example rotating or swapping words in a sentence). Their approach uses a GAN to be able to generalize in terms of better data-augmentation and to increase their performance on different datasets such as Cifar10 and the ACE relation extraction task. Sixt et al. (2016) also learns a sequence of transformations with generative networks from the GAN family, but they use a 3D model as input and create an augmented view of it. Our approach is similar by using generative networks for data-augmentation but we do not attempt to learn transformations.
+
+Instead, we use the generated data to assess if the generative model has been able to generalize over the distribution of the data.
+
+The evaluation of generative networks is discussed in Lucas Theis & Bethge (2016). The authors show that different metrics (as Parzen windows, Nearest Neighbor or Log likelihood) applied to generative models can lead to different results. Good results in one application of a generative model can not be used as evidence of good performance in another application. Their conclusion is that evaluation based on sample visual quality is a bad indicator for the entropy of samples. Conversely, the log-likelihood can be used to produce samples with high entropy but does not assure good visual quality. The method we propose can both estimate the quality and the entropy of samples as we will show in Section 3.
+
+The quality of the internal representation of a generator can also be estimated with a discriminator. In Radford et al. (2015) they use the discriminator of a ACGAN as feature extractor for evaluating the quality of unsupervised representation learning algorithms. They apply the feature extractor on supervised datasets and evaluate the performance of linear models fitted on top of these features. They experiment a good accuracy on Cifar10 thanks to this method. This approach gives insight on how the discriminator estimates if an image is true or false. If the discriminator has good enough feature extractors for classification, it means that the generator samples are hard to be discriminated from samples from the true distribution. It assess indirectly the quality of the generator. This method is however applicable only if a deep convolutional neural networks is used as discriminator and can not be applied, e.g., on variational auto-encoders. The principal difference between a discriminator and our classifier is that it is not involved in the training process. In our approach, the generator is completely independent from the classifier and therefore there is no bias from the classifier in the generator.
+
+Parzen windows estimate is a method to estimate the unknown probability density function $f$ of a probability distribution $P$ . This method uses a mixture of simpler probability density functions, called kernels, as approximates for $f$ . In general, a popular kernel used is an isotropic Gaussian centered on a given data point with a small variance (the variance is an hyper parameter here). The idea, like other methods based on Kernel Density Estimation, is to have a small window on each data point such that we apply some smoothing over the function we try to approximate. However, even if the number of samples is high, Parzen windows estimator can be still very far from the true likelihood as shown in Lucas Theis & Bethge (2016), and thus cannot be a good approach to evaluate if the data distribution learned by a model is close to the original one.
+
+Multi-scale structural similarity (MS-SIM, Wang et al. (2003)) is a measurement that gives a way to incorporate image details at different resolutions in order to compare two images. This similarity is generally used in the context of image compression to compare image before and after compression. In Odena et al. (2017) the authors use this similarity to estimate the variability inside a class. They randomly sample two images of a certain class and measure the MS-SIM. If the result is high, then images are considered different. By operating this process several times, the similarity should give an insight on the entropy of $P ( X | Y )$ (X a data point and $\mathrm { Y }$ its class): if the MS-SIM gives high result, the entropy is high; otherwise, the entropy is low. However, it can not estimate if the sample comes from one or several modes of the distribution $P ( X | Y )$ . For example, if we want to generate images of cats, the MS-SIM similarity can not differentiate a generator that produces different kinds of black cats from a network that produces different cats of different colors. In our method, if the generator is able to generate in only one mode of $P ( X | Y )$ , the score will be low in the testing phase.
+
+Another approach that aims to evaluate a generative model by using a conditional distribution learned by a classifier is the inception score (Salimans et al., 2016; Odena et al., 2017). The authors use a pretrained inception classifier model to get the conditional label distribution $P ( { Y \vert } X )$ over the generated samples. They proposed the following score in order to evaluate a generative model:
+
+$$
+\exp ( \mathbb { E } _ { X } K L ( P _ { d a t a } ( Y | X ) \parallel P ( Y ) ) ,
+$$
+
+When the score is high, the generator produces samples on varied classes (Cross entropy of $P ( { Y } | { X } ) , P ( { Y } )$ is high) and the samples look like real images from the original dataset (entropy of $P ( { Y \vert } X )$ is low ). Inception score can be seen as a measurement of the variability of the generated data while penalizing the uncertainty of $P ( { Y \vert } X )$ . Unfortunately, it does not estimate if the samples are varied inside a certain class (the entropy of $p ( X | Y ) )$ . Our approach imposes a high entropy of $P ( Y )$ and gives an unbiased indicator about the entropy of both $P ( { Y \vert } X )$ and $P ( X | Y )$ .
+
+# 3 METHODS
+
+We evaluate generators in a supervised training setup. We have a dataset $D$ composed of pair of examples $( x , y )$ where $x$ is a data point and $y$ the label associated to this data point. The dataset is split in three parts $D _ { t r a i n } , D _ { v a l i d }$ and $D _ { t e s t }$ . Our method needs a generative model that can sample conditionally from any given label $y$ . This conditional generative model is thus trained on $D _ { t r a i n }$ . Once the training of this model is done, we sample random labels and use the generative model to get a new dataset $D _ { g e n }$ . Then, we mix the true training set $D _ { t r a i n }$ with the new generated data $D _ { g e n }$ . The ratio of generated data into the whole mixture $D _ { m i x }$ is called $\tau$ . $\tau$ is used as the probability to sample a generated batch rather than a batch of true data during training. $D _ { m i x }$ is used to train a classifier on the classes of the dataset. This classifier is evaluated at each epoch over the portion $D _ { v a l i d }$ of the dataset. Once, we get the best model over $D _ { v a l i d }$ , we compute the score of this classifier over the test set $D _ { t e s t }$ . We compare the results from training on $D _ { m i x }$ with a baseline. The baseline is the score of the same classifier model trained only on training data $D _ { t r a i n }$ . Our conditional generative model evaluation can be seen as a ratio between those two scores. We can summarize our method as follows:
+
+1. Train a conditional generative model over $n$ sample of $D _ { t r a i n }$   
+2. Mix the samples $D _ { g e n }$ generated by this model with $D _ { t r a i n }$ under a probability $\tau$ into $D _ { m i x }$   
+3. Train a discriminative model over $D _ { m i x }$   
+4. Train a discriminative model over $D _ { t r a i n }$ using the $n$ same sample as for the generator training   
+5. Select a classifiers over a valid set $D _ { v a l i d }$ .   
+6. Compare the score of those classifiers over a test set $D _ { t e s t }$ .
+
+By iterating this method on diverse values of $\tau$ and $n$ we can evaluate the quality of a generator given a dataset.
+
+# 4 EXPERIMENTS
+
+# 4.1 EXPERIMENTAL PROTOCOL
+
+Often, generative models are presented on popular datasets like MNIST. Fashion-MNIST (Xiao et al., 2017) can also serve as a direct drop-in replacement for the original MNIST dataset. This dataset is however more complex than MNIST as images have a higher variability. Thus, we use these datasets in order to evaluate different generative models.
+
+We used two different methods in order to get conditional generative models. The first uses traditional generative neural network which can not produce labeled data. In order to associate each generated sample to a label, we train one generator for each specific class $y$ on $D _ { t r a i n }$ . This makes us able to label any generated sample. Once the training of those generators is done, we mix the samples obtained by each generator in order to produce $D _ { g e n }$ . For the experiments, we compare two generative models is this setting: a standard VAE and a WGAN. The second method uses conditional generative models which can generate samples in all classes while controlling the class of a particular sample. Conditional models can thus generate various labeled samples and produce a whole dataset $D _ { g e n }$ directly. In this last case, we ran our experiments on CVAE and CGAN. Once the dataset $D _ { g e n }$ is generated, we mixed it with the real dataset $D _ { t r a i n }$ . As we can generate as much data as we want, we experimented different ratios between real datasets and generated datasets. We call $\tau$ the probability of sampling from $D _ { g e n }$ . We made experiments with different values for $\tau = [ 0 . 0 0 0 , 0 . \bar { 1 } 2 5 , 0 . 2 5 \bar { 0 } , 0 . 3 7 5 , \bar { 0 } . 5 0 \bar { 0 } , 0 . 6 2 5 , \bar { 0 } . 7 5 0 , 0 . 8 7 5 , 1 . 0 0 \bar { 0 } ] .$ $\tau = 0$ implies that we use only data from $D _ { t r a i n }$ . In this specific setting, we compare the effectiveness of the data augmentation with generated samples versus classic data augmentation as isotropic Gaussian noise with an optimized variance or a random pixel dropout with a probability $\alpha$ of putting a pixel to 0. We also train a classifier without any data augmentation as baseline.
+
+We use a standard CNN with a softmax output as classifier to predict the labels on this mixture of samples. On each epoch we evaluate this classifier over a validation set $D _ { v a l i d }$ . Then, we choose the classifier that performs best on this validation set. We use early stopping to stop the training if the accuracy does not improve anymore on the validation set after 50 epochs. The classifier is then tested on $D _ { t e s t }$ . We train a classifier for each value of $\tau$ . We assess the quality of the generative model by comparing the test score of this classifier when $\tau = 0$ versus the best test score of the classifier with $\tau > 0$ . The result gives an indication on how the learned distribution from $D _ { t r a i n }$ fits and generalizes the distribution from $D _ { t r a i n }$ . In order to be able to compare results from different generators on a given dataset, we always use the same classifier architecture.
+
+![](images/203f196e0eaa3b481878d1d1ce48cf70072d1cf46ee7f54b255ca20b975ad17c.jpg)  
+Figure 1: Representation of test accuracy when $\tau$ increases on fashion-MNIST (MNIST in appendix) with different amounts of training data.
+
+To be able to estimate the impact of learning from generative settings versus discriminative ones, we have made variable the amount of data in $D _ { t r a i n }$ used to train both generator and classifier. Thus, we repeat all our experiments for the following amount of data samples: [100, 500, 1000, 5000, 10000, 50000]. This allows us to measure the regularization capacity of the generated samples over the classifier’s training. We interpret this regularization capacity of those samples as a capacity of generalization.
+
+# 4.2 RESULTS
+
+In Figure 1, we present the test accuracy when $\tau$ increase. When $\tau = 0$ there is no generated data, this is the result of the baseline without data augmentation. Our interpretation of the figure is that if the accuracy is better than baseline with a low $\tau$ $( \mathit { \Theta } < 0 . 5 )$ it means that the generator is able to generalize by learning meaningful informations about the dataset. When $\tau > 0 . 5$ if the accuracy is maintained it means the generated data can replace the dataset in most parts of the distribution. When $\tau = 1$ there is no more original data, the classifier is thus trained only on generated samples. If the accuracy is still better than the baseline, it means that the generator has fit the training distribution (and eventually has learned to generalize if this score is high over the test set).
+
+![](images/49ec4ba23b51ce2ef6dc8b2b43e516f6e5fb55b4f8d50566adaf900e38702c04.jpg)  
+(a) Relative accuracy wrt. baseline on mnist for (b) Relative accuracy wrt. baseline on fashiondifferent models mnist for different models   
+Figure 2: Representation of the data augmentation capacity of each generative models. For each number of training example we show the maximum accuracy a generative model can achieve by tuning $\tau$ . We also show results when tuning hyper-parameter for data augmentation method.
+
+Following this interpretation, Figure 1 allows us to compare different generative neural networks on fashion-MNIST. For example we can see that VAE (Figure 1b) and CVAE (Figure 1a) are able to maintain the baseline accuracy when we do data augmentation with generated samples for different amount of data $n$ . But the accuracy goes down when there are only generated data, which means that these generative models in those settings did not fit the whole distribution. CGAN has the same kind of behavior but the degradation is worse when $\tau = 1$ . However WGAN is able improve the accuracy for $\tau < 0 . 5$ and to maintain it even when $\tau = 1$ . Figure 1 also allows to estimate the entropy of $P ( X )$ . As a deep neural network can not be trained efficiently without various data samples, if the the accuracy on test set is good with only generated data $\mathit { \Pi } _ { \mathcal { T } } = 0 \mathit { \Pi } _ { \mathcal { r } }$ ), necessary the entropy of $P ( X )$ is high as the entropy of $P ( X | Y )$ . This condition is sufficient but not necessary to assess the quality of a generator on a given task.
+
+Our result shows that training one WGAN per class is the best solution to fit the complete distribution, whatever the number of training data used. The bad results on CGAN can be explained by the difficulties to train this model and it’s instability.
+
+Figure 2 represents the best accuracy that each model can achieve when $\tau$ is tuned. It shows that, aside from CGAN on MNIST, all generator can be used to increase accuracy, whatever the number of training data. In this context $\tau$ can be seen as a tunable hyper parameter for data augmentation. Figure 2 also shows that the capacity of generalization is particularly effective when the number of example is low. This can be explained by the fact that when the data number increases the need of data augmentation decrease.
+
+$$
+\Psi _ { G } = \frac { 1 } { n } \sum _ { n } m a x _ { \tau } [ ( a c c _ { n } ( G , \tau ) ) ] - a c c _ { n } ( \tau = 0 ) ,
+$$
+
+The results shown in Table 1 summarize Figure 2 for each generator $G$ by a numerical value $\Psi _ { G }$ (Eq. 2). We call $\Psi _ { G }$ the data augmentation capacity of a generator. It is computed, for a given generator $G$ , by the mean of the differences between the accuracy on mixture with $\tau$ tuned and the baseline accuracy for each number $n$ of training data. The different number of data is arbitrary chosen. In our experiment we compute it with $[ 0 . 2 \%$ , $1 \%$ , $2 \%$ , $1 0 \%$ , $2 0 \%$ , $1 0 0 \%$ , ] of the train set of the dataset. The important thing is to operate at different scales to estimate how the generative model is able to generalize.
+
+The results of Table 1 indicates if a generative model is globally able to perform data augmentation for a given data set. A positive result indicates that the generative model is able to generalize well on the dataset at different sizes. In case of low amount of data, it is better to refer to Figure 2 to choose the best model for that specific case.
+
+Table 1: Results Table : $\Psi _ { G }$ values   
+
+<table><tr><td rowspan=1 colspan=1>Datasets</td><td rowspan=1 colspan=1>Gaussian</td><td rowspan=1 colspan=1>Random Dropout</td><td rowspan=1 colspan=1>VAE</td><td rowspan=1 colspan=1>CVAE</td><td rowspan=1 colspan=1>WGAN</td><td rowspan=1 colspan=1>CGAN</td></tr><tr><td rowspan=1 colspan=1>MNIST</td><td rowspan=1 colspan=1>0.0024</td><td rowspan=1 colspan=1>0.0027</td><td rowspan=1 colspan=1>0.0123</td><td rowspan=1 colspan=1>0.0116</td><td rowspan=1 colspan=1>0.0474</td><td rowspan=1 colspan=1>0.0027</td></tr><tr><td rowspan=1 colspan=1>Fashion MNIST</td><td rowspan=1 colspan=1>0.0014</td><td rowspan=1 colspan=1>0.0018</td><td rowspan=1 colspan=1>0.0047</td><td rowspan=1 colspan=1>0.0041</td><td rowspan=1 colspan=1>0.0357</td><td rowspan=1 colspan=1>-0.0021</td></tr></table>
+
+# 5 DISCUSSION
+
+We presented a method to estimate how well a generative model has learned to generalize in a conditional setting. Using generated samples as data augmentation in order to improve a discriminative model is a well known technique. However, assessing the quality of a generative model with a discriminative model seems to have been less explored. As we have shown, this evaluation is meaningful to measure how well the generative model can sample data points that are probable under the true distribution we want to approximate. In this paper we applied this method on images samples. It means that we can correlate our measure with a visually assess of the samples quality as the generator outputs are in pixel space. Our assessment method can also be used in other spaces as long as labeled data are available.
+
+The relative benefits of discriminative and generative models have been studied in $\mathrm { N g }$ & Jordan (2002). They found that for a small number of training examples $n$ , a generative model will be less prone to overfitting. A discriminative model on a small number of examples risks to learn some spurious nodes that will penalize the generalization capacity. Our results are coherent with their conclusion as the data augmentation induced by generated data is particularly effective when $n$ is low.
+
+Our evaluation was performed on several datasets, generative models, different ratios and amounts of training data. With the current results, WGAN seems to be the most efficient solution. However, this result should be confirmed by experiments on other datasets, generative models and with different types of discriminative models to get a more general comparison. This will be explored in further experiments.
+
+As presented in White (2016), the sampling method of a generator can be adapted, which can have an impact on the data produced. A way to boost the performance of a generator can be to focus on improving the sampling phase instead of the model design or the training. An extension of this work could be to look into the impact of several sampling techniques on the performance of a generative model.
+
+# 6 CONCLUSION
+
+This paper introduces a new method to assess and compare the performances of generative models on various labeled datasets. By training a classifier on several mixture of generated and real data we can estimate the ability of a generative model to generalize. When addition of generated data into the training set achieved better data augmentation than traditional data augmentation as Gaussian noise or random dropout, it demonstrates the ability of generative models to create meaningful samples. By varying the number of training data, we compute a data augmentation capacity $\Psi _ { G }$ for each model on MNIST and fashion-MNIST datasets. $\Psi _ { G }$ is a global estimation of the generalization capacity of a generative model on a given dataset. The results presented here are produced on image datasets but this method can be used on all kinds of datasets or generative models as long as labeled data is available.
+
+# REFERENCES
+
+Martin Arjovsky, Soumith Chintala, and Leon Bottou. Wasserstein generative adversarial networks.´ In Proceedings of the 34th International Conference on Machine Learning, 2017.
+
+Emily L Denton, Soumith Chintala, arthur szlam, and Rob Fergus. Deep generative image models using a laplacian pyramid of adversarial networks. In Advances in Neural Information Processing Systems 28. 2015.
+
+Ian Goodfellow, Jean Pouget-Abadie, Mehdi Mirza, Bing Xu, David Warde-Farley, Sherjil Ozair, Aaron Courville, and Yoshua Bengio. Generative adversarial nets. In Advances in Neural Information Processing Systems 27. 2014.
+
+Diederik P Kingma and Max Welling. Auto-encoding variational bayes. In Proceedings of the 2nd International Conference on Learning Representations (ICLR), 2014.
+
+B. M. Lake, R. Salakhutdinov, and J. B. Tenenbaum. Human-level concept learning through probabilistic program induction. Science, 2015.
+
+Aaron van den Oord Lucas Theis and Matthias Bethge. A note on the evaluation of generative ¨ models. In Proceedings of the 4th International Conference on Learning Representations (ICLR), 2016.
+
+Mehdi Mirza and Simon Osindero. Conditional generative adversarial nets. CoRR, abs/1411.1784, 2014.
+
+Andrew Y Ng and Michael I Jordan. On discriminative vs. generative classifiers: A comparison of logistic regression and naive bayes. In Advances in neural information processing systems, 2002.
+
+Anh Nguyen, Jeff Clune, Yoshua Bengio, Alexey Dosovitskiy, and Jason Yosinski. Plug & play generative networks: Conditional iterative generation of images in latent space. In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, 2017.
+
+Augustus Odena, Christopher Olah, and Jonathon Shlens. Conditional image synthesis with auxiliary classifier GANs. In Proceedings of the 34th International Conference on Machine Learning, 2017.
+
+Alec Radford, Luke Metz, and Soumith Chintala. Unsupervised representation learning with deep convolutional generative adversarial networks. CoRR, abs/1511.06434, 2015.
+
+Alexander J Ratner, Henry R Ehrenberg, Zeshan Hussain, Jared Dunnmon, and Christopher Re.´ Learning to compose domain-specific transformations for data augmentation. stat, 2017.
+
+Danilo Jimenez Rezende, Shakir Mohamed, and Daan Wierstra. Stochastic backpropagation and approximate inference in deep generative models. In Proceedings of the 31th International Conference on Machine Learning,, 2014.
+
+Tim Salimans, Ian J. Goodfellow, Wojciech Zaremba, Vicki Cheung, Alec Radford, and Xi Chen. Improved techniques for training gans. CoRR, abs/1606.03498, 2016.
+
+Leon Sixt, Benjamin Wild, and Tim Landgraf. Rendergan: Generating realistic labeled data. CoRR, abs/1611.01331, 2016.
+
+Kihyuk Sohn, Honglak Lee, and Xinchen Yan. Learning structured output representation using deep conditional generative models. In Advances in Neural Information Processing Systems 28, 2015.
+
+Z Wang, E P Simoncelli, and A C Bovik. Multiscale structural similarity for image quality assessment. In Proc 37th Asilomar Conf on Signals, Systems and Computers, 2003.
+
+T. White. Sampling Generative Networks. ArXiv e-prints, September 2016.
+
+Han Xiao, Kashif Rasul, and Roland Vollgraf. Fashion-mnist: a novel image dataset for benchmarking machine learning algorithms, 2017.
+
+# A ADDITIONAL RESULTS
+
+![](images/172e954e19a082588a6212e6c53874988a74b56efd8e72f184ab9efe38852e41.jpg)  
+(c) VAE samples when trained with (d) VAE samples when trained with 500 images 1000 images   
+Figure 3: This figure presents samples from VAEs when trained with different amounts of data. It represents the overfitting capacity of a VAE. All four samples set look good, but for example, the top left trained with only 50 different data often produce similar images (as the samples on top right trained with 100 images). When the number of training images increases the variability seems good afterwards but as we can see in Figure $_ \mathrm { 4 c }$ when $\tau = 1$ the generator generalizes better the distribution when n is $< 1 0 0 0$ than when $> 1 0 0 0$ is high.
+
+![](images/439eeed9799f3f92e624bd200c6a10a30fe30c4e5df56c3bea8add5da29dd861.jpg)  
+Figure 4: Relative accuracy improvement between the baseline trained on original data and the accuracy with generated or noise data augmentation in training. $\tau$ is the ratio between the number of generated data and the total number of data used for training.
+
+![](images/d3f24e2f0234780496f4199cd1780a10954dc5d378e04b22941e08e7fda1d5ed.jpg)  
+Figure 5: Test accuracy on each model for different values of $\tau$ and different amounts of training data.
+
+![](images/38674be7b9468c15f1c71ac365ce7448bc87d14d737a59ea6eb899d7fdd62779.jpg)  
+Figure 6: Test accuracy when $\tau$ increases with different amounts of training data on MNIST.

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+        "text": "EVALUATION OF GENERATIVE NETWORKS THROUGH THEIR DATA AUGMENTATION CAPACITY ",
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+        "text": "Anonymous authors Paper under double-blind review ",
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+        "text": "ABSTRACT ",
+        "text_level": 1,
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+        ],
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+        "text": "Generative networks are known to be difficult to assess. Recent works on generative models, especially on generative adversarial networks, produce nice samples of varied categories of images. But the validation of their quality is highly dependent on the method used. A good generator should generate data which contain meaningful and varied information and that fit the distribution of a dataset. This paper presents a new method to assess a generator. Our approach is based on training a classifier with a mixture of real and generated samples. We train a generative model over a labeled training set, then we use this generative model to sample new data points that we mix with the original training data. This mixture of real and generated data is thus used to train a classifier which is afterwards tested on a given labeled test dataset. We compare this result with the score of the same classifier trained on the real training data mixed with noise. By computing the classifier’s accuracy with different ratios of samples from both distributions (real and generated) we are able to estimate if the generator successfully fits and is able to generalize the distribution of the dataset. Our experiments compare different generators from the VAE and GAN framework on MNIST and fashion MNIST dataset. ",
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+    {
+        "type": "text",
+        "text": "1 INTRODUCTION AND MOTIVATION ",
+        "text_level": 1,
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+        "text": "Generative network approaches have been widely used to generate samples in recent years. Methods such as GAN (Goodfellow et al., 2014), WGAN (Arjovsky et al., 2017), CGAN (Mirza & Osindero, 2014), CVAE (Sohn et al., 2015) and VAE (Kingma & Welling, 2014) have produced nice samples on various image datasets such as MNIST, bedrooms (Radford et al., 2015) or imageNet (Nguyen et al., 2017). ",
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+        "text": "One commonly accepted tool to evaluate a generative model trained on images is visual assessment to validate the realistic character of samples. One case of this method is called ’visual Turing tests’, in which samples are visualized by humans who try to guess if the images are generated or not. It has been used to assess generative models of images from ImageNet (Denton et al., 2015) and also on digit images (Lake et al., 2015). Salimans et al. (2016) proposes to automate this method with the inception score, which replaces the human judgment by the use of a pretrained classifier to assess the variability of the samples with an entropy measure of the predicted classes and the confidence in the prediction. Unfortunately, those two methods do not indicate if the generator collapses to a particular mode of the data distribution. Log-likelihood based evaluation metrics were widely used to evaluate generative models but as shown in Lucas Theis & Bethge (2016), those evaluations can be misleading in high dimensional cases. ",
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+        "type": "text",
+        "text": "The solution we propose to estimate both sample quality and global fit of the data distribution is to incorporate generated data into the training phase of a classifier before evaluating it. Using generated samples for training has several advantages over using only the original dataset. First, it can make training more efficient when the amount of data is low. As shown in $\\mathrm { N g } \\ \\&$ Jordan (2002), where the conditional distribution $P ( { Y \\vert } X )$ (X represents the samples and Y the classes) learned by a generative model is compared to the same conditional distribution learned by a discriminative model, the generative model performs better in learning this conditional distribution by regularizing the model when the amount of data is low. Secondly, once the generative model is trained, it can sample as much images as needed and can produce interpolations between two images which will induce less risk of overfitting on the data. Other works use generative models for data augmentation (Ratner et al., 2017) or to produce labeled data (Sixt et al., 2016) in order to improve the training of discriminative models, but their intention is not to use it to evaluate or compare generative neural networks. ",
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+        "text": "",
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+        "type": "text",
+        "text": "Our method evaluates the quality of a generative model by assessing its capacity to fit the real distribution of the data. For this purpose, we use the samples generated by a given trained generative model. Our work aims to show how this data augmentation can benefit the training of a classifier and how we can use this benefit as an evaluation tool in order to assess a generative model. This method evaluates whether the information of the original distribution is still present in the generated data and whether the generator is able to produce new samples that are eventually close to unseen data. We compare classifiers trained over mixtures of generated and real data with varying ratios and with varying total amounts of data. This allows us to compare generative models in various data settings (i.e., when there is few or many data points). ",
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+        "type": "text",
+        "text": "The next section will present the related work on generative models, the exploitation of the generated samples and their evaluation. We then present our generative model evaluation framework before presenting experimental results on several generative models with different datasets. ",
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+        "type": "text",
+        "text": "2 RELATED WORK ",
+        "text_level": 1,
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+        "text": "2.1 GENERATIVE MODELS ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "The variational auto-encoder (VAE) framework (Kingma & Welling, 2014), (Rezende et al., 2014) is a particular kind of auto-encoder which has control over its latent space, in which each variable is a sample from a prior distribution, often chosen as an univariate normal distribution $\\mathbf { N } ( 0 , I )$ (where $I$ is the identity matrix). The VAE learns to map this low dimensional latent space to the observation space. This characteristic makes the VAE an interesting option for generating new data after training. The particularity of the latent space comes from the minimization of the KL divergence between the distribution of the latent space and the prior $\\mathbf { N } ( 0 , I )$ . For the sake of simplicity, in this paper we will speak about the decoder of the VAE as a generator. ",
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+        "type": "text",
+        "text": "Generative adversarial networks (Goodfellow et al., 2014) are a framework of models that learn by a game between two networks: a generator that learns to produce images from a distribution $P$ and a discriminator which learns to discriminate between generated and true images. The generator wants to fool the discriminator and the discriminator wants to beat the generator. This class of generative models can produce visually realistic samples from diverse datasets but they suffer from instabilities in their training. Some recent approaches such as Wasserstein GAN (WGAN) (Arjovsky et al., 2017) try to address those issues by enforcing a Lipschitz constraint on the discriminator. ",
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+    {
+        "type": "text",
+        "text": "Conditional neural networks (Sohn et al., 2015) and in particular Conditional Variational Autoencoders (CVAE) or Conditional Generative adversarial networks (CGAN) (Mirza & Osindero, 2014) are a class of generative models that have control over the sample’s class. By imposing a label during training, a conditional generative network can generate from any class and thus produces labeled data. The conditional approach has been used to improve the quality of generative networks and make them more discriminative (Odena et al., 2017). They are particularly adapted for our setup because we need to generate labeled data to train our classifiers. ",
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+        "type": "text",
+        "text": "2.2 EXPLOITATION AND EVALUATION OF GENERATED SAMPLES ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "In Ratner et al. (2017), a generator is used to perform data augmentation. Instead of designing a composition of fine tuned transformations for this objective, the authors use adversarial training to learn a sequence of incremental operations (for example rotating or swapping words in a sentence). Their approach uses a GAN to be able to generalize in terms of better data-augmentation and to increase their performance on different datasets such as Cifar10 and the ACE relation extraction task. Sixt et al. (2016) also learns a sequence of transformations with generative networks from the GAN family, but they use a 3D model as input and create an augmented view of it. Our approach is similar by using generative networks for data-augmentation but we do not attempt to learn transformations. ",
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+    {
+        "type": "text",
+        "text": "Instead, we use the generated data to assess if the generative model has been able to generalize over the distribution of the data. ",
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+    {
+        "type": "text",
+        "text": "The evaluation of generative networks is discussed in Lucas Theis & Bethge (2016). The authors show that different metrics (as Parzen windows, Nearest Neighbor or Log likelihood) applied to generative models can lead to different results. Good results in one application of a generative model can not be used as evidence of good performance in another application. Their conclusion is that evaluation based on sample visual quality is a bad indicator for the entropy of samples. Conversely, the log-likelihood can be used to produce samples with high entropy but does not assure good visual quality. The method we propose can both estimate the quality and the entropy of samples as we will show in Section 3. ",
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+    {
+        "type": "text",
+        "text": "The quality of the internal representation of a generator can also be estimated with a discriminator. In Radford et al. (2015) they use the discriminator of a ACGAN as feature extractor for evaluating the quality of unsupervised representation learning algorithms. They apply the feature extractor on supervised datasets and evaluate the performance of linear models fitted on top of these features. They experiment a good accuracy on Cifar10 thanks to this method. This approach gives insight on how the discriminator estimates if an image is true or false. If the discriminator has good enough feature extractors for classification, it means that the generator samples are hard to be discriminated from samples from the true distribution. It assess indirectly the quality of the generator. This method is however applicable only if a deep convolutional neural networks is used as discriminator and can not be applied, e.g., on variational auto-encoders. The principal difference between a discriminator and our classifier is that it is not involved in the training process. In our approach, the generator is completely independent from the classifier and therefore there is no bias from the classifier in the generator. ",
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+    {
+        "type": "text",
+        "text": "Parzen windows estimate is a method to estimate the unknown probability density function $f$ of a probability distribution $P$ . This method uses a mixture of simpler probability density functions, called kernels, as approximates for $f$ . In general, a popular kernel used is an isotropic Gaussian centered on a given data point with a small variance (the variance is an hyper parameter here). The idea, like other methods based on Kernel Density Estimation, is to have a small window on each data point such that we apply some smoothing over the function we try to approximate. However, even if the number of samples is high, Parzen windows estimator can be still very far from the true likelihood as shown in Lucas Theis & Bethge (2016), and thus cannot be a good approach to evaluate if the data distribution learned by a model is close to the original one. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Multi-scale structural similarity (MS-SIM, Wang et al. (2003)) is a measurement that gives a way to incorporate image details at different resolutions in order to compare two images. This similarity is generally used in the context of image compression to compare image before and after compression. In Odena et al. (2017) the authors use this similarity to estimate the variability inside a class. They randomly sample two images of a certain class and measure the MS-SIM. If the result is high, then images are considered different. By operating this process several times, the similarity should give an insight on the entropy of $P ( X | Y )$ (X a data point and $\\mathrm { Y }$ its class): if the MS-SIM gives high result, the entropy is high; otherwise, the entropy is low. However, it can not estimate if the sample comes from one or several modes of the distribution $P ( X | Y )$ . For example, if we want to generate images of cats, the MS-SIM similarity can not differentiate a generator that produces different kinds of black cats from a network that produces different cats of different colors. In our method, if the generator is able to generate in only one mode of $P ( X | Y )$ , the score will be low in the testing phase. ",
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+    {
+        "type": "text",
+        "text": "Another approach that aims to evaluate a generative model by using a conditional distribution learned by a classifier is the inception score (Salimans et al., 2016; Odena et al., 2017). The authors use a pretrained inception classifier model to get the conditional label distribution $P ( { Y \\vert } X )$ over the generated samples. They proposed the following score in order to evaluate a generative model: ",
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+    {
+        "type": "equation",
+        "img_path": "images/275399b6e033c09069f8125a4d29f9e2d36b4af9d31f0525f88a9e2a7a66ddd8.jpg",
+        "text": "$$\n\\exp ( \\mathbb { E } _ { X } K L ( P _ { d a t a } ( Y | X ) \\parallel P ( Y ) ) ,\n$$",
+        "text_format": "latex",
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+        "page_idx": 2
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+    {
+        "type": "text",
+        "text": "When the score is high, the generator produces samples on varied classes (Cross entropy of $P ( { Y } | { X } ) , P ( { Y } )$ is high) and the samples look like real images from the original dataset (entropy of $P ( { Y \\vert } X )$ is low ). Inception score can be seen as a measurement of the variability of the generated data while penalizing the uncertainty of $P ( { Y \\vert } X )$ . Unfortunately, it does not estimate if the samples are varied inside a certain class (the entropy of $p ( X | Y ) )$ . Our approach imposes a high entropy of $P ( Y )$ and gives an unbiased indicator about the entropy of both $P ( { Y \\vert } X )$ and $P ( X | Y )$ . ",
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+    {
+        "type": "text",
+        "text": "3 METHODS ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "We evaluate generators in a supervised training setup. We have a dataset $D$ composed of pair of examples $( x , y )$ where $x$ is a data point and $y$ the label associated to this data point. The dataset is split in three parts $D _ { t r a i n } , D _ { v a l i d }$ and $D _ { t e s t }$ . Our method needs a generative model that can sample conditionally from any given label $y$ . This conditional generative model is thus trained on $D _ { t r a i n }$ . Once the training of this model is done, we sample random labels and use the generative model to get a new dataset $D _ { g e n }$ . Then, we mix the true training set $D _ { t r a i n }$ with the new generated data $D _ { g e n }$ . The ratio of generated data into the whole mixture $D _ { m i x }$ is called $\\tau$ . $\\tau$ is used as the probability to sample a generated batch rather than a batch of true data during training. $D _ { m i x }$ is used to train a classifier on the classes of the dataset. This classifier is evaluated at each epoch over the portion $D _ { v a l i d }$ of the dataset. Once, we get the best model over $D _ { v a l i d }$ , we compute the score of this classifier over the test set $D _ { t e s t }$ . We compare the results from training on $D _ { m i x }$ with a baseline. The baseline is the score of the same classifier model trained only on training data $D _ { t r a i n }$ . Our conditional generative model evaluation can be seen as a ratio between those two scores. We can summarize our method as follows: ",
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+    {
+        "type": "text",
+        "text": "1. Train a conditional generative model over $n$ sample of $D _ { t r a i n }$   \n2. Mix the samples $D _ { g e n }$ generated by this model with $D _ { t r a i n }$ under a probability $\\tau$ into $D _ { m i x }$   \n3. Train a discriminative model over $D _ { m i x }$   \n4. Train a discriminative model over $D _ { t r a i n }$ using the $n$ same sample as for the generator training   \n5. Select a classifiers over a valid set $D _ { v a l i d }$ .   \n6. Compare the score of those classifiers over a test set $D _ { t e s t }$ . ",
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+    {
+        "type": "text",
+        "text": "By iterating this method on diverse values of $\\tau$ and $n$ we can evaluate the quality of a generator given a dataset. ",
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+    {
+        "type": "text",
+        "text": "4 EXPERIMENTS ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "4.1 EXPERIMENTAL PROTOCOL ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "Often, generative models are presented on popular datasets like MNIST. Fashion-MNIST (Xiao et al., 2017) can also serve as a direct drop-in replacement for the original MNIST dataset. This dataset is however more complex than MNIST as images have a higher variability. Thus, we use these datasets in order to evaluate different generative models. ",
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+        "type": "text",
+        "text": "We used two different methods in order to get conditional generative models. The first uses traditional generative neural network which can not produce labeled data. In order to associate each generated sample to a label, we train one generator for each specific class $y$ on $D _ { t r a i n }$ . This makes us able to label any generated sample. Once the training of those generators is done, we mix the samples obtained by each generator in order to produce $D _ { g e n }$ . For the experiments, we compare two generative models is this setting: a standard VAE and a WGAN. The second method uses conditional generative models which can generate samples in all classes while controlling the class of a particular sample. Conditional models can thus generate various labeled samples and produce a whole dataset $D _ { g e n }$ directly. In this last case, we ran our experiments on CVAE and CGAN. Once the dataset $D _ { g e n }$ is generated, we mixed it with the real dataset $D _ { t r a i n }$ . As we can generate as much data as we want, we experimented different ratios between real datasets and generated datasets. We call $\\tau$ the probability of sampling from $D _ { g e n }$ . We made experiments with different values for $\\tau = [ 0 . 0 0 0 , 0 . \\bar { 1 } 2 5 , 0 . 2 5 \\bar { 0 } , 0 . 3 7 5 , \\bar { 0 } . 5 0 \\bar { 0 } , 0 . 6 2 5 , \\bar { 0 } . 7 5 0 , 0 . 8 7 5 , 1 . 0 0 \\bar { 0 } ] .$ $\\tau = 0$ implies that we use only data from $D _ { t r a i n }$ . In this specific setting, we compare the effectiveness of the data augmentation with generated samples versus classic data augmentation as isotropic Gaussian noise with an optimized variance or a random pixel dropout with a probability $\\alpha$ of putting a pixel to 0. We also train a classifier without any data augmentation as baseline. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "We use a standard CNN with a softmax output as classifier to predict the labels on this mixture of samples. On each epoch we evaluate this classifier over a validation set $D _ { v a l i d }$ . Then, we choose the classifier that performs best on this validation set. We use early stopping to stop the training if the accuracy does not improve anymore on the validation set after 50 epochs. The classifier is then tested on $D _ { t e s t }$ . We train a classifier for each value of $\\tau$ . We assess the quality of the generative model by comparing the test score of this classifier when $\\tau = 0$ versus the best test score of the classifier with $\\tau > 0$ . The result gives an indication on how the learned distribution from $D _ { t r a i n }$ fits and generalizes the distribution from $D _ { t r a i n }$ . In order to be able to compare results from different generators on a given dataset, we always use the same classifier architecture. ",
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+    {
+        "type": "image",
+        "img_path": "images/203f196e0eaa3b481878d1d1ce48cf70072d1cf46ee7f54b255ca20b975ad17c.jpg",
+        "image_caption": [
+            "Figure 1: Representation of test accuracy when $\\tau$ increases on fashion-MNIST (MNIST in appendix) with different amounts of training data. "
+        ],
+        "image_footnote": [],
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+        "type": "text",
+        "text": "",
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+        "type": "text",
+        "text": "To be able to estimate the impact of learning from generative settings versus discriminative ones, we have made variable the amount of data in $D _ { t r a i n }$ used to train both generator and classifier. Thus, we repeat all our experiments for the following amount of data samples: [100, 500, 1000, 5000, 10000, 50000]. This allows us to measure the regularization capacity of the generated samples over the classifier’s training. We interpret this regularization capacity of those samples as a capacity of generalization. ",
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+    {
+        "type": "text",
+        "text": "4.2 RESULTS ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "In Figure 1, we present the test accuracy when $\\tau$ increase. When $\\tau = 0$ there is no generated data, this is the result of the baseline without data augmentation. Our interpretation of the figure is that if the accuracy is better than baseline with a low $\\tau$ $( \\mathit { \\Theta } < 0 . 5 )$ it means that the generator is able to generalize by learning meaningful informations about the dataset. When $\\tau > 0 . 5$ if the accuracy is maintained it means the generated data can replace the dataset in most parts of the distribution. When $\\tau = 1$ there is no more original data, the classifier is thus trained only on generated samples. If the accuracy is still better than the baseline, it means that the generator has fit the training distribution (and eventually has learned to generalize if this score is high over the test set). ",
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+    {
+        "type": "image",
+        "img_path": "images/49ec4ba23b51ce2ef6dc8b2b43e516f6e5fb55b4f8d50566adaf900e38702c04.jpg",
+        "image_caption": [
+            "(a) Relative accuracy wrt. baseline on mnist for (b) Relative accuracy wrt. baseline on fashiondifferent models mnist for different models ",
+            "Figure 2: Representation of the data augmentation capacity of each generative models. For each number of training example we show the maximum accuracy a generative model can achieve by tuning $\\tau$ . We also show results when tuning hyper-parameter for data augmentation method. "
+        ],
+        "image_footnote": [],
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+    {
+        "type": "text",
+        "text": "Following this interpretation, Figure 1 allows us to compare different generative neural networks on fashion-MNIST. For example we can see that VAE (Figure 1b) and CVAE (Figure 1a) are able to maintain the baseline accuracy when we do data augmentation with generated samples for different amount of data $n$ . But the accuracy goes down when there are only generated data, which means that these generative models in those settings did not fit the whole distribution. CGAN has the same kind of behavior but the degradation is worse when $\\tau = 1$ . However WGAN is able improve the accuracy for $\\tau < 0 . 5$ and to maintain it even when $\\tau = 1$ . Figure 1 also allows to estimate the entropy of $P ( X )$ . As a deep neural network can not be trained efficiently without various data samples, if the the accuracy on test set is good with only generated data $\\mathit { \\Pi } _ { \\mathcal { T } } = 0 \\mathit { \\Pi } _ { \\mathcal { r } }$ ), necessary the entropy of $P ( X )$ is high as the entropy of $P ( X | Y )$ . This condition is sufficient but not necessary to assess the quality of a generator on a given task. ",
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+    {
+        "type": "text",
+        "text": "Our result shows that training one WGAN per class is the best solution to fit the complete distribution, whatever the number of training data used. The bad results on CGAN can be explained by the difficulties to train this model and it’s instability. ",
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+    {
+        "type": "text",
+        "text": "Figure 2 represents the best accuracy that each model can achieve when $\\tau$ is tuned. It shows that, aside from CGAN on MNIST, all generator can be used to increase accuracy, whatever the number of training data. In this context $\\tau$ can be seen as a tunable hyper parameter for data augmentation. Figure 2 also shows that the capacity of generalization is particularly effective when the number of example is low. This can be explained by the fact that when the data number increases the need of data augmentation decrease. ",
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+    {
+        "type": "equation",
+        "img_path": "images/6350fe225b8f4719390b5ce174958ef32191ed057caa2c4f6b298afa81b12042.jpg",
+        "text": "$$\n\\Psi _ { G } = \\frac { 1 } { n } \\sum _ { n } m a x _ { \\tau } [ ( a c c _ { n } ( G , \\tau ) ) ] - a c c _ { n } ( \\tau = 0 ) ,\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "The results shown in Table 1 summarize Figure 2 for each generator $G$ by a numerical value $\\Psi _ { G }$ (Eq. 2). We call $\\Psi _ { G }$ the data augmentation capacity of a generator. It is computed, for a given generator $G$ , by the mean of the differences between the accuracy on mixture with $\\tau$ tuned and the baseline accuracy for each number $n$ of training data. The different number of data is arbitrary chosen. In our experiment we compute it with $[ 0 . 2 \\%$ , $1 \\%$ , $2 \\%$ , $1 0 \\%$ , $2 0 \\%$ , $1 0 0 \\%$ , ] of the train set of the dataset. The important thing is to operate at different scales to estimate how the generative model is able to generalize. ",
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+    {
+        "type": "text",
+        "text": "The results of Table 1 indicates if a generative model is globally able to perform data augmentation for a given data set. A positive result indicates that the generative model is able to generalize well on the dataset at different sizes. In case of low amount of data, it is better to refer to Figure 2 to choose the best model for that specific case. ",
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+        "type": "table",
+        "img_path": "images/d8bb7ffb109cfa1338683f47a613b66dd62ca67f682a7545da599c55c5c540fb.jpg",
+        "table_caption": [
+            "Table 1: Results Table : $\\Psi _ { G }$ values "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=1 colspan=1>Datasets</td><td rowspan=1 colspan=1>Gaussian</td><td rowspan=1 colspan=1>Random Dropout</td><td rowspan=1 colspan=1>VAE</td><td rowspan=1 colspan=1>CVAE</td><td rowspan=1 colspan=1>WGAN</td><td rowspan=1 colspan=1>CGAN</td></tr><tr><td rowspan=1 colspan=1>MNIST</td><td rowspan=1 colspan=1>0.0024</td><td rowspan=1 colspan=1>0.0027</td><td rowspan=1 colspan=1>0.0123</td><td rowspan=1 colspan=1>0.0116</td><td rowspan=1 colspan=1>0.0474</td><td rowspan=1 colspan=1>0.0027</td></tr><tr><td rowspan=1 colspan=1>Fashion MNIST</td><td rowspan=1 colspan=1>0.0014</td><td rowspan=1 colspan=1>0.0018</td><td rowspan=1 colspan=1>0.0047</td><td rowspan=1 colspan=1>0.0041</td><td rowspan=1 colspan=1>0.0357</td><td rowspan=1 colspan=1>-0.0021</td></tr></table>",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "5 DISCUSSION ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            195,
+            312,
+            212
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "We presented a method to estimate how well a generative model has learned to generalize in a conditional setting. Using generated samples as data augmentation in order to improve a discriminative model is a well known technique. However, assessing the quality of a generative model with a discriminative model seems to have been less explored. As we have shown, this evaluation is meaningful to measure how well the generative model can sample data points that are probable under the true distribution we want to approximate. In this paper we applied this method on images samples. It means that we can correlate our measure with a visually assess of the samples quality as the generator outputs are in pixel space. Our assessment method can also be used in other spaces as long as labeled data are available. ",
+        "bbox": [
+            173,
+            227,
+            825,
+            353
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "The relative benefits of discriminative and generative models have been studied in $\\mathrm { N g }$ & Jordan (2002). They found that for a small number of training examples $n$ , a generative model will be less prone to overfitting. A discriminative model on a small number of examples risks to learn some spurious nodes that will penalize the generalization capacity. Our results are coherent with their conclusion as the data augmentation induced by generated data is particularly effective when $n$ is low. ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Our evaluation was performed on several datasets, generative models, different ratios and amounts of training data. With the current results, WGAN seems to be the most efficient solution. However, this result should be confirmed by experiments on other datasets, generative models and with different types of discriminative models to get a more general comparison. This will be explored in further experiments. ",
+        "bbox": [
+            174,
+            450,
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+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "As presented in White (2016), the sampling method of a generator can be adapted, which can have an impact on the data produced. A way to boost the performance of a generator can be to focus on improving the sampling phase instead of the model design or the training. An extension of this work could be to look into the impact of several sampling techniques on the performance of a generative model. ",
+        "bbox": [
+            174,
+            526,
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+            597
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "6 CONCLUSION ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            617,
+            318,
+            633
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "This paper introduces a new method to assess and compare the performances of generative models on various labeled datasets. By training a classifier on several mixture of generated and real data we can estimate the ability of a generative model to generalize. When addition of generated data into the training set achieved better data augmentation than traditional data augmentation as Gaussian noise or random dropout, it demonstrates the ability of generative models to create meaningful samples. By varying the number of training data, we compute a data augmentation capacity $\\Psi _ { G }$ for each model on MNIST and fashion-MNIST datasets. $\\Psi _ { G }$ is a global estimation of the generalization capacity of a generative model on a given dataset. The results presented here are produced on image datasets but this method can be used on all kinds of datasets or generative models as long as labeled data is available. ",
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
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+    {
+        "type": "text",
+        "text": "A ADDITIONAL RESULTS ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "image",
+        "img_path": "images/172e954e19a082588a6212e6c53874988a74b56efd8e72f184ab9efe38852e41.jpg",
+        "image_caption": [
+            "(c) VAE samples when trained with (d) VAE samples when trained with 500 images 1000 images ",
+            "Figure 3: This figure presents samples from VAEs when trained with different amounts of data. It represents the overfitting capacity of a VAE. All four samples set look good, but for example, the top left trained with only 50 different data often produce similar images (as the samples on top right trained with 100 images). When the number of training images increases the variability seems good afterwards but as we can see in Figure $_ \\mathrm { 4 c }$ when $\\tau = 1$ the generator generalizes better the distribution when n is $< 1 0 0 0$ than when $> 1 0 0 0$ is high. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            267,
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+        "page_idx": 8
+    },
+    {
+        "type": "image",
+        "img_path": "images/439eeed9799f3f92e624bd200c6a10a30fe30c4e5df56c3bea8add5da29dd861.jpg",
+        "image_caption": [
+            "Figure 4: Relative accuracy improvement between the baseline trained on original data and the accuracy with generated or noise data augmentation in training. $\\tau$ is the ratio between the number of generated data and the total number of data used for training. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            238,
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+        "page_idx": 9
+    },
+    {
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+        "img_path": "images/d3f24e2f0234780496f4199cd1780a10954dc5d378e04b22941e08e7fda1d5ed.jpg",
+        "image_caption": [
+            "Figure 5: Test accuracy on each model for different values of $\\tau$ and different amounts of training data. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            243,
+            151,
+            740,
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+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "image",
+        "img_path": "images/38674be7b9468c15f1c71ac365ce7448bc87d14d737a59ea6eb899d7fdd62779.jpg",
+        "image_caption": [
+            "Figure 6: Test accuracy when $\\tau$ increases with different amounts of training data on MNIST. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            243,
+            337,
+            741,
+            667
+        ],
+        "page_idx": 11
+    }
+]

parse/train/S1CChZ-CZ/S1CChZ-CZ.md

@@ -0,0 +1,292 @@
+# ASK THE RIGHT QUESTIONS: ACTIVE QUESTION REFORMULATION WITH REINFORCEMENT LEARNING
+
+Christian Buck, Jannis Bulian, Massimiliano Ciaramita, Wojciech Gajewski,   
+Andrea Gesmundo, Neil Houlsby, Wei Wang   
+Google   
+{cbuck,jbulian,massi,wgaj,agesmundo,neilhoulsby,wangwe}@google.com
+
+# ABSTRACT
+
+We frame Question Answering (QA) as a Reinforcement Learning task, an approach that we call Active Question Answering. We propose an agent that sits between the user and a black box QA system and learns to reformulate questions to elicit the best possible answers. The agent probes the system with, potentially many, natural language reformulations of an initial question and aggregates the returned evidence to yield the best answer. The reformulation system is trained end-to-end to maximize answer quality using policy gradient. We evaluate on SearchQA, a dataset of complex questions extracted from Jeopardy!. The agent outperforms a stateof-the-art base model, playing the role of the environment, and other benchmarks. We also analyze the language that the agent has learned while interacting with the question answering system. We find that successful question reformulations look quite different from natural language paraphrases. The agent is able to discover non-trivial reformulation strategies that resemble classic information retrieval techniques such as term re-weighting (tf-idf) and stemming.
+
+# 1 INTRODUCTION
+
+Web and social media have become primary sources of information. Users’ expectations and information seeking activities co-evolve with the increasing sophistication of these resources. Beyond navigation, document retrieval, and simple factual question answering, users seek direct answers to complex and compositional questions. Such search sessions may require multiple iterations, critical assessment, and synthesis (Marchionini, 2006).
+
+The productivity of natural language yields a myriad of ways to formulate a question (Chomsky, 1965). In the face of complex information needs, humans overcome uncertainty by reformulating questions, issuing multiple searches, and aggregating responses. Inspired by humans’ ability to ask the right questions, we present an agent that learns to carry out this process for the user. The agent sits between the user and a backend QA system that we refer to as ‘the environment’. We call the agent AQA, as it implements an active question answering strategy. AQA aims to maximize the chance of getting the correct answer by sending a reformulated question to the environment. The agent seeks to find the best answer by asking many questions and aggregating the returned evidence. The internals of the environment are not available to the agent, so it must learn to probe a black-box optimally using only question strings. The key component of the AQA agent is a sequence-to-sequence model trained with reinforcement learning (RL) using a reward based on the answer returned by the environment. The second component to AQA combines the evidence from interacting with the environment using a convolutional neural network to select an answer.
+
+We evaluate on a dataset of Jeopardy! questions, SearchQA (Dunn et al., 2017). These questions are hard to answer by design because they use convoluted language, e.g., Travel doesn’t seem to be an issue for this sorcerer & onetime surgeon; astral projection & teleportation are no prob (answer: Doctor Strange). Thus SearchQA tests the ability of AQA to reformulate questions such that the QA system has the best chance of returning the correct answer. AQA improves over the performance of a deep network built for QA, BiDAF (Seo et al., 2017a), which has produced state-of-the-art results on multiple tasks, by $1 1 . 4 \%$ absolute F1, a $32 \%$ relative F1 improvement. Additionally, AQA outperforms other competitive heuristic query reformulation benchmarks.
+
+AQA defines an instance of machine-machine communication. One side of the conversation, the AQA agent, is trying to adapt its language to improve the response from the other side, the QA environment. To shed some light on this process we perform a qualitative analysis of the language generated by the AQA agent. By evaluating on MSCOCO (Lin et al., 2014), we find that the agent’s question reformulations diverge significantly from natural language paraphrases. Remarkably, though, the agent is able to learn non-trivial and transparent policies. In particular, the agent is able to discover classic IR query operations such as term re-weighting, resembling tf-idf, and morphological simplification/stemming. A possible reason being that current machine comprehension tasks involve the ranking of short textual snippets, thus incentivizing relevance, more than deep language understanding.
+
+# 2 RELATED WORK
+
+Lin & Pantel (2001) learned patterns of question variants by comparing dependency parsing trees. Duboue & Chu-Carroll (2006) showed that MT-based paraphrases can be useful in principle by providing significant headroom in oracle-based estimations of QA performance. Recently, Berant & Liang (2014) used paraphrasing to augment the training of a semantic parser by expanding through the paraphrases as a latent representation. Bilingual corpora and MT have been used to generate paraphrases by pivoting through a second language. Recent work uses neural translation models and multiple pivots (Mallinson et al., 2017). In contrast, our approach does not use pivoting and is, to our knowledge, the first direct neural paraphrasing system. Riezler et al. (2007) propose phrase-based paraphrasing for query expansion. In contrast with this line of work, our goal is to generate full question reformulations while optimizing directly the end-to-end target performance metrics.
+
+Reinforcement learning is gaining traction in natural language understanding across many problems. For example, Narasimhan et al. (2015) use RL to learn control policies for multi-user dungeon games where the state of the game is summarized by a textual description, and Li et al. (2016) use RL for dialogue generation. Policy gradient methods have been investigated recently for MT and other sequence-to-sequence problems. They alleviate limitations inherent to the word-level optimization of the cross-entropy loss, allowing the use of sequence-level reward functions, like BLEU. Reward functions based on language models and reconstruction errors are used to bootstrap MT with fewer resources (Xia et al., 2016). RL training can also prevent exposure bias; an inconsistency between training and inference time stemming from the fact that the model never sees its own mistakes during training (Ranzato et al., 2015). We also use policy gradient to optimize our agent, however, we use end-to-end question answering quality as the reward.
+
+Uses of policy gradient for QA include Liang et al. (2017), who train a semantic parser to query a knowledge base, and Seo et al. (2017b) who propose query reduction networks that transform a query to answer questions that involve multi-hop common sense reasoning. The work of Nogueira & Cho (2016) is most related to ours. They identify a document containing an answer to a question by following links on a graph. Evaluating on a set of questions from the game Jeopardy!, they learn to walk the Wikipedia graph until they reach the predicted answer. In a follow-up, Nogueira & Cho (2017) improve document retrieval with an approach inspired by relevance feedback in combination with RL. They reformulate a query by adding terms from documents retrieved from a search engine for the original query. Our work differs in that we generate complete sequence reformulations rather than adding single terms, and we target question-answering rather than document retrieval.
+
+Active QA is also related to recent research on fact-checking: Wu et al. (2017) propose to perturb database queries in order to estimate the support of quantitative claims. In Active QA questions are perturbed semantically with a similar purpose, although directly at the surface natural language form.
+
+# 3 ACTIVE QUESTION ANSWERING MODEL
+
+Figure 1 shows the Active Question Answering (AQA) agent-environment setup. The AQA model interacts with a black-box environment. AQA queries it with many versions of a question, and finally returns the best of the answers found. An episode starts with an original question $q _ { 0 }$ . The agent then
+
+![](images/42ba155712e96f20018733a7e2f0a66dd2a7495f5ff2c80de1bf6e3f96049b86.jpg)  
+Figure 1: The AQA agent-environment setup. In the downward pass the agent reformulates the question and sends variants to the QA system. In the upward pass the final answer is selected.
+
+generates a set of reformulations $\{ q _ { i } \} _ { i = 1 } ^ { N }$ . These are sent to the environment which returns answers $\{ a _ { i } \} _ { i = 1 } ^ { N }$ . The selection model then picks the best from these candidates.
+
+# 3.1 QUESTION-ANSWERING ENVIRONMENT
+
+For the QA environment, we use a competitive neural question answering model, BiDirectional Attention Flow (BiDAF) (Seo et al., 2017a). BiDAF is an extractive QA system, it selects answers from contiguous spans of a given document. Given a question, the environment returns an answer and, during training, a reward. The reward may be any quality metric for the returned answer, we use token-level F1 score. Note that the reward for each answer $a _ { i }$ is computed against the original question $q _ { 0 }$ . We assume that the environment is opaque; the agent has no access to its parameters, activations or gradients. This setting enables one, in principle, to also interact with other information sources, possibly providing feedback in different modes such as images and structured data from knowledge bases. However, without propagating gradients through the environment we lose information, feedback on the quality of the question reformulations is noisy, presenting a challenge for training.
+
+# 3.2 REFORMULATION MODEL
+
+The reformulator is a sequence-to-sequence model, as is popular for neural machine translation. We build upon the implementation of Britz et al. (2017). The major departure from the standard MT setting is that our model reformulates utterances in the same language. Unlike in MT, there is little high quality training data available for monolingual paraphrasing. Effective training of highly parametrized neural networks relies on an abundance of data. We address this challenge by first pre-training on a related task, multilingual translation, and then using signals produced during the interaction with the environment for adaptation.
+
+# 3.3 ANSWER SELECTION MODEL
+
+During training, we have access to the reward for the answer returned for each reformulation $q _ { i }$ However, at test time we must predict the best answer $a ^ { * }$ . The selection model selects the best answer from the set $\{ a _ { i } \} _ { i = 1 } ^ { N }$ observed during the interaction by predicting the difference of the F1 score to the average F1 of all variants. We use pre-trained embeddings for the tokens of query, rewrite, and answer. For each, we add a 1-dimensional CNN followed by max-pooling. The three resulting vectors are then concatenated and passed through a feed-forward network which produces the output.
+
+# 4 TRAINING
+
+# 4.1 QUESTION ANSWERING ENVIRONMENT
+
+We train a model on the training set for the QA task at hand, see Section 5.4 for details. Afterwards, BiDAF becomes the black-box environment and its parameters are not updated further. In principle,
+
+we could train both the agent and the environment jointly to further improve performance. However, this is not our desired task: our aim is for the agent to learn to communicate using natural language with an environment over which is has no control.
+
+# 4.2 POLICY GRADIENT TRAINING OF THE REFORMULATION MODEL
+
+For a given question $q _ { 0 }$ , we want to return the best possible answer $a ^ { * }$ , maximizing a reward $a ^ { * } = \operatorname { a r g m a x } _ { a } R ( a | q _ { 0 } )$ . Typically, $R$ is the token level F1 score on the answer. The answer $a = f ( q )$ is an unknown function of a question $q$ , computed by the environment. The reward is computed with respect to the original question $q _ { 0 }$ while the answer is provided for $q$ . The question is generated according to a policy $\pi _ { \theta }$ where $\theta$ are the policy’s parameters $q \sim \pi _ { \theta } ( { \bf \nabla } \cdot | q _ { 0 } )$ . The policy, in this case, a sequence-to-sequence model, assigns a probability
+
+$$
+\pi _ { \boldsymbol { \theta } } ( q | q _ { 0 } ) = \prod _ { t = 1 } ^ { T } p ( w _ { t } | w _ { 1 } , \dots , w _ { t - 1 } , q _ { 0 } )
+$$
+
+to any possible question $q = w _ { 1 } , \ldots , w _ { T }$ , where $T$ is the length of $q$ with tokens $w _ { t } \in V$ from a fixed vocabulary $V$ . The goal is to maximize the expected reward of the answer returned under the policy, $\mathbb { E } _ { q \sim \pi _ { \theta } ( \cdot | q _ { 0 } ) } [ R ( f ( q ) ) ]$ . We optimize the reward directly with respect to parameters of the policy using Policy Gradient methods (Sutton $\&$ Barto, 1998). The expected reward cannot be computed in closed form, so we compute an unbiased estimate with Monte Carlo sampling,
+
+$$
+\mathbb { E } _ { q \sim \pi _ { \theta } ( \cdot | q _ { 0 } ) } [ R ( f ( q ) ) ] \approx \frac { 1 } { N } \sum _ { i = 1 } ^ { N } R ( f ( q _ { i } ) ) , \quad q _ { i } \sim \pi _ { \theta } ( \cdot | q _ { 0 } )
+$$
+
+To compute gradients for training we use REINFORCE (Williams & Peng, 1991),
+
+$$
+\begin{array} { r l } {  { \nabla \mathbb { E } _ { q \sim \pi _ { \theta } ( \cdot \vert q _ { 0 } ) } [ R ( f ( q ) ) ] = \mathbb { E } _ { q \sim \pi _ { \theta } ( \cdot \vert q _ { 0 } ) } \nabla _ { \theta } \log ( \pi _ { \theta } ( q \vert q _ { 0 } ) ) R ( f ( q ) ) } } \\ & { \approx \frac { 1 } { N } \sum _ { i = 1 } ^ { N } \nabla _ { \theta } \log ( \pi ( q _ { i } \vert q _ { 0 } ) ) R ( f ( q _ { i } ) ) , \quad q _ { i } \sim \pi _ { \theta } ( \cdot \vert q _ { 0 } ) } \end{array}
+$$
+
+This estimator is often found to have high variance, leading to unstable training (Greensmith et al., 2004). We reduce the variance by subtracting the following baseline reward: $B ( q _ { 0 } ) =$ $\mathbb { E } _ { q \sim \pi _ { \theta } ( \cdot | q _ { 0 } ) } [ R ( f ( q ) ) ]$ . This expectation is also computed by sampling from the policy given $q _ { 0 }$ .
+
+We often observed collapse onto a sub-optimal deterministic policy. To address this we use entropy regularization
+
+$$
+H [ \pi _ { \boldsymbol { \theta } } ( q | q _ { 0 } ) ] = - \sum _ { t = 1 } ^ { T } \sum _ { w _ { t } \in V } p _ { \boldsymbol { \theta } } ( w _ { t } | w _ { < t } , q _ { 0 } ) \log p _ { \boldsymbol { \theta } } ( w _ { t } | w _ { < t } , q _ { 0 } )
+$$
+
+This final objective is:
+
+$$
+\mathbb { E } _ { q \sim \pi _ { \theta } ( \cdot | q _ { 0 } ) } [ R ( f ( q ) ) - B ( q _ { 0 } ) ] + \lambda H [ \pi ( q | q _ { 0 } ) ] ,
+$$
+
+where $\lambda$ is the regularization weight.
+
+# 4.3 ANSWER SELECTION
+
+Unlike the reformulation policy, we train the answer with either beam search or sampling. We can produce many rewrites of a single question from our reformulation system. We issue each rewrite to the QA environment, yielding a set of (query, rewrite, answer) tuples from which we need to pick the best instance. We train another neural network to pick the best answer from the candidates. We frame the task as binary classification, distinguishing between above and below average performance. In training, we compute the F1 score of the answer for every instance. If the rewrite produces an answer with an F1 score greater than the average score of the other rewrites the instance is assigned a positive label. We ignore questions where all rewrites yield equally good/bad answers. We evaluated FFNNs, LSTMs, and CNNs and found that the performance of all systems was comparable. We choose a CNN which offers good computational efficiency and accuracy (cf. 3.3).
+
+# 4.4 PRETRAINING OF THE REFORMULATION MODEL
+
+We pre-train the policy by building a paraphrasing Neural MT model that can translate from English to English. While parallel corpora are available for many language pairs, English-English corpora are scarce. We first produce a multilingual translation system that translates between several languages (Johnson et al., 2016). This allows us to use available bilingual corpora. Multilingual training requires nothing more than adding two special tokens to every line which indicate the source and target languages. The encoder-decoder architecture of the translation model remains unchanged.
+
+As Johnson et al. (2016) show, this model can be used for zero-shot translation, i.e. to translate between language pairs for which it has seen no training examples. For example, after training English-Spanish, English-French, French-English, and Spanish-English the model has learned a single encoder that encodes English, Spanish, and French and a decoder for the same three languages. Thus, we can use the same model for French-Spanish, Spanish-French and also English-English translation by adding the respective tokens to the source. Johnson et al. (2016) note that zero-shot translation usually performs worse than bridging, an approach that uses the model twice: first, to translate into a pivot language, then into the target language. However, the performance gap can be closed by running a few training steps for the desired language pair. Thus, we first train on multilingual data, then on a small corpus of monolingual data.
+
+# 5 EXPERIMENTS
+
+# 5.1 QUESTION ANSWERING DATA AND BIDAF TRAINING
+
+SearchQA (Dunn et al., 2017) is a dataset built starting from a set of Jeopardy! clues. Clues are obfuscated queries such as This ‘Father of Our Country’ didn’t really chop down a cherry tree. Each clue is associated with the correct answer, e.g. George Washington, and a list of snippets from Google’s top search results. SearchQA contains over 140k question/answer pairs and $6 . 9 \mathbf { M }$ snippets. We train our model on the pre-defined training split, perform model selection and tuning on the validation split and report results on the validation and test splits. The training, validation and test sets contain 99,820, 13,393 and 27,248 examples, respectively.
+
+We train BiDAF directly on the SearchQA training data. We join snippets to form the context from which BiDAF selects answer spans. For performance reasons, we limit the context to the top 10 snippets. This corresponds to finding the answer on the first page of Google results. The results are only mildly affected by this limitation, for $10 \%$ of the questions, there is no answer in this shorter context. These data points are all counted as losses. We trained with the Adam optimizer for 4500 steps, using learning rate 0.001, batch size 60.
+
+# 5.2 QUESTION REFORMULATOR TRAINING
+
+For the pre-training of the reformulator, we use the multilingual United Nations Parallel Corpus v1.0 (Ziemski et al., 2016). This dataset contains 11.4M sentences which are fully aligned across six UN languages: Arabic, English, Spanish, French, Russian, and Chinese. From all bilingual pairs, we produce a multilingual training corpus of 30 language pairs. This yields 340M training examples which we use to train the zero-shot neural MT system (Johnson et al., 2016). We tokenize our data using 16k sentence pieces.1 Following Britz et al. (2017) we use a bidirectional LSTM as the encoder and a 4-layer stacked LSTM with attention as the decoder. The model converged after training on 400M instances using the Adam optimizer with a learning rate of 0.001 and batch size of 128.
+
+The model trained as described above has poor quality. For example, for the question What month, day and year did Super Bowl 50 take place?, the top rewrite is What month and year goes back to the morning and year?. To improve quality, we resume training on a smaller monolingual dataset, extracted from the Paralex database of question paraphrases (Fader et al., 2013).2 Unfortunately, this data contains many noisy pairs. We filter many of these pairs out by keeping only those where the Jaccard coefficient between the sets of source and target terms is above 0.5. Further, since the number of paraphrases for each question can vary significantly, we keep at most 4 paraphrases for each question. After processing, we are left with about $1 . 5 \mathbf { M }$ pairs out of the original 35M. The refined model has visibly better quality than the zero-shot one; for the example question above it generates What year did superbowl take place?. We also tried training on the monolingual pairs alone. As in (Johnson et al., 2016), the quality was in between the multilingual and refined models.
+
+After pre-training the reformulator, we switch the optimizer from Adam to SGD and train for $1 0 0 \mathrm { k }$ RL steps of batch size 64 with a low learning rate of 0.001. We use an entropy regularization weight of $\lambda = 0 . 0 0 1$ . For a stopping criterion, we monitor the reward from the best single rewrite, generated via greedy decoding, on the validation set. In contrast to our initial training which we ran on GPUs, this training phase is dominated by the latency of the QA system and we run inference and updates on CPU and the BiDAF environment on GPU.
+
+# 5.3 TRAINING THE ANSWER SELECTOR
+
+For the selection model we use supervised learning: first, we train the reformulator, then we generate $N = 2 0$ rewrites for each question in the SearchQA training and validation sets. After sending these to the environment we have about 2M (question, rewrite, answer) triples. We remove queries where all rewrites yield identical rewards, which removes about half of the training data. We use pre-trained 100-dimensional embeddings (Pennington et al., 2014) for the tokens. Our CNN-based selection model encodes the three strings into 100-dimensional vectors using a 1D CNN with kernel width 3 and output dimension 100 over the embedded tokens, followed by max-pooling. The vectors are then concatenated and passed through a feed-forward network which produces the binary output, indicating whether the triple performs below or above average, relative to the other reformulations and respective answers.
+
+We use the training portion of the SearchQA data thrice, first for the initial training of the BiDAF model, then for the reinforcement-learning based tuning of the reformulator, and finally for the training of the selector. We carefully monitored that this didn’t cause severe overfitting. BiDAF alone has a generalization gap between the training and validation set errors of $3 . 4 \mathrm { F } 1$ . This gap remains virtually identical after training the rewriter. After training the CNN, AQA-Full has a slightly larger gap of $3 . 9 \mathrm { F } 1$ . We conclude that training AQA on BiDAF’s training set causes very little additional overfitting. We use the test set only for evaluation of the final model.
+
+# 5.4 BASELINES AND BENCHMARKS
+
+As a baseline, we report the results of the modified pointer network, called Attention Sum Reader (ASR), developed for SearchQA (Dunn et al., 2017).We also report the performance of the BiDAF environment used without the reformulator to answer the original question.
+
+We evaluate against several benchmarks. First, following Kumaran & Carvalho (2009), we implement a system (MI-SubQuery) that generates reformulation candidates by enumerating all subqueries of the original SearchQA query and then keeps the top $N$ ranked by mutual information.3 From this set, we pick the highest scoring one as the top hypothesis to be used as a single rewrite. We also use the whole set to train a CNN answer selector for this specific source of rewrites. In this way, we can compare systems fairly both in single prediction or ensemble prediction modes. Additionally, we evaluate against another source of reformulations: the zero-shot monolingual NMT system trained on the U.N. corpus and Paralex (Base-NMT), without reinforcement learning. As with the MI-SubQuery benchmark, we evaluate the Base-NMT system both as a single reformulation predictor and as a source of $N$ best rewrites, for which we train a dedicated CNN answer selector. We also report human performance on SearchQA, based on a sample of the test set, from (Dunn et al., 2017).
+
+# 5.5 RESULTS
+
+We evaluate several variants of AQA. For each query $q$ in the evaluation we generate a list of reformulations $q _ { i }$ , for $i = 1 \dots N$ , from the AQA reformulator trained as described in Section 4. We set $N = 2 0$ in these experiments, the same value is used for the benchmarks. In AQA TopHyp we use the top hypothesis generated by the sequence model, $q _ { 1 }$ . In AQA Voting we use BiDAF scores for a heuristic weighted voting scheme to implement deterministic selection. Let $a$ be the answer returned by BiDAF for query $q$ , with an associated score $s ( a )$ . We pick the answer according to argmaxa $\scriptstyle \sum _ { a ^ { \prime } = a } s ( a ^ { \prime } )$ . In AQA MaxConf we select the answer with the single highest BiDAF score across question reformulations. Finally, AQA CNN identifies the complete system with the learned CNN model described in Section 3.
+
+Table 1: Results table for the experiments on SearchQA. Two-sample $t$ -tests between the AQA results and either the Base-NMT or the MI-SubQuery results show that differences in F1 and Exact Match scores are statistically significant, $p < 1 \bar { 0 } ^ { - 4 }$ , for both Top Hypothesis and CNN predictions. The difference between Base-NMT and MI-SubQuery is also significant for Top Hypothesis predictions.   
+
+<table><tr><td></td><td></td><td colspan="2">Baseline</td><td colspan="2">MI-SubQuery</td><td colspan="2">Base-NMT</td><td colspan="4">AQA</td><td></td></tr><tr><td></td><td></td><td></td><td>BAAR</td><td>TopHyp</td><td></td><td>TopHyp</td><td></td><td>TopHyp</td><td>Voting</td><td>MaxConf</td><td></td><td>Human</td></tr><tr><td></td><td>EM</td><td>1</td><td>31.7</td><td>24.1</td><td>37.5</td><td>26.0</td><td>37.5</td><td>32.0</td><td>33.6</td><td>35.5</td><td>40.5</td><td>1</td></tr><tr><td>Dev</td><td>F1</td><td>24.2</td><td>37.9</td><td>29.9</td><td>44.5</td><td>32.2</td><td>44.8</td><td>38.2</td><td>40.5</td><td>42.0</td><td>47.4</td><td>=</td></tr><tr><td></td><td>EM</td><td>1</td><td>28.6</td><td>23.2</td><td>35.8</td><td>24.8</td><td>35.7</td><td>30.6</td><td>33.3</td><td>33.8</td><td>38.7</td><td>43.9</td></tr><tr><td>Test</td><td>F1</td><td>22.8</td><td>34.6</td><td>29.0</td><td>42.8</td><td>31.0</td><td>42.9</td><td>36.8</td><td>39.3</td><td>40.2</td><td>45.6</td><td>1</td></tr></table>
+
+Table 1 shows the results. We report exact match (EM) and F1 metrics, computed on token level between the predicted answer and the gold answer. We present results on the full validation and test sets (referred to as $n$ -gram in (Dunn et al., 2017)). Overall, SearchQA appears to be harder than other recent QA tasks such as SQuAD (Rajpurkar et al., 2016), for both machines and humans. BiDAF’s performance drops by $4 0 \ : \mathrm { F 1 }$ points on SearchQA compared to SQuAD. However, BiDAF is still competitive on SeachQA, improving over the Attention Sum Reader network by 13.7 F1 points.
+
+Using the top hypothesis already yields an improvement of $2 . 2 \mathrm { F } 1$ on the test set. This demonstrates that even the reformulator alone is capable to produce questions more easily answered by the environment. When generating a single prediction, both MI-SubQuery and Base-NMT benchmarks perform worse than BiDAF. Heuristic selection via both Voting and Max Conf yields a further performance boost. Both heuristics draw upon the intuition that when BiDAF is confident in its answer it is more likely to be correct, and that multiple instances of the same answer provide positive evidence (for MaxConf, the max operation implicitly rewards having an answer scored with respect to multiple questions). Finally, a trained selection function improves performance further, yielding an absolute increase of $1 1 . 4 \mathrm { F } 1$ points ( $32 \%$ relative) over BiDAF with the original questions. In terms of exact match score, this more than closes half the gap between BiDAF and human performance. The benchmarks improve considerably when they generate $N$ candidates, and paired with a dedicated CNN selector. This is not surprising as it provides an ensemble prediction setup. However, the AQA CNN system outperforms both MI-SubQuery and Base-NMT in all conditions by about $3 \%$ .
+
+Finally, we consider the maximum performance possible that could be achieved by picking the answer with the highest F1 score from the set of those returned for all available reformulations. Here we find that the different sources of rewrites provide comparable headroom: the oracle Exact Match is near 50, while the oracle F1 is close to 58.
+
+# 6 ANALYSIS OF THE AGENT’S LANGUAGE
+
+The AQA agent can learn several types of sub-optimal policies. For example, it can converge to deterministic policies by learning to emit the same, meaningless, reformulation for any input question. This strategy can lead to local optima because the environment has built in strong priors on what looks like a likely answer, even ignoring the input question. Hence, convergence to non-negligible performance is easy. Entropy regularization typically fixes this behavior. Too much weight on the entropy regularizer, on the other hand, might yield random policies. A more competitive sub-optimal policy is one that generates minimal changes to the input, in order to stay close to the original question. This is a successful strategy because the environment has been trained on the original questions alone, which leads to baseline performance.
+
+![](images/9aab172b023b9ff732b608e7c3ec294d8183d89ca1894d60c382e8e3f664e4ee.jpg)  
+Figure 2: Boxplot summaries of the statistics collected for all types of questions. Two-sample $t \cdot$ -tests performed on all pairs in each box confirm that the differences in means are significant $p \bar { < } 1 0 ^ { - 3 }$ .
+
+It seems quite remarkable then that AQA is able to learn non-trivial reformulation policies, that differ significantly from all of the above. One can think of the policy as a language for formulating questions that the agent has developed while engaging in a machine-machine communication with the environment. In this section we look deeper into the agent’s language.
+
+# 6.1 GENERAL PROPERTIES
+
+We analyze input questions and reformulations on the development partition of SearchQA to gain insights on how the agent’s language evolves during training via policy gradient. It is important to note that in the SearchQA dataset the original Jeopardy! clues have been preprocessed by lowercasing and stop word removal. The resulting preprocessed clues that form the sources (inputs) for the sequence-to-sequence reformulation model resemble more keyword-based search queries than grammatical questions. For example, the clue Gandhi was deeply influenced by this count who wrote "War and Peace" is simplified to gandhi deeply influenced count wrote war peace.
+
+The (preprocessed) SearchQA questions contain 9.6 words on average. They contain few repeated terms, computed as the mean term frequency (TF) per question. The average is 1.03, but for most of the queries $( 7 5 \% )$ TF is 1.0. We also compute the median document frequency (DF) per query, where the document is the context from which the answer is selected, as a measure of how informative a term is.4 As another measure of query performance, we also compute Query Clarity (QC) (Cronen-Townsend et al., 2002).5 Figure 2 summarizes statistics of the questions and rewrites.
+
+We first consider the top hypothesis generated by the pre-trained NMT reformulation system, before reinforcement learning (Base-NMT). The Base-NMT rewrites differ greatly from their sources. They are shorter, 6.3 words on average, and have even fewer repeated terms (1.01). Interestingly, these reformulations are mostly syntactically well-formed questions. For example, the clue above becomes Who influenced count wrote war?.6 Base-NMT improves structural language quality by properly reinserting dropped function words and wh-phrases. We also verified the increased fluency by using a large language model and found that the Base-NMT rewrites are $50 \%$ more likely than the original questions. While more fluent, the Base-NMT rewrites involve lower DF terms. This is probably due to a domain mismatch between SearchQA and the NMT training corpus. The query clarity of the Base-NMT rewrites is also degraded as a result of the transduction process.
+
+We next consider the top hypothesis generated by the AQA question reformulator (AQA-QR) after the policy gradient training. The AQA-QR rewrites are those whose corresponding answers are evaluated as AQA TopHyp in Table 1. These single rewrites alone outperform the original SearchQA queries by $2 \%$ on the test set. We analyze the top hypothesis instead of the final output of the full AQA agent to avoid confounding effects from the answer selection step. These rewrites look different from both the Base-NMT and the SearchQA ones. For the example above AQA-QR’s top hypothesis is What is name gandhi gandhi influence wrote peace peace?. Surprisingly, $9 9 . 8 \%$ start with the prefix What is name. The second most frequent is What country is (81 times), followed by What is is (70) and What state (14). This is puzzling as it occurs in only 9 Base-NMT rewrites, and never in the original SearchQA questions. We speculate it might be related to the fact that virtually all answers involve names, of named entities (Micronesia) or generic concepts (pizza).
+
+AQA-QR’s rewrites seem less fluent than both the SearchQA and the Base-MT counterparts. In terms of language model probability, they are less likely than both SearchQA and Base-NMT. However, they have more repeated terms (1.2 average TF), are significantly longer (11.9) than in Base-NMT and contain more informative context terms than SearchQA questions (lower DF). Also, the translation process does not affect query clarity much. Finally, we find that AQA-QR’s reformulations contain morphological variants in $12 . 5 \%$ of cases. The number of questions that contain multiple tokens with the same stem doubles from SearchQA to AQA-QR. Singular forms are preferred over plurals. Morphological simplification is useful because it increases the chance that a word variant in the question matches the context.
+
+# 6.2 PARAPHRASING QUALITY
+
+We also investigate the general paraphrasing abilities of our model, focusing on the relation between paraphrasing quality and QA quality. To tease apart the relationship between paraphrasing and reformulation for QA we evaluated 3 variants of the reformulator:
+
+Base-NMT This is the model used to initialize RL training of the agent. Trained first on the multilingual U.N. corpus, then on the Paralex corpus, as detailed in Section 5.2.
+
+Base-NMT-NoParalex This is the model above trained solely on the multilingual U.N. corpus, without the Paralex monolingual corpus.
+
+Base-NMT $^ { \cdot } +$ Quora This is the same as Base-NMT, additionally trained on the Quora dataset7 which contains $1 5 0 \mathrm { k }$ duplicate questions.
+
+Following Prakash et al. (2016), we evaluate all models on the MSCOCO8 (Lin et al., 2014) validation set (VAL2014). This dataset consists of images with 5 captions each, of which we select a random one as the source and the other four as references. We use beam search, to compute the top hypothesis and report uncased, moses-tokenized BLEU using multeval9 (Clark et al., 2011). Please note, that the MSCOCO data is only used for evaluation purposes. Examples of all systems can be found in Appendix C.
+
+The Base-NMT model performs at 11.4 BLEU (see Table 1 for the QA eval numbers). In contrast, Base-NMT-NoParalex performs poorly at 5.0 BLEU. Limiting training to the multilingual data alone also degrades QA performance: the scores of the Top Hypothesis are at least 5 points lower in all metrics and CNN scores are 2-3 points lower.
+
+By training on additional monolingual data, the Base-NMT $^ +$ Quora model improves BLEU score slightly to 11.6. End-to-end QA performance also improves marginally, the maximum delta with respect to Base-NMT under all conditions is $+ 0 . 5$ points, but the difference is not statistically significant. Thus, adding the Quora training does not have a significant effect. This might be due to the fact that most of the improvement is captured by training on the larger Paralex data set. Improving raw paraphrasing quality as well as reformulation fluency helps AQA up to a point. However, they are only partially aligned with the main task, which is QA performance. The AQA-QR reformulator has a BLEU score of 8.6, well below both Base-NMT models trained on monolingual data. Yet, AQA-QR significantly outperforms all others in the QA task. Training the agent starting from the Base-NMT $\scriptstyle +$ Quora model yielded comparable results as starting from Base-NMT.
+
+# 6.3 DISCUSSION
+
+Recently, Lewis et al. (2017) trained chatbots that negotiate via language utterances in order to complete a task. They report that the agent’s language diverges from human language if there is no incentive for fluency in the reward function. Our findings seem related. The fact that the questions reformulated by AQA do not resemble natural language is not due to the keyword-like SearchQA input questions, because Base-NMT is capable of producing more fluent questions from the same input. AQA learns to re-weight terms by focusing on informative (lower document frequency), query-specific (high query clarity), terms while increasing term frequency (TF) via duplication. At the same time it learns to modify surface forms in ways akin to stemming and morphological analysis.
+
+Some of the techniques seem to adapt to the specific properties of current deep QA architectures such as character-based modeling and attention. Sometimes AQA learns to generate semantically nonsensical, novel, surface term variants; e.g., it might transform the adjective dense to densey. The only justification for this is that such forms can be still exploited by the character-based BiDAF question encoder. Finally, repetitions can directly increase the chances of alignment in the attention components.
+
+We hypothesize that, while there is no incentive for the model to use human language due to the nature of the task, AQA learns to ask BiDAF questions by optimizing a language that increases the likelihood of BiDAF ranking better the candidate answers. Jia & Liang (2017) argue that reading comprehension systems are not capable of significant language understanding and fail easily in adversarial settings. We speculate that current machine comprehension tasks involve mostly pattern matching and relevance modeling. As a consequence deep QA systems might implement sophisticated ranking systems trained to sort snippets of text from the context. As such, they resemble document retrieval systems which incentivizes the (re-)discovery of IR techniques, such as tf-idf re-weighting and stemming, that have been successful for decades (Baeza-Yates & Ribeiro-Neto, 1999).
+
+# 7 CONCLUSION
+
+We propose a new framework to improve question answering. We call it active question answering (AQA), as it aims to improve answering by systematically perturbing input questions. We investigated a first system of this kind that has three components: a question reformulator, a black box QA system, and a candidate answer aggregator. The reformulator and aggregator form a trainable agent that seeks to elicit the best answers from the QA system. Importantly, the agent may only query the environment with natural language questions. Experimental results prove that the approach is highly effective and that the agent is able to learn non-trivial and somewhat interpretable reformulation policies.
+
+For future work, we will continue developing active question answering, investigating the sequential, iterative aspects of information seeking tasks, framed as end-to-end RL problems, thus, closing the loop between the reformulator and the selector.
+
+# 8 ACKNOWLEDGEMENTS
+
+We would like to thank the anonymous reviewers for their valuable comments and suggestions. We would also like to thank Jyrki Alakuijala, Gábor Bártok, Alexey Gronskiy, Rodrigo Nogueira and Hugo Penedones for insightful discussions and technical feedback.
+
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+Rodrigo Nogueira and Kyunghyun Cho. End-to-end goal-driven web navigation. In Proceedings of NIPS, 2016.
+
+Rodrigo Nogueira and Kyunghyun Cho. Task-oriented query reformulation with reinforcement learning. In Proceedings of EMNLP, 2017.
+
+Jeffrey Pennington, Richard Socher, and Christopher Manning. Glove: Global vectors for word representation. In Proceedings of EMNLP, pp. 1532–1543, 2014.
+
+Aaditya Prakash, Sadid A. Hasan, Kathy Lee, Vivek Datla, Ashequl Qadir, Joey Liu, and Oladimeji Farri. Neural paraphrase generation with stacked residual LSTM networks. In Proceedings of the 26th International Conference on Computational Linguistics (COLING), pp. 2923–2934, 2016.
+
+Pranav Rajpurkar, Jian Zhang, Konstantin Lopyrev, and Percy Liang. SQuAD: $^ { 1 0 0 , 0 0 0 + }$ Questions for Machine Comprehension of Text. In Proceedings of EMNLP, pp. 2383–2392, 2016.
+
+Marc’Aurelio Ranzato, Sumit Chopra, Michael Auli, and Wojciech Zaremba. Sequence level training with recurrent neural networks. https://arxiv.org/abs/1511.06732, 2015.
+
+Stefan Riezler, Alexander Vasserman, Ioannis Tsochantaridis, Vibhu Mittal, and Yi Liu. Statistical machine translation for query expansion in answer retrieval. In Proceedings of ACL, 2007.
+
+Minjoon Seo, Aniruddha Kembhavi, Ali Farhadi, and Hannaneh Hajishirzi. Bidirectional Attention Flow for Machine Comprehension. In Proceedings of ICLR, 2017a.
+
+Minjoon Seo, Sewon Min, Ali Farhadi, and Hannaneh Hajishirzi. Query-reduction networks for question answering. In Proceedings of ICLR 2017, ICLR 2017, 2017b.
+
+Richard S. Sutton and Andrew G. Barto. Introduction to Reinforcement Learning. MIT Press, 1998.
+
+Ronald J. Williams and Jing Peng. Function optimization using connectionist reinforcement learning algorithms. Connection Science, 3(3):241–268, 1991.
+
+You Wu, Pankaj K. Agarwal, Chengkai Li, Jun Yang, and Cong Yu. Computational fact checking through query perturbations. ACM Trans. Database Syst., 42(1):4:1–4:41, 2017.
+
+Yingce Xia, Di He, Tao Qin, Liwei Wang, Nenghai Yu, Tie-Yan Liu, and Wei-Ying Ma. Dual learning for machine translation. In Proceedings of the 30th International Conference on Neural Information Processing Systems, 2016.
+
+Michał Ziemski, Marcin Junczys-Dowmunt, and Bruno Poliquen. The united nations parallel corpus v1.0. In Proceedings of LREC, 2016.
+
+# A REFORMULATION EXAMPLES
+
+Table 2: Results of the qualitative analysis on SearchQA. For the original Jeopardy! questions we give the reference answer, otherwise the answer given by BiDAF.   
+
+<table><tr><td>Model</td><td>Query</td><td>Reference /Answer from BiDAF (F1)</td></tr><tr><td>Jeopardy!</td><td>People of this nation AKA Nippon wrote with a brush,so painting became the preferred form of artistic expression</td><td> japan</td></tr><tr><td>SearchQA</td><td>people nation aka nippon wrote brush ,painting became preferred</td><td>japan (1.0)</td></tr><tr><td></td><td>form artistic expression nippon brush preferred</td><td> julian (0)</td></tr><tr><td>MI Base-NMT</td><td>Aka nippon written form artistic expression?</td><td> julian (0)</td></tr><tr><td>AQA-QR</td><td>What is name did people nation aka nippon wrote brush expres-</td><td>japan (1.0)</td></tr><tr><td>AQA-Full</td><td>sion？ people nation aka nippon wrote brush ,painting became preferred</td><td> japan (1.0)</td></tr><tr><td>Jeopardy!</td><td>form artistic expression Michael Caine &amp; Steve Martin teamed up as Lawrence &amp; Freddy,</td><td>dirty rotten scoundrels</td></tr><tr><td>SearchQA</td><td>a couple of these,the title of a1988 flm michael caine steve martin teamed lawrence freddy,couple,title</td><td>dirty rotten scoundrels (1.0)</td></tr><tr><td>MI</td><td>1988 film caine teamed freddy</td><td>dirty rotten scoundrels (1.0)</td></tr><tr><td>Base-NMT</td><td>Who was lawrence of michael caine steve martin?</td><td>rain man 1988 best picture fikkle [...25 tokens]</td></tr><tr><td>AQA-QR</td><td>What is name is name is name michael caine steve martin teamed</td><td>(0.18) dirty rotten scoundrels (1.0)</td></tr><tr><td>AQA-Full</td><td>lawrence freddy and title 1988 film? What is name is name where name is name michael caine steve martin teamed lawrence freddy and title 1988 film key 2000 ?</td><td>dirty rotten scoundrels (1.0)</td></tr><tr><td>Jeopardy!</td><td>Used underwater,ammonia gelatin is a waterproof type of this ex-</td><td>dynamite</td></tr><tr><td>SearchQA</td><td>plosive used underwater,ammonia gelatin waterproof type explosive</td><td></td></tr><tr><td>MI</td><td>ammonia gelatin waterproof</td><td>nitroglycerin (0) nitroglycerin (0)</td></tr><tr><td>Base-NMT</td><td>Where is ammonia gelatin waterproof? nitroglycerin (0)</td><td></td></tr><tr><td>AQA-QR</td><td>What is name is used under water with ammonia gelatin water wa- terproof type explosive?</td><td>nitroglycerin (0)</td></tr><tr><td>AQA-Full</td><td>used underwater,ammonia gelatin waterproof type explosive</td><td>nitroglycerin (0)</td></tr><tr><td>Jeopardy!</td><td>The Cleveland Peninsula is about 40 miles northwest of Ketchikan in this state</td><td>alaska</td></tr><tr><td>SearchQA</td><td>cleveland peninsula 4O miles northwest ketchikan state</td><td>alaska &#x27;s community information summary says [. 113 tokens] (0.02)</td></tr><tr><td>MI Base-NMT</td><td>cleveland peninsula ketchikan The cleveland peninsula 40 miles?</td><td>alaska &#x27;s dec 16,1997[..132 tokens](0.01) ketchikan alaskalocatednorthwesttip</td></tr><tr><td>AQA-QR</td><td>What is name is cleveland peninsula state northwest state state</td><td>[... 46 tokens] (0.04) alaska (1.0)</td></tr><tr><td>AQA-Full</td><td>state? What is name are cleveland peninsula state northwest state state</td><td>alaska (1.0)</td></tr><tr><td></td><td>state ？</td><td></td></tr><tr><td>Jeopardy! SearchQA</td><td>Tess Ocean,Tinker Bell, Charlotte the Spider tess ocean,tinker bell,charlotte spider</td><td>julia roberts julia roberts tv com charlotte spider[..87 tokens]</td></tr><tr><td>MI</td><td>tess tinker spider</td><td>(0.04) julia roberts tvcom charlotte spider.. 119 tokens]</td></tr><tr><td>Base-NMT</td><td>What ocean tess tinker bell?</td><td>(0.01) julia roberts american actress producer made</td></tr><tr><td>AQA-QR</td><td>What is name tess ocean tinker bellink charlotte spider?</td><td>[.. 206 tokens] (0.02) julia roberts (1.0)</td></tr><tr><td>AQA-Full</td><td>What is name is name tess ocean tinker bell spider contain char- lotte spider contain hump around the world winter au to finish au de mon moist</td><td> julia roberts (1.0)</td></tr><tr><td>Jeopardy!</td><td>During the Tertiary Period,India plowed into Eurasia &amp; this high- est mountain range was formed</td><td>himalayas</td></tr><tr><td>SearchQA</td><td>tertiary period，india plowed eurasia highest mountain range formed</td><td>himalayas (1.0)</td></tr><tr><td>MI</td><td>tertiary plowed eurasia</td><td>himalayas (1.0)</td></tr><tr><td>Base-NMT</td><td>What is eurasia highest mountain range?</td><td>himalayas (1.0)</td></tr><tr><td>AQA-QR</td><td>What is name were tertiary period in india plowed eurasia?</td><td>himalayas (1.0)</td></tr><tr><td>AQA-Full</td><td>tertiary period，india plowed eurasia highest mountain range formed</td><td>himalayas (1.0)</td></tr><tr><td></td><td></td><td></td></tr><tr><td>Jeopardy!</td><td>The melody heard here is from the opera about Serse,better known to us as this &quot;X&quot;-rated Persian king</td><td>xerxes</td></tr><tr><td>SearchQA</td><td>melody heard opera serse,better known us X rated persian king</td><td>gilbert sullivan (0)</td></tr><tr><td>MI</td><td>melody opera persian</td><td>gilbert sullivan (0)</td></tr><tr><td>Base-NMT</td><td>Melody heard opera serse thing?</td><td>gilbert sullivan (0)</td></tr><tr><td>AQA-QR</td><td>What is name melody heard opera serse is better persian king?</td><td>gilbert sullivan (0)</td></tr><tr><td>AQA-Full</td><td>What is name is name melody heard opera serse is better persian king persian K ?</td><td>gilbert sullivan (0)</td></tr></table>
+
+# B EXAMPLES OF RANKING LOSSES
+
+Table 3: Examples of queries where none of the methods produce the right answer, but the Oracle model can.   
+
+<table><tr><td>Model</td><td>Query</td><td>Answer (F1)</td></tr><tr><td>SearchQA</td><td>ancient times ,pentelic quarries major source building material athens</td><td>parthenon (0.0)</td></tr><tr><td>AQA-QR AQA-CNN</td><td>What is name an Ancient History Warry quarry material athens? ancient times ,pentelic quarries major source building material</td><td>parthenon (0.0) parthenon (0.0)</td></tr><tr><td>AQA-Oracle</td><td>athens What is name is name an Ancient Romes For pentelic quarry</td><td>marble (1.0)</td></tr><tr><td>SearchQA</td><td>material athens measure athens? 1949 1999 germany &#x27;s bundestag legislature met city</td><td>berlin (0.0)</td></tr><tr><td>AQA-QR</td><td>What is name is name 1999 germanys bundestag legislature met city city?</td><td>berlin (0.0)</td></tr><tr><td>AQA-CNN</td><td>What is name is name 1999 germany germany bundestag legisla- ture city city?</td><td>berlin (0.0)</td></tr><tr><td>AQA-Oracle</td><td>What is name is name a 1999 germany germany legislature met city city?</td><td>bonn (1.0)</td></tr><tr><td>SearchQA</td><td>utah jazz retired 12 jersey</td><td>karl malone (0.0)</td></tr><tr><td>AQA-QR</td><td>What is name did utah jazz retired jersey jersey?</td><td>karl malone (0.0)</td></tr><tr><td>AQA-CNN</td><td>What is name did utah jazz retired jersey jersey?</td><td>karl malone (0.0)</td></tr><tr><td>AQA-Oracle</td><td>What is name is name where utah jazz is retired jersey jersey?</td><td> john stockton (1.0)</td></tr><tr><td>SearchQA</td><td>swing intersection count basie capital virginia</td><td></td></tr><tr><td>AQA-QR</td><td>What is name for swing intersection count basie capital virginia?</td><td>chicago (0.0)</td></tr><tr><td>AQA-CNN</td><td></td><td>chicago (0.0)</td></tr><tr><td></td><td>swing intersection count basie capital virginia</td><td>chicago (0.0)</td></tr><tr><td>AQA-Oracle SearchQA</td><td>What is name swing swing intersection capital virginia?</td><td>richmond (1.0)</td></tr><tr><td></td><td>huge upset virginia , jim webb beat senator prospective presiden- tial candidate</td><td>hillary clinton (0.0)</td></tr><tr><td>AQA-QR</td><td>What is name is name huge upset virginia jug webb beat senator?</td><td>jim webb (0.0)</td></tr><tr><td>AQA-CNN AQA-Oracle</td><td>What is name is name huge upset virginia jug webb beat senator world candidate?</td><td> jim webb (0.0)</td></tr></table>
+
+# C PARAPHRASING EXAMPLES
+
+Table 4: Paraphrasing examples on captions from MSCOCO (Lin et al., 2014).   
+
+<table><tr><td>Source Reference</td><td>a man riding a wave on top of a surfboard. a man on a surfboard in the crest of a wave.</td></tr><tr><td></td><td></td></tr><tr><td>Base-NMT</td><td>a man on top of a surfboard?</td></tr><tr><td>Base-NMT-NoParalex</td><td>amankind arises away froma floor.</td></tr><tr><td>Base-NMT+Quora</td><td>it possible to riding a wave on a surfboard?</td></tr><tr><td>AQA-QR</td><td>what is name an average man riding a wave at onfboard?</td></tr><tr><td>Source</td><td>a person riding some skis through a snowy mountain trail.</td></tr><tr><td>Reference</td><td>a man on skis is standing by some trees.</td></tr><tr><td>Base-NMT</td><td>what is some skis through a snowy mountain trail?</td></tr><tr><td>Base-NMT-NoParalex</td><td>the person rising somekises through a snow blood.</td></tr><tr><td>Base-NMT+Quora</td><td>what is a skis mountain trail?</td></tr><tr><td>AQA-QR</td><td>what is name an average person riding some one snowy mountain trail?</td></tr><tr><td>Source</td><td> a group of people all taking a bite out of the same sandwich.</td></tr><tr><td>Reference</td><td>a group of people eating rom a large sandwich.</td></tr><tr><td>Base-NMT</td><td>what do people call a bite out of the same sandwich?</td></tr><tr><td>Base-NMT-NoParalex</td><td>a group of people who takes a bargained from the same index.</td></tr><tr><td>Base-NMT+Quora AQA-QR</td><td>what is the group of people who take bite out of the same sandwich?</td></tr><tr><td></td><td>what is name a group of people with bite out of sandwich?</td></tr><tr><td>Source</td><td>a partially lit asian person holding a phone</td></tr><tr><td>Reference</td><td>a man standing next to a sign talking on a phone.</td></tr><tr><td>Base-NMT</td><td>what is a partical that holds a phone?</td></tr><tr><td>Base-NMT-NoParalex</td><td>plus little as an associate person</td></tr><tr><td>Base-NMT+Quora</td><td>what is a partially little asian person?</td></tr><tr><td>AQA-QR</td><td>what is name is associated with a partially active asian person holding a phone?</td></tr><tr><td>Source</td><td>a car driving through a very flooded street</td></tr><tr><td>Reference</td><td>a car is driving through a flooded street.</td></tr><tr><td>Base-NMT</td><td>what car driving through a flooded street?</td></tr><tr><td>Base-NMT-NoParalex</td><td>analysing through a heat street</td></tr><tr><td>Base-NMT+Quora</td><td>i driving through a flooded street?</td></tr><tr><td>AQA-QR</td><td>what is name is given to a car with a car driving through a flooded street?</td></tr><tr><td>Source</td><td>a man is holding a tennis racket on a court</td></tr><tr><td>Reference</td><td>a man holding a racquet on top of a tennis court.</td></tr><tr><td>Base-NMT</td><td>how do you hold a tennis racket?</td></tr><tr><td>Base-NMT-NoParalex</td><td>a man to hold a threshold on a court</td></tr><tr><td>Base-NMT+Quora</td><td>what is a tennis racket?</td></tr><tr><td>AQA-QR</td><td>what is name an average man make a tennis racket on a court?</td></tr><tr><td>Source</td><td>a pizza on a pan that is half gone</td></tr><tr><td>Reference</td><td>a table with a half eaten pizza and glasses if beer.</td></tr><tr><td>Base-NMT</td><td>what is half gone in a pizza?</td></tr><tr><td>Base-NMT-NoParalex</td><td>average on a size</td></tr><tr><td>Base-NMT+Quora</td><td>is a pizza on a pan that is half gone?</td></tr><tr><td>AQA-QR</td><td>what is name a pizza on a pizza pan that is half gone?</td></tr><tr><td>Source</td><td>a boy holding a glove while on a baseball field.</td></tr><tr><td>Reference</td><td>a young boy bending over holding a catchers mitt.</td></tr><tr><td>Base-NMT</td><td>how do you hold a glove on a baseball field?</td></tr><tr><td>Base-NMT-NoParalex</td><td>a cell holding a globe ata later field.</td></tr><tr><td>Base-NMT+Quora</td><td>is it possible to hold a glove while on a baseball feld?</td></tr><tr><td>AQA-QR</td><td>what is name an boy make up a glove while on baseball feld?</td></tr><tr><td>Source</td><td>a street light with a red arrow pointing left.</td></tr><tr><td>Reference</td><td>a tagged traffic light displays ared arrow.</td></tr><tr><td>Base-NMT</td><td>a street light with red arrow pointing?</td></tr><tr><td>Base-NMT-NoParalex</td><td></td></tr><tr><td>Base-NMT+Quora</td><td>a strength with a skilling point. what is the street light of a red arrow pointing?</td></tr><tr><td>AQA-QR</td><td>what is name a street light with red arrow pointing left?</td></tr></table>

parse/train/S1CChZ-CZ/S1CChZ-CZ_content_list.json

@@ -0,0 +1,1609 @@
+[
+    {
+        "type": "text",
+        "text": "ASK THE RIGHT QUESTIONS: ACTIVE QUESTION REFORMULATION WITH REINFORCEMENT LEARNING ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            98,
+            692,
+            171
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Christian Buck, Jannis Bulian, Massimiliano Ciaramita, Wojciech Gajewski,   \nAndrea Gesmundo, Neil Houlsby, Wei Wang   \nGoogle   \n{cbuck,jbulian,massi,wgaj,agesmundo,neilhoulsby,wangwe}@google.com ",
+        "bbox": [
+            186,
+            193,
+            761,
+            251
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "ABSTRACT ",
+        "text_level": 1,
+        "bbox": [
+            454,
+            286,
+            544,
+            301
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "We frame Question Answering (QA) as a Reinforcement Learning task, an approach that we call Active Question Answering. We propose an agent that sits between the user and a black box QA system and learns to reformulate questions to elicit the best possible answers. The agent probes the system with, potentially many, natural language reformulations of an initial question and aggregates the returned evidence to yield the best answer. The reformulation system is trained end-to-end to maximize answer quality using policy gradient. We evaluate on SearchQA, a dataset of complex questions extracted from Jeopardy!. The agent outperforms a stateof-the-art base model, playing the role of the environment, and other benchmarks. We also analyze the language that the agent has learned while interacting with the question answering system. We find that successful question reformulations look quite different from natural language paraphrases. The agent is able to discover non-trivial reformulation strategies that resemble classic information retrieval techniques such as term re-weighting (tf-idf) and stemming. ",
+        "bbox": [
+            233,
+            319,
+            766,
+            513
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "1 INTRODUCTION ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            546,
+            336,
+            561
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Web and social media have become primary sources of information. Users’ expectations and information seeking activities co-evolve with the increasing sophistication of these resources. Beyond navigation, document retrieval, and simple factual question answering, users seek direct answers to complex and compositional questions. Such search sessions may require multiple iterations, critical assessment, and synthesis (Marchionini, 2006). ",
+        "bbox": [
+            174,
+            578,
+            825,
+            648
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "The productivity of natural language yields a myriad of ways to formulate a question (Chomsky, 1965). In the face of complex information needs, humans overcome uncertainty by reformulating questions, issuing multiple searches, and aggregating responses. Inspired by humans’ ability to ask the right questions, we present an agent that learns to carry out this process for the user. The agent sits between the user and a backend QA system that we refer to as ‘the environment’. We call the agent AQA, as it implements an active question answering strategy. AQA aims to maximize the chance of getting the correct answer by sending a reformulated question to the environment. The agent seeks to find the best answer by asking many questions and aggregating the returned evidence. The internals of the environment are not available to the agent, so it must learn to probe a black-box optimally using only question strings. The key component of the AQA agent is a sequence-to-sequence model trained with reinforcement learning (RL) using a reward based on the answer returned by the environment. The second component to AQA combines the evidence from interacting with the environment using a convolutional neural network to select an answer. ",
+        "bbox": [
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+            655,
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "We evaluate on a dataset of Jeopardy! questions, SearchQA (Dunn et al., 2017). These questions are hard to answer by design because they use convoluted language, e.g., Travel doesn’t seem to be an issue for this sorcerer & onetime surgeon; astral projection & teleportation are no prob (answer: Doctor Strange). Thus SearchQA tests the ability of AQA to reformulate questions such that the QA system has the best chance of returning the correct answer. AQA improves over the performance of a deep network built for QA, BiDAF (Seo et al., 2017a), which has produced state-of-the-art results on multiple tasks, by $1 1 . 4 \\%$ absolute F1, a $32 \\%$ relative F1 improvement. Additionally, AQA outperforms other competitive heuristic query reformulation benchmarks. ",
+        "bbox": [
+            174,
+            840,
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "AQA defines an instance of machine-machine communication. One side of the conversation, the AQA agent, is trying to adapt its language to improve the response from the other side, the QA environment. To shed some light on this process we perform a qualitative analysis of the language generated by the AQA agent. By evaluating on MSCOCO (Lin et al., 2014), we find that the agent’s question reformulations diverge significantly from natural language paraphrases. Remarkably, though, the agent is able to learn non-trivial and transparent policies. In particular, the agent is able to discover classic IR query operations such as term re-weighting, resembling tf-idf, and morphological simplification/stemming. A possible reason being that current machine comprehension tasks involve the ranking of short textual snippets, thus incentivizing relevance, more than deep language understanding. ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "2 RELATED WORK ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            301,
+            339,
+            316
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Lin & Pantel (2001) learned patterns of question variants by comparing dependency parsing trees. Duboue & Chu-Carroll (2006) showed that MT-based paraphrases can be useful in principle by providing significant headroom in oracle-based estimations of QA performance. Recently, Berant & Liang (2014) used paraphrasing to augment the training of a semantic parser by expanding through the paraphrases as a latent representation. Bilingual corpora and MT have been used to generate paraphrases by pivoting through a second language. Recent work uses neural translation models and multiple pivots (Mallinson et al., 2017). In contrast, our approach does not use pivoting and is, to our knowledge, the first direct neural paraphrasing system. Riezler et al. (2007) propose phrase-based paraphrasing for query expansion. In contrast with this line of work, our goal is to generate full question reformulations while optimizing directly the end-to-end target performance metrics. ",
+        "bbox": [
+            174,
+            334,
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+            473
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Reinforcement learning is gaining traction in natural language understanding across many problems. For example, Narasimhan et al. (2015) use RL to learn control policies for multi-user dungeon games where the state of the game is summarized by a textual description, and Li et al. (2016) use RL for dialogue generation. Policy gradient methods have been investigated recently for MT and other sequence-to-sequence problems. They alleviate limitations inherent to the word-level optimization of the cross-entropy loss, allowing the use of sequence-level reward functions, like BLEU. Reward functions based on language models and reconstruction errors are used to bootstrap MT with fewer resources (Xia et al., 2016). RL training can also prevent exposure bias; an inconsistency between training and inference time stemming from the fact that the model never sees its own mistakes during training (Ranzato et al., 2015). We also use policy gradient to optimize our agent, however, we use end-to-end question answering quality as the reward. ",
+        "bbox": [
+            174,
+            478,
+            825,
+            632
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Uses of policy gradient for QA include Liang et al. (2017), who train a semantic parser to query a knowledge base, and Seo et al. (2017b) who propose query reduction networks that transform a query to answer questions that involve multi-hop common sense reasoning. The work of Nogueira & Cho (2016) is most related to ours. They identify a document containing an answer to a question by following links on a graph. Evaluating on a set of questions from the game Jeopardy!, they learn to walk the Wikipedia graph until they reach the predicted answer. In a follow-up, Nogueira & Cho (2017) improve document retrieval with an approach inspired by relevance feedback in combination with RL. They reformulate a query by adding terms from documents retrieved from a search engine for the original query. Our work differs in that we generate complete sequence reformulations rather than adding single terms, and we target question-answering rather than document retrieval. ",
+        "bbox": [
+            174,
+            637,
+            825,
+            776
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Active QA is also related to recent research on fact-checking: Wu et al. (2017) propose to perturb database queries in order to estimate the support of quantitative claims. In Active QA questions are perturbed semantically with a similar purpose, although directly at the surface natural language form. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "3 ACTIVE QUESTION ANSWERING MODEL ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "Figure 1 shows the Active Question Answering (AQA) agent-environment setup. The AQA model interacts with a black-box environment. AQA queries it with many versions of a question, and finally returns the best of the answers found. An episode starts with an original question $q _ { 0 }$ . The agent then ",
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+    {
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+        "img_path": "images/42ba155712e96f20018733a7e2f0a66dd2a7495f5ff2c80de1bf6e3f96049b86.jpg",
+        "image_caption": [
+            "Figure 1: The AQA agent-environment setup. In the downward pass the agent reformulates the question and sends variants to the QA system. In the upward pass the final answer is selected. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "generates a set of reformulations $\\{ q _ { i } \\} _ { i = 1 } ^ { N }$ . These are sent to the environment which returns answers $\\{ a _ { i } \\} _ { i = 1 } ^ { N }$ . The selection model then picks the best from these candidates. ",
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+    {
+        "type": "text",
+        "text": "3.1 QUESTION-ANSWERING ENVIRONMENT ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "For the QA environment, we use a competitive neural question answering model, BiDirectional Attention Flow (BiDAF) (Seo et al., 2017a). BiDAF is an extractive QA system, it selects answers from contiguous spans of a given document. Given a question, the environment returns an answer and, during training, a reward. The reward may be any quality metric for the returned answer, we use token-level F1 score. Note that the reward for each answer $a _ { i }$ is computed against the original question $q _ { 0 }$ . We assume that the environment is opaque; the agent has no access to its parameters, activations or gradients. This setting enables one, in principle, to also interact with other information sources, possibly providing feedback in different modes such as images and structured data from knowledge bases. However, without propagating gradients through the environment we lose information, feedback on the quality of the question reformulations is noisy, presenting a challenge for training. ",
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+    {
+        "type": "text",
+        "text": "3.2 REFORMULATION MODEL ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "The reformulator is a sequence-to-sequence model, as is popular for neural machine translation. We build upon the implementation of Britz et al. (2017). The major departure from the standard MT setting is that our model reformulates utterances in the same language. Unlike in MT, there is little high quality training data available for monolingual paraphrasing. Effective training of highly parametrized neural networks relies on an abundance of data. We address this challenge by first pre-training on a related task, multilingual translation, and then using signals produced during the interaction with the environment for adaptation. ",
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+    {
+        "type": "text",
+        "text": "3.3 ANSWER SELECTION MODEL ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "During training, we have access to the reward for the answer returned for each reformulation $q _ { i }$ However, at test time we must predict the best answer $a ^ { * }$ . The selection model selects the best answer from the set $\\{ a _ { i } \\} _ { i = 1 } ^ { N }$ observed during the interaction by predicting the difference of the F1 score to the average F1 of all variants. We use pre-trained embeddings for the tokens of query, rewrite, and answer. For each, we add a 1-dimensional CNN followed by max-pooling. The three resulting vectors are then concatenated and passed through a feed-forward network which produces the output. ",
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+    {
+        "type": "text",
+        "text": "4 TRAINING ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "4.1 QUESTION ANSWERING ENVIRONMENT ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "We train a model on the training set for the QA task at hand, see Section 5.4 for details. Afterwards, BiDAF becomes the black-box environment and its parameters are not updated further. In principle, ",
+        "bbox": [
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+        "page_idx": 2
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+    {
+        "type": "text",
+        "text": "we could train both the agent and the environment jointly to further improve performance. However, this is not our desired task: our aim is for the agent to learn to communicate using natural language with an environment over which is has no control. ",
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "4.2 POLICY GRADIENT TRAINING OF THE REFORMULATION MODEL ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "For a given question $q _ { 0 }$ , we want to return the best possible answer $a ^ { * }$ , maximizing a reward $a ^ { * } = \\operatorname { a r g m a x } _ { a } R ( a | q _ { 0 } )$ . Typically, $R$ is the token level F1 score on the answer. The answer $a = f ( q )$ is an unknown function of a question $q$ , computed by the environment. The reward is computed with respect to the original question $q _ { 0 }$ while the answer is provided for $q$ . The question is generated according to a policy $\\pi _ { \\theta }$ where $\\theta$ are the policy’s parameters $q \\sim \\pi _ { \\theta } ( { \\bf \\nabla } \\cdot | q _ { 0 } )$ . The policy, in this case, a sequence-to-sequence model, assigns a probability ",
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+        "page_idx": 3
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+    {
+        "type": "equation",
+        "img_path": "images/06bbb7a767304328d8c412e33c7337acd435b8acf8e32b668976582370bd1657.jpg",
+        "text": "$$\n\\pi _ { \\boldsymbol { \\theta } } ( q | q _ { 0 } ) = \\prod _ { t = 1 } ^ { T } p ( w _ { t } | w _ { 1 } , \\dots , w _ { t - 1 } , q _ { 0 } )\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "to any possible question $q = w _ { 1 } , \\ldots , w _ { T }$ , where $T$ is the length of $q$ with tokens $w _ { t } \\in V$ from a fixed vocabulary $V$ . The goal is to maximize the expected reward of the answer returned under the policy, $\\mathbb { E } _ { q \\sim \\pi _ { \\theta } ( \\cdot | q _ { 0 } ) } [ R ( f ( q ) ) ]$ . We optimize the reward directly with respect to parameters of the policy using Policy Gradient methods (Sutton $\\&$ Barto, 1998). The expected reward cannot be computed in closed form, so we compute an unbiased estimate with Monte Carlo sampling, ",
+        "bbox": [
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+    {
+        "type": "equation",
+        "img_path": "images/8ce2466179045315c5a9dfc5d79741546e24f9fb197f54ea3f212a3263a31452.jpg",
+        "text": "$$\n\\mathbb { E } _ { q \\sim \\pi _ { \\theta } ( \\cdot | q _ { 0 } ) } [ R ( f ( q ) ) ] \\approx \\frac { 1 } { N } \\sum _ { i = 1 } ^ { N } R ( f ( q _ { i } ) ) , \\quad q _ { i } \\sim \\pi _ { \\theta } ( \\cdot | q _ { 0 } )\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "To compute gradients for training we use REINFORCE (Williams & Peng, 1991), ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/ea858ee1040232cca4adf8a1c864e79ad0d3a6d6037d636d02e82a8a0e9f965f.jpg",
+        "text": "$$\n\\begin{array} { r l } {  { \\nabla \\mathbb { E } _ { q \\sim \\pi _ { \\theta } ( \\cdot \\vert q _ { 0 } ) } [ R ( f ( q ) ) ] = \\mathbb { E } _ { q \\sim \\pi _ { \\theta } ( \\cdot \\vert q _ { 0 } ) } \\nabla _ { \\theta } \\log ( \\pi _ { \\theta } ( q \\vert q _ { 0 } ) ) R ( f ( q ) ) } } \\\\ & { \\approx \\frac { 1 } { N } \\sum _ { i = 1 } ^ { N } \\nabla _ { \\theta } \\log ( \\pi ( q _ { i } \\vert q _ { 0 } ) ) R ( f ( q _ { i } ) ) , \\quad q _ { i } \\sim \\pi _ { \\theta } ( \\cdot \\vert q _ { 0 } ) } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "This estimator is often found to have high variance, leading to unstable training (Greensmith et al., 2004). We reduce the variance by subtracting the following baseline reward: $B ( q _ { 0 } ) =$ $\\mathbb { E } _ { q \\sim \\pi _ { \\theta } ( \\cdot | q _ { 0 } ) } [ R ( f ( q ) ) ]$ . This expectation is also computed by sampling from the policy given $q _ { 0 }$ . ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "We often observed collapse onto a sub-optimal deterministic policy. To address this we use entropy regularization ",
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+        "page_idx": 3
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+    {
+        "type": "equation",
+        "img_path": "images/dd8c78286d294d9c8d08c6a320080edbd8a8642e2b4e3b7b17f82bed01396eef.jpg",
+        "text": "$$\nH [ \\pi _ { \\boldsymbol { \\theta } } ( q | q _ { 0 } ) ] = - \\sum _ { t = 1 } ^ { T } \\sum _ { w _ { t } \\in V } p _ { \\boldsymbol { \\theta } } ( w _ { t } | w _ { < t } , q _ { 0 } ) \\log p _ { \\boldsymbol { \\theta } } ( w _ { t } | w _ { < t } , q _ { 0 } )\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "This final objective is: ",
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+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/2ccc5d36ebe0cf38f4d9a776711238f15ebad78ba448c60e16141e0173691d89.jpg",
+        "text": "$$\n\\mathbb { E } _ { q \\sim \\pi _ { \\theta } ( \\cdot | q _ { 0 } ) } [ R ( f ( q ) ) - B ( q _ { 0 } ) ] + \\lambda H [ \\pi ( q | q _ { 0 } ) ] ,\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 3
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+    {
+        "type": "text",
+        "text": "where $\\lambda$ is the regularization weight. ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "4.3 ANSWER SELECTION ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Unlike the reformulation policy, we train the answer with either beam search or sampling. We can produce many rewrites of a single question from our reformulation system. We issue each rewrite to the QA environment, yielding a set of (query, rewrite, answer) tuples from which we need to pick the best instance. We train another neural network to pick the best answer from the candidates. We frame the task as binary classification, distinguishing between above and below average performance. In training, we compute the F1 score of the answer for every instance. If the rewrite produces an answer with an F1 score greater than the average score of the other rewrites the instance is assigned a positive label. We ignore questions where all rewrites yield equally good/bad answers. We evaluated FFNNs, LSTMs, and CNNs and found that the performance of all systems was comparable. We choose a CNN which offers good computational efficiency and accuracy (cf. 3.3). ",
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "4.4 PRETRAINING OF THE REFORMULATION MODEL ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "We pre-train the policy by building a paraphrasing Neural MT model that can translate from English to English. While parallel corpora are available for many language pairs, English-English corpora are scarce. We first produce a multilingual translation system that translates between several languages (Johnson et al., 2016). This allows us to use available bilingual corpora. Multilingual training requires nothing more than adding two special tokens to every line which indicate the source and target languages. The encoder-decoder architecture of the translation model remains unchanged. ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "As Johnson et al. (2016) show, this model can be used for zero-shot translation, i.e. to translate between language pairs for which it has seen no training examples. For example, after training English-Spanish, English-French, French-English, and Spanish-English the model has learned a single encoder that encodes English, Spanish, and French and a decoder for the same three languages. Thus, we can use the same model for French-Spanish, Spanish-French and also English-English translation by adding the respective tokens to the source. Johnson et al. (2016) note that zero-shot translation usually performs worse than bridging, an approach that uses the model twice: first, to translate into a pivot language, then into the target language. However, the performance gap can be closed by running a few training steps for the desired language pair. Thus, we first train on multilingual data, then on a small corpus of monolingual data. ",
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+    {
+        "type": "text",
+        "text": "5 EXPERIMENTS ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "5.1 QUESTION ANSWERING DATA AND BIDAF TRAINING ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "SearchQA (Dunn et al., 2017) is a dataset built starting from a set of Jeopardy! clues. Clues are obfuscated queries such as This ‘Father of Our Country’ didn’t really chop down a cherry tree. Each clue is associated with the correct answer, e.g. George Washington, and a list of snippets from Google’s top search results. SearchQA contains over 140k question/answer pairs and $6 . 9 \\mathbf { M }$ snippets. We train our model on the pre-defined training split, perform model selection and tuning on the validation split and report results on the validation and test splits. The training, validation and test sets contain 99,820, 13,393 and 27,248 examples, respectively. ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "We train BiDAF directly on the SearchQA training data. We join snippets to form the context from which BiDAF selects answer spans. For performance reasons, we limit the context to the top 10 snippets. This corresponds to finding the answer on the first page of Google results. The results are only mildly affected by this limitation, for $10 \\%$ of the questions, there is no answer in this shorter context. These data points are all counted as losses. We trained with the Adam optimizer for 4500 steps, using learning rate 0.001, batch size 60. ",
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "5.2 QUESTION REFORMULATOR TRAINING ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "For the pre-training of the reformulator, we use the multilingual United Nations Parallel Corpus v1.0 (Ziemski et al., 2016). This dataset contains 11.4M sentences which are fully aligned across six UN languages: Arabic, English, Spanish, French, Russian, and Chinese. From all bilingual pairs, we produce a multilingual training corpus of 30 language pairs. This yields 340M training examples which we use to train the zero-shot neural MT system (Johnson et al., 2016). We tokenize our data using 16k sentence pieces.1 Following Britz et al. (2017) we use a bidirectional LSTM as the encoder and a 4-layer stacked LSTM with attention as the decoder. The model converged after training on 400M instances using the Adam optimizer with a learning rate of 0.001 and batch size of 128. ",
+        "bbox": [
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+        "page_idx": 4
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+    {
+        "type": "text",
+        "text": "The model trained as described above has poor quality. For example, for the question What month, day and year did Super Bowl 50 take place?, the top rewrite is What month and year goes back to the morning and year?. To improve quality, we resume training on a smaller monolingual dataset, extracted from the Paralex database of question paraphrases (Fader et al., 2013).2 Unfortunately, this data contains many noisy pairs. We filter many of these pairs out by keeping only those where the Jaccard coefficient between the sets of source and target terms is above 0.5. Further, since the number of paraphrases for each question can vary significantly, we keep at most 4 paraphrases for each question. After processing, we are left with about $1 . 5 \\mathbf { M }$ pairs out of the original 35M. The refined model has visibly better quality than the zero-shot one; for the example question above it generates What year did superbowl take place?. We also tried training on the monolingual pairs alone. As in (Johnson et al., 2016), the quality was in between the multilingual and refined models. ",
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "After pre-training the reformulator, we switch the optimizer from Adam to SGD and train for $1 0 0 \\mathrm { k }$ RL steps of batch size 64 with a low learning rate of 0.001. We use an entropy regularization weight of $\\lambda = 0 . 0 0 1$ . For a stopping criterion, we monitor the reward from the best single rewrite, generated via greedy decoding, on the validation set. In contrast to our initial training which we ran on GPUs, this training phase is dominated by the latency of the QA system and we run inference and updates on CPU and the BiDAF environment on GPU. ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "5.3 TRAINING THE ANSWER SELECTOR ",
+        "text_level": 1,
+        "bbox": [
+            178,
+            252,
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+            265
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "For the selection model we use supervised learning: first, we train the reformulator, then we generate $N = 2 0$ rewrites for each question in the SearchQA training and validation sets. After sending these to the environment we have about 2M (question, rewrite, answer) triples. We remove queries where all rewrites yield identical rewards, which removes about half of the training data. We use pre-trained 100-dimensional embeddings (Pennington et al., 2014) for the tokens. Our CNN-based selection model encodes the three strings into 100-dimensional vectors using a 1D CNN with kernel width 3 and output dimension 100 over the embedded tokens, followed by max-pooling. The vectors are then concatenated and passed through a feed-forward network which produces the binary output, indicating whether the triple performs below or above average, relative to the other reformulations and respective answers. ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "We use the training portion of the SearchQA data thrice, first for the initial training of the BiDAF model, then for the reinforcement-learning based tuning of the reformulator, and finally for the training of the selector. We carefully monitored that this didn’t cause severe overfitting. BiDAF alone has a generalization gap between the training and validation set errors of $3 . 4 \\mathrm { F } 1$ . This gap remains virtually identical after training the rewriter. After training the CNN, AQA-Full has a slightly larger gap of $3 . 9 \\mathrm { F } 1$ . We conclude that training AQA on BiDAF’s training set causes very little additional overfitting. We use the test set only for evaluation of the final model. ",
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "5.4 BASELINES AND BENCHMARKS ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "As a baseline, we report the results of the modified pointer network, called Attention Sum Reader (ASR), developed for SearchQA (Dunn et al., 2017).We also report the performance of the BiDAF environment used without the reformulator to answer the original question. ",
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+    {
+        "type": "text",
+        "text": "We evaluate against several benchmarks. First, following Kumaran & Carvalho (2009), we implement a system (MI-SubQuery) that generates reformulation candidates by enumerating all subqueries of the original SearchQA query and then keeps the top $N$ ranked by mutual information.3 From this set, we pick the highest scoring one as the top hypothesis to be used as a single rewrite. We also use the whole set to train a CNN answer selector for this specific source of rewrites. In this way, we can compare systems fairly both in single prediction or ensemble prediction modes. Additionally, we evaluate against another source of reformulations: the zero-shot monolingual NMT system trained on the U.N. corpus and Paralex (Base-NMT), without reinforcement learning. As with the MI-SubQuery benchmark, we evaluate the Base-NMT system both as a single reformulation predictor and as a source of $N$ best rewrites, for which we train a dedicated CNN answer selector. We also report human performance on SearchQA, based on a sample of the test set, from (Dunn et al., 2017). ",
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+    {
+        "type": "text",
+        "text": "5.5 RESULTS ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "We evaluate several variants of AQA. For each query $q$ in the evaluation we generate a list of reformulations $q _ { i }$ , for $i = 1 \\dots N$ , from the AQA reformulator trained as described in Section 4. We set $N = 2 0$ in these experiments, the same value is used for the benchmarks. In AQA TopHyp we use the top hypothesis generated by the sequence model, $q _ { 1 }$ . In AQA Voting we use BiDAF scores for a heuristic weighted voting scheme to implement deterministic selection. Let $a$ be the answer returned by BiDAF for query $q$ , with an associated score $s ( a )$ . We pick the answer according to argmaxa $\\scriptstyle \\sum _ { a ^ { \\prime } = a } s ( a ^ { \\prime } )$ . In AQA MaxConf we select the answer with the single highest BiDAF score across question reformulations. Finally, AQA CNN identifies the complete system with the learned CNN model described in Section 3. ",
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+    {
+        "type": "table",
+        "img_path": "images/279d4cfd004c893d3f31e79ca2724804c0d540a1da1902a5c6bc6e5a7aa4d7fd.jpg",
+        "table_caption": [
+            "Table 1: Results table for the experiments on SearchQA. Two-sample $t$ -tests between the AQA results and either the Base-NMT or the MI-SubQuery results show that differences in F1 and Exact Match scores are statistically significant, $p < 1 \\bar { 0 } ^ { - 4 }$ , for both Top Hypothesis and CNN predictions. The difference between Base-NMT and MI-SubQuery is also significant for Top Hypothesis predictions. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td></td><td></td><td colspan=\"2\">Baseline</td><td colspan=\"2\">MI-SubQuery</td><td colspan=\"2\">Base-NMT</td><td colspan=\"4\">AQA</td><td></td></tr><tr><td></td><td></td><td></td><td>BAAR</td><td>TopHyp</td><td></td><td>TopHyp</td><td></td><td>TopHyp</td><td>Voting</td><td>MaxConf</td><td></td><td>Human</td></tr><tr><td></td><td>EM</td><td>1</td><td>31.7</td><td>24.1</td><td>37.5</td><td>26.0</td><td>37.5</td><td>32.0</td><td>33.6</td><td>35.5</td><td>40.5</td><td>1</td></tr><tr><td>Dev</td><td>F1</td><td>24.2</td><td>37.9</td><td>29.9</td><td>44.5</td><td>32.2</td><td>44.8</td><td>38.2</td><td>40.5</td><td>42.0</td><td>47.4</td><td>=</td></tr><tr><td></td><td>EM</td><td>1</td><td>28.6</td><td>23.2</td><td>35.8</td><td>24.8</td><td>35.7</td><td>30.6</td><td>33.3</td><td>33.8</td><td>38.7</td><td>43.9</td></tr><tr><td>Test</td><td>F1</td><td>22.8</td><td>34.6</td><td>29.0</td><td>42.8</td><td>31.0</td><td>42.9</td><td>36.8</td><td>39.3</td><td>40.2</td><td>45.6</td><td>1</td></tr></table>",
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+        "type": "text",
+        "text": "Table 1 shows the results. We report exact match (EM) and F1 metrics, computed on token level between the predicted answer and the gold answer. We present results on the full validation and test sets (referred to as $n$ -gram in (Dunn et al., 2017)). Overall, SearchQA appears to be harder than other recent QA tasks such as SQuAD (Rajpurkar et al., 2016), for both machines and humans. BiDAF’s performance drops by $4 0 \\ : \\mathrm { F 1 }$ points on SearchQA compared to SQuAD. However, BiDAF is still competitive on SeachQA, improving over the Attention Sum Reader network by 13.7 F1 points. ",
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+    {
+        "type": "text",
+        "text": "Using the top hypothesis already yields an improvement of $2 . 2 \\mathrm { F } 1$ on the test set. This demonstrates that even the reformulator alone is capable to produce questions more easily answered by the environment. When generating a single prediction, both MI-SubQuery and Base-NMT benchmarks perform worse than BiDAF. Heuristic selection via both Voting and Max Conf yields a further performance boost. Both heuristics draw upon the intuition that when BiDAF is confident in its answer it is more likely to be correct, and that multiple instances of the same answer provide positive evidence (for MaxConf, the max operation implicitly rewards having an answer scored with respect to multiple questions). Finally, a trained selection function improves performance further, yielding an absolute increase of $1 1 . 4 \\mathrm { F } 1$ points ( $32 \\%$ relative) over BiDAF with the original questions. In terms of exact match score, this more than closes half the gap between BiDAF and human performance. The benchmarks improve considerably when they generate $N$ candidates, and paired with a dedicated CNN selector. This is not surprising as it provides an ensemble prediction setup. However, the AQA CNN system outperforms both MI-SubQuery and Base-NMT in all conditions by about $3 \\%$ . ",
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+    {
+        "type": "text",
+        "text": "Finally, we consider the maximum performance possible that could be achieved by picking the answer with the highest F1 score from the set of those returned for all available reformulations. Here we find that the different sources of rewrites provide comparable headroom: the oracle Exact Match is near 50, while the oracle F1 is close to 58. ",
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+    },
+    {
+        "type": "text",
+        "text": "6 ANALYSIS OF THE AGENT’S LANGUAGE ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "The AQA agent can learn several types of sub-optimal policies. For example, it can converge to deterministic policies by learning to emit the same, meaningless, reformulation for any input question. This strategy can lead to local optima because the environment has built in strong priors on what looks like a likely answer, even ignoring the input question. Hence, convergence to non-negligible performance is easy. Entropy regularization typically fixes this behavior. Too much weight on the entropy regularizer, on the other hand, might yield random policies. A more competitive sub-optimal policy is one that generates minimal changes to the input, in order to stay close to the original question. This is a successful strategy because the environment has been trained on the original questions alone, which leads to baseline performance. ",
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+    },
+    {
+        "type": "image",
+        "img_path": "images/9aab172b023b9ff732b608e7c3ec294d8183d89ca1894d60c382e8e3f664e4ee.jpg",
+        "image_caption": [
+            "Figure 2: Boxplot summaries of the statistics collected for all types of questions. Two-sample $t \\cdot$ -tests performed on all pairs in each box confirm that the differences in means are significant $p \\bar { < } 1 0 ^ { - 3 }$ . "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "It seems quite remarkable then that AQA is able to learn non-trivial reformulation policies, that differ significantly from all of the above. One can think of the policy as a language for formulating questions that the agent has developed while engaging in a machine-machine communication with the environment. In this section we look deeper into the agent’s language. ",
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+    },
+    {
+        "type": "text",
+        "text": "6.1 GENERAL PROPERTIES",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "We analyze input questions and reformulations on the development partition of SearchQA to gain insights on how the agent’s language evolves during training via policy gradient. It is important to note that in the SearchQA dataset the original Jeopardy! clues have been preprocessed by lowercasing and stop word removal. The resulting preprocessed clues that form the sources (inputs) for the sequence-to-sequence reformulation model resemble more keyword-based search queries than grammatical questions. For example, the clue Gandhi was deeply influenced by this count who wrote \"War and Peace\" is simplified to gandhi deeply influenced count wrote war peace. ",
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+    },
+    {
+        "type": "text",
+        "text": "The (preprocessed) SearchQA questions contain 9.6 words on average. They contain few repeated terms, computed as the mean term frequency (TF) per question. The average is 1.03, but for most of the queries $( 7 5 \\% )$ TF is 1.0. We also compute the median document frequency (DF) per query, where the document is the context from which the answer is selected, as a measure of how informative a term is.4 As another measure of query performance, we also compute Query Clarity (QC) (Cronen-Townsend et al., 2002).5 Figure 2 summarizes statistics of the questions and rewrites. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "We first consider the top hypothesis generated by the pre-trained NMT reformulation system, before reinforcement learning (Base-NMT). The Base-NMT rewrites differ greatly from their sources. They are shorter, 6.3 words on average, and have even fewer repeated terms (1.01). Interestingly, these reformulations are mostly syntactically well-formed questions. For example, the clue above becomes Who influenced count wrote war?.6 Base-NMT improves structural language quality by properly reinserting dropped function words and wh-phrases. We also verified the increased fluency by using a large language model and found that the Base-NMT rewrites are $50 \\%$ more likely than the original questions. While more fluent, the Base-NMT rewrites involve lower DF terms. This is probably due to a domain mismatch between SearchQA and the NMT training corpus. The query clarity of the Base-NMT rewrites is also degraded as a result of the transduction process. ",
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "We next consider the top hypothesis generated by the AQA question reformulator (AQA-QR) after the policy gradient training. The AQA-QR rewrites are those whose corresponding answers are evaluated as AQA TopHyp in Table 1. These single rewrites alone outperform the original SearchQA queries by $2 \\%$ on the test set. We analyze the top hypothesis instead of the final output of the full AQA agent to avoid confounding effects from the answer selection step. These rewrites look different from both the Base-NMT and the SearchQA ones. For the example above AQA-QR’s top hypothesis is What is name gandhi gandhi influence wrote peace peace?. Surprisingly, $9 9 . 8 \\%$ start with the prefix What is name. The second most frequent is What country is (81 times), followed by What is is (70) and What state (14). This is puzzling as it occurs in only 9 Base-NMT rewrites, and never in the original SearchQA questions. We speculate it might be related to the fact that virtually all answers involve names, of named entities (Micronesia) or generic concepts (pizza). ",
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "AQA-QR’s rewrites seem less fluent than both the SearchQA and the Base-MT counterparts. In terms of language model probability, they are less likely than both SearchQA and Base-NMT. However, they have more repeated terms (1.2 average TF), are significantly longer (11.9) than in Base-NMT and contain more informative context terms than SearchQA questions (lower DF). Also, the translation process does not affect query clarity much. Finally, we find that AQA-QR’s reformulations contain morphological variants in $12 . 5 \\%$ of cases. The number of questions that contain multiple tokens with the same stem doubles from SearchQA to AQA-QR. Singular forms are preferred over plurals. Morphological simplification is useful because it increases the chance that a word variant in the question matches the context. ",
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "6.2 PARAPHRASING QUALITY ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "We also investigate the general paraphrasing abilities of our model, focusing on the relation between paraphrasing quality and QA quality. To tease apart the relationship between paraphrasing and reformulation for QA we evaluated 3 variants of the reformulator: ",
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Base-NMT This is the model used to initialize RL training of the agent. Trained first on the multilingual U.N. corpus, then on the Paralex corpus, as detailed in Section 5.2. ",
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Base-NMT-NoParalex This is the model above trained solely on the multilingual U.N. corpus, without the Paralex monolingual corpus. ",
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Base-NMT $^ { \\cdot } +$ Quora This is the same as Base-NMT, additionally trained on the Quora dataset7 which contains $1 5 0 \\mathrm { k }$ duplicate questions. ",
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Following Prakash et al. (2016), we evaluate all models on the MSCOCO8 (Lin et al., 2014) validation set (VAL2014). This dataset consists of images with 5 captions each, of which we select a random one as the source and the other four as references. We use beam search, to compute the top hypothesis and report uncased, moses-tokenized BLEU using multeval9 (Clark et al., 2011). Please note, that the MSCOCO data is only used for evaluation purposes. Examples of all systems can be found in Appendix C. ",
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "The Base-NMT model performs at 11.4 BLEU (see Table 1 for the QA eval numbers). In contrast, Base-NMT-NoParalex performs poorly at 5.0 BLEU. Limiting training to the multilingual data alone also degrades QA performance: the scores of the Top Hypothesis are at least 5 points lower in all metrics and CNN scores are 2-3 points lower. ",
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "By training on additional monolingual data, the Base-NMT $^ +$ Quora model improves BLEU score slightly to 11.6. End-to-end QA performance also improves marginally, the maximum delta with respect to Base-NMT under all conditions is $+ 0 . 5$ points, but the difference is not statistically significant. Thus, adding the Quora training does not have a significant effect. This might be due to the fact that most of the improvement is captured by training on the larger Paralex data set. Improving raw paraphrasing quality as well as reformulation fluency helps AQA up to a point. However, they are only partially aligned with the main task, which is QA performance. The AQA-QR reformulator has a BLEU score of 8.6, well below both Base-NMT models trained on monolingual data. Yet, AQA-QR significantly outperforms all others in the QA task. Training the agent starting from the Base-NMT $\\scriptstyle +$ Quora model yielded comparable results as starting from Base-NMT. ",
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+        "page_idx": 8
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+    {
+        "type": "text",
+        "text": "",
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+    },
+    {
+        "type": "text",
+        "text": "6.3 DISCUSSION ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Recently, Lewis et al. (2017) trained chatbots that negotiate via language utterances in order to complete a task. They report that the agent’s language diverges from human language if there is no incentive for fluency in the reward function. Our findings seem related. The fact that the questions reformulated by AQA do not resemble natural language is not due to the keyword-like SearchQA input questions, because Base-NMT is capable of producing more fluent questions from the same input. AQA learns to re-weight terms by focusing on informative (lower document frequency), query-specific (high query clarity), terms while increasing term frequency (TF) via duplication. At the same time it learns to modify surface forms in ways akin to stemming and morphological analysis. ",
+        "bbox": [
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+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Some of the techniques seem to adapt to the specific properties of current deep QA architectures such as character-based modeling and attention. Sometimes AQA learns to generate semantically nonsensical, novel, surface term variants; e.g., it might transform the adjective dense to densey. The only justification for this is that such forms can be still exploited by the character-based BiDAF question encoder. Finally, repetitions can directly increase the chances of alignment in the attention components. ",
+        "bbox": [
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+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "We hypothesize that, while there is no incentive for the model to use human language due to the nature of the task, AQA learns to ask BiDAF questions by optimizing a language that increases the likelihood of BiDAF ranking better the candidate answers. Jia & Liang (2017) argue that reading comprehension systems are not capable of significant language understanding and fail easily in adversarial settings. We speculate that current machine comprehension tasks involve mostly pattern matching and relevance modeling. As a consequence deep QA systems might implement sophisticated ranking systems trained to sort snippets of text from the context. As such, they resemble document retrieval systems which incentivizes the (re-)discovery of IR techniques, such as tf-idf re-weighting and stemming, that have been successful for decades (Baeza-Yates & Ribeiro-Neto, 1999). ",
+        "bbox": [
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+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "7 CONCLUSION ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "We propose a new framework to improve question answering. We call it active question answering (AQA), as it aims to improve answering by systematically perturbing input questions. We investigated a first system of this kind that has three components: a question reformulator, a black box QA system, and a candidate answer aggregator. The reformulator and aggregator form a trainable agent that seeks to elicit the best answers from the QA system. Importantly, the agent may only query the environment with natural language questions. Experimental results prove that the approach is highly effective and that the agent is able to learn non-trivial and somewhat interpretable reformulation policies. ",
+        "bbox": [
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+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "For future work, we will continue developing active question answering, investigating the sequential, iterative aspects of information seeking tasks, framed as end-to-end RL problems, thus, closing the loop between the reformulator and the selector. ",
+        "bbox": [
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+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "8 ACKNOWLEDGEMENTS ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "We would like to thank the anonymous reviewers for their valuable comments and suggestions. We would also like to thank Jyrki Alakuijala, Gábor Bártok, Alexey Gronskiy, Rodrigo Nogueira and Hugo Penedones for insightful discussions and technical feedback. ",
+        "bbox": [
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+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "REFERENCES ",
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+        "bbox": [
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+        "type": "text",
+        "text": "A REFORMULATION EXAMPLES ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            103,
+            452,
+            117
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "table",
+        "img_path": "images/3abc2f99f6f890e39757a1d741f0ed35f25518f84177c93b3e4a132d34bc5210.jpg",
+        "table_caption": [
+            "Table 2: Results of the qualitative analysis on SearchQA. For the original Jeopardy! questions we give the reference answer, otherwise the answer given by BiDAF. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Model</td><td>Query</td><td>Reference /Answer from BiDAF (F1)</td></tr><tr><td>Jeopardy!</td><td>People of this nation AKA Nippon wrote with a brush,so painting became the preferred form of artistic expression</td><td> japan</td></tr><tr><td>SearchQA</td><td>people nation aka nippon wrote brush ,painting became preferred</td><td>japan (1.0)</td></tr><tr><td></td><td>form artistic expression nippon brush preferred</td><td> julian (0)</td></tr><tr><td>MI Base-NMT</td><td>Aka nippon written form artistic expression?</td><td> julian (0)</td></tr><tr><td>AQA-QR</td><td>What is name did people nation aka nippon wrote brush expres-</td><td>japan (1.0)</td></tr><tr><td>AQA-Full</td><td>sion？ people nation aka nippon wrote brush ,painting became preferred</td><td> japan (1.0)</td></tr><tr><td>Jeopardy!</td><td>form artistic expression Michael Caine &amp; Steve Martin teamed up as Lawrence &amp; Freddy,</td><td>dirty rotten scoundrels</td></tr><tr><td>SearchQA</td><td>a couple of these,the title of a1988 flm michael caine steve martin teamed lawrence freddy,couple,title</td><td>dirty rotten scoundrels (1.0)</td></tr><tr><td>MI</td><td>1988 film caine teamed freddy</td><td>dirty rotten scoundrels (1.0)</td></tr><tr><td>Base-NMT</td><td>Who was lawrence of michael caine steve martin?</td><td>rain man 1988 best picture fikkle [...25 tokens]</td></tr><tr><td>AQA-QR</td><td>What is name is name is name michael caine steve martin teamed</td><td>(0.18) dirty rotten scoundrels (1.0)</td></tr><tr><td>AQA-Full</td><td>lawrence freddy and title 1988 film? What is name is name where name is name michael caine steve martin teamed lawrence freddy and title 1988 film key 2000 ?</td><td>dirty rotten scoundrels (1.0)</td></tr><tr><td>Jeopardy!</td><td>Used underwater,ammonia gelatin is a waterproof type of this ex-</td><td>dynamite</td></tr><tr><td>SearchQA</td><td>plosive used underwater,ammonia gelatin waterproof type explosive</td><td></td></tr><tr><td>MI</td><td>ammonia gelatin waterproof</td><td>nitroglycerin (0) nitroglycerin (0)</td></tr><tr><td>Base-NMT</td><td>Where is ammonia gelatin waterproof? nitroglycerin (0)</td><td></td></tr><tr><td>AQA-QR</td><td>What is name is used under water with ammonia gelatin water wa- terproof type explosive?</td><td>nitroglycerin (0)</td></tr><tr><td>AQA-Full</td><td>used underwater,ammonia gelatin waterproof type explosive</td><td>nitroglycerin (0)</td></tr><tr><td>Jeopardy!</td><td>The Cleveland Peninsula is about 40 miles northwest of Ketchikan in this state</td><td>alaska</td></tr><tr><td>SearchQA</td><td>cleveland peninsula 4O miles northwest ketchikan state</td><td>alaska &#x27;s community information summary says [. 113 tokens] (0.02)</td></tr><tr><td>MI Base-NMT</td><td>cleveland peninsula ketchikan The cleveland peninsula 40 miles?</td><td>alaska &#x27;s dec 16,1997[..132 tokens](0.01) ketchikan alaskalocatednorthwesttip</td></tr><tr><td>AQA-QR</td><td>What is name is cleveland peninsula state northwest state state</td><td>[... 46 tokens] (0.04) alaska (1.0)</td></tr><tr><td>AQA-Full</td><td>state? What is name are cleveland peninsula state northwest state state</td><td>alaska (1.0)</td></tr><tr><td></td><td>state ？</td><td></td></tr><tr><td>Jeopardy! SearchQA</td><td>Tess Ocean,Tinker Bell, Charlotte the Spider tess ocean,tinker bell,charlotte spider</td><td>julia roberts julia roberts tv com charlotte spider[..87 tokens]</td></tr><tr><td>MI</td><td>tess tinker spider</td><td>(0.04) julia roberts tvcom charlotte spider.. 119 tokens]</td></tr><tr><td>Base-NMT</td><td>What ocean tess tinker bell?</td><td>(0.01) julia roberts american actress producer made</td></tr><tr><td>AQA-QR</td><td>What is name tess ocean tinker bellink charlotte spider?</td><td>[.. 206 tokens] (0.02) julia roberts (1.0)</td></tr><tr><td>AQA-Full</td><td>What is name is name tess ocean tinker bell spider contain char- lotte spider contain hump around the world winter au to finish au de mon moist</td><td> julia roberts (1.0)</td></tr><tr><td>Jeopardy!</td><td>During the Tertiary Period,India plowed into Eurasia &amp; this high- est mountain range was formed</td><td>himalayas</td></tr><tr><td>SearchQA</td><td>tertiary period，india plowed eurasia highest mountain range formed</td><td>himalayas (1.0)</td></tr><tr><td>MI</td><td>tertiary plowed eurasia</td><td>himalayas (1.0)</td></tr><tr><td>Base-NMT</td><td>What is eurasia highest mountain range?</td><td>himalayas (1.0)</td></tr><tr><td>AQA-QR</td><td>What is name were tertiary period in india plowed eurasia?</td><td>himalayas (1.0)</td></tr><tr><td>AQA-Full</td><td>tertiary period，india plowed eurasia highest mountain range formed</td><td>himalayas (1.0)</td></tr><tr><td></td><td></td><td></td></tr><tr><td>Jeopardy!</td><td>The melody heard here is from the opera about Serse,better known to us as this &quot;X&quot;-rated Persian king</td><td>xerxes</td></tr><tr><td>SearchQA</td><td>melody heard opera serse,better known us X rated persian king</td><td>gilbert sullivan (0)</td></tr><tr><td>MI</td><td>melody opera persian</td><td>gilbert sullivan (0)</td></tr><tr><td>Base-NMT</td><td>Melody heard opera serse thing?</td><td>gilbert sullivan (0)</td></tr><tr><td>AQA-QR</td><td>What is name melody heard opera serse is better persian king?</td><td>gilbert sullivan (0)</td></tr><tr><td>AQA-Full</td><td>What is name is name melody heard opera serse is better persian king persian K ?</td><td>gilbert sullivan (0)</td></tr></table>",
+        "bbox": [
+            173,
+            172,
+            823,
+            912
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "B EXAMPLES OF RANKING LOSSES ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            102,
+            483,
+            118
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "table",
+        "img_path": "images/f3def862bc5b4d4b16f23d2d95530f94ed309cda01b027d92eee03e7f8a7975e.jpg",
+        "table_caption": [
+            "Table 3: Examples of queries where none of the methods produce the right answer, but the Oracle model can. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Model</td><td>Query</td><td>Answer (F1)</td></tr><tr><td>SearchQA</td><td>ancient times ,pentelic quarries major source building material athens</td><td>parthenon (0.0)</td></tr><tr><td>AQA-QR AQA-CNN</td><td>What is name an Ancient History Warry quarry material athens? ancient times ,pentelic quarries major source building material</td><td>parthenon (0.0) parthenon (0.0)</td></tr><tr><td>AQA-Oracle</td><td>athens What is name is name an Ancient Romes For pentelic quarry</td><td>marble (1.0)</td></tr><tr><td>SearchQA</td><td>material athens measure athens? 1949 1999 germany &#x27;s bundestag legislature met city</td><td>berlin (0.0)</td></tr><tr><td>AQA-QR</td><td>What is name is name 1999 germanys bundestag legislature met city city?</td><td>berlin (0.0)</td></tr><tr><td>AQA-CNN</td><td>What is name is name 1999 germany germany bundestag legisla- ture city city?</td><td>berlin (0.0)</td></tr><tr><td>AQA-Oracle</td><td>What is name is name a 1999 germany germany legislature met city city?</td><td>bonn (1.0)</td></tr><tr><td>SearchQA</td><td>utah jazz retired 12 jersey</td><td>karl malone (0.0)</td></tr><tr><td>AQA-QR</td><td>What is name did utah jazz retired jersey jersey?</td><td>karl malone (0.0)</td></tr><tr><td>AQA-CNN</td><td>What is name did utah jazz retired jersey jersey?</td><td>karl malone (0.0)</td></tr><tr><td>AQA-Oracle</td><td>What is name is name where utah jazz is retired jersey jersey?</td><td> john stockton (1.0)</td></tr><tr><td>SearchQA</td><td>swing intersection count basie capital virginia</td><td></td></tr><tr><td>AQA-QR</td><td>What is name for swing intersection count basie capital virginia?</td><td>chicago (0.0)</td></tr><tr><td>AQA-CNN</td><td></td><td>chicago (0.0)</td></tr><tr><td></td><td>swing intersection count basie capital virginia</td><td>chicago (0.0)</td></tr><tr><td>AQA-Oracle SearchQA</td><td>What is name swing swing intersection capital virginia?</td><td>richmond (1.0)</td></tr><tr><td></td><td>huge upset virginia , jim webb beat senator prospective presiden- tial candidate</td><td>hillary clinton (0.0)</td></tr><tr><td>AQA-QR</td><td>What is name is name huge upset virginia jug webb beat senator?</td><td>jim webb (0.0)</td></tr><tr><td>AQA-CNN AQA-Oracle</td><td>What is name is name huge upset virginia jug webb beat senator world candidate?</td><td> jim webb (0.0)</td></tr></table>",
+        "bbox": [
+            171,
+            181,
+            797,
+            588
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "C PARAPHRASING EXAMPLES ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            102,
+            437,
+            118
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "table",
+        "img_path": "images/a486e8a5948a74684ea7871519072c9e4af2c726bfb6a736929eedf3ae021d0d.jpg",
+        "table_caption": [
+            "Table 4: Paraphrasing examples on captions from MSCOCO (Lin et al., 2014). "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Source Reference</td><td>a man riding a wave on top of a surfboard. a man on a surfboard in the crest of a wave.</td></tr><tr><td></td><td></td></tr><tr><td>Base-NMT</td><td>a man on top of a surfboard?</td></tr><tr><td>Base-NMT-NoParalex</td><td>amankind arises away froma floor.</td></tr><tr><td>Base-NMT+Quora</td><td>it possible to riding a wave on a surfboard?</td></tr><tr><td>AQA-QR</td><td>what is name an average man riding a wave at onfboard?</td></tr><tr><td>Source</td><td>a person riding some skis through a snowy mountain trail.</td></tr><tr><td>Reference</td><td>a man on skis is standing by some trees.</td></tr><tr><td>Base-NMT</td><td>what is some skis through a snowy mountain trail?</td></tr><tr><td>Base-NMT-NoParalex</td><td>the person rising somekises through a snow blood.</td></tr><tr><td>Base-NMT+Quora</td><td>what is a skis mountain trail?</td></tr><tr><td>AQA-QR</td><td>what is name an average person riding some one snowy mountain trail?</td></tr><tr><td>Source</td><td> a group of people all taking a bite out of the same sandwich.</td></tr><tr><td>Reference</td><td>a group of people eating rom a large sandwich.</td></tr><tr><td>Base-NMT</td><td>what do people call a bite out of the same sandwich?</td></tr><tr><td>Base-NMT-NoParalex</td><td>a group of people who takes a bargained from the same index.</td></tr><tr><td>Base-NMT+Quora AQA-QR</td><td>what is the group of people who take bite out of the same sandwich?</td></tr><tr><td></td><td>what is name a group of people with bite out of sandwich?</td></tr><tr><td>Source</td><td>a partially lit asian person holding a phone</td></tr><tr><td>Reference</td><td>a man standing next to a sign talking on a phone.</td></tr><tr><td>Base-NMT</td><td>what is a partical that holds a phone?</td></tr><tr><td>Base-NMT-NoParalex</td><td>plus little as an associate person</td></tr><tr><td>Base-NMT+Quora</td><td>what is a partially little asian person?</td></tr><tr><td>AQA-QR</td><td>what is name is associated with a partially active asian person holding a phone?</td></tr><tr><td>Source</td><td>a car driving through a very flooded street</td></tr><tr><td>Reference</td><td>a car is driving through a flooded street.</td></tr><tr><td>Base-NMT</td><td>what car driving through a flooded street?</td></tr><tr><td>Base-NMT-NoParalex</td><td>analysing through a heat street</td></tr><tr><td>Base-NMT+Quora</td><td>i driving through a flooded street?</td></tr><tr><td>AQA-QR</td><td>what is name is given to a car with a car driving through a flooded street?</td></tr><tr><td>Source</td><td>a man is holding a tennis racket on a court</td></tr><tr><td>Reference</td><td>a man holding a racquet on top of a tennis court.</td></tr><tr><td>Base-NMT</td><td>how do you hold a tennis racket?</td></tr><tr><td>Base-NMT-NoParalex</td><td>a man to hold a threshold on a court</td></tr><tr><td>Base-NMT+Quora</td><td>what is a tennis racket?</td></tr><tr><td>AQA-QR</td><td>what is name an average man make a tennis racket on a court?</td></tr><tr><td>Source</td><td>a pizza on a pan that is half gone</td></tr><tr><td>Reference</td><td>a table with a half eaten pizza and glasses if beer.</td></tr><tr><td>Base-NMT</td><td>what is half gone in a pizza?</td></tr><tr><td>Base-NMT-NoParalex</td><td>average on a size</td></tr><tr><td>Base-NMT+Quora</td><td>is a pizza on a pan that is half gone?</td></tr><tr><td>AQA-QR</td><td>what is name a pizza on a pizza pan that is half gone?</td></tr><tr><td>Source</td><td>a boy holding a glove while on a baseball field.</td></tr><tr><td>Reference</td><td>a young boy bending over holding a catchers mitt.</td></tr><tr><td>Base-NMT</td><td>how do you hold a glove on a baseball field?</td></tr><tr><td>Base-NMT-NoParalex</td><td>a cell holding a globe ata later field.</td></tr><tr><td>Base-NMT+Quora</td><td>is it possible to hold a glove while on a baseball feld?</td></tr><tr><td>AQA-QR</td><td>what is name an boy make up a glove while on baseball feld?</td></tr><tr><td>Source</td><td>a street light with a red arrow pointing left.</td></tr><tr><td>Reference</td><td>a tagged traffic light displays ared arrow.</td></tr><tr><td>Base-NMT</td><td>a street light with red arrow pointing?</td></tr><tr><td>Base-NMT-NoParalex</td><td></td></tr><tr><td>Base-NMT+Quora</td><td>a strength with a skilling point. what is the street light of a red arrow pointing?</td></tr><tr><td>AQA-QR</td><td>what is name a street light with red arrow pointing left?</td></tr></table>",
+        "bbox": [
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+        "page_idx": 14
+    }
+]

parse/train/fy4ZBWxYbIo/fy4ZBWxYbIo.md

@@ -0,0 +1,239 @@
+# A Workflow for Offline Model-Free Robotic Reinforcement Learning
+
+Aviral Kumar?,1, Anikait Singh?,1, Stephen $\mathbf { T i a n } ^ { 1 }$ , Chelsea $\mathbf { F i n n ^ { 2 } }$ , Sergey Levine1 1 UC Berkeley, 2 Stanford University (∗ Equal Contribution) aviralk@berkeley.edu, asap7772@berkeley.edu
+
+Abstract: Offline reinforcement learning (RL) enables learning control policies by utilizing only prior experience, without any online interaction. This can allow robots to acquire generalizable skills from large and diverse datasets, without any costly or unsafe online data collection. Despite recent algorithmic advances in offline RL, applying these methods to real-world problems has proven challenging. Although offline RL methods can learn from prior data, there is no clear and well-understood process for making various design choices, from model architecture to algorithm hyperparameters, without actually evaluating the learned policies online. In this paper, our aim is to develop a practical workflow for using offline RL analogous to the relatively well-understood workflows for supervised learning problems. To this end, we devise a set of metrics and conditions that can be tracked over the course of offline training, and can inform the practitioner about how the algorithm and model architecture should be adjusted to improve final performance. Our workflow is derived from a conceptual understanding of the behavior of conservative offline RL algorithms and cross-validation in supervised learning. We demonstrate the efficacy of this workflow in producing effective policies without any online tuning, both in several simulated robotic learning scenarios and for three tasks on two distinct real robots, focusing on learning manipulation skills with raw image observations with sparse binary rewards. Explanatory video and additional content can be found at sites.google.com/view/offline-rl-workflow.
+
+Keywords: workflow, offline RL, offline tuning
+
+# 1 Introduction
+
+Offline reinforcement learning (RL) can in principle make it possible to convert existing large datasets of robotic experience into effective policies, without the need for costly or dangerous online interaction for each training run. While offline RL algorithms have improved significantly [1, 2, 3, 4, 5], applying such methods to real-world robotic control problems presents a number of major challenges. In standard online RL, any intermediate policy found during training is executed in the environment to collect more experience, which naturally allows for an evaluation of the policy performance. This ability to evaluate intermediate policies lets practitioners use “brute
+
+![](images/93b250387a2633b12ccbb56017457a77edb5b8bb68d22310145c16c8258e9664.jpg)  
+Figure 1: Our proposed workflow aims to detect overfitting and underfitting, and provides guidelines for addressing these issues via policy selection, regularization, and architecture design. We evaluate this workflow on two real-world robotic systems and simulation domains, and we find it to be effective.
+
+force” to evaluate the effects of various design factors, such as model capacity and expressivity, the number of training steps, and so forth, and facilitates comparatively straightforward tuning. In contrast, offline RL methods do not have access to real-world on-policy rollouts for evaluating the learned policy. Thus, in order for these methods to be truly practical for real-world applications, we not only require effective algorithms, but also an effective workflow: a set of protocols and metrics that can be used to reliably and consistently adjust model capacity, regularization, etc in offline RL to obtain policies with good performance, without requiring real-world rollouts for tuning.
+
+A number of prior works have studied model selection in offline RL by utilizing off-policy evaluation (OPE) methods [6] to estimate policy performance. These methods can be based either on model or value learning [7, 8, 9, 10] or importance sampling [6, 11, 12, 13]. However, developing reliable OPE methods is itself an open problem, and modern OPE methods themselves suffer from hyperparameter selection challenges (see Fu et al. [14] for an empirical study). Moreover, accurate off-policy evaluation is likely not necessary to simply tune algorithms for best performance – we do not need a precise estimate of how good our policy is, but rather a workflow that enables us to best improve it by adjusting various algorithm hyperparameters.
+
+In this paper, we devise a practical workflow for selecting regularizers, model architectures, and policy checkpoints for offline RL methods in robotic learning settings. We focus on a specific class of conservative offline RL algorithms [15, 2] that regularize the Q-function, but also show that our workflow can be effectively applied to policy constraint methods [16]. Our aim is not to focus on complete off-policy evaluation or to devise a new approach for off-policy evaluation, but rather to adopt a strategy similar to the one in supervised learning. Analogously to how supervised learning practitioners can detect overfitting and underfitting by tracking training and validation losses, and then adjust hyperparameters based on these metrics, our workflow (see Figure 1 for a schematic) first defines and characterizes overfitting and underfitting, proposes metrics and conditions that users can track to determine if an offline RL exhibits overfitting or underfitting, and then utilizes these metrics to inform design decisions pertaining to neural net architectures, regularization, and early stopping. This protocol is intended to act as a “user’s manual” for a practitioner, with guidelines for how to modify algorithm parameters for best results without real-world evaluation rollouts.
+
+The primary contribution of this paper is a simple yet effective workflow for robotic offline RL. We propose metrics and protocols to assist practitioners in selecting policy checkpoints, regularization parameters, and model architectures for conservative offline RL algorithms such as CQL [2] and BRAC [16]. We empirically verify the efficacy of our proposed workflow on simulated robotic manipulation problems as well as three real-world robotic manipulation problems on two different robots, with diverse objects, pixel observations, and sparse binary reward supervision. Experimentally, we evaluate our method on two real-world robots (the Sawyer and WidowX robots), and one realistic simulated tasks. Our approach is effective in all of these cases, and on two tasks with the Sawyer robot that initially fail completely, our workflow improves the success rate to $70 \%$ .
+
+# 2 Preliminaries, Background, and Definitions
+
+The goal in RL is to optimize the infinite horizon discounted return $\begin{array} { r } { R = \sum _ { t = 0 } ^ { \infty } \gamma ^ { t } r ( \mathbf { s } _ { t } , \mathbf { a } _ { t } ) } \end{array}$ , where $r ( s , a )$ represents the reward function evaluated at a state-action pair $( \mathbf { s } , \mathbf { a } )$ . We operate in the offline RL setting and are provided with a fixed dataset $\mathcal { D } = \{ ( \mathbf { s } , \mathbf { a } , r ( \mathbf { s } , \mathbf { a } ) , \mathbf { s } ^ { \prime } ) \}$ , consisting of transition tuples obtained from rollouts under a behavior policy $\pi _ { \beta } ( \mathbf { a } | \mathbf { s } )$ . Our goal is to obtain the best possible policy by only training on this fixed offline dataset $\mathcal { D }$ , with no access to online rollouts. We focus on conservative offline RL algorithms that modify the $\mathrm { Q }$ -function to penalize distributional shift, with most experiments on CQL [2], though we also adapt our workflow to BRAC [16] in Appendix F.1.
+
+Conservative Q-learning (CQL). The actor-critic formulation of CQL trains a Q-function $Q _ { \boldsymbol { \theta } } ( \mathbf { s } , \mathbf { a } )$ with a separate policy $\pi _ { \phi } ( \mathbf { a } | \mathbf { s } )$ , which maximizes the expected $\mathrm { Q }$ -value $\begin{array} { r } { \mathbb { E } _ { \mathbf { s } \sim \mathcal { D } , \mathbf { a } \sim \pi _ { \phi } } \left[ Q _ { \theta } ( \mathbf { s } , \mathbf { a } ) \right] } \end{array}$ like other standard actor-critic deep RL methods [17, 18, 19]. However, in addition to the standard TD error ${ \mathcal { L } } _ { \mathrm { T D } } ( \theta )$ (in blue below), CQL applies a regularizer ${ \mathcal { R } } ( \theta )$ (in red below) to prevent overestimation of $\mathrm { Q }$ -values for out-of-distribution (OOD) actions. This term minimizes the $\mathrm { Q }$ -values under a distribution $\mu ( \mathbf { a } | \mathbf { s } )$ , which is automatically chosen to pick actions a with high Q-values $Q _ { \boldsymbol { \theta } } ( \mathbf { s } , \mathbf { a } )$ , and counterbalances this term by maximizing the values of the actions in the dataset:
+
+$$
+\begin{array} { r l } { \underset { \theta } { \mathrm { m i n } } \ : \ : \ : } & { { } \ : \ : \ : \ : \ : \ : \ : \ : \ : \ : \ : \ : \ : \ : \ : \ : \ : \ : \ : \ : \ : \ : \ : \ : \ : \ : \ : \ : \ : \ : \ : \ : \ : \ : \ : \ : \ : \ : } \\ { \mathrm { m i n } \ : \ : \ : \ : \ : } & { { } \ : \ : \ : \ : \ : \ : \ : \ : \ : \ : \ : \ : \ : \ : \ : \ : \ : } & { \mathrm { ~ \ : ~ \ : \ : \ : \ : } \ : \ : \ : } \end{array}
+$$
+
+where $B ^ { \pi } \bar { Q } ( \mathbf { s } , \mathbf { a } )$ is the Bellman backup operator with a delayed target Q-function, $\bar { Q }$ : $B ^ { \pi } \bar { Q } ( { \bf s } , { \bf a } ) : =$ $r ( \mathbf { s } , \mathbf { a } ) + \gamma \mathbb { E } _ { \mathbf { a ^ { \prime } } \sim \pi ( \mathbf { a ^ { \prime } } | \mathbf { s ^ { \prime } } ) } [ \bar { Q } ( \mathbf { s ^ { \prime } } , \mathbf { a ^ { \prime } } ) ]$ . In practice, CQL computes $\mu ( \mathbf { a } | \mathbf { s } )$ using actions sampled from the policy $\pi _ { \phi } ( \mathbf { a } | \mathbf { s } )$ . More discussion of CQL is in Appendix B. In this paper, we will utilize CQL as a base algorithm that our workflow intends to tune, but we also extend it to BRAC.
+
+Overfitting and underfitting in CQL. Conservative offline RL algorithms [2, 20] like CQL can be sensitive to design choices, including number of gradient steps for training [21, 22] and network capacity. These challenges are also present in supervised learning, but supervised learning methods benefit from a simple and powerful workflow that involves using training error and validation error to characterize overfitting and underfitting. A practitioner can then make tuning choices based on these characterizations. To derive an analogous workflow for offline RL, we first ask: what do overfitting and underfitting actually mean for the case of conservative offline RL?
+
+To define overfitting and underfitting generically for any conservative offline RL method, we consider an abstract optimization formulation for such methods [2]:
+
+$$
+\pi ^ { * } : = \arg \operatorname* { m a x } _ { \pi } ~ J _ { \mathcal { D } } ( \pi ) - \alpha D ( \pi , \pi _ { \beta } )
+$$
+
+(Conservative offline RL).
+
+$J _ { \mathcal { D } } ( \pi )$ denotes the average return of policy $\pi$ in the empirical MDP induced by the transitions in the offline dataset $\mathcal { D }$ , and $D ( \pi , \pi _ { \beta } )$ denotes a closeness constraint to the behavior policy, effectively applied by the offline RL method. Our definition of conservative offline RL requires that this
+
+Table 1: Summary of train error, test error and our definitions of overfitting and underfitting in supervised learning and conservative offline RL methods. We will propose metrics to measure these phenomena in a purely offline manner and recommend how to tune the underlying method accordingly.   
+
+<table><tr><td>Quantity</td><td>Supervised Learning</td><td>Conservative Offline RL</td></tr><tr><td>Test error</td><td>Loss L evaluated on test data,Dtest</td><td>Performance of policy,J(π)</td></tr><tr><td>Train error</td><td>Loss L evaluated on train data,Dtrain</td><td>Objective in Equations 2,1</td></tr><tr><td>Overfitting</td><td>L(Dtrain) low,L(Dval) high,Dval is a validation set drawn i.i.d.as Dtrain</td><td>Training objective in Equation l is ex- tremely low,low value of J(π)</td></tr><tr><td>Underfitting</td><td>high value of train error L(Dtrain)</td><td>Training objective in Equation 1 is ex- tremely high,low value of J(π)</td></tr></table>
+
+divergence be computed in expectation over the state visitation distribution of the learned policy $\pi$ in the empirical MDP as discussed in Appendix F.1. For example, Equation 1 translates to utilizing $\begin{array} { r } { D _ { \mathrm { C Q L } } ( p , \bar { q } ) : = \sum _ { \mathbf { x } } p ( \mathbf { x } ) ( p ( \mathbf { x } ) / q ( \mathbf { x } ) - \bar { 1 } ) } \end{array}$ in Equation 2 (see Theorem 3.5 in Kumar et al. [2] for a proof). The training loss is discussed in Equations 1 and 2 and the test loss is equal to the negative of the actual return $J ( \pi )$ of the learned policy. Analogously to supervised learning, we can use the notion of train and test error to define overfitting and underfitting in offline RL, as discussed in Table 1. However, note that the conditions summarized in Table 1 are not measurable completely offline. Precisely estimating if a run of an offline RL method overfits or underfits requires evaluating the learned policy via interaction with the real-world environment. In Section 3, our goal will be to devise offline metrics for characterizing overfitting that do not have this requirement. We will tailor our study specifically towards CQL, though we extend it to BRAC in Appendix F.1. A similar procedure could be devised for other offline RL methods, but we leave this for future work.
+
+# 3 Detecting Overfitting and Underfitting in Conservative Offline RL
+
+In standard supervised learning, we can determine if a method overfits or underfits by comparing the training loss to the same loss function evaluated on a held-out validation dataset, which serves as a “proxy” test dataset. In contrast, the return of the learned policy $J ( \pi )$ in RL does not have a direct proxy that can be computed offline. Thus, our goal is to identify offline metrics and conditions that allow us to measure overfitting and underfitting in conservative offline RL, with a focus on CQL. We also adapt these conditions to BRAC [16], a policy-constraint method in Appendix F.2.
+
+Detecting overfitting in CQL. Our definition of overfitting (Table 1) corresponds to a low value for the training loss (Equation 1), but poor actual policy performance $J ( \pi )$ . To detect this, we analyze the time series of the estimated Q-values averaged over the dataset samples $( \mathbf { s } , \mathbf { a } , r , \mathbf { s } ^ { \prime } ) \in \mathcal { D }$ over the course of training with a large number of gradient steps. A run is labeled as overfitting if we see that the expected dataset Q-value exhibits a non-monotonic trend: if the average Q-values first increase and then decrease as shown in the figure on the right. Additionally, we would see that training loss in Equation 1 eventually becomes very low. Why do we see such a trend in the average dataset $\mathbf { Q }$ -value? Since CQL selectively penalizes the average Qvalue under the distribution $\mu ( \mathbf { a } | \mathbf { s } )$ supported on actions with large Q-values, we would expect the Q-values on states from the dataset s $\sim \mathcal { D }$ and the learned $\mathbf { a } \sim \pi ( \cdot | \mathbf { s } )$ to be small since the policy is trained to maximize the Q-function as well. This in turn would lead to an eventual reduction in the average Q-value on dataset actions, $\mathbb { E } _ { \mathbf { s } , \mathbf { a } \sim \mathcal { D } } [ Q _ { \theta } ( \mathbf { s } , \mathbf { a } ) ]$ . This would be visible after sufficiently many steps of training, when values have propagated via Bellman backups in Equation 1 giving rise to the non-monotonic trend. If such a trend is observed, this raises two questions, as we discuss next.
+
+![](images/a834916e9152742e6ff0bc3376e3cee94d030ab397761d3b7ccebf41e7204020.jpg)
+
+What does a low average $\varrho$ -value $\mathbb { E } _ { \mathbf { s } , \mathbf { a } \sim \mathcal { D } } [ Q _ { \theta } ( \mathbf { s } , \mathbf { a } ) ]$ imply about $J ( \pi )$ ? We show in Appendix A that, in principle, CQL training (Equation 1) should never learn Q-values smaller than the dataset Monte-Carlo return, and the $\mathrm { Q }$ -values should increase unless the learned policy $\pi$ is better than $\pi _ { \beta }$ . Intuitively, this is because the objective in Equation 1 aims to also maximize the average dataset
+
+Q-value and thus the Q-values for the behavior policy are not underestimated in expectation. Now, if the policy optimizer finds a policy that attains a smaller learned Q-value than the dataset return, the policy can always be updated further towards the behavior policy so as to raise the Q-value. Therefore, Q-values can only decrease when the policy found by CQL is better than the behavior policy. We formalize this intuition in Appendix A in Theorem A.1. Thus, a low $\mathrm { Q }$ -value on $( \mathbf { s } , \mathbf { a } ) \in$ $\mathcal { D }$ indicates that the Q-function predicts extremely small Q-values on actions sampled from $\mu ( \mathbf { a } | \mathbf { s } )$ . Typically, this would mean the highest Q-value actions a at a state $\mathbf { s } \in \mathcal { D }$ are those sampled from the offline dataset, drawn from the behavior policy. Thus, policy optimization, which aims to maximize the Q-value, would make $\pi ( \mathbf { a } | \mathbf { s } )$ closer to the behavior policy $\pi _ { \beta } ( \mathbf { a } | \mathbf { s } )$ on $\mathbf { s } \in \mathcal { D }$ .
+
+Which training checkpoint is likely to attain the best policy performance? Tracking overfitting in supervised learning is important for selecting the best-performing checkpoint, before overfitting becomes severe. Analogously, we can compare the average dataset Q-value across different checkpoints within the same run to pick the best policy. Since CQL aims to increase the average dataset Q-value (Equation 1), we would expect Q-values to initially increase, until learning starts to overfit and the average dataset $\mathrm { Q }$ -value starts decreasing. We should therefore select the latest checkpoint that corresponds to a peak in the estimated dataset Q-value. A visual illustration of this idea is shown in the figure on the previous page, where the checkpoint marked by the green line is recommended to be chosen. In summary, (a) to detect overfitting we can track:
+
+Metric 3.1 (Overfitting). A low average data $Q$ -value $\mathbb { E } _ { \mathbf { s } , \mathbf { a } \sim \mathcal { D } } [ Q _ { \theta } ( \mathbf { s } , \mathbf { a } ) ]$ that decreases with more gradient steps on Equation 1 indicates that the offline RL algorithm is overfitting.
+
+and (b) further, given a run that exhibits overfitting, our principle for policy selection is given by:
+
+Guideline 3.1 (Policy selection). If a run overfits (per Metric 3.1), select the checkpoint that attains the highest average dataset $Q$ -value before overfitting for deployment.
+
+Finally, for actor-critic algorithms [18] that update the actor slower than the critic, the next policy checkpoint after the peak in the average dataset Q-value appears must be selected. In most of our experiments, we find that simply utilizing the policy checkpoint at the point of the peak in the Qvalue also leads to good results making this a rare concern, but in some cases, utilizing the next checkpoint after the Q-value peak performs better empirically.
+
+Detecting underfitting in CQL. Next, we turn to devising a procedure to detect underfitting. As summarized in Table 1, underfitting occurs when the RL algorithm is unable to minimize the training objective in Equation 1 effectively. Therefore, large values for the TD error, the CQL regularizer, or both imply underfitting. A large value for the CQL regularizer, ${ \mathcal { R } } ( \theta )$ , indicates an overestimation of Q-values relative to their true value [2] and thus, unlike the overfitting regime, we would not expect the average learned $\mathrm { Q }$ -value to decrease with more training. Thus, one approach to predict underfitting is to track both the TD error, ${ \mathcal { L } } _ { \mathrm { T D } } ( \theta )$ , and the CQL regularizer, ${ \mathcal { R } } ( \theta )$ , and check if the value of even one of these quantities is large. More discussion is provided in Appendix A.
+
+![](images/4b1e144e763ab56eff446401eff0f116a6cc810b779d08af3c7a767883b102d8.jpg)
+
+How do we determine if the $\mathbf { \nabla } ^ { T D }$ error and the CQL regularizer are “large”? In order to determine if the error of a particular run is large, we can rerun the base CQL algorithm but with models of higher capacity, which does not necessarily correspond to the function approximator size, as we will discuss in Section 4. For each model, we record the corresponding training errors and check if the training TD error and CQL regularizer value are reduced with capacity increase. If increasing capacity leads to a reduction in the loss without exhibiting the overfitting signs described previously, then we are in an underfitting regime. Another approach to answer the question is to utilize the value of the TD error $\left( \mathcal { L } _ { \mathrm { T D } } ( \theta ) \right)$ and the task horizon $( 1 / ( 1 - \gamma ) )$ to estimate the overall error in the learned Q-values against the actual Q-value, which is equal to $\dot { \mathcal { L } } _ { \mathrm { T D } } ( \theta ) / ( 1 - \gamma )$ [23] (see Appendix A). If this overall error spans the range of allowed Q-values on the task – which could be inferred based on the structure of the reward function in the task – then we can say that the algorithm is underfitting.
+
+Metric 3.2 (Underfitting). Compute the values of the training $T D$ error, ${ \mathcal { L } } _ { \mathrm { T D } } ( \theta )$ and CQL regularizer, ${ \mathcal { R } } ( \theta )$ for the current run and another identical run with increased model capacity. If the training errors reduce with increasing model capacity, the original run was underfitting.
+
+# 4 Addressing Overfitting and Underfitting in Conservative Offline RL
+
+The typical workflow for supervised learning not only identifies overfitting and underfitting, but also guides the practitioner how to adjust their method so as to alleviate it (e.g., by modifying regularization or model capacity), thus improving performance. Can we devise similar guidelines to address overfitting and underfitting with conservative offline RL? Here, we discuss some ways to adjust regularization and model capacity to alleviate these phenomena.
+
+Capacity-decreasing regularization for overfitting. As we observed in Section 3, the mechanism behind extremely low $\mathbf { Q }$ -values on the dataset is that CQL training minimizes $\mathrm { Q }$ -values on actions sampled from $\mu ( \mathbf { a } | \mathbf { s } )$ . Two possible approaches to preventing over-minimization of these values are (1) applying regularization such as dropout [24] on Q-function layers, similar to supervised learning, and (2) enforcing that representations of the learned Q-function match a pre-specified target for all state-action tuples. For (2), we can apply techniques such as a variational information bottleneck (VIB) [25, 26] regularizer on the learned representations, $\phi ( \mathbf { s } )$ . Formally, let $( \mathbf { s } , \mathbf { a } )$ denote a stateaction pair. Instead of predicting a deterministic $\phi ( \mathbf { s } ) \in \mathbb { R } ^ { d }$ (Figure 10), we modify the Q-network to predict two distinct vectors, $\phi _ { m } ( \mathbf { s } ) \in \mathbb { R } ^ { d }$ and $\phi _ { \Sigma } ( \mathbf { s } ) \in \mathbb { R } ^ { d }$ , and sample $\phi ( \mathbf { s } )$ randomly from a Gaussian centered at $\phi _ { m }$ with covariance $\phi _ { \Sigma }$ , i.e., $\phi ( \mathbf { s } ) \sim \mathcal { N } ( \phi _ { m } ( \mathbf { s } ) , \mathrm { d i a g } ( \phi _ { \Sigma } ( \mathbf { s } ) )$ . VIB then regularizes $\mathcal { N } ( \phi _ { m } ( \mathbf { s } ) , \mathrm { d i a g } ( \phi _ { \Sigma } ( \mathbf { s } ) )$ to be close to a prior distribution, $\mathcal { N } ( 0 , \mathbb { I } )$ :
+
+$$
+\operatorname* { m i n } _ { \theta } \ \mathcal { L } _ { \mathrm { C Q L } } ( \theta ) + \beta \mathbb { E } _ { \mathrm { s } \sim \mathcal { D } } \left[ \mathrm { D } _ { \mathrm { K L } } \left( \mathcal { N } ( \phi _ { m } ( \mathbf { s } ) , \mathrm { d i a g } ( \phi _ { \Sigma } ( \mathbf { s } ) ) ) \ | | \mathcal { N } ( 0 , \mathbb { I } ) \right) \right] \quad ( \mathrm { V I B ~ r e g u l a r i z e r } ) ,
+$$
+
+Guideline 4.1. To address overfitting, we recommend using some form of capacity-decreasing regularization on the $Q$ -function, such as dropout or the VIB regularizer shown in Equation 3.
+
+Capacity-increasing techniques for underfitting. To address underfitting, we need to increase model capacity to improve optimization of the training objective. Analogous to supervised learning, model capacity can be increased by using more expressive neural nets (e.g., ResNets [27], transformers [28]) for representing the learned policy. We use ResNets in our experiments (Figure 10). However, the RL setting presents an additional challenge with capacity: while larger models in principle have more capacity, recent work [29, 21, 22] has shown that utilizing larger networks to represent Q-functions does not always improve its capacity in practice, because TD-based RL methods introduce an “implicit under-parameterization” effect that can result in aliased (i.e., similar) internal representations for different state-action inputs, even for very large neural networks that can express the true Q-function effectively. To address this issue, these works apply a “capacityincreasing” regularizer to Q-function training. For instance, we can use the DR3 regularizer [22], which penalizes the dot product of $\phi ( \mathbf { s } )$ and $\phi ( \mathbf { s } ^ { \prime } )$ for a transition $( \mathbf { s } , \mathbf { a } , \mathbf { s } ^ { \prime } ) \in \mathcal { D }$ , and hence reduces aliasing. This objective is given by:
+
+$$
+\operatorname* { m i n } _ { \theta } \ \mathcal { L } _ { \mathrm { C Q L } } ( \theta ) + \beta \mathbb { E } _ { { \mathbf s } , { \mathbf a } , { \mathbf s } ^ { \prime } \sim \mathcal { D } } \left[ \left| \phi ( { \mathbf s } ) ^ { \top } \phi ( { \mathbf s } ^ { \prime } ) \right| \right] \qquad ( { \mathrm { D R 3 ~ r e g u l a r i z e r ~ } } [ 2 2 ] ) ,
+$$
+
+Guideline 4.2. To address underfitting, we recommend using some capacity-increasing regularization on the Q-function and the policy either in conjunction or separately. Examples: (1) bigger policy networks (e.g., ResNets), (2) DR3 regularizer on the Q-network.
+
+# 5 Evaluation of Our Workflow Metrics and Protocols in Simulation
+
+Next, we empirically validate the workflow proposed in Sections 3 and 4 on a suite of simulated robotic manipulation domains that mimic real-robot scenarios, from image observations with sparse binary rewards. We will examine how applying the workflow in Section 3 to detect overfitting or underfitting and then utilizing the strategies in Section 4 affects the performance of offline RL methods. An improved performance would indicate the efficacy of our workflow in making successful design decisions without any online tuning.
+
+![](images/feadf98282550c14dd6939d9b1aaebdb526316856a1034070ef48af1c82a1480.jpg)
+
+Experimental setup. We use the environments from Singh et al. [3] to design offline RL tasks and datasets that we use for our empirical analysis. We consider two tasks: (1) a pick and place task and (2) a grasping object from a drawer task. Examples of trajectories in both of these simulated domains are shown in Figure 2 and are detailed in Appendix D. Briefly, the pick and place task consists of a 6-DoF WidowX robot in front of a tray with an object. The goal is to put the object inside the tray. A non-zero reward of $+ 1$ is provided only when the object has been placed in the box. The offline dataset for this task consists of trajectories that grasp an object with a $3 5 \%$ success and other trajectories that place an object with a $40 \%$ success. Our second task is a grasping from drawer task where the WidowX robot is placed in front of a drawer and multiple objects. The robot can open or close the drawer, grasp objects from inside the drawer or on the table, and place them anywhere in the scene. The goal is to close the top drawer, then open the bottom drawer and take the object out. Only if the object has been taken out, a reward of $+ 1$ is obtained. The offline dataset consists of trajectories with a $3 0 { - } 4 0 \%$ success rate for opening and closing a drawer and other trajectories with only $40 \%$ placing success. We use $\alpha = 1 . 0$ for CQL training in all experiments, which is directly taken from prior work [3], without any tuning. However, too low or too high $\alpha$ values will inhibit the effectiveness of regular CQL and we first need to tune $\alpha$ as discussed in Appendix G. More details are provided in Appendix D.
+
+![](images/7149b62299170ecd7a26a8a110d939daa58337b186bfd1671932f83c2c649963.jpg)  
+Figure 3: Policy performance (Top) and average dataset Q-values of CQL (bottom) with varying number of trajectories. Vertical bands indicate regions around the peak in average $\mathrm { Q }$ -value and observe that these regions correspond to policies with good actual performance.
+
+Scenario #1: Variable amount of training data. Our first scenario consists of the simulated tasks discussed above with a variable number of trajectories in the training data (50, 100, 500, 10000). We run CQL and track metrics 3.1 and 3.2 in each case. Observe in Figure 3 (bottom) that with fewer trajectories, the average dataset Q-value $\mathbb { E } _ { \mathbf { s } , \mathbf { a } \sim \mathcal { D } } [ Q _ { \theta } ( \mathbf { s } , \mathbf { a } ) ]$ first rises, and then drops. This matches the description of overfitting in Section 3. Observe in Figure 4 (left) that, at the same time, the value of the CQL regularizer is very low, which is not consistent with what we expect of underfitting. Thus, we can conclude that these conditions exhibit overfitting, especially with 50 and 100 trajectories. The vertical dashed lines indicate the checkpoints that would be selected for evaluation per Guideline 3.1. We further visualize the performance of the chosen checkpoints against the actual return of each intermediate policy in Figure 3 (top). Note that this value is obtained by rolling out the learned policy, and would not be available in a realistic offline RL setting, but is provided only for analysis. Selecting the checkpoint based on Guideline 3.1 leads us to select a model with close to the peak performance over the training process, validating the efficacy of Guideline 3.1.
+
+Since we detected overfitting by following our workflow, we now aim to address it by using the VIB regularizer in the setting with 100 trajectories. As shown in Figure 4 (right), applying this regularizer not only alleviates the drop in Q-values after many training steps, but allows us to pick later checkpoints in training which perform better than base CQL on both the tasks. This validates that overfitting, as detected via our workflow, can be effectively mitigated by decreasing capacity, in this case by using VIB. We evaluate dropout, $\ell _ { 1 }$ and $\ell _ { 2 }$ regularization schemes in Appendix J.
+
+![](images/c8b831ad68078d018a8733761f0da0593a9224d909869ecb852f0ffd8ee06f24.jpg)  
+Figure 4: Left: CQL regularizer attains low values, especially with 50 and 100 trajectories in the pick and place task, Right: Using VIB mitigates overfitting, giving rise to a stable trend in $\mathrm { Q }$ -values and better performance which does not degrade with more training steps.
+
+Scenario #2: Multiple training objects. Our second test scenario consists of the pick and place task, modified to include a variable number of object types (1, 5, 10, 20, 35). Handling more objects requires higher capacity, since each object has a different shape and appearance. In each case, CQL is provided with 5000 trajectories. Following our workflow from Section 3, we first compute the average dataset Q-value and the training TD error. We observe in Figure 5 that, unlike in Scenario #1, Q-values do not generally decrease when trained for many steps, suggesting that the Q-function is likely not overfitting. To check for underfitting, we visualize the training TD error and find that, with 10, 20 and 35 objects, TD error magnitudes are in the range of [1.0, 2.0], which suggests a overall Q-value error of [30.0, 60.0] since the task horizon is 30. On an absolute scale, this error magnitude is large: since the rewards are $_ { 0 / 1 }$ , the range of difference between actual Q-values for any two policies is at most 30, which suggests that the error magnitude in the runs in Figure 5 are high. Hence, we conclude that this scenario generally exhibits underfitting with more objects. Indeed this trend is reflected in the policy performance that we plot for analysis in Figure 5: note that the policy return decreases with an increased number of objects, and the policy performance initially increases and saturates at a suboptimal value.
+
+![](images/abb86c89607619cd53397449a92fc47f01701017a73f7df2599b8c654f14d0ca.jpg)  
+Figure 5: Performance (left), TD error (middle) and average dataset Qvalues (right) for the pick and place task with a variable number of objects. Note that while the learned Q-values increase and stabilize, the TD error values in scenarios with more than 10 objects are large (1.0-2.0). Correspondingly, the performance generally decreases as the number of objects increases.
+
+![](images/b4ce278afd05bdaaa57ddb722d6767ecad2d352a23720d7d7f144183cc608616.jpg)  
+Figure 6: Correcting underfitting by applying our workflow for 35 objects.
+
+To address underfitting in the multi-object case, we apply the proposed capacity-increasing measures to the 35-object task (results for 10 and 20 object settings are in Appendix I). We use a more expressive ResNet architecture for the policy and the DR3 regularizer for the Q-function together. Observe in the figure on the right that this combination (shown in red) improves policy performance in this setting (compared to green), which validates our workflow protocol for addressing underfitting.
+
+# 6 Tuning CQL for Real-World Robotic Manipulation
+
+Having evaluated the efficacy of our proposed workflow in simulation, we now utilize our workflow to tune CQL for real-world robotic manipulation. We test in two setups that require the robot to learn from sparse binary rewards and image observations. The settings differ in robot platform, task specification, and dataset size. Additional results and robot videos are at the following website: https://sites. google.com/view/offline-rl-workflow
+
+Sawyer manipulation tasks [30]. First, we train a
+
+![](images/566e1138148ac542d1676690513ab2bab4af3cbfec1257dc70a864019d00681c.jpg)  
+Figure 7: Real-world tasks. Successful rollouts of CQL tuned with our workflow from Sections 3 & 4. Top to bottom: Sawyer lid on pot, Sawyer drawer opening, WidowX pick-place task.
+
+Sawyer robot in a tabletop setting to perform two tasks: (1) placing the lid onto a pot and (2) opening a drawer. The robot must perform these tasks in the presence of visual distractor objects, as shown in Figure 7. We directly use the dataset of 100 trajectories for each task collected by Khazatsky et al. [30] for our experiments so as to mimic the real-world use case of leveraging existing data with offline RL. We use four-dimensional actions with 3D end-effector velocity control in xyz-space and 1D gripper open/close action. More details regarding the setup are provided in Appendix D.
+
+We run default CQL on these tasks and track the average Q-value, TD error, and CQL regularizer value. As shown in Figure 8, the average Q-value does not decrease over training, and the TD error (and CQL regularizer shown in Appendix E.2) is large. Per our discussion in Section 3, this indicates underfitting. Following our guidelines from Section 4, we utilize a more expressive ResNet policy (Figure 10), which increases the number of total convolutional layers from 3 to 9. We observe that this reduces the values of both the TD error Figure 8 and CQL regularizer (Appendix E.2) on both tasks. We
+
+![](images/d5317090e1ad32bc153c1daf1e4d3802dfab85d171f13b7f15890473be1358cc.jpg)  
+Figure 8: Average Q-value and TD error on Sawyer tasks as model capacity increases. Q-values increase over training with lower capacity ruling out overfitting and increasing model capacity leads to a reduction in TD error indicating the presence of underfitting.
+
+then evaluate the learned policy over 12 trials conducted with different sets of distractor objects, including ones that are unseen during training. While the policy trained using base CQL is unable to successfully complete either task even once attaining a score of 0/12 on both tasks, the run that uses ResNet attains a significantly better success rate of 9/12 on the put lid on pot task and 8/12 on the drawer opening task, equal to $7 0 . 8 \%$ success rate on average.
+
+![](images/b9dc053e0a74e8660fec33a5ef35e25d9ae5b1e03b4824f6ff4428e483bb79a7.jpg)  
+Figure 9: Q-values (left) and performance of CQL with (middle) and without (right) the variational information bottleneck correction for overfitting on the real-world widowX pick and place task. Since the Q-values start to decrease with more training, our workflow detects that CQL is overfitting. Using our policy selection guideline (Guideline 3.1) enables us to choose checkpoint 50 marked with the green vertical dashed line (right) which performs well. Further, addressing overfitting by applying the VIB regularizer stabilizes the Q-values (brown) which do not decrease unlike base CQL (blue) (left). Finally, applying the VIB regularizer improves performance and reduces sensitivity to policy selection (middle).
+
+WidowX pick and place task. In our second setting, we tune CQL on a pick and place task with a WidowX 250 robotic arm, shown in Figure 7. The dataset consists of 200 trajectories collected by running a noisy scripted policy (Appendix D) with $3 5 \%$ success. We run CQL on this task and track the average Q-values, which we find initially increase and then decrease (Figure 9 (left; labeled as “Q-values”)), indicating overfitting. We then evaluate our policy selection scheme, which in this case suggests deploying checkpoint 50, the immediate checkpoint after the peak in Q-values. To see if this checkpoint is effective, we evaluate the performance of a few other policy checkpoints (for analysis only) and plot this performance trend in Figure 9 (right) as a dashed line. Observe that indeed the checkpoint found by our workflow attains the highest success rate (7/9) compared to other checkpoints, which only succeed $\leq 4 / 9$ times.
+
+Since overfitting is detected, we now turn to addressing overfitting by adding the VIB regularizer (Equation 3) during training. As shown in Figure 9 (left), the Q-values obtained after the addition of this regularizer (shown in brown; labeled “Q-values (VIB)”) are now stable and do not decrease over the course of training and so we can choose any policy for evaluation. We evaluate multiple policies, for visualization pur
+
+<table><tr><td rowspan=1 colspan=5>Real-world WidowX pick and place</td></tr><tr><td rowspan=1 colspan=1>MethodEpoch</td><td rowspan=1 colspan=1>50</td><td rowspan=1 colspan=1>75</td><td rowspan=1 colspan=1>100</td><td rowspan=1 colspan=1>200</td></tr><tr><td rowspan=1 colspan=1>CQL</td><td rowspan=1 colspan=1>7/9</td><td rowspan=1 colspan=1>4/9</td><td rowspan=1 colspan=1>4/9</td><td rowspan=1 colspan=1>2/9</td></tr><tr><td rowspan=1 colspan=1>CQL + VIB</td><td rowspan=1 colspan=1>3/9</td><td rowspan=1 colspan=1>8/9</td><td rowspan=1 colspan=1>7/9</td><td rowspan=1 colspan=1>7/9</td></tr></table>
+
+Table 2: Performance of various policy checkpoints of CQL and $\mathrm { C Q L + V I B }$ on the real WidowX pick and place task (bold entry denotes the checkpoint selected by our workflow). Note that when overfitting is corrected via VIB, multiple checkpoints perform well.
+
+poses only, in Figure 9 (middle), we find that all of them attain a $\geq 7 / 9$ success, comparable or better than the base CQL algorithm (Figure 9 (right)). This indicates that addressing overfitting not only leads to some gains in performance but also greatly simplifies policy selection as all checkpoints perform similarly and well. Table 2 summarizes these results below, where the bold entries denote the checkpoints found by our policy selection rule. These results indicate the effectiveness of our workflow in tuning CQL by addressing overfitting and underfitting on multiple real robot platforms.
+
+# 7 Discussion
+
+While offline RL algorithms have improved significantly, applying these methods to real-world robotic domains is still challenging due to little guidance on tuning them. In this paper, we devise a workflow for algorithms such as CQL and BRAC, which consists of a set of metrics and conditions that can be tracked by a practitioner over the course of offline training to detect overfitting and underfitting, and recommendations to addresses the observed challenges. Applying our workflow both in simulation and the real world shows strong performance benefits. While our proposed workflow is an initial step towards practical robotic offline RL and is based on our best conceptual understanding of certain offline RL algorithms, these guidelines are heuristic. To some extent this is unavoidable, since a workflow is a set of guidelines and recommendations, rather than a rigid algorithm. Regardless of how theoretically justified it is, in the end, its value is determined by its ability to produce good results. We believe the breadth of tasks considered, which consist of two different real robots and multiple simulated tasks, indicates its broad applicability. However, deriving theoretical guarantees regarding workflows of this type is an important direction for future research.
+
+# Acknowledgements
+
+We thank Ilya Kostrikov, Avi Singh, Ashvin Nair, Alexander Khazatsky, Albert Yu, Jedrzej Orbik, and Jonathan Yang for their help with setting up and debugging various aspects of the experimental setup as well as for providing us with offline datasets we could test our workflow on. We thank Dibya Ghosh, anonymous reviewers, and the area chair from CoRL for constructive feedback on an earlier version of this paper. AK thanks George Tucker and Rishabh Agarwal for valuable discussions. This research was funded by the DARPA Assued Autonomy Program and compute support from Google and Microsoft Azure.
+
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parse/train/fy4ZBWxYbIo/fy4ZBWxYbIo_content_list.json

@@ -0,0 +1,1056 @@
+[
+    {
+        "type": "text",
+        "text": "A Workflow for Offline Model-Free Robotic Reinforcement Learning ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Aviral Kumar?,1, Anikait Singh?,1, Stephen $\\mathbf { T i a n } ^ { 1 }$ , Chelsea $\\mathbf { F i n n ^ { 2 } }$ , Sergey Levine1 1 UC Berkeley, 2 Stanford University (∗ Equal Contribution) aviralk@berkeley.edu, asap7772@berkeley.edu ",
+        "bbox": [
+            218,
+            176,
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Abstract: Offline reinforcement learning (RL) enables learning control policies by utilizing only prior experience, without any online interaction. This can allow robots to acquire generalizable skills from large and diverse datasets, without any costly or unsafe online data collection. Despite recent algorithmic advances in offline RL, applying these methods to real-world problems has proven challenging. Although offline RL methods can learn from prior data, there is no clear and well-understood process for making various design choices, from model architecture to algorithm hyperparameters, without actually evaluating the learned policies online. In this paper, our aim is to develop a practical workflow for using offline RL analogous to the relatively well-understood workflows for supervised learning problems. To this end, we devise a set of metrics and conditions that can be tracked over the course of offline training, and can inform the practitioner about how the algorithm and model architecture should be adjusted to improve final performance. Our workflow is derived from a conceptual understanding of the behavior of conservative offline RL algorithms and cross-validation in supervised learning. We demonstrate the efficacy of this workflow in producing effective policies without any online tuning, both in several simulated robotic learning scenarios and for three tasks on two distinct real robots, focusing on learning manipulation skills with raw image observations with sparse binary rewards. Explanatory video and additional content can be found at sites.google.com/view/offline-rl-workflow. ",
+        "bbox": [
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Keywords: workflow, offline RL, offline tuning ",
+        "bbox": [
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "1 Introduction ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            580,
+            312,
+            595
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Offline reinforcement learning (RL) can in principle make it possible to convert existing large datasets of robotic experience into effective policies, without the need for costly or dangerous online interaction for each training run. While offline RL algorithms have improved significantly [1, 2, 3, 4, 5], applying such methods to real-world robotic control problems presents a number of major challenges. In standard online RL, any intermediate policy found during training is executed in the environment to collect more experience, which naturally allows for an evaluation of the policy performance. This ability to evaluate intermediate policies lets practitioners use “brute",
+        "bbox": [
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+        "page_idx": 0
+    },
+    {
+        "type": "image",
+        "img_path": "images/93b250387a2633b12ccbb56017457a77edb5b8bb68d22310145c16c8258e9664.jpg",
+        "image_caption": [
+            "Figure 1: Our proposed workflow aims to detect overfitting and underfitting, and provides guidelines for addressing these issues via policy selection, regularization, and architecture design. We evaluate this workflow on two real-world robotic systems and simulation domains, and we find it to be effective. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "force” to evaluate the effects of various design factors, such as model capacity and expressivity, the number of training steps, and so forth, and facilitates comparatively straightforward tuning. In contrast, offline RL methods do not have access to real-world on-policy rollouts for evaluating the learned policy. Thus, in order for these methods to be truly practical for real-world applications, we not only require effective algorithms, but also an effective workflow: a set of protocols and metrics that can be used to reliably and consistently adjust model capacity, regularization, etc in offline RL to obtain policies with good performance, without requiring real-world rollouts for tuning. ",
+        "bbox": [
+            174,
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "A number of prior works have studied model selection in offline RL by utilizing off-policy evaluation (OPE) methods [6] to estimate policy performance. These methods can be based either on model or value learning [7, 8, 9, 10] or importance sampling [6, 11, 12, 13]. However, developing reliable OPE methods is itself an open problem, and modern OPE methods themselves suffer from hyperparameter selection challenges (see Fu et al. [14] for an empirical study). Moreover, accurate off-policy evaluation is likely not necessary to simply tune algorithms for best performance – we do not need a precise estimate of how good our policy is, but rather a workflow that enables us to best improve it by adjusting various algorithm hyperparameters. ",
+        "bbox": [
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "In this paper, we devise a practical workflow for selecting regularizers, model architectures, and policy checkpoints for offline RL methods in robotic learning settings. We focus on a specific class of conservative offline RL algorithms [15, 2] that regularize the Q-function, but also show that our workflow can be effectively applied to policy constraint methods [16]. Our aim is not to focus on complete off-policy evaluation or to devise a new approach for off-policy evaluation, but rather to adopt a strategy similar to the one in supervised learning. Analogously to how supervised learning practitioners can detect overfitting and underfitting by tracking training and validation losses, and then adjust hyperparameters based on these metrics, our workflow (see Figure 1 for a schematic) first defines and characterizes overfitting and underfitting, proposes metrics and conditions that users can track to determine if an offline RL exhibits overfitting or underfitting, and then utilizes these metrics to inform design decisions pertaining to neural net architectures, regularization, and early stopping. This protocol is intended to act as a “user’s manual” for a practitioner, with guidelines for how to modify algorithm parameters for best results without real-world evaluation rollouts. ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "The primary contribution of this paper is a simple yet effective workflow for robotic offline RL. We propose metrics and protocols to assist practitioners in selecting policy checkpoints, regularization parameters, and model architectures for conservative offline RL algorithms such as CQL [2] and BRAC [16]. We empirically verify the efficacy of our proposed workflow on simulated robotic manipulation problems as well as three real-world robotic manipulation problems on two different robots, with diverse objects, pixel observations, and sparse binary reward supervision. Experimentally, we evaluate our method on two real-world robots (the Sawyer and WidowX robots), and one realistic simulated tasks. Our approach is effective in all of these cases, and on two tasks with the Sawyer robot that initially fail completely, our workflow improves the success rate to $70 \\%$ . ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "2 Preliminaries, Background, and Definitions ",
+        "text_level": 1,
+        "bbox": [
+            173,
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "The goal in RL is to optimize the infinite horizon discounted return $\\begin{array} { r } { R = \\sum _ { t = 0 } ^ { \\infty } \\gamma ^ { t } r ( \\mathbf { s } _ { t } , \\mathbf { a } _ { t } ) } \\end{array}$ , where $r ( s , a )$ represents the reward function evaluated at a state-action pair $( \\mathbf { s } , \\mathbf { a } )$ . We operate in the offline RL setting and are provided with a fixed dataset $\\mathcal { D } = \\{ ( \\mathbf { s } , \\mathbf { a } , r ( \\mathbf { s } , \\mathbf { a } ) , \\mathbf { s } ^ { \\prime } ) \\}$ , consisting of transition tuples obtained from rollouts under a behavior policy $\\pi _ { \\beta } ( \\mathbf { a } | \\mathbf { s } )$ . Our goal is to obtain the best possible policy by only training on this fixed offline dataset $\\mathcal { D }$ , with no access to online rollouts. We focus on conservative offline RL algorithms that modify the $\\mathrm { Q }$ -function to penalize distributional shift, with most experiments on CQL [2], though we also adapt our workflow to BRAC [16] in Appendix F.1. ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Conservative Q-learning (CQL). The actor-critic formulation of CQL trains a Q-function $Q _ { \\boldsymbol { \\theta } } ( \\mathbf { s } , \\mathbf { a } )$ with a separate policy $\\pi _ { \\phi } ( \\mathbf { a } | \\mathbf { s } )$ , which maximizes the expected $\\mathrm { Q }$ -value $\\begin{array} { r } { \\mathbb { E } _ { \\mathbf { s } \\sim \\mathcal { D } , \\mathbf { a } \\sim \\pi _ { \\phi } } \\left[ Q _ { \\theta } ( \\mathbf { s } , \\mathbf { a } ) \\right] } \\end{array}$ like other standard actor-critic deep RL methods [17, 18, 19]. However, in addition to the standard TD error ${ \\mathcal { L } } _ { \\mathrm { T D } } ( \\theta )$ (in blue below), CQL applies a regularizer ${ \\mathcal { R } } ( \\theta )$ (in red below) to prevent overestimation of $\\mathrm { Q }$ -values for out-of-distribution (OOD) actions. This term minimizes the $\\mathrm { Q }$ -values under a distribution $\\mu ( \\mathbf { a } | \\mathbf { s } )$ , which is automatically chosen to pick actions a with high Q-values $Q _ { \\boldsymbol { \\theta } } ( \\mathbf { s } , \\mathbf { a } )$ , and counterbalances this term by maximizing the values of the actions in the dataset: ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "equation",
+        "img_path": "images/14481712837aa8b4041c0afbae4d0a62161a885580f62a118ffd01fc68c921ec.jpg",
+        "text": "$$\n\\begin{array} { r l } { \\underset { \\theta } { \\mathrm { m i n } } \\ : \\ : \\ : } & { { } \\ : \\ : \\ : \\ : \\ : \\ : \\ : \\ : \\ : \\ : \\ : \\ : \\ : \\ : \\ : \\ : \\ : \\ : \\ : \\ : \\ : \\ : \\ : \\ : \\ : \\ : \\ : \\ : \\ : \\ : \\ : \\ : \\ : \\ : \\ : \\ : \\ : \\ : } \\\\ { \\mathrm { m i n } \\ : \\ : \\ : \\ : \\ : } & { { } \\ : \\ : \\ : \\ : \\ : \\ : \\ : \\ : \\ : \\ : \\ : \\ : \\ : \\ : \\ : \\ : \\ : } & { \\mathrm { ~ \\ : ~ \\ : \\ : \\ : \\ : } \\ : \\ : \\ : } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            163,
+            741,
+            810,
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "where $B ^ { \\pi } \\bar { Q } ( \\mathbf { s } , \\mathbf { a } )$ is the Bellman backup operator with a delayed target Q-function, $\\bar { Q }$ : $B ^ { \\pi } \\bar { Q } ( { \\bf s } , { \\bf a } ) : =$ $r ( \\mathbf { s } , \\mathbf { a } ) + \\gamma \\mathbb { E } _ { \\mathbf { a ^ { \\prime } } \\sim \\pi ( \\mathbf { a ^ { \\prime } } | \\mathbf { s ^ { \\prime } } ) } [ \\bar { Q } ( \\mathbf { s ^ { \\prime } } , \\mathbf { a ^ { \\prime } } ) ]$ . In practice, CQL computes $\\mu ( \\mathbf { a } | \\mathbf { s } )$ using actions sampled from the policy $\\pi _ { \\phi } ( \\mathbf { a } | \\mathbf { s } )$ . More discussion of CQL is in Appendix B. In this paper, we will utilize CQL as a base algorithm that our workflow intends to tune, but we also extend it to BRAC. ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Overfitting and underfitting in CQL. Conservative offline RL algorithms [2, 20] like CQL can be sensitive to design choices, including number of gradient steps for training [21, 22] and network capacity. These challenges are also present in supervised learning, but supervised learning methods benefit from a simple and powerful workflow that involves using training error and validation error to characterize overfitting and underfitting. A practitioner can then make tuning choices based on these characterizations. To derive an analogous workflow for offline RL, we first ask: what do overfitting and underfitting actually mean for the case of conservative offline RL? ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "To define overfitting and underfitting generically for any conservative offline RL method, we consider an abstract optimization formulation for such methods [2]: ",
+        "bbox": [
+            174,
+            126,
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/5f87e2d9edb1ebc744585eaf9bbbefae1189640155a200a885d288ec1f092126.jpg",
+        "text": "$$\n\\pi ^ { * } : = \\arg \\operatorname* { m a x } _ { \\pi } ~ J _ { \\mathcal { D } } ( \\pi ) - \\alpha D ( \\pi , \\pi _ { \\beta } )\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "(Conservative offline RL). ",
+        "bbox": [
+            504,
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "$J _ { \\mathcal { D } } ( \\pi )$ denotes the average return of policy $\\pi$ in the empirical MDP induced by the transitions in the offline dataset $\\mathcal { D }$ , and $D ( \\pi , \\pi _ { \\beta } )$ denotes a closeness constraint to the behavior policy, effectively applied by the offline RL method. Our definition of conservative offline RL requires that this ",
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "table",
+        "img_path": "images/1e432827fa691c00d934499d561870f9ab21e128eee2a69b3018b096d8e40450.jpg",
+        "table_caption": [
+            "Table 1: Summary of train error, test error and our definitions of overfitting and underfitting in supervised learning and conservative offline RL methods. We will propose metrics to measure these phenomena in a purely offline manner and recommend how to tune the underlying method accordingly. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Quantity</td><td>Supervised Learning</td><td>Conservative Offline RL</td></tr><tr><td>Test error</td><td>Loss L evaluated on test data,Dtest</td><td>Performance of policy,J(π)</td></tr><tr><td>Train error</td><td>Loss L evaluated on train data,Dtrain</td><td>Objective in Equations 2,1</td></tr><tr><td>Overfitting</td><td>L(Dtrain) low,L(Dval) high,Dval is a validation set drawn i.i.d.as Dtrain</td><td>Training objective in Equation l is ex- tremely low,low value of J(π)</td></tr><tr><td>Underfitting</td><td>high value of train error L(Dtrain)</td><td>Training objective in Equation 1 is ex- tremely high,low value of J(π)</td></tr></table>",
+        "bbox": [
+            377,
+            193,
+            812,
+            281
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "divergence be computed in expectation over the state visitation distribution of the learned policy $\\pi$ in the empirical MDP as discussed in Appendix F.1. For example, Equation 1 translates to utilizing $\\begin{array} { r } { D _ { \\mathrm { C Q L } } ( p , \\bar { q } ) : = \\sum _ { \\mathbf { x } } p ( \\mathbf { x } ) ( p ( \\mathbf { x } ) / q ( \\mathbf { x } ) - \\bar { 1 } ) } \\end{array}$ in Equation 2 (see Theorem 3.5 in Kumar et al. [2] for a proof). The training loss is discussed in Equations 1 and 2 and the test loss is equal to the negative of the actual return $J ( \\pi )$ of the learned policy. Analogously to supervised learning, we can use the notion of train and test error to define overfitting and underfitting in offline RL, as discussed in Table 1. However, note that the conditions summarized in Table 1 are not measurable completely offline. Precisely estimating if a run of an offline RL method overfits or underfits requires evaluating the learned policy via interaction with the real-world environment. In Section 3, our goal will be to devise offline metrics for characterizing overfitting that do not have this requirement. We will tailor our study specifically towards CQL, though we extend it to BRAC in Appendix F.1. A similar procedure could be devised for other offline RL methods, but we leave this for future work. ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "3 Detecting Overfitting and Underfitting in Conservative Offline RL ",
+        "text_level": 1,
+        "bbox": [
+            173,
+            517,
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "In standard supervised learning, we can determine if a method overfits or underfits by comparing the training loss to the same loss function evaluated on a held-out validation dataset, which serves as a “proxy” test dataset. In contrast, the return of the learned policy $J ( \\pi )$ in RL does not have a direct proxy that can be computed offline. Thus, our goal is to identify offline metrics and conditions that allow us to measure overfitting and underfitting in conservative offline RL, with a focus on CQL. We also adapt these conditions to BRAC [16], a policy-constraint method in Appendix F.2. ",
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Detecting overfitting in CQL. Our definition of overfitting (Table 1) corresponds to a low value for the training loss (Equation 1), but poor actual policy performance $J ( \\pi )$ . To detect this, we analyze the time series of the estimated Q-values averaged over the dataset samples $( \\mathbf { s } , \\mathbf { a } , r , \\mathbf { s } ^ { \\prime } ) \\in \\mathcal { D }$ over the course of training with a large number of gradient steps. A run is labeled as overfitting if we see that the expected dataset Q-value exhibits a non-monotonic trend: if the average Q-values first increase and then decrease as shown in the figure on the right. Additionally, we would see that training loss in Equation 1 eventually becomes very low. Why do we see such a trend in the average dataset $\\mathbf { Q }$ -value? Since CQL selectively penalizes the average Qvalue under the distribution $\\mu ( \\mathbf { a } | \\mathbf { s } )$ supported on actions with large Q-values, we would expect the Q-values on states from the dataset s $\\sim \\mathcal { D }$ and the learned $\\mathbf { a } \\sim \\pi ( \\cdot | \\mathbf { s } )$ to be small since the policy is trained to maximize the Q-function as well. This in turn would lead to an eventual reduction in the average Q-value on dataset actions, $\\mathbb { E } _ { \\mathbf { s } , \\mathbf { a } \\sim \\mathcal { D } } [ Q _ { \\theta } ( \\mathbf { s } , \\mathbf { a } ) ]$ . This would be visible after sufficiently many steps of training, when values have propagated via Bellman backups in Equation 1 giving rise to the non-monotonic trend. If such a trend is observed, this raises two questions, as we discuss next. ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "image",
+        "img_path": "images/a834916e9152742e6ff0bc3376e3cee94d030ab397761d3b7ccebf41e7204020.jpg",
+        "image_caption": [],
+        "image_footnote": [],
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "What does a low average $\\varrho$ -value $\\mathbb { E } _ { \\mathbf { s } , \\mathbf { a } \\sim \\mathcal { D } } [ Q _ { \\theta } ( \\mathbf { s } , \\mathbf { a } ) ]$ imply about $J ( \\pi )$ ? We show in Appendix A that, in principle, CQL training (Equation 1) should never learn Q-values smaller than the dataset Monte-Carlo return, and the $\\mathrm { Q }$ -values should increase unless the learned policy $\\pi$ is better than $\\pi _ { \\beta }$ . Intuitively, this is because the objective in Equation 1 aims to also maximize the average dataset ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Q-value and thus the Q-values for the behavior policy are not underestimated in expectation. Now, if the policy optimizer finds a policy that attains a smaller learned Q-value than the dataset return, the policy can always be updated further towards the behavior policy so as to raise the Q-value. Therefore, Q-values can only decrease when the policy found by CQL is better than the behavior policy. We formalize this intuition in Appendix A in Theorem A.1. Thus, a low $\\mathrm { Q }$ -value on $( \\mathbf { s } , \\mathbf { a } ) \\in$ $\\mathcal { D }$ indicates that the Q-function predicts extremely small Q-values on actions sampled from $\\mu ( \\mathbf { a } | \\mathbf { s } )$ . Typically, this would mean the highest Q-value actions a at a state $\\mathbf { s } \\in \\mathcal { D }$ are those sampled from the offline dataset, drawn from the behavior policy. Thus, policy optimization, which aims to maximize the Q-value, would make $\\pi ( \\mathbf { a } | \\mathbf { s } )$ closer to the behavior policy $\\pi _ { \\beta } ( \\mathbf { a } | \\mathbf { s } )$ on $\\mathbf { s } \\in \\mathcal { D }$ . ",
+        "bbox": [
+            173,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Which training checkpoint is likely to attain the best policy performance? Tracking overfitting in supervised learning is important for selecting the best-performing checkpoint, before overfitting becomes severe. Analogously, we can compare the average dataset Q-value across different checkpoints within the same run to pick the best policy. Since CQL aims to increase the average dataset Q-value (Equation 1), we would expect Q-values to initially increase, until learning starts to overfit and the average dataset $\\mathrm { Q }$ -value starts decreasing. We should therefore select the latest checkpoint that corresponds to a peak in the estimated dataset Q-value. A visual illustration of this idea is shown in the figure on the previous page, where the checkpoint marked by the green line is recommended to be chosen. In summary, (a) to detect overfitting we can track: ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Metric 3.1 (Overfitting). A low average data $Q$ -value $\\mathbb { E } _ { \\mathbf { s } , \\mathbf { a } \\sim \\mathcal { D } } [ Q _ { \\theta } ( \\mathbf { s } , \\mathbf { a } ) ]$ that decreases with more gradient steps on Equation 1 indicates that the offline RL algorithm is overfitting. ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "and (b) further, given a run that exhibits overfitting, our principle for policy selection is given by: ",
+        "bbox": [
+            173,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Guideline 3.1 (Policy selection). If a run overfits (per Metric 3.1), select the checkpoint that attains the highest average dataset $Q$ -value before overfitting for deployment. ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Finally, for actor-critic algorithms [18] that update the actor slower than the critic, the next policy checkpoint after the peak in the average dataset Q-value appears must be selected. In most of our experiments, we find that simply utilizing the policy checkpoint at the point of the peak in the Qvalue also leads to good results making this a rare concern, but in some cases, utilizing the next checkpoint after the Q-value peak performs better empirically. ",
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+        "text": "Detecting underfitting in CQL. Next, we turn to devising a procedure to detect underfitting. As summarized in Table 1, underfitting occurs when the RL algorithm is unable to minimize the training objective in Equation 1 effectively. Therefore, large values for the TD error, the CQL regularizer, or both imply underfitting. A large value for the CQL regularizer, ${ \\mathcal { R } } ( \\theta )$ , indicates an overestimation of Q-values relative to their true value [2] and thus, unlike the overfitting regime, we would not expect the average learned $\\mathrm { Q }$ -value to decrease with more training. Thus, one approach to predict underfitting is to track both the TD error, ${ \\mathcal { L } } _ { \\mathrm { T D } } ( \\theta )$ , and the CQL regularizer, ${ \\mathcal { R } } ( \\theta )$ , and check if the value of even one of these quantities is large. More discussion is provided in Appendix A. ",
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+        "type": "text",
+        "text": "How do we determine if the $\\mathbf { \\nabla } ^ { T D }$ error and the CQL regularizer are “large”? In order to determine if the error of a particular run is large, we can rerun the base CQL algorithm but with models of higher capacity, which does not necessarily correspond to the function approximator size, as we will discuss in Section 4. For each model, we record the corresponding training errors and check if the training TD error and CQL regularizer value are reduced with capacity increase. If increasing capacity leads to a reduction in the loss without exhibiting the overfitting signs described previously, then we are in an underfitting regime. Another approach to answer the question is to utilize the value of the TD error $\\left( \\mathcal { L } _ { \\mathrm { T D } } ( \\theta ) \\right)$ and the task horizon $( 1 / ( 1 - \\gamma ) )$ to estimate the overall error in the learned Q-values against the actual Q-value, which is equal to $\\dot { \\mathcal { L } } _ { \\mathrm { T D } } ( \\theta ) / ( 1 - \\gamma )$ [23] (see Appendix A). If this overall error spans the range of allowed Q-values on the task – which could be inferred based on the structure of the reward function in the task – then we can say that the algorithm is underfitting. ",
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+        "type": "text",
+        "text": "Metric 3.2 (Underfitting). Compute the values of the training $T D$ error, ${ \\mathcal { L } } _ { \\mathrm { T D } } ( \\theta )$ and CQL regularizer, ${ \\mathcal { R } } ( \\theta )$ for the current run and another identical run with increased model capacity. If the training errors reduce with increasing model capacity, the original run was underfitting. ",
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+    {
+        "type": "text",
+        "text": "4 Addressing Overfitting and Underfitting in Conservative Offline RL ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "The typical workflow for supervised learning not only identifies overfitting and underfitting, but also guides the practitioner how to adjust their method so as to alleviate it (e.g., by modifying regularization or model capacity), thus improving performance. Can we devise similar guidelines to address overfitting and underfitting with conservative offline RL? Here, we discuss some ways to adjust regularization and model capacity to alleviate these phenomena. ",
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+        "text": "Capacity-decreasing regularization for overfitting. As we observed in Section 3, the mechanism behind extremely low $\\mathbf { Q }$ -values on the dataset is that CQL training minimizes $\\mathrm { Q }$ -values on actions sampled from $\\mu ( \\mathbf { a } | \\mathbf { s } )$ . Two possible approaches to preventing over-minimization of these values are (1) applying regularization such as dropout [24] on Q-function layers, similar to supervised learning, and (2) enforcing that representations of the learned Q-function match a pre-specified target for all state-action tuples. For (2), we can apply techniques such as a variational information bottleneck (VIB) [25, 26] regularizer on the learned representations, $\\phi ( \\mathbf { s } )$ . Formally, let $( \\mathbf { s } , \\mathbf { a } )$ denote a stateaction pair. Instead of predicting a deterministic $\\phi ( \\mathbf { s } ) \\in \\mathbb { R } ^ { d }$ (Figure 10), we modify the Q-network to predict two distinct vectors, $\\phi _ { m } ( \\mathbf { s } ) \\in \\mathbb { R } ^ { d }$ and $\\phi _ { \\Sigma } ( \\mathbf { s } ) \\in \\mathbb { R } ^ { d }$ , and sample $\\phi ( \\mathbf { s } )$ randomly from a Gaussian centered at $\\phi _ { m }$ with covariance $\\phi _ { \\Sigma }$ , i.e., $\\phi ( \\mathbf { s } ) \\sim \\mathcal { N } ( \\phi _ { m } ( \\mathbf { s } ) , \\mathrm { d i a g } ( \\phi _ { \\Sigma } ( \\mathbf { s } ) )$ . VIB then regularizes $\\mathcal { N } ( \\phi _ { m } ( \\mathbf { s } ) , \\mathrm { d i a g } ( \\phi _ { \\Sigma } ( \\mathbf { s } ) )$ to be close to a prior distribution, $\\mathcal { N } ( 0 , \\mathbb { I } )$ : ",
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+        "type": "equation",
+        "img_path": "images/9db0d18bcf179c6a8325f93c9a8d0886149e29c38e5c7dbf3d677141313eeab5.jpg",
+        "text": "$$\n\\operatorname* { m i n } _ { \\theta } \\ \\mathcal { L } _ { \\mathrm { C Q L } } ( \\theta ) + \\beta \\mathbb { E } _ { \\mathrm { s } \\sim \\mathcal { D } } \\left[ \\mathrm { D } _ { \\mathrm { K L } } \\left( \\mathcal { N } ( \\phi _ { m } ( \\mathbf { s } ) , \\mathrm { d i a g } ( \\phi _ { \\Sigma } ( \\mathbf { s } ) ) ) \\ | | \\mathcal { N } ( 0 , \\mathbb { I } ) \\right) \\right] \\quad ( \\mathrm { V I B ~ r e g u l a r i z e r } ) ,\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "Guideline 4.1. To address overfitting, we recommend using some form of capacity-decreasing regularization on the $Q$ -function, such as dropout or the VIB regularizer shown in Equation 3. ",
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+        "type": "text",
+        "text": "Capacity-increasing techniques for underfitting. To address underfitting, we need to increase model capacity to improve optimization of the training objective. Analogous to supervised learning, model capacity can be increased by using more expressive neural nets (e.g., ResNets [27], transformers [28]) for representing the learned policy. We use ResNets in our experiments (Figure 10). However, the RL setting presents an additional challenge with capacity: while larger models in principle have more capacity, recent work [29, 21, 22] has shown that utilizing larger networks to represent Q-functions does not always improve its capacity in practice, because TD-based RL methods introduce an “implicit under-parameterization” effect that can result in aliased (i.e., similar) internal representations for different state-action inputs, even for very large neural networks that can express the true Q-function effectively. To address this issue, these works apply a “capacityincreasing” regularizer to Q-function training. For instance, we can use the DR3 regularizer [22], which penalizes the dot product of $\\phi ( \\mathbf { s } )$ and $\\phi ( \\mathbf { s } ^ { \\prime } )$ for a transition $( \\mathbf { s } , \\mathbf { a } , \\mathbf { s } ^ { \\prime } ) \\in \\mathcal { D }$ , and hence reduces aliasing. This objective is given by: ",
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+    {
+        "type": "equation",
+        "img_path": "images/55e6e675c67ec1e3e4745a24fa9f244367408982eb82461b79c4a5d6cace2f4e.jpg",
+        "text": "$$\n\\operatorname* { m i n } _ { \\theta } \\ \\mathcal { L } _ { \\mathrm { C Q L } } ( \\theta ) + \\beta \\mathbb { E } _ { { \\mathbf s } , { \\mathbf a } , { \\mathbf s } ^ { \\prime } \\sim \\mathcal { D } } \\left[ \\left| \\phi ( { \\mathbf s } ) ^ { \\top } \\phi ( { \\mathbf s } ^ { \\prime } ) \\right| \\right] \\qquad ( { \\mathrm { D R 3 ~ r e g u l a r i z e r ~ } } [ 2 2 ] ) ,\n$$",
+        "text_format": "latex",
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+        "type": "text",
+        "text": "Guideline 4.2. To address underfitting, we recommend using some capacity-increasing regularization on the Q-function and the policy either in conjunction or separately. Examples: (1) bigger policy networks (e.g., ResNets), (2) DR3 regularizer on the Q-network. ",
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+        "type": "text",
+        "text": "5 Evaluation of Our Workflow Metrics and Protocols in Simulation ",
+        "text_level": 1,
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+        "text": "Next, we empirically validate the workflow proposed in Sections 3 and 4 on a suite of simulated robotic manipulation domains that mimic real-robot scenarios, from image observations with sparse binary rewards. We will examine how applying the workflow in Section 3 to detect overfitting or underfitting and then utilizing the strategies in Section 4 affects the performance of offline RL methods. An improved performance would indicate the efficacy of our workflow in making successful design decisions without any online tuning. ",
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+        "type": "text",
+        "text": "Experimental setup. We use the environments from Singh et al. [3] to design offline RL tasks and datasets that we use for our empirical analysis. We consider two tasks: (1) a pick and place task and (2) a grasping object from a drawer task. Examples of trajectories in both of these simulated domains are shown in Figure 2 and are detailed in Appendix D. Briefly, the pick and place task consists of a 6-DoF WidowX robot in front of a tray with an object. The goal is to put the object inside the tray. A non-zero reward of $+ 1$ is provided only when the object has been placed in the box. The offline dataset for this task consists of trajectories that grasp an object with a $3 5 \\%$ success and other trajectories that place an object with a $40 \\%$ success. Our second task is a grasping from drawer task where the WidowX robot is placed in front of a drawer and multiple objects. The robot can open or close the drawer, grasp objects from inside the drawer or on the table, and place them anywhere in the scene. The goal is to close the top drawer, then open the bottom drawer and take the object out. Only if the object has been taken out, a reward of $+ 1$ is obtained. The offline dataset consists of trajectories with a $3 0 { - } 4 0 \\%$ success rate for opening and closing a drawer and other trajectories with only $40 \\%$ placing success. We use $\\alpha = 1 . 0$ for CQL training in all experiments, which is directly taken from prior work [3], without any tuning. However, too low or too high $\\alpha$ values will inhibit the effectiveness of regular CQL and we first need to tune $\\alpha$ as discussed in Appendix G. More details are provided in Appendix D. ",
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+        "img_path": "images/7149b62299170ecd7a26a8a110d939daa58337b186bfd1671932f83c2c649963.jpg",
+        "image_caption": [
+            "Figure 3: Policy performance (Top) and average dataset Q-values of CQL (bottom) with varying number of trajectories. Vertical bands indicate regions around the peak in average $\\mathrm { Q }$ -value and observe that these regions correspond to policies with good actual performance. "
+        ],
+        "image_footnote": [],
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+    {
+        "type": "text",
+        "text": "Scenario #1: Variable amount of training data. Our first scenario consists of the simulated tasks discussed above with a variable number of trajectories in the training data (50, 100, 500, 10000). We run CQL and track metrics 3.1 and 3.2 in each case. Observe in Figure 3 (bottom) that with fewer trajectories, the average dataset Q-value $\\mathbb { E } _ { \\mathbf { s } , \\mathbf { a } \\sim \\mathcal { D } } [ Q _ { \\theta } ( \\mathbf { s } , \\mathbf { a } ) ]$ first rises, and then drops. This matches the description of overfitting in Section 3. Observe in Figure 4 (left) that, at the same time, the value of the CQL regularizer is very low, which is not consistent with what we expect of underfitting. Thus, we can conclude that these conditions exhibit overfitting, especially with 50 and 100 trajectories. The vertical dashed lines indicate the checkpoints that would be selected for evaluation per Guideline 3.1. We further visualize the performance of the chosen checkpoints against the actual return of each intermediate policy in Figure 3 (top). Note that this value is obtained by rolling out the learned policy, and would not be available in a realistic offline RL setting, but is provided only for analysis. Selecting the checkpoint based on Guideline 3.1 leads us to select a model with close to the peak performance over the training process, validating the efficacy of Guideline 3.1. ",
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+        "type": "text",
+        "text": "Since we detected overfitting by following our workflow, we now aim to address it by using the VIB regularizer in the setting with 100 trajectories. As shown in Figure 4 (right), applying this regularizer not only alleviates the drop in Q-values after many training steps, but allows us to pick later checkpoints in training which perform better than base CQL on both the tasks. This validates that overfitting, as detected via our workflow, can be effectively mitigated by decreasing capacity, in this case by using VIB. We evaluate dropout, $\\ell _ { 1 }$ and $\\ell _ { 2 }$ regularization schemes in Appendix J. ",
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+        "image_caption": [
+            "Figure 4: Left: CQL regularizer attains low values, especially with 50 and 100 trajectories in the pick and place task, Right: Using VIB mitigates overfitting, giving rise to a stable trend in $\\mathrm { Q }$ -values and better performance which does not degrade with more training steps. "
+        ],
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+        "text": "Scenario #2: Multiple training objects. Our second test scenario consists of the pick and place task, modified to include a variable number of object types (1, 5, 10, 20, 35). Handling more objects requires higher capacity, since each object has a different shape and appearance. In each case, CQL is provided with 5000 trajectories. Following our workflow from Section 3, we first compute the average dataset Q-value and the training TD error. We observe in Figure 5 that, unlike in Scenario #1, Q-values do not generally decrease when trained for many steps, suggesting that the Q-function is likely not overfitting. To check for underfitting, we visualize the training TD error and find that, with 10, 20 and 35 objects, TD error magnitudes are in the range of [1.0, 2.0], which suggests a overall Q-value error of [30.0, 60.0] since the task horizon is 30. On an absolute scale, this error magnitude is large: since the rewards are $_ { 0 / 1 }$ , the range of difference between actual Q-values for any two policies is at most 30, which suggests that the error magnitude in the runs in Figure 5 are high. Hence, we conclude that this scenario generally exhibits underfitting with more objects. Indeed this trend is reflected in the policy performance that we plot for analysis in Figure 5: note that the policy return decreases with an increased number of objects, and the policy performance initially increases and saturates at a suboptimal value. ",
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+    {
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+        "img_path": "images/abb86c89607619cd53397449a92fc47f01701017a73f7df2599b8c654f14d0ca.jpg",
+        "image_caption": [
+            "Figure 5: Performance (left), TD error (middle) and average dataset Qvalues (right) for the pick and place task with a variable number of objects. Note that while the learned Q-values increase and stabilize, the TD error values in scenarios with more than 10 objects are large (1.0-2.0). Correspondingly, the performance generally decreases as the number of objects increases. "
+        ],
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+        "image_caption": [
+            "Figure 6: Correcting underfitting by applying our workflow for 35 objects. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        "type": "text",
+        "text": "To address underfitting in the multi-object case, we apply the proposed capacity-increasing measures to the 35-object task (results for 10 and 20 object settings are in Appendix I). We use a more expressive ResNet architecture for the policy and the DR3 regularizer for the Q-function together. Observe in the figure on the right that this combination (shown in red) improves policy performance in this setting (compared to green), which validates our workflow protocol for addressing underfitting. ",
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+        "type": "text",
+        "text": "6 Tuning CQL for Real-World Robotic Manipulation ",
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+        "type": "text",
+        "text": "Having evaluated the efficacy of our proposed workflow in simulation, we now utilize our workflow to tune CQL for real-world robotic manipulation. We test in two setups that require the robot to learn from sparse binary rewards and image observations. The settings differ in robot platform, task specification, and dataset size. Additional results and robot videos are at the following website: https://sites. google.com/view/offline-rl-workflow ",
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+        "text": "Sawyer manipulation tasks [30]. First, we train a ",
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+    {
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+        "img_path": "images/566e1138148ac542d1676690513ab2bab4af3cbfec1257dc70a864019d00681c.jpg",
+        "image_caption": [
+            "Figure 7: Real-world tasks. Successful rollouts of CQL tuned with our workflow from Sections 3 & 4. Top to bottom: Sawyer lid on pot, Sawyer drawer opening, WidowX pick-place task. "
+        ],
+        "image_footnote": [],
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+    {
+        "type": "text",
+        "text": "Sawyer robot in a tabletop setting to perform two tasks: (1) placing the lid onto a pot and (2) opening a drawer. The robot must perform these tasks in the presence of visual distractor objects, as shown in Figure 7. We directly use the dataset of 100 trajectories for each task collected by Khazatsky et al. [30] for our experiments so as to mimic the real-world use case of leveraging existing data with offline RL. We use four-dimensional actions with 3D end-effector velocity control in xyz-space and 1D gripper open/close action. More details regarding the setup are provided in Appendix D. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "We run default CQL on these tasks and track the average Q-value, TD error, and CQL regularizer value. As shown in Figure 8, the average Q-value does not decrease over training, and the TD error (and CQL regularizer shown in Appendix E.2) is large. Per our discussion in Section 3, this indicates underfitting. Following our guidelines from Section 4, we utilize a more expressive ResNet policy (Figure 10), which increases the number of total convolutional layers from 3 to 9. We observe that this reduces the values of both the TD error Figure 8 and CQL regularizer (Appendix E.2) on both tasks. We ",
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+    {
+        "type": "image",
+        "img_path": "images/d5317090e1ad32bc153c1daf1e4d3802dfab85d171f13b7f15890473be1358cc.jpg",
+        "image_caption": [
+            "Figure 8: Average Q-value and TD error on Sawyer tasks as model capacity increases. Q-values increase over training with lower capacity ruling out overfitting and increasing model capacity leads to a reduction in TD error indicating the presence of underfitting. "
+        ],
+        "image_footnote": [],
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+    {
+        "type": "text",
+        "text": "then evaluate the learned policy over 12 trials conducted with different sets of distractor objects, including ones that are unseen during training. While the policy trained using base CQL is unable to successfully complete either task even once attaining a score of 0/12 on both tasks, the run that uses ResNet attains a significantly better success rate of 9/12 on the put lid on pot task and 8/12 on the drawer opening task, equal to $7 0 . 8 \\%$ success rate on average. ",
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+        "image_caption": [
+            "Figure 9: Q-values (left) and performance of CQL with (middle) and without (right) the variational information bottleneck correction for overfitting on the real-world widowX pick and place task. Since the Q-values start to decrease with more training, our workflow detects that CQL is overfitting. Using our policy selection guideline (Guideline 3.1) enables us to choose checkpoint 50 marked with the green vertical dashed line (right) which performs well. Further, addressing overfitting by applying the VIB regularizer stabilizes the Q-values (brown) which do not decrease unlike base CQL (blue) (left). Finally, applying the VIB regularizer improves performance and reduces sensitivity to policy selection (middle). "
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+        "text": "WidowX pick and place task. In our second setting, we tune CQL on a pick and place task with a WidowX 250 robotic arm, shown in Figure 7. The dataset consists of 200 trajectories collected by running a noisy scripted policy (Appendix D) with $3 5 \\%$ success. We run CQL on this task and track the average Q-values, which we find initially increase and then decrease (Figure 9 (left; labeled as “Q-values”)), indicating overfitting. We then evaluate our policy selection scheme, which in this case suggests deploying checkpoint 50, the immediate checkpoint after the peak in Q-values. To see if this checkpoint is effective, we evaluate the performance of a few other policy checkpoints (for analysis only) and plot this performance trend in Figure 9 (right) as a dashed line. Observe that indeed the checkpoint found by our workflow attains the highest success rate (7/9) compared to other checkpoints, which only succeed $\\leq 4 / 9$ times. ",
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+        "text": "Since overfitting is detected, we now turn to addressing overfitting by adding the VIB regularizer (Equation 3) during training. As shown in Figure 9 (left), the Q-values obtained after the addition of this regularizer (shown in brown; labeled “Q-values (VIB)”) are now stable and do not decrease over the course of training and so we can choose any policy for evaluation. We evaluate multiple policies, for visualization pur",
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+        "table_body": "<table><tr><td rowspan=1 colspan=5>Real-world WidowX pick and place</td></tr><tr><td rowspan=1 colspan=1>MethodEpoch</td><td rowspan=1 colspan=1>50</td><td rowspan=1 colspan=1>75</td><td rowspan=1 colspan=1>100</td><td rowspan=1 colspan=1>200</td></tr><tr><td rowspan=1 colspan=1>CQL</td><td rowspan=1 colspan=1>7/9</td><td rowspan=1 colspan=1>4/9</td><td rowspan=1 colspan=1>4/9</td><td rowspan=1 colspan=1>2/9</td></tr><tr><td rowspan=1 colspan=1>CQL + VIB</td><td rowspan=1 colspan=1>3/9</td><td rowspan=1 colspan=1>8/9</td><td rowspan=1 colspan=1>7/9</td><td rowspan=1 colspan=1>7/9</td></tr></table>",
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+        "text": "Table 2: Performance of various policy checkpoints of CQL and $\\mathrm { C Q L + V I B }$ on the real WidowX pick and place task (bold entry denotes the checkpoint selected by our workflow). Note that when overfitting is corrected via VIB, multiple checkpoints perform well. ",
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+        "text": "poses only, in Figure 9 (middle), we find that all of them attain a $\\geq 7 / 9$ success, comparable or better than the base CQL algorithm (Figure 9 (right)). This indicates that addressing overfitting not only leads to some gains in performance but also greatly simplifies policy selection as all checkpoints perform similarly and well. Table 2 summarizes these results below, where the bold entries denote the checkpoints found by our policy selection rule. These results indicate the effectiveness of our workflow in tuning CQL by addressing overfitting and underfitting on multiple real robot platforms. ",
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+        "type": "text",
+        "text": "7 Discussion ",
+        "text_level": 1,
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+        "text": "While offline RL algorithms have improved significantly, applying these methods to real-world robotic domains is still challenging due to little guidance on tuning them. In this paper, we devise a workflow for algorithms such as CQL and BRAC, which consists of a set of metrics and conditions that can be tracked by a practitioner over the course of offline training to detect overfitting and underfitting, and recommendations to addresses the observed challenges. Applying our workflow both in simulation and the real world shows strong performance benefits. While our proposed workflow is an initial step towards practical robotic offline RL and is based on our best conceptual understanding of certain offline RL algorithms, these guidelines are heuristic. To some extent this is unavoidable, since a workflow is a set of guidelines and recommendations, rather than a rigid algorithm. Regardless of how theoretically justified it is, in the end, its value is determined by its ability to produce good results. We believe the breadth of tasks considered, which consist of two different real robots and multiple simulated tasks, indicates its broad applicability. However, deriving theoretical guarantees regarding workflows of this type is an important direction for future research. ",
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+        "type": "text",
+        "text": "Acknowledgements ",
+        "text_level": 1,
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+        "text": "We thank Ilya Kostrikov, Avi Singh, Ashvin Nair, Alexander Khazatsky, Albert Yu, Jedrzej Orbik, and Jonathan Yang for their help with setting up and debugging various aspects of the experimental setup as well as for providing us with offline datasets we could test our workflow on. We thank Dibya Ghosh, anonymous reviewers, and the area chair from CoRL for constructive feedback on an earlier version of this paper. AK thanks George Tucker and Rishabh Agarwal for valuable discussions. This research was funded by the DARPA Assued Autonomy Program and compute support from Google and Microsoft Azure. ",
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+        "type": "text",
+        "text": "References ",
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