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+# VARIANCE REDUCTION FOR POLICY GRADIENT WITH ACTION-DEPENDENT FACTORIZED BASELINES
+
+Cathy $\mathbf { W } \mathbf { u } ^ { 1 }$ , Aravind Rajeswaran2, Yan Duan13, Vikash Kumar2, Alexandre M Bayen14, Sham Kakade2, Igor Mordatch3, Pieter Abbeel13 cathywu@eecs.berkeley.edu, aravraj@cs.washington.edu, rockyduan@eecs.berkeley.edu, vikash@cs.washington.edu, bayen@berkeley.edu, sham@cs.washington.edu, igor.mordatch@gmail.com, pabbeel@cs.berkeley.edu
+
+1 Department of EECS, UC Berkeley   
+2 Department of CSE, University of Washington   
+3 OpenAI   
+4 Institute for Transportation Studies, UC Berkeley
+
+# ABSTRACT
+
+Policy gradient methods have enjoyed great success in deep reinforcement learning but suffer from high variance of gradient estimates. The high variance problem is particularly exasperated in problems with long horizons or high-dimensional action spaces. To mitigate this issue, we derive a bias-free action-dependent baseline for variance reduction which fully exploits the structural form of the stochastic policy itself and does not make any additional assumptions about the MDP. We demonstrate and quantify the benefit of the action-dependent baseline through both theoretical analysis as well as numerical results, including an analysis of the suboptimality of the optimal state-dependent baseline. The result is a computationally efficient policy gradient algorithm, which scales to high-dimensional control problems, as demonstrated by a synthetic 2000-dimensional target matching task. Our experimental results indicate that action-dependent baselines allow for faster learning on standard reinforcement learning benchmarks and highdimensional hand manipulation and synthetic tasks. Finally, we show that the general idea of including additional information in baselines for improved variance reduction can be extended to partially observed and multi-agent tasks.
+
+# 1 INTRODUCTION
+
+Deep reinforcement learning has achieved impressive results in recent years in domains such as video games from raw visual inputs (Mnih et al., 2015), board games (Silver et al., 2016), simulated control tasks (Schulman et al., 2016; Lillicrap et al., 2016; Rajeswaran et al., 2017a), and robotics (Levine et al., 2016). An important class of methods behind many of these success stories are policy gradient methods (Williams, 1992; Sutton et al., 2000; Kakade, 2002; Schulman et al., 2015; Mnih et al., 2016), which directly optimize parameters of a stochastic policy through local gradient information obtained by interacting with the environment using the current policy. Policy gradient methods operate by increasing the log probability of actions proportional to the future rewards influenced by these actions. On average, actions which perform better will acquire higher probability, and the policy’s expected performance improves.
+
+A critical challenge of policy gradient methods is the high variance of the gradient estimator. This high variance is caused in part due to difficulty in credit assignment to the actions which affected the future rewards. Such issues are further exacerbated in long horizon problems, where assigning credits properly becomes even more challenging. To reduce variance, a “baseline” is often employed, which allows us to increase or decrease the log probability of actions based on whether they perform better or worse than the average performance when starting from the same state. This is particularly useful in long horizon problems, since the baseline helps with temporal credit assignment by removing the influence of future actions from the total reward. A better baseline, which predicts the average performance more accurately, will lead to lower variance of the gradient estimator.
+
+The key insight of this paper is that when the individual actions produced by the policy can be decomposed into multiple factors, we can incorporate this additional information into the baseline to further reduce variance. In particular, when these factors are conditionally independent given the current state, we can compute a separate baseline for each factor, whose value can depend on all quantities of interest except that factor. This serves to further help credit assignment by removing the influence of other factors on the rewards, thereby reducing variance. In other words, information about the other factors can provide a better evaluation of how well a specific factor performs. Such factorized policies are very common, with some examples listed below.
+
+• In continuous control and robotics tasks, multivariate Gaussian policies with a diagonal covariance matrix are often used. In such cases, each action coordinate can be considered a factor. Similarly, factorized categorical policies are used in game domains like board games and Atari. • In multi-agent and distributed systems, each agent deploys its own policy, and thus the actions of each agent can be considered a factor of the union of all actions (by all agents). This is particularly useful in the recent emerging paradigm of centralized learning and decentralized execution (Foerster et al., 2017; Lowe et al., 2017). In contrast to the previous example, where factorized policies are a common design choice, in these problems they are dictated by the problem setting.
+
+We demonstrate that action-dependent baselines consistently improve the performance compared to baselines that use only state information. The relative performance gain is task-specific, but in certain tasks, we observe significant speed-up in the learning process. We evaluate our proposed method on standard benchmark continuous control tasks, as well as on a high-dimensional door opening task with a five-fingered hand, a synthetic high-dimensional target matching task, on a blind peg insertion POMDP task, and a multi-agent communication task. We believe that our method will facilitate further applications of reinforcement learning methods in domains with extremely highdimensional actions, including multi-agent systems. Videos and additional results of the paper are available at https://sites.google.com/view/ad-baselines.
+
+# 2 RELATED WORKS
+
+Three main classes of methods for reinforcement learning include value-based methods (Watkins & Dayan, 1992), policy-based methods (Williams, 1992; Kakade, 2002; Schulman et al., 2015), and actor-critic methods (Konda & Tsitsiklis, 2000; Peters & Schaal, 2008; Mnih et al., 2016). Valuebased and actor-critic methods usually compute a gradient of the objective through the use of critics, which are often biased, unless strict compatibility conditions are met (Sutton et al., 2000; Konda & Tsitsiklis, 2000). Such conditions are rarely satisfied in practice due to the use of stochastic gradient methods and powerful function approximators. In comparison, policy gradient methods are able to compute an unbiased gradient, but suffer from high variance. Policy gradient methods are therefore usually less sample efficient, but can be more stable than critic-based methods (Duan et al., 2016).
+
+A large body of work has investigated variance reduction techniques for policy gradient methods. One effective method to reduce variance without introducing bias is through using a baseline, which has been widely studied (Sutton & Barto, 1998; Weaver & Tao, 2001; Greensmith et al., 2004; Schulman et al., 2016). However, fully exploiting the factorizability of the policy probability distribution to further reduce variance has not been studied. Recently, methods like Q-Prop (Gu et al., 2017) make use of an action-dependent control variate, a technique commonly used in Monte Carlo methods and recently adopted for RL. Since Q-Prop utilizes off-policy data, it has the potential to be more sample efficient than pure on-policy methods. However, Q-prop is significantly more computationally expensive, since it needs to perform a large number of gradient updates on the critic using the off-policy data, thus not suitable with fast simulators. In contrast, our formulation of action-dependent baselines has little computational overhead, and improves the sample efficiency compared to on-policy methods with state-only baseline.
+
+The idea of using additional information in the baseline or critic has also been studied in other contexts. Methods such as Guided Policy Search (Levine et al., 2016; Mordatch et al., 2015) and variants train policies that act on high-dimensional observations like images, but use a low dimensional encoding of the problem like joint positions during the training process. Recent efforts in multi-agent systems (Foerster et al., 2017; Lowe et al., 2017) also use additional information in the centralized training phase to speed-up learning. However, using the structure in the policy parameterization itself to enhance the learning speed, as we do in this work, has not been explored.
+
+# 3 PRELIMINARIES
+
+In this section, we establish the notations used throughout this paper, as well as basic results for policy gradient methods, and variance reduction via baselines.
+
+# 3.1 NOTATION
+
+This paper assumes a discrete-time Markov decision process (MDP), defined by $( S , \mathcal { A } , \mathcal { P } , r , \rho _ { 0 } , \gamma )$ , in which $S \subseteq \mathbb { R } ^ { n }$ is an $n$ -dimensional state space, $\mathcal { A } \subseteq \mathbb { R } ^ { m }$ an $m$ -dimensional action space, $\mathcal { P } : \mathcal { S } \times \mathcal { A } \times \mathcal { S }  \mathbb { R } _ { + }$ a transition probability function, $r : S \times \mathcal { A }  \mathbb { R }$ a bounded reward function, $\rho _ { 0 } : { \mathcal { S } }  \mathbb { R } _ { + }$ an initial state distribution, and $\gamma \in ( 0 , 1 ]$ a discount factor. The presented models are based on the optimization of a stochastic policy $\pi _ { \theta } : \mathcal { S } \times \mathcal { A } \to \mathbb { R } _ { + }$ parameterized by $\theta$ . Let $\eta ( \pi _ { \theta } )$ denote its expected return: $\begin{array} { r } { \eta ( \pi _ { \theta } ) = \mathbb { E } _ { \tau } [ \sum _ { t = 0 } ^ { \infty } \gamma ^ { t } r ( s _ { t } , a _ { t } ) ] } \end{array}$ , where ${ \boldsymbol \tau } = ( s _ { 0 } , a _ { 0 } , \ldots )$ denotes the whole trajectory, $s _ { 0 } \sim \rho _ { 0 } ( s _ { 0 } )$ , $a _ { t } \sim \pi _ { \theta } ( a _ { t } | s _ { t } )$ , and $s _ { t + 1 } \sim \mathcal { P } ( s _ { t + 1 } | s _ { t } , a _ { t } )$ for all $t$ . Our goal is to find the optimal policy arg $\operatorname* { m a x } _ { \theta } \eta ( \pi _ { \theta } )$ . We will use $\hat { Q } ( s _ { t } , a _ { t } )$ to describe samples of cumulative discounted return, and $Q ( a _ { t } , s _ { t } )$ to describe a function approximation of $\hat { Q } ( s _ { t } , a _ { t } )$ . We will use ”Q-function” when describing an abstract action-value function.
+
+For a partially observable Markov decision process (POMDP), two more components are required, namely $\Omega$ , a set of observations, and $\mathcal { O } : \mathcal { S } \times \Omega  \mathbb { R } _ { \ge 0 }$ , the observation probability distribution. In the fully observable case, $\Omega \equiv s$ . Though the analysis in this article is written for policies over states, the same analysis can be done for policies over observations.
+
+# 3.2 THE SCORE FUNCTION (SF) ESTIMATOR
+
+An important technique used in the derivation of the policy gradient is known as the score function (SF) estimator (Williams, 1992), which also comes up in the justification of baselines. Suppose that we want to estimate $\nabla _ { \theta } \mathbb { E } _ { x } [ f ( x ) ]$ where $x \sim p _ { \theta } ( x )$ , and the family of distributions $\{ p _ { \theta } ( x ) : \theta \in \Theta \}$ has common support. Further suppose that $\log p _ { \theta } ( x )$ is continuous in $\theta$ . In this case we have
+
+$$
+\begin{array} { l } { \nabla _ { \theta } \mathbb { E } _ { x } [ f ( x ) ] = \nabla _ { \theta } \displaystyle \int p _ { \theta } ( x ) f ( x ) d x = \int p _ { \theta } ( x ) \frac { \nabla _ { \theta } p _ { \theta } ( x ) } { p _ { \theta } ( x ) } f ( x ) d x } \\ { \displaystyle = \int p _ { \theta } ( x ) \nabla _ { \theta } \log p _ { \theta } ( x ) f ( x ) d x = \mathbb { E } _ { x } \left[ \nabla _ { \theta } \log p _ { \theta } ( x ) f ( x ) \right] . } \end{array}
+$$
+
+# 3.3 POLICY GRADIENT
+
+The Policy Gradient Theorem (Sutton et al., 2000) states that
+
+$$
+\nabla _ { \theta } \eta ( \pi _ { \theta } ) = \mathbb { E } _ { \tau } \left[ \sum _ { t = 0 } ^ { \infty } \nabla _ { \theta } \log \pi _ { \theta } ( a _ { t } | s _ { t } ) \sum _ { t ^ { \prime } = t } ^ { \infty } \gamma ^ { t ^ { \prime } - t } r _ { t ^ { \prime } } \right] .
+$$
+
+For convenience, define $\begin{array} { r } { \rho _ { \pi } ( s ) = \sum _ { t = 0 } ^ { \infty } \gamma ^ { t } p ( s _ { t } = s ) } \end{array}$ as the state visitation frequency, and $\begin{array} { r } { \hat { Q } ( s _ { t } , a _ { t } ) = \sum _ { t ^ { \prime } = t } ^ { \infty } \gamma ^ { t ^ { \prime } - t } r _ { t ^ { \prime } } } \end{array}$ . We can rewrite the above equation (with abuse of notation) as
+
+$$
+\begin{array} { r } { \nabla _ { \theta } \eta ( \pi _ { \theta } ) = \mathbb { E } _ { \rho _ { \pi } , \pi } \left[ \nabla _ { \theta } \log \pi _ { \theta } ( a _ { t } | s _ { t } ) \hat { Q } ( s _ { t } , a _ { t } ) \right] . } \end{array}
+$$
+
+It is further shown that we can reduce the variance of this gradient estimator without introducing bias by subtracting off a quantity dependent on $s _ { t }$ from $\hat { Q } ( s _ { t } , a _ { t } )$ (Williams, 1992; Greensmith et al., 2004). See Appendix A for a derivation of the optimal state-dependent baseline.
+
+$$
+\nabla _ { \theta } \eta ( \pi _ { \theta } ) = \mathbb { E } _ { \rho _ { \pi } , \pi } \left[ \nabla _ { \theta } \log \pi _ { \theta } ( a _ { t } | s _ { t } ) \left( \hat { Q } ( s _ { t } , a _ { t } ) - b ( s _ { t } ) \right) \right]
+$$
+
+This is valid because, applying the SF estimator in the opposite direction, we have
+
+$$
+\mathbb { E } _ { a _ { t } } \left[ \nabla _ { \theta } \log \pi _ { \theta } ( a _ { t } | s _ { t } ) b ( s _ { t } ) \right] = \nabla _ { \theta } \mathbb { E } _ { a _ { t } } \left[ b ( s _ { t } ) \right] = 0
+$$
+
+In practice there can be rich internal structure in the policy parameterization. For example, for continuous control tasks, a very common parameterization is to make $\pi _ { \boldsymbol { \theta } } \big ( \boldsymbol { a } _ { t } | \boldsymbol { s } _ { t } \big )$ a multivariate Gaussian with diagonal variance, in which case each dimension $a _ { t } ^ { i }$ of the action $a _ { t }$ is conditionally independent of other dimensions, given the current state $s _ { t }$ . Another example is when the policy outputs a tuple of discrete actions with factorized categorical distributions. In the following subsections, we show that such structure can be exploited to further reduce the variance of the gradient estimator without introducing bias by changing the form of the baseline. Then, we derive the optimal action-dependent baseline for a class of problems and analyze the suboptimality of non-optimal baselines in terms of variance reduction. We then propose several practical baselines for implementation purposes. We conclude the section with the overall policy gradient algorithm with action-dependent baselines for factorized policies. We provide an exposition for situations when the conditional independence assumption does not hold, such as for stochastic policies with general covariance structures, in Appendix E, and for compatibility with other variance reduction techniques in Appendix F.
+
+# 4.1 BASELINES FOR POLICIES WITH CONDITIONALLY INDEPENDENT FACTORS
+
+In the following, we analyze action-dependent baselines for policies with conditionally independent factors. For example, multivariate Gaussian policies with a diagonal covariance structure are commonly used in continuous control tasks. Assuming an $m$ -dimensional action space, we have $\begin{array} { r } { \pi _ { \theta } ( a _ { t } | s _ { t } ) ^ { \bullet } = \prod _ { i = 1 } ^ { m } \pi _ { \theta } ( a _ { t } ^ { i } | s _ { t } ) } \end{array}$ . Hence
+
+$$
+\nabla _ { \theta } \eta ( \pi _ { \theta } ) = \mathbb { E } _ { \rho _ { \pi } , \pi } \left[ \nabla _ { \theta } \log \pi _ { \theta } ( a _ { t } | s _ { t } ) \hat { Q } ( s _ { t } , a _ { t } ) \right] = \mathbb { E } _ { \rho _ { \pi } , \pi } \left[ \sum _ { i = 1 } ^ { m } \nabla _ { \theta } \log \pi _ { \theta } ( a _ { t } ^ { i } | s _ { t } ) \hat { Q } ( s _ { t } , a _ { t } ) \right]
+$$
+
+In this case, we can set $b _ { i }$ , the baseline for the $i$ th factor, to depend on all other actions in addition to the state. Let $a _ { t } ^ { - i }$ denote all dimensions other than $i$ in $a _ { t }$ and denote the $i$ th baseline by $b _ { i } ( s _ { t } , a _ { t } ^ { - i } )$ . Due to conditional independence and the score function estimator, we have
+
+$$
+\begin{array} { r l } & { \mathbb { E } _ { a _ { t } } \left[ \nabla _ { \theta } \log \pi _ { \theta } ( a _ { t } ^ { i } | s _ { t } ) b _ { i } ( s _ { t } , a _ { t } ^ { - i } ) \right] = \mathbb { E } _ { a _ { t } ^ { - i } } \left[ \nabla _ { \theta } \mathbb { E } _ { a _ { t } ^ { i } } \left[ b _ { i } ( s _ { t } , a _ { t } ^ { - i } ) \right] \right] = 0 } \end{array}
+$$
+
+Hence we can use the following gradient estimator
+
+$$
+\nabla _ { \theta } \eta ( \pi _ { \theta } ) = \mathbb { E } _ { \rho _ { \pi } , \pi } \left[ \sum _ { i = 1 } ^ { m } \nabla _ { \theta } \log \pi _ { \theta } ( a _ { t } ^ { i } | s _ { t } ) \left( \hat { Q } ( s _ { t } , a _ { t } ) - b _ { i } ( s _ { t } , a _ { t } ^ { - i } ) \right) \right]
+$$
+
+This is compatible with advantage function form of the policy gradient (Schulman et al., 2016):
+
+$$
+\nabla _ { \theta } \eta ( \pi _ { \theta } ) = \mathbb { E } _ { \rho _ { \pi } , \pi } \left[ \sum _ { i = 1 } ^ { m } \nabla _ { \theta } \log \pi _ { \theta } ( a _ { t } ^ { i } | s _ { t } ) \hat { A } _ { i } ( s _ { t } , a _ { t } ) \right]
+$$
+
+where ${ \hat { A } } _ { i } ( s _ { t } , a _ { t } ) = Q ( s _ { t } , a _ { t } ) - b _ { i } \big ( s _ { t } , a _ { t } ^ { - i } \big )$ . Note that the policy gradient now comprises of $m$ component policy gradient terms, each with a different advantage term.
+
+In Appendix E, we show that the methodology also applies to general policy structures (for example, a Gaussian policy with a general covariance structure), where the conditional independence assumption does not hold. The result is bias-free albeit different baselines.
+
+# 4.2 OPTIMAL ACTION-DEPENDENT BASELINE
+
+In this section, we derive the optimal action-dependent baseline and show that it is better than the state-only baseline. We seek the optimal baseline to minimize the variance of the policy gradient estimate. First, we write out the variance of the policy gradient under any action-dependent baseline. Let us define $z _ { i } : = \nabla _ { \theta } \log \pi _ { \theta } ( a _ { t } ^ { i } | s _ { t } )$ and the component policy gradient:
+
+$$
+\begin{array} { r } { \nabla \eta _ { i } ( \pi _ { \theta } ) : = \mathbb { E } _ { \rho _ { \pi } , \pi } \left[ \nabla _ { \theta } \log \pi _ { \theta } ( a _ { t } ^ { i } | s _ { t } ) \left( \hat { Q } ( s _ { t } , a _ { t } ) - b _ { i } ( s _ { t } , a _ { t } ^ { - i } ) \right) \right] . } \end{array}
+$$
+
+For simplicity of exposition, we make the following assumption:
+
+$$
+\nabla _ { \theta } \log \pi _ { \theta } ( a _ { t } ^ { i } | s _ { t } ) ^ { T } \nabla _ { \theta } \log \pi _ { \theta } ( a _ { t } ^ { j } | s _ { t } ) \equiv z _ { i } ^ { T } z _ { j } = 0 , \quad \forall i \neq j
+$$
+
+which translates to meaning that different subsets of parameters strongly influence different action dimensions or factors. We note that this assumption is primarily for the theoretical analysis to be clean, and is not required to run the algorithm in practice. In particular, even without this assumption, the proposed baseline is bias-free. When the assumption holds, the optimal actiondependent baseline can be analyzed thoroughly. Some examples where these assumptions do hold include multi-agent settings where the policies are conditionally independent by construction, cases where the policy acts based on independent components (Cao et al., 2007) of the observation space, and cases where different function approximators are used to control different actions or synergies (Todorov & Ghahramani, 2004; Todorov et al., 2005) without weight sharing.
+
+The optimal action-dependent baseline is then derived to be:
+
+$$
+b _ { i } ^ { * } ( s _ { t } , a _ { t } ^ { - i } ) = \frac { \mathbb { E } _ { a _ { t } ^ { i } } \left[ \nabla _ { \theta } \log \pi _ { \theta } ( a _ { t } ^ { i } | s _ { t } ) ^ { T } \nabla _ { \theta } \log \pi _ { \theta } ( a _ { t } ^ { i } | s _ { t } ) \hat { Q } ( s _ { t } , a _ { t } ) \right] } { \mathbb { E } _ { a _ { t } ^ { i } } \left[ \nabla _ { \theta } \log \pi _ { \theta } ( a _ { t } ^ { i } | s _ { t } ) ^ { T } \nabla _ { \theta } \log \pi _ { \theta } ( a _ { t } ^ { i } | s _ { t } ) \right] } .
+$$
+
+See Appendix B for the full derivation. Since the optimal action-dependent baseline is different for different action coordinates, it is outside the family of state-dependent baselines barring pathological cases.
+
+# 4.3 SUBOPTIMALITY OF THE OPTIMAL STATE-DEPENDENT BASELINE
+
+How much do we reduce variance over a traditional baseline that only depends on state? We use the following notation:
+
+$$
+\begin{array} { r l } & { Z _ { i } : = Z _ { i } \bigl ( s _ { t } , a _ { t } ^ { - i } \bigr ) = \mathbb { E } _ { a _ { t } ^ { i } } \left[ \nabla _ { \theta } \log \pi _ { \theta } \bigl ( a _ { t } ^ { i } \big | s _ { t } \bigr ) ^ { T } \nabla _ { \theta } \log \pi _ { \theta } \bigl ( a _ { t } ^ { i } \big | s _ { t } \bigr ) \right] } \\ & { Y _ { i } : = Y _ { i } \bigl ( s _ { t } , a _ { t } ^ { - i } \bigr ) = \mathbb { E } _ { a _ { t } ^ { i } } \left[ \nabla _ { \theta } \log \pi _ { \theta } \bigl ( a _ { t } ^ { i } \big | s _ { t } \bigr ) ^ { T } \nabla _ { \theta } \log \pi _ { \theta } \bigl ( a _ { t } ^ { i } \big | s _ { t } \bigr ) \hat { Q } \bigl ( s _ { t } , a _ { t } \bigr ) \right] } \end{array}
+$$
+
+Then, using Equation (51) (Appendix C), we show the following improvement with the optimal action-dependent baseline:
+
+$$
+I _ { b = b ^ { * } ( s ) } = \sum _ { i } \mathbb { E } _ { \rho _ { \pi } , a _ { t } ^ { - i } } \left[ \frac { 1 } { Z _ { i } } \left( \frac { Z _ { i } } { \sum _ { j } Z _ { j } } \sum _ { j } Y _ { j } - Y _ { i } \right) ^ { 2 } \right]
+$$
+
+See Appendices $\textrm { C }$ and $\mathbf { D }$ for the full derivation. We conclude that the optimal action-dependent baseline does not degenerate into the optimal state-dependent baseline. Equation (15) states that the variance difference is a weighted sum of the deviation of the per-component score-weighted marginalized Q (denoted $Y _ { i }$ ) from the component weight (based on score only, not Q) of the overall aggregated marginalized Q values (denoted $\textstyle \sum _ { j } Y _ { j } )$ . This suggests that the difference is particularly large when the Q function is highly sensitive to the actions, especially along those directions that influence the gradient the most. Our empirical results in Section 5 additionally demonstrate the benefit of action-dependent over state-only baselines.
+
+# 4.4 MARGINALIZATION OF THE GLOBAL ACTION-VALUE FUNCTION
+
+Using the previous theory, we now consider various baselines that could be used in practice and their associated computational cost.
+
+Marginalized Q baseline Even though the optimal state-only baseline is known, it is rarely used in practice (Duan et al., 2016). Rather, for both computational and conceptual benefit, the choice of $b ( s _ { t } ) = \mathbb { E } _ { a _ { t } } [ \hat { Q } ( s _ { t } , a _ { t } ) ] = V ( s _ { t } )$ is often used. Similarly, we propose to use $b _ { i } ( s _ { t } , a _ { t } ^ { - i } ) ~ =$ $\mathbb { E } _ { a _ { t } ^ { i } } \left[ \hat { Q } ( s _ { t } , a _ { t } ) \right]$ which is the action-dependent analogue. In particular, when log probability of each policy factor is loosely correlated with the action-value function, then the proposed baseline is close to the optimal baseline.
+
+$$
+I _ { b = \mathbb { E } _ { a _ { t } ^ { i } } [ \hat { Q } ( a _ { t } , s _ { t } ) ] } = \sum _ { i } \mathbb { E } _ { \rho _ { \tau , a _ { t } ^ { - i } } } \left[ Z _ { i } \left( \mathbb { E } _ { a ^ { i } } \left[ \hat { Q } ( a _ { t } , s _ { t } ) \right] - \frac { \mathbb { E } _ { a _ { t } ^ { i } } \left[ z _ { i } ^ { T } z _ { i } \hat { Q } ( s _ { t } , a _ { t } ) \right] } { \mathbb { E } _ { a _ { t } ^ { i } } \left[ z _ { i } ^ { T } z _ { i } \right] } \right) ^ { 2 } \right] \approx 0
+$$
+
+This has the added benefit of requiring learning only one function approximator, for estimating $Q ( s _ { t } , a _ { t } )$ , and implicitly using it to obtain the baselines for each action coordinate. That is, $Q ( s _ { t } , a _ { t } )$ is a function approximating samples $\hat { Q } ( s _ { t } , a _ { t } )$ .
+
+Monte Carlo marginalized Q baseline After fitting $Q _ { \pi _ { \theta } } ( s _ { t } , a _ { t } )$ we can obtain the baselines through Monte Carlo estimates:
+
+$$
+b _ { i } ( s _ { t } , a _ { t } ^ { - i } ) = \frac { 1 } { M } \sum _ { j = 0 } ^ { M } Q _ { \pi _ { \theta } } ( s _ { t } , ( a _ { t } ^ { - i } , \alpha _ { j } ) )
+$$
+
+where $\alpha _ { j } \sim \pi _ { \theta } ( a _ { t } ^ { i } | s _ { t } )$ are samples of the action coordinate $i$ . In general, any function may be used to aggregate the samples, so long as it does not depend on the sample value $a _ { t } ^ { i }$ . For instance, for discrete action dimensions, the sample max can be computed instead of the mean.
+
+Mean marginalized Q baseline Though we reduced the computational burden from learning $m$ functions to one function, the use of Monte Carlo samples can still be computationally expensive. In particular, when using deep neural networks to approximate the Q-function, forward propagation through the network can be even more computationally expensive than stepping through a fast simulator (e.g. MuJoCo). In such settings, we further propose the following more computationally practical baseline:
+
+$$
+b _ { i } ( s _ { t } , a _ { t } ^ { - i } ) = Q _ { \pi _ { \theta } } ( s _ { t } , ( a _ { t } ^ { - i } , \bar { a } _ { t } ^ { i } ) )
+$$
+
+where ${ \bar { a } } _ { t } ^ { i } = \mathbb { E } _ { \pi _ { \theta } } \left[ a _ { t } ^ { i } \right]$ is the average action for coordinate $i$ .
+
+# 4.5 FINAL ALGORITHM
+
+The final practical algorithm for fully factorized policies is as follows.
+
+Require: number of iterations $N$ , batch size $B$ , initial policy parameters $\theta$ Initialize action-value function estimate $Q _ { \pi _ { \theta } } ( s _ { t } , a _ { t } ) \equiv 0$ and policy $\pi _ { \theta }$ for $j$ in $\{ 1 , \ldots , N \}$ do Collect samples: (st, at)t∈{1,...,B} Compute baseline: $b _ { i } \big ( s _ { t } , a _ { t } ^ { - i } \big ) = \mathbb { E } _ { a _ { t } ^ { i } } \left[ \hat { Q } \big ( s _ { t } , a _ { t } \big ) \right]$ for $i \in \{ 1 , \ldots , m \}$ [e.g. Equations (17- 18)] Compute advantages: $\hat { A } _ { i } \big ( s _ { t } , a _ { t } \big ) : = \hat { Q } \big ( s _ { t } , a _ { t } \big ) - b _ { i } \big ( s _ { t } , a _ { t } ^ { - i } \big ) , \forall t$ Perform a policy update step on $\theta$ using $\hat { A } _ { i } ( s _ { t } , a _ { t } )$ [Equation (9)] Update action-value function approximation with current batch: $Q _ { \pi _ { \theta } } ( s _ { t } , a _ { t } )$ end for
+
+Computing the baseline can be done with either proposed technique in Section 4.4. A similar algorithm can be written for general policies (Appendix E), which makes no assumptions on the conditional independence across action dimensions.
+
+# 5 EXPERIMENTS AND RESULTS
+
+Continuous control benchmarks Firstly, we present the results of the proposed action-dependent baselines on popular benchmark tasks. These tasks have been widely studied in the deep reinforcement learning community (Duan et al., 2016; Gu et al., 2017; Lillicrap et al., 2016; Rajeswaran et al., 2017b). The studied tasks include the hopper, half-cheetah, and ant locomotion tasks simulated in MuJoCo (Todorov et al., 2012).1 In addition to these tasks, we also consider a door opening task with a high-dimensional multi-fingered hand, introduced in Rajeswaran et al. (2017a) to study the effectiveness of the proposed approach in high-dimensional tasks. Figure 1 presents the learning curves on these tasks. We compare the action-dependent baseline with a baseline that uses only information about the states, which is the most common approach in the literature. We observe that the action-dependent baselines perform consistently better.
+
+A popular baseline parameterization choice is a linear function on a small number of non-linear features of the state (Duan et al., 2016), especially for policy gradient methods. In this work, to enable a fair comparison, we use a Random Fourier Feature representation for the baseline (Rahimi & Recht, 2007; Rajeswaran et al., 2017b). The features are constructed as: $\begin{array} { r } { y ( x ) = \sin ( \frac { 1 } { \nu } \dot { P } x + \phi ) } \end{array}$ where $P$ is a matrix with each element independently drawn from the standard normal distribution, $\phi$ is a random phase shift in $[ - \pi , \pi )$ and, and $\nu$ is a bandwidth parameter. These features approximate the RKHS features under an RBF kernel. Using these features, the baseline is parameterized as $b = w ^ { T } y ( x )$ where $x$ are the appropriate inputs to the baseline, and $w$ are trainable parameters. $P$ and $\phi$ are not trained in this parameterization. Such a representation was chosen for two reasons: (a) we wish to have the same number of trainable parameters for all the baseline architectures, and not have more parameters in the action-dependent case (which has a larger number of inputs to the baseline); (b) since the final representation is linear, it is possible to accurately estimate the optimal parameters with a Newton step, thereby alleviating the results from confounding optimization issues. For policy optimization, we use a variant of the natural policy gradient method as described in Rajeswaran et al. (2017b). See Appendix G for further experimental details.
+
+![](images/4f6f9b0677d452543e58624d6addaac67db78c90cb61bdab751046a80ca1c0e0.jpg)  
+Figure 1: Comparison between value function baseline and action-conditioned baseline on various continuous control tasks. Action-dependent baseline performs consistently better across all the tasks.
+
+Choice of action-dependent baseline form Next, we study the influence of computing the baseline by using empirical averages sampled from the Q-function versus using the mean-action of the action-coordinate for computing the baseline (both described in 4.4). In our experiments, as shown in Figure 2 we find that the two variants perform comparably, with the latter performing slightly better towards the end of the learning process. This suggests that though sampling from the Q-function might provide a better estimate of the conditional expectation in theory, function approximation from finite samples injects errors that may degrade the quality of estimates. In particular, sub-sampling from the Q-function is likely to produce better results if the learned Q-function is accurate for a large fraction of the action space, but getting such high quality approximations might be hard in practice.
+
+High-dimensional action spaces Intuitively, the benefit of the action-dependent baseline can be greater for higher dimensional problems. We show this effect on a simple synthetic example called $m$ -DimTargetMatching. The example is a one-step MDP comprising of a single state, ${ \mathcal { S } } = \{ 0 \}$ , an $m$ -dimensional action space, $\mathcal { A } = \mathbb { R } ^ { m }$ , and a fixed vector $c \in \mathbb { R } ^ { m }$ . The reward is given as the negative squared $\ell _ { 2 }$ loss of the action vector, $r ( s , a ) = - \| a - c \| _ { 2 } ^ { 2 }$ . The optimal action is thus to match the given vector by selecting $a = c$ . The results for the demonstrative example are shown in Table 1, which shows that the action-dependent baseline successfully improves convergence more for higher dimensional problems than lower dimensional problems. Due to the lack of state information, the linear baseline reduces to whitening the returns. The action-dependent baseline, on the other hand, allows the learning algorithm to assess the advantage of each individual action dimension by utilizing information from all other action dimensions. Additionally, this experiment demonstrates that our algorithm scales well computationally to high-dimensional problems.
+
+![](images/a486d1bf8132b5168e69a386db3b3697361a6947520c20b92cb8338ea5029422.jpg)  
+Figure 2: Variants of the action-dependent baseline that use: (i) sampling from the Q-function to estimate the conditional expectation; (ii) Using the mean action to form a linear approximation to the conditional expectation. We find that both variants perform comparably, with the latter being more computationally efficient.
+
+<table><tr><td rowspan="2">Action dimensions</td><td colspan="4">Solve time (iterations)</td><td rowspan="2">Solution threshold</td></tr><tr><td>Action-dependent</td><td>State-dependent</td><td>Delta</td><td>% speed improvement</td></tr><tr><td>12</td><td>45.6</td><td>45.6</td><td>0</td><td>0.0%</td><td>-0.01</td></tr><tr><td>100</td><td>136</td><td>150</td><td>14</td><td>9.3%</td><td>-0.25</td></tr><tr><td>400</td><td>268.2</td><td>304</td><td>35.8</td><td>11.8%</td><td>-0.99</td></tr><tr><td>2000</td><td>595.5</td><td>671.5</td><td>76</td><td>11.3%</td><td>-4.96</td></tr></table>
+
+Table 1: Shown are the results for the synthetic high-dimensional target matching task (5 seeds), for 12 to 2000 dimensional action spaces. At high dimensions, the linear feature action-dependent baseline provides notable and consistent variance reduction, as compared to a linear feature baseline, resulting in around $10 \%$ faster convergence. For the corresponding learning curves, see Appendix G.
+
+Partially observable and multi-agent tasks Finally, we also consider the extension of the core idea of using global information, by studying a POMDP task and a multi-agent task. We use the blind peg-insertion task which is widely studied in the robot learning literature (Montgomery & Levine, 2016). The task requires the robot to insert the peg into the hole (slot), but the robot is blind to the location of the hole. Thus, we expect a searching behavior to emerge from the robot, where it learns that the hole is present on the table and performs appropriate sweeping motions till it is able to find the hole. In this case, we consider a baseline that is given access to the location of the hole. We observe that a baseline with this additional information enables faster learning. For the multi-agent setting, we analyze a two-agent particle environment task in which the goal is for each agent to reach their goal, where their goal is known by the other agent and they have a continuous communication channel. Similar training procedures have been employed in recent related works Lowe et al. (2017); Levine et al. (2016). Figure 3 shows that including the inclusion of information from other agents into the action-dependent baseline improves the training performance, indicating that variance reduction may be key for multi-agent reinforcement learning.
+
+# 6 CONCLUSION
+
+An action-dependent baseline enables using additional signals beyond the state to achieve bias-free variance reduction. In this work, we consider both conditionally independent policies and general policies, and derive an optimal action-dependent baseline. We provide analysis of the variance
+
+![](images/17c0b9d38054dff9df4e8e1e7ae4e4ebb46bab73f446a6b1d14fd602586dadcc.jpg)
+
+![](images/83c05cab4874ca65a212c2f239094c9ca56de45cfeb0ab8f22b2ab21aeb5c119.jpg)
+
+(a) Success percentage on the blind peg insertion task. The policy still acts on the observations and does not know the hole location. However, the baseline has access to this goal information, in addition to the observations and action, and helps to speed up the learning. By comparison, in blue, the baseline has access only to the observations and actions.
+
+(b) Training curve for multi-agent communication task with two agents. Two policies are simultaneously trained, one for each agent. Each policy acts on the observations of its respective agent only. However, the shared baseline has access to the other agent’s state and action, in addition to its own state and action, and results in considerably faster training. By comparison, in blue, the independent learners baseline has access to only a single agent’s state and action.
+
+Figure 3: Experiments with additional information in the baseline.
+
+reduction improvement over non-optimal baselines, including the traditional optimal baseline that only depends on state. We additionally propose several practical action-dependent baselines which perform well on a variety of continuous control tasks and synthetic high-dimensional action problems. The use of additional signals beyond the local state generalizes to other problem settings, for instance in POMDP and multi-agent tasks. In future work, we propose to investigate related methods in such settings on large-scale problems.
+
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+
+We provide a derivation of the optimal state-dependent baseline, which minimizes the variance of the policy gradient estimate, and is based in (Greensmith et al., 2004, Theorem 8). More precisely, we minimize the trace of the covariance of the policy gradient; that is, the sum of the variance of the components of the vectors. Recall the policy gradient expression with a state-dependent baseline:
+
+$$
+\nabla _ { \theta } \eta ( \pi _ { \theta } ) : = \mathbb { E } _ { \rho _ { \pi } , \pi } \left[ \nabla _ { \theta } \log \pi _ { \theta } ( a _ { t } | s _ { t } ) \left( \hat { Q } ( s _ { t } , a _ { t } ) - b ( s _ { t } ) \right) \right]
+$$
+
+Denote $g$ to be the associated random variable, that is, $\nabla _ { \theta } \eta ( \pi _ { \theta } ) = \mathbb { E } _ { \rho _ { \pi } , \pi } [ g ]$ :
+
+$$
+\begin{array} { r } { g : = \nabla _ { \boldsymbol { \theta } } \log \pi _ { \boldsymbol { \theta } } ( a _ { t } | s _ { t } ) \left( \hat { Q } ( s _ { t } , a _ { t } ) - b ( s _ { t } ) \right) , \quad a _ { t } \sim \pi _ { \boldsymbol { \theta } } ( a _ { t } | s _ { t } ) , s _ { t } \sim \rho _ { \pi } ( s _ { t } ) } \end{array}
+$$
+
+The variance of the policy gradient is:
+
+$$
+\begin{array} { r l } & { \mathrm { V a r } ( g ) = \mathbb { E } _ { \rho _ { \pi } , \pi } \left[ ( g - \mathbb { E } _ { \rho _ { \pi } , \pi } \left[ g \right] ) ^ { T } ( g - \mathbb { E } _ { \rho _ { \pi } , \pi } \left[ g \right] ) \right] } \\ & { \quad \quad \quad = \mathbb { E } _ { \rho _ { \pi } , \pi } \left[ \nabla _ { \theta } \log \pi _ { \theta } ( a _ { t } | s _ { t } ) ^ { T } \nabla _ { \theta } \log \pi _ { \theta } ( a _ { t } | s _ { t } ) \right] b ( s _ { t } ) ^ { 2 } } \\ & { \quad \quad \quad - 2 \mathbb { E } _ { \rho _ { \pi } , \pi } \left[ \nabla _ { \theta } \log \pi _ { \theta } ( a _ { t } | s _ { t } ) ^ { T } \nabla _ { \theta } \log \pi _ { \theta } ( a _ { t } | s _ { t } ) \hat { Q } ( s _ { t } , a _ { t } ) \right] b ( s _ { t } ) } \end{array}
+$$
+
+Note that $\mathbb { E } \left[ \eta ( \pi _ { \theta } ) \right] )$ contains a bias-free term, by the score function argument, which then does not affect the minimizer. Terms which do not depend on $b ( s _ { t } )$ also do not affect the minimizer.
+
+$$
+\begin{array} { r l } & { \frac { \partial } { \partial b } \left[ \mathrm { V a r } ( g ) \right] = 0 } \\ & { \quad \quad \quad = 2 \mathbb { E } _ { \rho _ { \pi } , \pi } \left[ \nabla _ { \theta } \log \pi _ { \theta } ( a _ { t } | s _ { t } ) ^ { T } \nabla _ { \theta } \log \pi _ { \theta } ( a _ { t } | s _ { t } ) \right] b ( s _ { t } ) } \\ & { \quad \quad \quad \quad - 2 \mathbb { E } _ { \rho _ { \pi } , \pi } \left[ \nabla _ { \theta } \log \pi _ { \theta } ( a _ { t } | s _ { t } ) ^ { T } \nabla _ { \theta } \log \pi _ { \theta } ( a _ { t } | s _ { t } ) \hat { Q } ( s _ { t } , a _ { t } ) \right] } \end{array}
+$$
+
+$$
+\implies b ^ { * } ( s _ { t } ) = \frac { \mathbb { E } _ { \rho _ { \pi } , \pi } \left[ \nabla _ { \theta } \log \pi _ { \theta } ( a _ { t } | s _ { t } ) ^ { T } \nabla _ { \theta } \log \pi _ { \theta } ( a _ { t } | s _ { t } ) \hat { Q } ( s _ { t } , a _ { t } ) \right] } { \mathbb { E } _ { \rho _ { \pi } , \pi } \left[ \nabla _ { \theta } \log \pi _ { \theta } ( a _ { t } | s _ { t } ) ^ { T } \nabla _ { \theta } \log \pi _ { \theta } ( a _ { t } | s _ { t } ) \right] }
+$$
+
+# B DERIVATION OF THE OPTIMAL ACTION-DEPENDENT BASELINE
+
+We derive the optimal action-dependent baseline, which minimizes the variance of the policy gradient estimate. First, we write out the variance of the policy gradient under any action-dependent baseline. Recall the following notations: we define $\bar { z } _ { i } : = \bar { \nabla } _ { \boldsymbol { \theta } } \log \pi _ { \boldsymbol { \theta } } ( a _ { t } ^ { i } | s _ { t } )$ and the component policy gradient:
+
+$$
+\begin{array} { r } { \nabla \eta _ { i } ( \pi _ { \theta } ) : = \mathbb { E } _ { \rho _ { \pi } , \pi } \left[ \nabla _ { \theta } \log \pi _ { \theta } ( a _ { t } ^ { i } | s _ { t } ) \left( \hat { Q } ( s _ { t } , a _ { t } ) - b _ { i } ( s _ { t } , a _ { t } ^ { - i } ) \right) \right] . } \end{array}
+$$
+
+Denote $g _ { i }$ to be the associated random variables:
+
+$$
+\begin{array} { r } { g _ { i } : = \nabla _ { \theta } \log \pi _ { \theta } ( a _ { t } ^ { i } | s _ { t } ) \left( \hat { Q } ( s _ { t } , a _ { t } ) - b _ { i } ( s _ { t } , a _ { t } ^ { - i } ) \right) , \quad a _ { t } \sim \pi _ { \theta } ( a _ { t } | s _ { t } ) , s _ { t } \sim \rho _ { \pi } ( s _ { t } ) , } \end{array}
+$$
+
+such that
+
+$$
+\nabla _ { \boldsymbol { \theta } } \eta ( \pi _ { \boldsymbol { \theta } } ) = \nabla _ { \boldsymbol { \theta } } \left[ \sum _ { i = 1 } ^ { m } \eta _ { i } ( \pi _ { \boldsymbol { \theta } } ) \right] = \mathbb { E } _ { \boldsymbol { \rho } _ { \pi } , \pi } \left[ \sum _ { i = 1 } ^ { m } g _ { i } \right] .
+$$
+
+Recall the following assumption:
+
+$$
+\begin{array} { r } { \nabla _ { \theta } \log \pi _ { \theta } ( a _ { t } ^ { i } | s _ { t } ) ^ { T } \nabla _ { \theta } \log \pi _ { \theta } ( a _ { t } ^ { j } | s _ { t } ) \equiv z _ { i } ^ { T } z _ { j } \approx 0 , \quad \forall i \ne j , } \end{array}
+$$
+
+which translates to meaning that different subsets of parameters strongly influence different action dimensions or factors. This is true in case of distributed systems by construction, and also true
+
+in a single agent system if different action coordinates are strongly influenced by different policy network channels. Under this assumption, we have:
+
+$$
+\begin{array} { r l } & { \displaystyle \mathrm { V a r } ( \sum _ { i = 1 } ^ { m } g _ { i } ) = \sum _ { i } \mathrm { V a r } ( g _ { i } ) + \sum _ { i } \sum _ { j \ne i } \mathrm { C o v } ( g _ { i } , g _ { j } ) } \\ & { \displaystyle \quad = \sum _ { i } \mathrm { V a r } ( g _ { i } ) + \sum _ { i } \sum _ { j \ne i } \mathbb { E } _ { \rho _ { \pi } , \pi } \left[ g _ { i } ^ { T } g _ { j } \right] - \mathbb { E } _ { \rho _ { \pi } , \pi } \left[ g _ { i } \right] ^ { T } \mathbb { E } _ { \rho _ { \pi } , \pi } \left[ g _ { j } \right] } \\ & { \displaystyle \quad = \sum _ { i } \mathrm { V a r } ( g _ { i } ) + 0 - \sum _ { i } \sum _ { j \ne i } \mathbb { E } _ { \rho _ { \pi } , \pi } \left[ g _ { i } \right] ^ { T } \mathbb { E } _ { \rho _ { \pi } , \pi } \left[ g _ { j } \right] \quad \mathrm { ( b y ~ E q u a t i o n ~ ( 3 1 ) ) } } \\ & { \displaystyle \quad = \sum _ { i } \mathrm { V a r } ( g _ { i } ) - \sum _ { i } \sum _ { j \ne i } M _ { i j } \quad \mathrm { ( b y ~ s c o r e ~ f u n c t i o n ~ e s t i m a t o r ) } } \end{array}
+$$
+
+where we denote the mean correction term $M _ { i j } : = \mathbb { E } _ { \rho _ { \pi } , \pi } \left[ z _ { i } \hat { Q } ( s _ { t } , a _ { t } ) \right] ^ { T } \mathbb { E } _ { \rho _ { \pi } , \pi } \left[ z _ { j } \hat { Q } ( s _ { t } , a _ { t } ) \right]$ . Also let $\begin{array} { r } { M = \sum _ { i } \sum _ { j } M _ { i j } } \end{array}$ . Note that $M$ does not depend on $\bar { b _ { i } } ( \cdot )$ , and thus does not affect the optimal value.
+
+The overall variance is minimized when each component variance is minimized. We now derive the optimal baselines $b _ { i } ^ { * } ( s _ { t } , a _ { t } ^ { - i } )$ which minimize each respective component.
+
+$$
+\begin{array} { r l } { \mathrm { V a r } ( g _ { \mathrm { t } } ) = \mathbb { E } _ { \rho _ { x } , \pi } [ z _ { i } ^ { T } z _ { i } ( \hat { Q } ( s _ { t } , a _ { t } ) - b _ { i } ( s _ { t } , a _ { t } ^ { - i } ) ) ^ { 2 } ] } & { { } } \\ { - \mathbb { E } _ { \rho _ { x } , \pi } [ z _ { i } ( \hat { Q } ( s _ { t } , a _ { t } ) - b _ { i } ( s _ { t } , a _ { t } ^ { - i } ) ) ] ^ { T } \mathbb { E } _ { \rho _ { x } , \pi } [ z _ { i } ( \hat { Q } ( s _ { t } , a _ { t } ) - b _ { i } ( s _ { t } , a _ { t } ^ { - i } ) ) ] } & { { } } \\ { = \mathbb { E } _ { \rho _ { x } , \pi } [ z _ { i } ^ { T } z _ { i } ( \hat { Q } ( s _ { t } , a _ { t } ) ^ { 2 } - 2 b _ { i } ( s _ { t } , a _ { t } ^ { - i } ) \hat { Q } ( s _ { t } , a _ { t } ) + b _ { i } ( s _ { t } , a _ { t } ^ { - i } ) ) ^ { 2 } ] } & { { } } \\ { - \mathbb { E } _ { \rho _ { x } , \pi } [ z _ { i } ( \hat { Q } ( s _ { t } , a _ { t } ) ) ] ^ { T } \mathbb { E } _ { \rho _ { x } , \pi } [ z _ { i } ( \hat { Q } ( s _ { t } , a _ { t } ) ) ] } & { { } } \\ { = \mathbb { E } _ { \rho _ { x } , \pi } [ z _ { i } ^ { T } z _ { i } \hat { Q } ( s _ { t } , a _ { t } ) ^ { 2 } ] } & { { } } \\  + \mathbb { E } _ { \rho _ { x } , \pi } [ z _ { i } ^ { T } z _ { i } \hat { Q } ( s _ { t } , a _ { t } ^ { - i } ) \mathbb { E } _ { \sigma _ { i } } [ z _ { i } ^ { T } z _ { i } \hat { Q } ( s _ { t } , a _ { t } ) ] + b _ { i } ( s _ { t } , a _ { t } ^ { - i } ) ^ { 2 } \mathbb { E } _  \rho \end{array}
+$$
+
+Having written down the expression for variance under any action-dependent baseline, we seek the optimal baseline that would minimize this variance.
+
+$$
+{ \frac { \partial } { \partial b _ { i } } } \left[ \operatorname { v a r } ( \sum _ { i } g _ { i } ) \right] = { \frac { \partial } { \partial b _ { i } } } \left[ \operatorname { V a r } ( g _ { i } ) \right] = 0
+$$
+
+$$
+\implies b _ { i } ^ { * } ( s _ { t } , a _ { t } ^ { - i } ) = \frac { \mathbb { E } _ { a _ { t } ^ { i } } \left[ z _ { i } ^ { T } z _ { i } \hat { Q } ( s _ { t } , a _ { t } ) \right] } { \mathbb { E } _ { a _ { t } ^ { i } } \left[ z _ { i } ^ { T } z _ { i } \right] }
+$$
+
+The optimal action-dependent baseline is:
+
+$$
+b _ { i } ^ { * } ( s _ { t } , a _ { t } ^ { - i } ) = \frac { \mathbb { E } _ { a _ { t } ^ { i } } \left[ \nabla _ { \theta } \log \pi _ { \theta } ( a _ { t } ^ { i } | s _ { t } ) ^ { T } \nabla _ { \theta } \log \pi _ { \theta } ( a _ { t } ^ { i } | s _ { t } ) \hat { Q } ( s _ { t } , a _ { t } ) \right] } { \mathbb { E } _ { a _ { t } ^ { i } } \left[ \nabla _ { \theta } \log \pi _ { \theta } ( a _ { t } ^ { i } | s _ { t } ) ^ { T } \nabla _ { \theta } \log \pi _ { \theta } ( a _ { t } ^ { i } | s _ { t } ) \right] }
+$$
+
+# C DERIVATION OF VARIANCE REDUCTION IMPROVEMENT
+
+We now turn to quantifying the reduction in variance of the policy gradient estimate under the optimal baseline derived above. Let $\textstyle \operatorname { V a r } ^ { * } ( \sum _ { i } g _ { i } )$ denote the variance resulting from the optimal action-dependent baseline, and let $\textstyle \operatorname { V a r } ( \sum _ { i } g _ { i } )$ denote the variance resulting from another baseline
+
+$\boldsymbol { b } = ( b _ { i } ( s _ { t } , a _ { t } ^ { - i } ) ) _ { i \in [ m ] }$ , which may be suboptimal or action-independent. Recall the notation:
+
+$$
+\begin{array} { r l } & { Z _ { i } : = Z _ { i } ( s _ { t } , a _ { t } ^ { - i } ) = \mathbb { E } _ { a _ { t } ^ { i } } \left[ \nabla _ { \theta } \log \pi _ { \theta } ( a _ { t } ^ { i } | s _ { t } ) ^ { T } \nabla _ { \theta } \log \pi _ { \theta } ( a _ { t } ^ { i } | s _ { t } ) \right] } \\ & { Y _ { i } : = Y _ { i } ( s _ { t } , a _ { t } ^ { - i } ) = \mathbb { E } _ { a _ { t } ^ { i } } \left[ \nabla _ { \theta } \log \pi _ { \theta } ( a _ { t } ^ { i } | s _ { t } ) ^ { T } \nabla _ { \theta } \log \pi _ { \theta } ( a _ { t } ^ { i } | s _ { t } ) \hat { Q } ( s _ { t } , a _ { t } ) \right] } \\ & { X _ { i } : = X _ { i } ( s _ { t } , a _ { t } ^ { - i } ) = \mathbb { E } _ { a _ { t } ^ { i } } \left[ \nabla _ { \theta } \log \pi _ { \theta } ( a _ { t } ^ { i } | s _ { t } ) ^ { T } \nabla _ { \theta } \log \pi _ { \theta } ( a _ { t } ^ { i } | s _ { t } ) \hat { Q } ( s _ { t } , a _ { t } ) ^ { 2 } \right] } \end{array}
+$$
+
+Finally, define the variance improvement $I _ { b } : = \mathrm { V a r } ( \textstyle \sum _ { i } g _ { i } ) - \mathrm { V a r } ^ { * } ( \textstyle \sum _ { i } g _ { i } )$ . Using these definitions, the variance can be re-written as:
+
+$$
+\mathrm { V a r } ( \sum _ { i } g _ { i } ) = \sum _ { i } \mathbb { E } _ { \rho _ { \pi } , a _ { t } ^ { - i } } \left[ X _ { i } - 2 b _ { i } ( s _ { t } , a _ { t } ^ { - i } ) Y _ { i } + b _ { i } ( s _ { t } , a _ { t } ^ { - i } ) ^ { 2 } Z _ { i } \right] - M
+$$
+
+Furthermore, the variance of the gradient with the optimal baseline can be written as
+
+$$
+\operatorname { V a r } ^ { * } ( \sum _ { i } g _ { i } ) = \sum _ { i } \mathbb { E } _ { \rho _ { \pi } , a _ { t } ^ { - i } } \left[ X _ { i } - { \frac { Y _ { i } ^ { 2 } } { Z _ { i } } } \right] - M
+$$
+
+The difference in variance can be calculated as:
+
+$$
+\begin{array} { r l } & { d _ { b } : = \displaystyle \sum _ { i } \bigg ( \mathbb { E } _ { \rho _ { s - \sigma _ { i } ^ { - 1 } } } \left[ X _ { i } - 2 b _ { i } ( s _ { k } , \sigma _ { i } ^ { - 1 } ) Y _ { i } + b _ { i } ( s _ { k } , \sigma _ { i } ^ { - 1 } ) ^ { 2 } Z _ { i } \right] - \big ( \mathbb { E } _ { \rho _ { s - \sigma _ { i } ^ { - 1 } } } \left[ X _ { i } - \frac { Y _ { i } ^ { 2 } } { Z _ { i } } \right] \big ) \bigg ) } \\ & { \quad = \displaystyle \sum _ { i } \mathbb { E } _ { \rho _ { s - \sigma _ { i } ^ { - 1 } } } \left[ - 2 b _ { i } ( s _ { k } , \sigma _ { i } ^ { - 1 } ) Y _ { i } + b _ { i } ( s _ { k } , \sigma _ { i } ^ { - 1 } ) ^ { 2 } Z _ { i } + \frac { Y _ { i } ^ { 2 } } { Z _ { i } } \right] } \\ & { \quad = \displaystyle \sum _ { i } \mathbb { E } _ { \rho _ { s - \sigma _ { i } ^ { - 1 } } } \left[ \left( b _ { i } ( s _ { k } , \sigma _ { i } ^ { - 1 } ) \sqrt { Z _ { i } } - \frac { Y _ { i } } { \sqrt { Z _ { i } } } \right) ^ { 2 } \right] } \\ & { \quad = \displaystyle \sum _ { i } \mathbb { E } _ { \rho _ { s - \sigma _ { i } ^ { - 1 } } } \left[ Z _ { \rho _ { s } } \left( b _ { i } ( s _ { k } , \sigma _ { i } ^ { - 1 } ) - \frac { Y _ { i } } { Z _ { i } } \right) ^ { 2 } \right] } \\ & { \quad = \displaystyle \sum _ { i } \mathbb { E } _ { \rho _ { s - \sigma _ { i } ^ { - 1 } } } \left[ Z _ { \rho _ { s } } \left( b _ { i } ( s _ { k } , \sigma _ { i } ^ { - 1 } ) - \frac { Y _ { i } } { Z _ { i } } \right) ^ { 2 } \right] } \\ & { \quad = \displaystyle \sum _ { i } \mathbb { E } _ { \rho _ { s - \sigma _ { i } ^ { - 1 } } } \left[ Z _ { \rho _ { s } } \left( b _ { i } ( s _ { k } , \sigma _ { i } ^ { - 1 } ) - b _ { i } ^ { * } ( s _ { k } , \sigma _ { i } ^ { - 1 } ) \right) ^ { 2 } \right] } \\ &  \quad \end{array}
+$$
+
+# D DERIVATION OF SUBOPTIMALITY OF THE OPTIMAL STATE-DEPENDENT BASELINE
+
+Using the notation from Appendix $\textrm { C }$ and working off of Equation (51), we have:
+
+$$
+\begin{array} { r l } & { I _ { b = b ^ { \star } ( s ) } : = \displaystyle \sum _ { i } \mathbb { E } _ { \rho _ { \tau } , a _ { t } ^ { - i } } \left[ Z _ { i } \left( b ^ { \star } ( s _ { t } ) - b _ { i } ^ { \star } ( s _ { t } , a _ { t } ^ { - i } ) \right) ^ { 2 } \right] } \\ & { \quad \quad \quad \quad = \displaystyle \sum _ { i } \mathbb { E } _ { \rho _ { \tau } , a _ { t } ^ { - i } } \left[ Z _ { i } \left( \frac { \sum _ { j } Y _ { j } } { \sum _ { j } Z _ { j } } - \frac { Y _ { i } } { Z _ { i } } \right) ^ { 2 } \right] } \\ & { \quad \quad \quad \quad = \displaystyle \sum _ { i } \mathbb { E } _ { \rho _ { \tau } , a _ { t } ^ { - i } } \left[ \frac { 1 } { Z _ { i } } \left( \frac { Z _ { i } } { \sum _ { j } Z _ { j } } \sum _ { j } Y _ { j } - Y _ { i } \right) ^ { 2 } \right] ^ { - 1 } } \end{array}
+$$
+
+# E BASELINES FOR GENERAL ACTIONS
+
+In the preceding derivations, we have assumed policy actions are conditionally independent across dimensions. In the more general case, we only assume that there are $m$ factors $\bar { a } _ { t } ^ { 1 }$ through $a _ { t } ^ { m }$
+
+which altogether forms the action $a _ { t }$ . Conditioned on $s _ { t }$ , the different factors form a certain directed acyclic graphical model (including the fully dependent case). Without loss of generality, we assume that the following factorization holds:
+
+$$
+\pi _ { \boldsymbol { \theta } } \big ( a _ { t } | s _ { t } \big ) = \prod _ { i = 1 } ^ { m } \pi _ { \boldsymbol { \theta } } \big ( a _ { t } ^ { i } | s _ { t } , a _ { t } ^ { f ( i ) } \big )
+$$
+
+where $f ( i )$ denotes the indices of the parents of the $i$ th factor. Let $D ( i )$ denote the indices of descendants of $i$ in the graphical model (including $i$ itself). In this case, we can set the $i$ th baseline to be $b _ { i } ( s _ { t } , a _ { t } ^ { [ m ] \setminus D ( i ) } )$ , where $[ m ] = \{ 1 , 2 , \dots , m \}$ . In other words, the ith baseline can depend on all other factors which the ith factor does not influence. The overall gradient estimator is given by
+
+$$
+\nabla _ { \theta } \eta ( \pi _ { \theta } ) = \mathbb { E } _ { \rho _ { \pi } , \pi } \left[ \sum _ { i = 1 } ^ { m } \nabla _ { \theta } \log \pi _ { \theta } ( a _ { t } ^ { i } | s _ { t } , a _ { t } ^ { f ( i ) } ) \left( \hat { Q } ( s _ { t } , a _ { t } ) - b _ { i } ( s _ { t } , a _ { t } ^ { [ m ] \setminus D ( i ) } ) \right) \right]
+$$
+
+In the most general case without any conditional independence assumptions, we have $f ( i ) =$ $\{ 1 , 2 , \ldots , i - 1 \}$ , and $D ( i ) = \{ i , i + 1 , . . . , m \}$ . The above equation reduces to
+
+$$
+\nabla _ { \theta } \eta ( \pi _ { \theta } ) = \mathbb { E } _ { \rho _ { \pi } , \pi } \left[ \sum _ { i = 1 } ^ { m } \nabla _ { \theta } \log \pi _ { \theta } ( a _ { t } ^ { i } | s _ { t } , a _ { t } ^ { 1 } , \dots , a _ { t } ^ { i - 1 } ) \left( \hat { Q } ( s _ { t } , a _ { t } ) - b _ { i } ( s _ { t } , a _ { t } ^ { 1 } , \dots , a _ { t } ^ { i - 1 } ) \right) \right]
+$$
+
+The above analysis for optimal baselines and variance suboptimality transfers also to the case of general actions.
+
+The applicability of our techniques to general action spaces may be of crucial importance for many application domains where the conditional independence assumption does not hold up, such as language tasks and other compositional domains. Even in continuous control tasks, such as hand manipulation, and many other tasks where it is common practice to use conditionally independent factorized policies, it is reasonable to expect training improvement from policies without a full conditionally independence structure.
+
+Computing action-dependent baselines for general actions The marginalization presented in Section 4.4 does not apply for the general action setting. Instead, $m$ individual baselines can be trained according to the factorization, and each of them can be fitted from data collected from the previous iteration. In the general case, this means fitting $m$ functions $b _ { i } ( s _ { t } , a _ { t } ^ { 1 } , \dots , a _ { t } ^ { i - 1 } )$ , for $i \in$ $\{ 1 , \ldots , m \}$ . The resulting method is described in Algorithm 2.
+
+There may also exist special cases like conditional independent actions, for which more efficient baseline constructions exist. A closely related example to the conditionally independent case is the case of block diagonal covariance structures (e.g. in multi-agent settings), where we may wish to instead learn an overall Q function and marginalize over block factors. Another interesting example to explore is sparse covariance structures.
+
+<table><tr><td>Algorithm 2 Policy gradient for general factorization policies using action-dependent baselines</td></tr><tr><td>Require: number of iterations N, batch size B,initial policy parameters θ</td></tr><tr><td>Initialize baselines bi(St, al a[m\D(i)) =0,for i ∈ {1,...,m} and policy π</td></tr><tr><td>for j in {1,...,N} do Collect samples: (St, at)teε{.,..,B}</td></tr><tr><td>Compute advantages: Ai(St,at):= Q(st,at)-bi(Sst,a[m\D(i),t</td></tr><tr><td>Perform a policy update step on θ using Ai(st, at） [Equation (57)]</td></tr><tr><td></td></tr><tr><td>Update baseline functions with current batch: bi(st,a[m\D(i) end for</td></tr></table>
+
+# F COMPATIBILITY WITH GAE
+
+Temporal Difference (TD) learning methods such as GAE (Schulman et al., 2016) allow us to smoothly interpolate between high-bias, low-variance estimates and low-bias, high-variance estimates of the policy gradient. These methods are based on the idea of being able to predict future returns, thereby bootstrapping the learning procedure. In particular, when using the value function as a baseline, we have $\bar { A } ( \bar { s } _ { t } , \bar { a } _ { t } ) = \mathbb { E } \left[ r _ { t } + \gamma V ( s _ { t + 1 } ) - \bar { V } ( s _ { t } ) \right] = [ r _ { t } + \gamma b ( \bar { s _ { t + 1 } } ) - b ( s _ { t } ) ] ]$ if $b ( s )$ is an unbiased estimator for $V ( s )$ . GAE uses an exponential averaging of such temporal difference terms over a trajectory to significantly reduce the variance of the advantage at the cost of a small bias (it allows us to pick where we want to be on the bias-variance curve). Similarly, if we use $b _ { i } ( s _ { t } , a _ { t } ^ { - i } )$ as an unbiased estimator for $\mathbb { E } _ { a _ { t } ^ { i } } [ \hat { Q } ( s _ { t } , a _ { t } ) ]$ , we have:
+
+$$
+\mathbb { E } _ { \pi , \boldsymbol { M } } \left[ r _ { t } + \gamma b _ { i } ( s _ { t + 1 } , a _ { t + 1 } ^ { - i } ) - b _ { i } ( s _ { t } , a _ { t } ^ { - i } ) \right] = \boldsymbol { Q } ( s _ { t } , a _ { t } ) - \mathbb { E } _ { a _ { t } ^ { i } } [ \hat { \boldsymbol { Q } } ( s _ { t } , a _ { t } ) ] = A _ { i } ( s _ { t } , a _ { t } )
+$$
+
+Thus, the temporal difference error with the action dependent baselines is an unbiased estimator for the advantage function as well. This allows us to use the GAE procedure to further reduce variance at the cost of some bias.
+
+The following study shows that action-dependent baselines are consistent with TD procedures with their temporal differences being estimates of the advantage function. Our results summarized in Figure 4 suggests that slightly biasing the gradient to reduce variance produces the best results, while high-bias estimates perform poorly. Prior work with baselines that utilize global information (Foerster et al., 2017) employ the high-bias variant. The results here suggest that there is potential to further improve upon those results by carefully studying the bias-variance trade-off.
+
+![](images/88235c570f63b7588c651b56f8672074ee73db1202b3409032193d6a8fcfdd0a.jpg)  
+Figure 4: We study the influence of $\lambda$ in GAE which allows us to trade off bias and variance as desired. High bias gradient corresponding to smaller values of $\lambda$ do not make progress after a while. High variance gradient $\lambda = 1$ ) has trouble learning initially. Allowing for a small bias to reduce the variance, corresponding to the intermediate $\lambda = 0 . 9 7$ produces the best overall result, consistent with the findings in Schulman et al. (2016).
+
+# G HIGH-DIMENSIONAL ACTION SPACES: TRAINING CURVES
+
+Figure 5 shows the resulting training curves for a synthetic high-dimensional target matching task, as described in Section 5. For higher dimensional action spaces (100 dimensions or greater), the action-dependent baseline consistently converges to the optimal solution $10 \%$ faster than the stateonly baseline.
+
+For reference, Figure 6 shows the result of the original high-dimensional action space experiment. Due to a discovered issue in the TensorFlow version of rllab, which results in training instability, both methods (action-dependent and state-dependent baselines) under-performed relative to the revised experiment (Figure 5), which uses a clean implementation based on the implementation referenced inRajeswaran et al. (2017b). The regression in training is most evident by the number of iterations required to solve the task; for instance, the old experiment could take as long as five times more iterations to solve the same task, even for a 12-dimensional task.
+
+# H EXPERIMENT DETAILS
+
+Parameters: Unless otherwise stated, the following parameters are used in the experiments in this work: $\gamma = 0 . 9 9 5 , \lambda _ { \mathrm { G A E } } = 0 . 9 7 , k l _ { \mathrm { d e s } }$ $k l _ { \mathrm { d e s i r e d } } = 0 . 0 2 5$ .
+
+![](images/0220e9d4e0a6a6436552f8cae7053882313b8a8de12f1fa8bff11671ebcc414d.jpg)  
+Figure 5: Shown is the learning curve for a synthetic high-dimensional target matching task (5 seeds), for 12 to 2000 dimensional action spaces. At high dimensions, the linear feature actiondependent baseline provides notable and consistent variance reduction, as compared to a linear feature state baseline. For 100, 400, and 2000 dimensions, our method converges $10 \%$ faster to the optimal solution.
+
+![](images/21ef4add1f2f546a8b53e9fdf47a24d710b6555634013548a3014966156b7a83.jpg)  
+Figure 6: Reference training curves for an early and invalidated version of the synthetic highdimensional target matching task (3 seeds).
+
+Policies: The policies used are 2-layer fully connected networks with hidden sizes $=$ (32, 32).
+
+Initialization: the policy is initialized with Xavier initialization except final layer weights are scaled down (by a factor of 100x). Note that since the baseline is linear (with RBF features) and estimated with a Newton step, the initialization is inconsequential.
+
+Per-experiment configuration: The following parameters in Table 2 are for both state-only and action-dependent versions of the experiments. The $m$ -DimTargetMatching experiments use a linear feature baseline. Table 3 details the dimensionality of the action space for each task.
+
+Table 2: Experiment details   
+
+<table><tr><td>Task</td><td>Benchmarks</td><td>Hand task</td><td>Peg Insertion</td><td>CommunicateTarget</td><td>m-DimTargetMatching</td></tr><tr><td>Trajectories</td><td>10</td><td>100</td><td>200</td><td>300</td><td>150</td></tr><tr><td>Horizon</td><td>1000</td><td>200</td><td>250</td><td>100</td><td>1</td></tr><tr><td>RBFfeatures</td><td>100</td><td>250</td><td>250</td><td>250</td><td>N/A</td></tr></table>
+
+Table 3: Action dimensionality of tasks   
+
+<table><tr><td>Task</td><td>Action dimensions</td></tr><tr><td>Hopper</td><td>3</td></tr><tr><td>HalfCheetah</td><td>6</td></tr><tr><td>Ant</td><td>8</td></tr><tr><td>Hand</td><td>30</td></tr><tr><td>Peg</td><td>7</td></tr><tr><td>CommunicateTarget m-DimTargetMatching</td><td>8 (4 per agent) m</td></tr></table>

parse/train/HygnDhEtvr/HygnDhEtvr.md

@@ -0,0 +1,440 @@
+# REINFORCEMENT LEARNING BASED GRAPH-TO-SEQUENCE MODEL FOR NATURAL QUESTION GENERATION
+
+Yu Chen Department of Computer Science Rensselaer Polytechnic Institute cheny39@rpi.edu
+
+Lingfei Wu˚   
+IBM Research   
+lwu@email.wm.edu
+
+Mohammed J. Zaki Department of Computer Science Rensselaer Polytechnic Institute zaki@cs.rpi.edu
+
+# ABSTRACT
+
+Natural question generation (QG) aims to generate questions from a passage and an answer. Previous works on QG either (i) ignore the rich structure information hidden in text, (ii) solely rely on cross-entropy loss that leads to issues like exposure bias and inconsistency between train/test measurement, or (iii) fail to fully exploit the answer information. To address these limitations, in this paper, we propose a reinforcement learning (RL) based graph-to-sequence (Graph2Seq) model for QG. Our model consists of a Graph2Seq generator with a novel Bidirectional Gated Graph Neural Network based encoder to embed the passage, and a hybrid evaluator with a mixed objective combining both cross-entropy and RL losses to ensure the generation of syntactically and semantically valid text. We also introduce an effective Deep Alignment Network for incorporating the answer information into the passage at both the word and contextual levels. Our model is end-to-end trainable and achieves new state-of-the-art scores, outperforming existing methods by a significant margin on the standard SQuAD benchmark.
+
+# 1 INTRODUCTION
+
+Natural question generation (QG) has many useful applications such as improving the question answering task (Chen et al., 2017; 2019a) by providing more training data (Tang et al., 2017; Yuan et al., 2017), generating practice exercises and assessments for educational purposes (Heilman & Smith, 2010; Danon & Last, 2017), and helping dialog systems to kick-start and continue a conversation with human users (Mostafazadeh et al., 2016). While many existing works focus on QG from images (Fan et al., 2018; Li et al., 2018) or knowledge bases (Serban et al., 2016; Elsahar et al., 2018), in this work, we focus on QG from text.
+
+Conventional methods (Mostow & Chen, 2009; Heilman & Smith, 2010; Heilman, 2011) for QG rely on heuristic rules or hand-crafted templates, leading to the issues of low generalizability and scalability. Recent attempts have been focused on exploiting Neural Network (NN) based approaches that do not require manually-designed rules and are end-to-end trainable. Encouraged by the huge success of neural machine translation, these approaches formulate the QG task as a sequence-tosequence (Seq2Seq) learning problem. Specifically, attention-based Seq2Seq models (Bahdanau et al., 2014; Luong et al., 2015) and their enhanced versions with copy (Vinyals et al., 2015; Gu et al., 2016) and coverage (Tu et al., 2016) mechanisms have been widely applied and show promising results on this task (Du et al., 2017; Zhou et al., 2017; Song et al., 2018a; Kumar et al., 2018a). However, these methods typically ignore the hidden structural information associated with a word sequence such as the syntactic parsing tree. Failing to utilize the rich text structure information beyond the simple word sequence may limit the effectiveness of these models for QG.
+
+It has been observed that in general, cross-entropy based sequence training has several limitations like exposure bias and inconsistency between train/test measurement (Ranzato et al., 2015; Wu et al., 2016). As a result, they do not always produce the best results on discrete evaluation metrics on sequence generation tasks such as text summarization (Paulus et al., 2017) or question generation (Song et al., 2017). To cope with these issues, some recent QG approaches (Song et al., 2017; Kumar et al., 2018b) directly optimize evaluation metrics using Reinforcement Learning (RL) (Williams, 1992). However, existing approaches usually only employ evaluation metrics like BLEU and ROUGE-L as rewards for RL training. More importantly, they fail to exploit other important metrics such as syntactic and semantic constraints for guiding high-quality text generation.
+
+Early works on neural QG did not take into account the answer information when generating a question. Recent works have started to explore various means of utilizing the answer information. When question generation is guided by the semantics of an answer, the resulting questions become more relevant and readable. Conceptually, there are three different ways to incorporate the answer information by simply marking the answer location in the passage (Zhou et al., 2017; Zhao et al., 2018; Liu et al., 2019), or using complex passage-answer matching strategies (Song et al., 2017), or separating answers from passages when applying a Seq2Seq model (Kim et al., 2018; Sun et al., 2018). However, they neglect potential semantic relations between passage words and answer words, and thus fail to explicitly model the global interactions among them in the embedding space.
+
+To address these aforementioned issues, in this paper, we present a novel reinforcement learning based generator-evaluator architecture that aims to: i) make full use of rich hidden structure information beyond the simple word sequence; ii) generate syntactically and semantically valid text while maintaining the consistency of train/test measurement; iii) model explicitly the global interactions of semantic relationships between passage and answer at both word-level and contextual-level.
+
+In particular, to achieve the first goal, we explore two different means to either construct a syntaxbased static graph or a semantics-aware dynamic graph from the text sequence, as well as its rich hidden structure information. Then, we design a graph-to-sequence (Graph2Seq) model based generator that encodes the graph representation of a text passage and decodes a question sequence using a Recurrent Neural Network (RNN). Our Graph2Seq model is based on a novel bidirectional gated graph neural network, which extends the gated graph neural network (Li et al., 2015) by considering both incoming and outgoing edges, and fusing them during the graph embedding learning.
+
+To achieve the second goal, we design a hybrid evaluator which is trained by optimizing a mixed objective function that combines both cross-entropy and RL loss. We use not only discrete evaluation metrics like BLEU, but also semantic metrics like word mover’s distance (Kusner et al., 2015) to encourage both syntactically and semantically valid text generation. To achieve the third goal, we propose a novel Deep Alignment Network (DAN) for effectively incorporating answer information into the passage at multiple granularity levels.
+
+Our main contributions are as follows:
+
+• We propose a novel RL-based Graph2Seq model for natural question generation. To the best of our knowledge, we are the first to introduce the Graph2Seq architecture for QG. • We explore both static and dynamic ways of constructing graph from text and are the first to systematically investigate their performance impacts on a GNN encoder. • The proposed model is end-to-end trainable, achieves new state-of-the-art scores, and outperforms existing methods by a significant margin on the standard SQuAD benchmark for QG. Our human evaluation study also corroborates that the questions generated by our model are more natural (semantically and syntactically) compared to other baselines.
+
+# 2 AN RL-BASED GENERATOR-EVALUATOR ARCHITECTURE
+
+In this section, we define the question generation task, and then present our RL-based Graph2Seq model for question generation. We first motivate the design, and then present the details of each component as shown in Fig. 1.
+
+![](images/ec860f3d8c6b732e9a8af64da76147a3c5a656f5530e291195724ae91bb138b5.jpg)  
+Figure 1: Overall architecture of the proposed model. Best viewed in color.
+
+# 2.1 PROBLEM FORMULATION
+
+The goal of question generation is to generate natural language questions based on a given form of data, such as knowledge base triples or tables (Bao et al., 2018), sentences (Du et al., 2017; Song et al., 2018a), or images (Li et al., 2018), where the generated questions need to be answerable from the input data. In this paper, we focus on QG from a given text passage, along with a target answer.
+
+We assume that a text passage is a collection of word tokens $X ^ { p } = \{ x _ { 1 } ^ { p } , x _ { 2 } ^ { p } , . . . , x _ { N } ^ { p } \}$ , and a target answer is also a collection of word tokens $X ^ { a } ~ = ~ \{ x _ { 1 } ^ { a } , x _ { 2 } ^ { a } , . . . , x _ { L } ^ { a } \}$ . The task of natural question generation is to generate the best natural language question consisting of a sequence of word tokens $\bar { \hat { Y } } = \{ y _ { 1 } , y _ { 2 } , . . . , y _ { T } \}$ which maximizes the conditional likelihood $\hat { Y } = \arg \operatorname* { m a x } _ { Y } P ( Y | X ^ { p } , X ^ { a } )$ . Here $N , L$ , and $T$ are the lengths of the passage, answer and question, respectively. We focus on the problem setting where we have a set of passage (and answers) and target questions pairs, to learn the mapping; existing QG approaches (Du et al., 2017; Song et al., 2018a; Zhao et al., 2018; Kim et al., 2018) make a similar assumption.
+
+# 2.2 DEEP ALIGNMENT NETWORK
+
+Answer information is crucial for generating relevant and high quality questions from a passage. Unlike previous methods that neglect potential semantic relations between passage and answer words, we explicitly model the global interactions among them in the embedding space. To this end, we propose a novel Deep Alignment Network (DAN) component for effectively incorporating answer information into the passage with multiple granularity levels. Specifically, we perform attentionbased soft-alignment at the word-level, as well as at the contextual-level, so that multiple levels of alignments can help learn hierarchical representations.
+
+![](images/b3fab1bcd4ba51aad085139b56e8f4f89a28919dde310f9b68a2a13800329c9d.jpg)  
+Figure 2: The attention-based soft-alignment mechanism.
+
+Let $\mathbf { X } ^ { p } \in \mathbb { R } ^ { F \times N }$ and $\tilde { \mathbf { X } } ^ { p } \in \mathbb { R } ^ { \tilde { F } _ { p } \times N }$ denote two embeddings associated with passage text. Similarly, let $\mathbf { X } ^ { a } \in \mathbb { R } ^ { F \times L }$ and $\tilde { \mathbf { X } } ^ { a } \in \mathbb { R } ^ { \tilde { F } _ { a } \times L }$ denote two embeddings associated with answer text. Conceptually, as shown in Fig. 2, the soft-alignment mechanism consists of three steps: i) compute the attention score $\beta _ { i , j }$ for each pair of passage word $\boldsymbol { x } _ { i } ^ { p }$ and answer word $\boldsymbol { x } _ { j } ^ { a }$ : ii) multiply the attention matrix $\beta$ with the answer embeddings $\widetilde { \mathbf { X } } ^ { a }$ to obtain the aligned answer embeddings $\mathbf { H } ^ { p }$ for the passage; iii) concatenate the resulting aligned answer embeddings $\mathbf { H } ^ { p }$ with the passage embeddings $\widetilde { \mathbf { X } } ^ { p }$ to get the final passage embeddings $\bar { \tilde { \mathbf { H } } } ^ { p } \in \mathbb { R } ^ { ( \widetilde { F } _ { p } + \widetilde { F } _ { a } ) \times N }$ .
+
+Formally, we define our soft-alignment function as following:
+
+$$
+\widetilde { \mathbf { H } } ^ { p } = \mathrm { A l i g n } ( \mathbf { X } ^ { p } , \mathbf { X } ^ { a } , \widetilde { \mathbf { X } } ^ { p } , \widetilde { \mathbf { X } } ^ { a } ) = \mathrm { C A T } ( \widetilde { \mathbf { X } } ^ { p } ; \mathbf { H } ^ { p } ) = \mathrm { C A T } ( \widetilde { \mathbf { X } } ^ { p } ; \widetilde { \mathbf { X } } ^ { a } \beta ^ { T } )
+$$
+
+where the matrix $\widetilde { \mathbf { H } } ^ { p }$ is the final passage embedding, the function CAT is a simple concatenation operation, and $\beta$ is a $N \times L$ attention score matrix, computed by
+
+$$
+\beta \propto \exp \Bigl ( \mathrm { R e L U } ( \mathbf { W X } ^ { p } ) ^ { T } \mathbf { R e L U } ( \mathbf { W X } ^ { a } ) \Bigr )
+$$
+
+where $\mathbf { W } \in \mathbb { R } ^ { d \times F }$ is a trainable weight matrix, with $d$ being the hidden state size and ReLU is the rectified linear unit (Nair & Hinton, 2010). After introducing the general soft-alignment mechanism, we next introduce how we do soft-alignment at both word-level and contextual-level.
+
+# 2.2.1 WORD-LEVEL ALIGNMENT
+
+In the word-level alignment stage, we first perform a soft-alignment between the passage and the answer based only on their pretrained GloVe embeddings and compute the final passage embeddings by $\widetilde { \bf H } ^ { p } = \mathrm { A l i g n } ( { \bf G } ^ { p } , { \bf G } ^ { a } , [ { \bf G } ^ { p } ; { \bf B } ^ { p } ; { \bf L } ^ { p } ] , { \bf G } ^ { a } )$ , where $\mathbf { G } ^ { p }$ , $\mathbf { B } ^ { p }$ , and $\mathbf { L } ^ { p }$ are the corresponding GloVe embedding (Pennington et al., 2014), BERT embedding (Devlin et al., 2018), and linguistic feature (i.e., case, NER and POS) embedding of the passage text, respectively. Then a bidirectional LSTM (Hochreiter & Schmidhuber, 1997) is applied to the final passage embeddings s $\widetilde { \mathbf { H } } ^ { p } = \{ \widetilde { \mathbf { h } } _ { i } ^ { p } \} _ { i = 1 } ^ { N }$ to obtain contextualized passage embeddings Hs p P RFˆN .
+
+On the other hand, for the answer text $\mathbf { X } ^ { a }$ , we simply concatenate its GloVe embedding $\mathbf { G } ^ { a }$ and its BERT embedding $\mathbf { B } ^ { a }$ to obtain its word embedding matrix $\mathbf { H } ^ { a } \in \mathbb { R } ^ { d ^ { \prime } \times L }$ . Another BiLSTM is then applied to the concatenated answer embedding sequence to obtain the contextualized answer embeddings Hs a P RFsˆL.
+
+# 2.2.2 CONTEXTUAL-LEVEL ALIGNMENT
+
+In the contextual-level alignment stage, we perform another soft-alignment based on the contextualized passage and answer embeddings. Similarly, we compute the aligned answer embedding, and concatenate it with the contextualized passage embedding to obtain the final passage embedding matrix $\operatorname { A l i g n } ( [ \mathbf { G } ^ { p } ; \mathbf { B } ^ { p } ; \bar { \mathbf { H } } ^ { p } ] , [ \mathbf { G } ^ { a } ; \mathbf { B } ^ { a } ; \bar { \mathbf { H } } ^ { \dot { a } } ] , \bar { \mathbf { H } } ^ { \omega } , \bar { \mathbf { H } } ^ { a } )$ . Finally, we apply another BiLSTM to the above concatenated embedding to get a $F \times N$ passage embedding matrix X.
+
+# 2.3 BIDIECTIONAL GRAPH-TO-SEQUENCE GENERATOR
+
+While RNNs are good at capturing local dependencies among consecutive words in text, GNNs have been shown to better utilize the rich hidden text structure information such as syntactic parsing ( $\mathrm { { X u } }$ et al., 2018b) or semantic parsing (Song et al., 2018b), and can model the global interactions (relations) among sequence words to further improve the representations. Therefore, unlike most of the existing methods that rely on RNNs to encode the input passage, we first construct a passage graph $\mathcal { G }$ from text where each passage word is treated as a graph node, and then employ a novel Graph2Seq model to encode the passage graph (and answer), and to decode the question sequence.
+
+# 2.3.1 PASSAGE GRAPH CONSTRUCTION
+
+Existing GNNs assume a graph structured input and directly consume it for computing the corresponding node embeddings. However, we need to construct a graph from the text. Although there are early attempts on constructing a graph from a sentence $\mathrm { { X u } }$ et al., 2018b), there is no clear answer as to the best way of representing text as a graph. We explore both static and dynamic graph construction approaches, and systematically investigate the performance differences between these two methods in the experimental section.
+
+Syntax-based static graph construction: We construct a directed and unweighted passage graph based on dependency parsing. For each sentence in a passage, we first get its dependency parse tree. We then connect neighboring dependency parse trees by connecting those nodes that are at a sentence boundary and next to each other in text.
+
+Semantics-aware dynamic graph construction: We dynamically build a directed and weighted graph to model semantic relationships among passage words. We make the process of building such a graph depend on not only the passage, but also on the answer. The graph construction procedure consists of three steps: i) we compute a dense adjacency matrix A for the passage graph by applying self-attention to the word-level passage embeddings $\widetilde { \mathbf { H } } ^ { p }$ , ii) a kNN-style graph sparsification strategy (Chen et al., 2019c) is adopted to obtain a sparse adjacency matrix $\bar { \mathbf { A } }$ , where we only keep the $K$ nearest neighbors (including itself) as well as the associated attention scores (i.e., the remaining attentions scores are masked off) for each node; and iii) inspired by BiLSTM over LSTM, we also compute two normalized adjacency matrices $\mathbf { A } ^ { + }$ and $\mathbf { A } ^ { \vdash }$ according to their incoming and outgoing directions, by applying softmax operation on the resulting sparse adjacency matrix $\bar { \mathbf { A } }$ and its transpose, respectively.
+
+$$
+\mathbf { A } = \operatorname { R e L U } ( \mathbf { U } \tilde { \mathbf { H } } ^ { p } ) ^ { T } \operatorname { R e L U } ( \mathbf { U } \tilde { \mathbf { H } } ^ { p } ) , \quad \bar { \mathbf { A } } = \operatorname { k N N } ( \mathbf { A } ) , \quad \mathbf { A } ^ { \ - + } , \mathbf { A } ^ { \ - - } = \operatorname { s o f t m a x } \bigl ( \{ \bar { \mathbf { A } } , \bar { \mathbf { A } } ^ { T } \} \bigr )
+$$
+
+where $\mathbf { U }$ is a $d \times ( \widetilde { F } _ { p } + \widetilde { F } _ { a } )$ trainable weight matrix. Note that the supervision signal is able to back-propagate through the graph sparsification operation as the $K$ nearest attention scores are kept.
+
+# 2.3.2 BIDIRECTIONAL GATED GRAPH NEURAL NETWORKS
+
+To effectively learn the graph embeddings from the constructed text graph, we propose a novel Bidirectional Gated Graph Neural Network (BiGGNN) which extends Gated Graph Sequence Neural Networks (Li et al., 2015) by learning node embeddings from both incoming and outgoing edges in an interleaved fashion when processing the directed passage graph. Similar idea has also been exploited in (Xu et al., 2018a), which extended another popular variant of GNNs - GraphSAGE (Hamilton et al., 2017). However, one of key difference between our BiGGNN and their bidirectional GraphSAGE is that we fuse the intermediate node embeddings from both incoming and outgoing directions in every iteration, whereas their model simply learns the node embeddings of each direction independently and concatenates them in the final step.
+
+In BiGGNN, node embeddings are initialized to the passage embeddings $\mathbf { X }$ returned by DAN. The same set of network parameters are shared at every hop of computation. At each computation hop, for every node in the graph, we apply an aggregation function which takes as input a set of incoming (or outgoing) neighboring node vectors and outputs a backward (or forward) aggregation vector. For the syntax-based static graph, we use a mean aggregator for simplicity although other operators such as max or attention (Velickovi ˇ c et al., 2017) could also be employed, ´
+
+$$
+\begin{array} { r l } & { \mathbf { h } _ { \mathbb { N } _ {  ( ( v ) } } ^ { k } = \mathbf { M } \mathrm { E A N } ( \{ \mathbf { h } _ { v } ^ { k - 1 } \} \cup \{ \mathbf { h } _ { u } ^ { k - 1 } , \forall u \in \mathbb { N } _ {  ( ( v ) } \} ) } \\ & { \mathbf { h } _ { \mathbb { N } _ { \vdash ( v ) } } ^ { k } = \mathbf { M } \mathrm { E A N } ( \{ \mathbf { h } _ { v } ^ { k - 1 } \} \cup \{ \mathbf { h } _ { u } ^ { k - 1 } , \forall u \in \mathbb { N } _ { \vdash ( v ) } \} ) } \end{array}
+$$
+
+For the semantics-aware dynamic graph we compute a weighted average for aggregation where the weights come from the normalized adjacency matrices $\mathbf { A } ^ { + }$ and $\mathbf { A } ^ { \vdash }$ , defined as,
+
+$$
+\mathbf { h } _ { \mathbb { N } _ { \prec ( v ) } } ^ { k } = \sum _ { \forall u \in \mathbb { N } _ { \prec ( v ) } } \mathbf { a } _ { v , u } ^ { - } \mathbf { h } _ { u } ^ { k - 1 } , \quad \mathbf { h } _ { \mathbb { N } _ { \vdash ( v ) } } ^ { k } = \sum _ { \forall u \in \mathbb { N } _ { \vdash ( v ) } } \mathbf { a } _ { v , u } ^ { - } \mathbf { h } _ { u } ^ { k - 1 }
+$$
+
+While ( $\mathrm { { X u } }$ et al., 2018a) learn separate node embeddings for both directions independently, we opt to fuse information aggregated in two directions at each hop, which we find works better in general.
+
+$$
+\mathbf { h } _ { \mathcal { N } _ { ( v ) } } ^ { k } = \mathrm { F u s e } ( \mathbf { h } _ { \mathcal { N } _ { \mathrm { - } | ( v ) } } ^ { k } , \mathbf { h } _ { \mathcal { N } _ { \mathrm { - } ( v ) } } ^ { k } )
+$$
+
+We design the fusion function as a gated sum of two information sources,
+
+$$
+\operatorname { F u s e } ( \mathbf { a } , \mathbf { b } ) = \mathbf { z } \odot \mathbf { a } + ( 1 - \mathbf { z } ) \odot \mathbf { b } , \quad \mathbf { z } = \sigma ( \mathbf { W } _ { z } [ \mathbf { a } ; \mathbf { b } ; \mathbf { a } \odot \mathbf { b } ; \mathbf { a } - \mathbf { b } ] + \mathbf { b } _ { z } )
+$$
+
+where $\odot$ is the component-wise multiplication, $\sigma$ is a sigmoid function, and $\mathbf { z }$ is a gating vector.
+
+Finally, a Gated Recurrent Unit (GRU) (Cho et al., 2014) is used to update the node embeddings by incorporating the aggregation information.
+
+$$
+\mathbf h _ { v } ^ { k } = \mathbf { G } \mathbf { R } \mathbf U ( \mathbf h _ { v } ^ { k - 1 } , \mathbf h _ { \mathcal N _ { ( v ) } } ^ { k } )
+$$
+
+After $n$ hops of GNN computation, where $n$ is a hyperparameter, we obtain the final state embedding $\mathbf { h } _ { v } ^ { n }$ for node $v$ . To compute the graph-level embedding, we first apply a linear projection to the node embeddings, and then apply max-pooling over all node embeddings to get a $d$ -dim vector $\mathbf { h } ^ { \mathcal { G } }$ .
+
+# 2.3.3 RNN DECODER
+
+On the decoder side, we adopt the same model architecture as other state-of-the-art Seq2Seq models where an attention-based (Bahdanau et al., 2014; Luong et al., 2015) LSTM decoder with copy (Vinyals et al., 2015; Gu et al., 2016) and coverage mechanisms (Tu et al., 2016) is employed. The decoder takes the graph-level embedding $\mathbf { h } ^ { \mathcal { G } }$ followed by two separate fully-connected layers as initial hidden states (i.e., $\mathbf { c } _ { 0 }$ and ${ \bf s } _ { 0 }$ ) and the node embeddings $\{ \bar { \mathbf { h } } _ { v } ^ { n } , \forall v \in \mathcal { \bar { G } } \}$ as the attention memory, and generates the output sequence one word at a time. The particular decoder used in this work closely follows (See et al., 2017). We refer the readers to Appendix A for more details.
+
+# 2.4 HYBRID EVALUATOR
+
+It has been observed that optimizing such cross-entropy based training objectives for sequence learning does not always produce the best results on discrete evaluation metrics (Ranzato et al., 2015; Wu et al., 2016; Paulus et al., 2017). Major limitations of this strategy include exposure bias and evaluation discrepancy between training and testing. To tackle these issues, some recent QG approaches (Song et al., 2017; Kumar et al., 2018b) directly optimize evaluation metrics using REINFORCE. We further use a mixed objective function with both syntactic and semantic constraints for guiding text generation. In particular, we present a hybrid evaluator with a mixed objective function that combines both cross-entropy loss and RL loss in order to ensure the generation of syntactically and semantically valid text.
+
+For the RL part, we employ the self-critical sequence training (SCST) algorithm (Rennie et al., 2017) to directly optimize the evaluation metrics. SCST is an efficient REINFORCE algorithm that utilizes the output of its own test-time inference algorithm to normalize the rewards it experiences. In SCST, at each training iteration, the model generates two output sequences: the sampled output $Y ^ { s }$ , produced by multinomial sampling, that is, each word $\boldsymbol y _ { t } ^ { s }$ is sampled according to the likelihood $P ( y _ { t } | X , y _ { < t } )$ predicted by the generator, and the baseline output $\hat { Y }$ , obtained by greedy search, that is, by maximizing the output probability distribution at each decoding step. We define $r ( Y )$ as the reward of an output sequence $Y$ , computed by comparing it to corresponding ground-truth sequence $Y ^ { * }$ with some reward metrics. The loss function is defined as:ÿ
+
+$$
+\mathcal { L } _ { r l } = \left( r ( \hat { Y } ) - r ( Y ^ { s } ) \right) \sum _ { t } \log P ( y _ { t } ^ { s } | X , y _ { < t } ^ { s } )
+$$
+
+As we can see, if the sampled output has a higher reward than the baseline one, we maximize its likelihood, and vice versa.
+
+One of the key factors for RL is to pick the proper reward function. To take syntactic and semantic constraints into account, we consider the following metrics as our reward functions:
+
+Evaluation metric as reward function: We use one of our evaluation metrics, BLEU-4, as our reward function $f _ { \mathrm { e v a l } }$ , which lets us directly optimize the model towards the evaluation metrics.
+
+Semantic metric as reward function: One drawback of some evaluation metrics like BLEU is that they do not measure meaning, but only reward systems that have exact n-gram matches in the reference system. To make our reward function more effective and robust, we additionally use word movers distance (WMD) as a semantic reward function $f _ { \mathrm { s e m } }$ . WMD is the state-of-the-art approach to measure the dissimilarity between two sentences based on word embeddings (Kusner et al., 2015). Following Gong et al. (2019), we take the negative of the WMD distance between a generated sequence and the ground-truth sequence and divide it by the sequence length as its semantic score.
+
+We define the final reward function as $r ( Y ) = f _ { \mathrm { e v a l } } ( Y , Y ^ { * } ) + \alpha f _ { \mathrm { s e m } } ( Y , Y ^ { * } )$ where $\alpha$ is a scalar.
+
+# 2.5 TRAINING AND TESTING
+
+We train our model in two stages. In the first state, we train the model using regular cross-entropy loss, defined as,
+
+$$
+\mathcal { L } _ { l m } = \sum _ { t } - \log P ( y _ { t } ^ { * } | X , y _ { < t } ^ { * } ) + \lambda \operatorname { c o v l o s s } _ { t }
+$$
+
+where ${ \boldsymbol y } _ { t } ^ { * }$ is the word at the ř $t$ -th position of the ground-truth output sequence and $\mathrm { c o v l o s s } _ { t }$ is the coverage loss defined as $\begin{array} { r } { \sum _ { i } m i n ( \bar { a } _ { i } ^ { t } , c _ { i } ^ { t } ) } \end{array}$ , with $a _ { i } ^ { t }$ being the $i$ -th element of the attention vector over the input sequence at time step $t$ . Scheduled teacher forcing (Bengio et al., 2015) is adopted to alleviate the exposure bias problem. In the second stage, we fine-tune the model by optimizing a mixed objective function combining both cross-entropy loss and RL loss, defined as,
+
+$$
+\mathscr { L } = \gamma \mathscr { L } _ { r l } + ( 1 - \gamma ) \mathscr { L } _ { l m }
+$$
+
+where $\gamma$ is a scaling factor controling the trade-off between cross-entropy loss and RL loss. During the testing phase, we use beam search to generate final predictions.
+
+# 3 EXPERIMENTS
+
+We evaluate our proposed model against state-of-the-art methods on the SQuAD dataset (Rajpurkar et al., 2016). Our full models have two variants $\mathrm { G 2 S } _ { s t a } + \mathrm { B E R T + R L }$ and $\mathrm { G 2 S } _ { d y n } + \mathrm { B E R T + R L }$ which adopts static graph construction or dynamic graph construction, respectively. For model settings and sensitivity analysis, please refer to Appendix B and C. The implementation of our model is publicly available at https://github.com/hugochan/RL-based-Graph2Seq-for-NQG.
+
+# 3.1 BASELINE METHODS
+
+We compare against the following baselines in our experiments: i) Transformer (Vaswani et al., 2017), ii) SeqCopyNet (Zhou et al., 2018), iii) $\mathrm { N Q G + + }$ (Zhou et al., 2017), iv) $\mathbf { M P Q G + R }$ (Song et al., 2017), v) AFPQA (Sun et al., 2018), vi) s2sa-at-mp-gsa (Zhao et al., 2018), vii) ASs2s (Kim et al., 2018), and viii) CGC-QG (Liu et al., 2019). Detailed descriptions of the baselines are provided in Appendix D. Experiments on baselines followed by \* are conducted using released code. Results of other baselines are taken from the corresponding papers, with unreported metrics marked as –.
+
+# 3.2 DATA AND METRICS
+
+SQuAD contains more than 100K questions posed by crowd workers on 536 Wikipedia articles. Since the test set of the original SQuAD is not publicly available, the accessible parts $( \approx 9 0 \% )$ are used as the entire dataset in our experiments. For fair comparison with previous methods, we evaluated our model on both data split-1 (Song et al., 2018a)1 that contains 75,500/17,934/11,805 (train/development/test) examples and data split-2 (Zhou et al., 2017) 2 that contains 86,635/8,965/8,964 examples.
+
+Following previous works, we use BLEU-4 (Papineni et al., 2002), METEOR (Banerjee & Lavie, 2005), ROUGE-L (Lin, 2004) and Q-BLEU1 (Nema & Khapra, 2018) as our evaluation metrics. Initially, BLEU-4 and METEOR were designed for evaluating machine translation systems and ROUGE-L was designed for evaluating text summarization systems. Recently, Q-BLEU1 was designed for better evaluating question generation systems, which was shown to correlate significantly better with human judgments compared to existing metrics.
+
+Besides automatic evaluation, we also conduct a human evaluation study on split-2. We ask human evaluators to rate generated questions from a set of anonymized competing systems based on whether they are syntactically correct, semantically correct and relevant to the passage. The rating scale is from 1 to 5, on each of the three categories. Evaluation scores from all evaluators are collected and averaged as final scores. Further details on human evaluation can be found in Appendix E.
+
+Table 1: Automatic evaluation results on the SQuAD test set.   
+
+<table><tr><td rowspan="2">Methods</td><td colspan="4">Split-1</td><td colspan="4">Split-2</td></tr><tr><td>BLEU-4 METEOR</td><td></td><td>ROUGE-L</td><td>Q-BLEU1</td><td>BLEU-4 METEOR</td><td></td><td>ROUGE-L</td><td>Q-BLEU1</td></tr><tr><td>Transformer</td><td>2.56</td><td>8.98</td><td>26.01</td><td>16.70</td><td>3.09</td><td>9.68</td><td>28.86</td><td>20.10</td></tr><tr><td>SeqCopyNet</td><td>1</td><td>1</td><td>1</td><td>1</td><td>13.02</td><td>1</td><td>44.00</td><td>1</td></tr><tr><td>NQG++</td><td>一</td><td>一</td><td>=</td><td>1</td><td>13.29</td><td>=</td><td>1</td><td>1</td></tr><tr><td>MPQG+R*</td><td>14.39</td><td>18.99</td><td>42.46</td><td>52.00</td><td>14.71</td><td>18.93</td><td>42.60</td><td>50.30</td></tr><tr><td>AFPQA</td><td></td><td></td><td></td><td>1</td><td>15.64</td><td>一</td><td></td><td>1</td></tr><tr><td>s2sa-at-mp-gsa</td><td>15.32</td><td>19.29</td><td>43.91</td><td>1</td><td>15.82</td><td>19.67</td><td>44.24</td><td>1</td></tr><tr><td>ASs2s</td><td>16.20</td><td>19.92</td><td>43.96</td><td>一</td><td>16.17</td><td>1</td><td>1</td><td></td></tr><tr><td>CGC-QG</td><td>1</td><td>1</td><td>1</td><td>1</td><td>17.55</td><td>21.24</td><td>44.53</td><td>1</td></tr><tr><td>G2Sdyn +BERT+RL</td><td>17.55</td><td>21.42</td><td>45.59</td><td>55.40</td><td>18.06</td><td>21.53</td><td>45.91</td><td>55.00</td></tr><tr><td>G2S sta+BERT+RL</td><td>17.94</td><td>21.76</td><td>46.02</td><td>55.60</td><td>18.30</td><td>21.70</td><td>45.98</td><td>55.20</td></tr></table>
+
+Table 2: Human evaluation results ( $\pm$ standard deviation) on the SQuAD split-2 test set. The rating scale is from 1 to 5 (higher scores indicate better results).   
+
+<table><tr><td>Methods</td><td>Syntactically correct</td><td>Semantically correct</td><td>Relevant</td></tr><tr><td>MPQG+R*</td><td>4.34 (0.15)</td><td>4.01 (0.23)</td><td>3.21 (0.31)</td></tr><tr><td>G2S sta+BERT+RL</td><td>4.41 (0.09)</td><td>4.31 (0.12)</td><td>3.79 (0.45)</td></tr><tr><td>Ground-truth</td><td>4.74 (0.14)</td><td>4.74 (0.19)</td><td>4.25 (0.38)</td></tr></table>
+
+# 3.3 EXPERIMENTAL RESULTS AND HUMAN EVALUATION
+
+Table 1 shows the automatic evaluation results comparing our proposed models against other stateof-the-art baseline methods. First of all, we can see that both of our full models $\mathrm { G 2 S } _ { s t a } + \mathrm { B E R T + R L }$ and $\mathrm { G 2 S } _ { d y n } + \mathrm { B E R T + R L }$ achieve the new state-of-the-art scores on both data splits and consistently outperform previous methods by a significant margin. This highlights that our RL-based Graph2Seq model, together with the deep alignment network, successfully addresses the three issues we highlighted in Sec. 1. Between these two variants, $\mathrm { G 2 S } _ { s t a } \mathrm { + B E R T + R L }$ outperforms $\mathrm { G } 2 \mathrm { S } _ { d y n } + \mathrm { B E R T + R L }$ on all the metrics. Also, unlike the baseline methods, our model does not rely on any hand-crafted rules or ad-hoc strategies, and is fully end-to-end trainable.
+
+As shown in Table 2, we conducted a human evaluation study to assess the quality of the questions generated by our model, the baseline method $\mathbf { M P Q G + R }$ , and the ground-truth data in terms of syntax, semantics and relevance metrics. We can see that our best performing model achieves good results even compared to the ground-truth, and outperforms the strong baseline method $\mathbf { M P Q G + R }$ . Our error analysis shows that main syntactic error occurs in repeated/unknown words in generated questions. Further, the slightly lower quality on semantics also impacts the relevance.
+
+# 3.4 ABLATION STUDY
+
+Table 3: Ablation study on the SQuAD split-2 test set.   
+
+<table><tr><td>Methods</td><td>BLEU-4</td><td>Methods</td><td>BLEU-4</td></tr><tr><td>G2Sdyn +BERT+RL</td><td>18.06</td><td>G2Sdyn</td><td>16.81</td></tr><tr><td>G2S sta +BERT+RL</td><td>18.30</td><td>G2S sta</td><td>16.96</td></tr><tr><td>G2S sta+BERT-fixed+RL</td><td>18.20</td><td>G2Sdyn w/o DAN</td><td>12.58</td></tr><tr><td>G2Sdyn+BERT</td><td>17.56</td><td>G2Ssta w/o DAN</td><td>12.62</td></tr><tr><td>G2S sta+BERT</td><td>18.02</td><td>G2Ssta w/o BiGGNN, w/ Seq2Seq</td><td>16.14</td></tr><tr><td>G2S sta+BERT-fixed</td><td>17.86</td><td>G2Ssta w/o BiGGNN, w/GCN</td><td>14.47</td></tr><tr><td>G2Sdyn+RL</td><td>17.18</td><td>G2Ssta w/ GGNN-forward</td><td>16.53</td></tr><tr><td>G2Ssta+RL</td><td>17.49</td><td>G2Ssta w/ GGNN-backward</td><td>16.75</td></tr></table>
+
+1https://www.cs.rochester.edu/˜lsong10/downloads/nqg_data.tgz 2https://res.qyzhou.me/redistribute.zip
+
+As shown in Table 3, we perform an ablation study to systematically assess the impact of different model components (e.g., BERT, RL, DAN, and BiGGNN) for two proposed full model variants (static vs dynamic) on the SQuAD split-2 test set. It confirms our finding that syntax-based static graph construction $( \mathbf { G } 2 \mathbf { S } _ { s t a } \mathbf { + } \mathbf { B } \mathbf { E } \mathbf { R } \mathbf { T } \mathbf { + } \mathbf { R } \mathbf { L } )$ ) performs better than semantics-aware dynamic graph construction $( \mathbf { G } 2 \mathbf { S } _ { d y n } \mathbf { + } \mathbf { B } \mathbf { E } \mathbf { R } \mathbf { T } \mathbf { + } \mathbf { R } \mathbf { L } )$ in almost every setting. However, it may be too early to conclude which one is the method of choice for QG. On the one hand, an advantage of static graph construction is that useful domain knowledge can be hard-coded into the graph, which can greatly benefit the downstream task. However, it might suffer if there is a lack of prior knowledge for a specific domain knowledge. On the other hand, dynamic graph construction does not need any prior knowledge about the hidden structure of text, and only relies on the attention matrix to capture these structured information, which provides an easy way to achieve a decent performance. One interesting direction is to explore effective ways of combining both static and dynamic graphs.
+
+By turning off the Deep Alignment Network (DAN), the BLEU-4 score of ${ \mathrm { G } } 2 { \mathrm { S } } _ { s t a }$ (similarly for ${ \mathrm { G } } 2 { \mathrm { S } } _ { d y n } )$ dramatically drops from $1 6 . 9 6 \%$ to $1 2 . 6 2 \%$ , which indicates the importance of answer information for QG and shows the effectiveness of DAN. This can also be verified by comparing the performance between the DAN-enhanced Seq2Seq model (16.14 BLEU-4 score) and other carefully designed answer-aware Seq2Seq baselines such as $\mathrm { N Q G + + }$ (13.29 BLEU-4 score), $\mathbf { M P Q G + R }$ (14.71 BLEU-4 score) and AFPQA (15.82 BLEU-4 score). Further experiments demonstrate that both word-level $\mathrm { \ G } 2 \mathrm { S } _ { s t a }$ w/ DAN-word only) and contextual-level $\mathrm { G } 2 \mathrm { S } _ { s t a }$ w/ DAN-contextual only) answer alignments in DAN are helpful.
+
+We can see the advantages of Graph2Seq learning over Seq2Seq learning on this task by comparing the performance between ${ \mathrm { G } } 2 { \mathrm { S } } _ { s t a }$ and Seq2Seq. Compared to Seq2Seq based QG methods that completely ignore hidden structure information in the passage, our Graph2Seq based method is aware of more hidden structure information such as semantic similarity between any pair of words that are not directly connected or syntactic relationships between two words captured in a dependency parsing tree. In our experiments, we also observe that doing both forward and backward message passing in the GNN encoder is beneficial. Surprisingly, using GCN (Kipf & Welling, 2016) as the graph encoder (and converting the input graph to an undirected graph) does not provide good performance. In addition, fine-tuning the model using REINFORCE can further improve the model performance in all settings (i.e., w/ and w/o BERT), which shows the benefits of directly optimizing the evaluation metrics. Besides, we find that the pretrained BERT embedding has a considerable impact on the performance and fine-tuning BERT embedding even further improves the performance, which demonstrates the power of large-scale pretrained language models.
+
+# 3.5 CASE STUDY
+
+Table 4: Generated questions on SQuAD split-2 test set. Target answers are underlined.   
+
+<table><tr><td>Passage: for the successful execution of a project, effective planning is essential . Gold: what is essential for the successful execution of a project ? G2Ssta W/o BiGGNN (Seq2Seq): what type of planning is essential for the project ? G2Ssta w/o DAN.: what type of planning is essential for the successful execution of a project ? G2S sta: what is essential for the successful execution of a project ? G2Ssta+BERT: what is essential for the successful execution of a project ? G2Ssta+BERT+RL:what is essential for the successful execution of a project ? G2Sdyn +BERT+RL: what is essential for the successful execution of a project ?</td></tr><tr><td>Passage: the church operates three hundred sixty schools and institutions overseas . Gold:how many schools and institutions does the church operate overseas ? G2S sta w/o BiGGNN (Seq2Seq): how many schools does the church have ? G2Ssta w/o DAN.: how many schools does the church have ? G2Ssta:how many schools and institutions does the church have ?</td></tr></table>
+
+In Table 4, we further show a few examples that illustrate the quality of generated text given a passage under different ablated systems. As we can see, incorporating answer information helps the model identify the answer type of the question to be generated, and thus makes the generated questions more relevant and specific. Also, we find our Graph2Seq model can generate more complete and valid questions compared to the Seq2Seq baseline. We think it is because a Graph2Seq model is able to exploit the rich text structure information better than a Seq2Seq model. Lastly, it shows that fine-tuning the model using REINFORCE can improve the quality of the generated questions.
+
+# 4 RELATED WORK
+
+# 4.1 NATURAL QUESTION GENERATION
+
+Early works (Mostow & Chen, 2009; Heilman & Smith, 2010) for QG focused on rule-based approaches that rely on heuristic rules or hand-crafted templates, with low generalizability and scalability. Recent attempts have focused on NN-based approaches that do not require manually-designed rules and are end-to-end trainable. Existing NN-based approaches (Du et al., 2017; Yao et al.; Zhou et al., 2018) rely on the Seq2Seq model with attention, copy or coverage mechanisms. In addition, various ways (Zhou et al., 2017; Song et al., 2017; Zhao et al., 2018) have been proposed to utilize the target answer for guiding the question generation. Some recent approaches (Song et al., 2017; Kumar et al., 2018b) aim at directly optimizing evaluation metrics using REINFORCE. Concurrent works have explored tackling the QG task with various semantics-enhanced rewards (Zhang & Bansal, 2019) or large-scale pretrained language models (Dong et al., 2019).
+
+However, the existing approaches for QG suffer from several limitations; they (i) ignore the rich structure information hidden in text, (ii) solely rely on cross-entropy loss that leads to issues like exposure bias and inconsistency between train/test measurement, and (iii) fail to fully exploit the answer information. To address these limitations, we propose a RL based Graph2Seq model augmented with a deep alignment network to effectively tackle the QG task. To the best of our knowledge, we are the first to introduce the Graph2Seq architecture to solve the question generation task.
+
+# 4.2 GRAPH NEURAL NETWORKS
+
+Over the past few years, graph neural networks (GNNs) (Kipf & Welling, 2016; Gilmer et al., 2017; Hamilton et al., 2017) have attracted increasing attention. Due to more recent advances in graph representation learning, a number of works have extended the widely used Seq2Seq architectures (Sutskever et al., 2014; Cho et al., 2014) to Graph2Seq architectures for machine translation, semantic parsing, AMR(SQL)-to-text, and online forums health stage prediction tasks (Bastings et al., 2017; Beck et al., 2018; Xu et al., 2018a;b;c; Song et al., 2018b; Gao et al., 2019). While the high-quality graph structure is crucial for the performance of GNN-based approaches, most existing works use syntax-based static graph structures when applied to textual data. Very recently, researchers have started exploring methods to automatically construct a graph of visual objects (Norcliffe-Brown et al., 2018) or words (Liu et al., 2018; Chen et al., 2019c;b) when applying GNNs to non-graph structured data. To the best of our knowledge, we are the first to investigate systematically the performance difference between syntactic-aware static graph construction and semantics-aware dynamic graph construction in the context of question generation.
+
+# 5 CONCLUSION
+
+We proposed a novel RL based Graph2Seq model for QG, where the answer information is utilized by an effective Deep Alignment Network and a novel bidirectional GNN is proposed to process the directed passage graph. On the SQuAD dataset, our method outperforms existing methods by a significant margin and achieves the new state-of-the-art results. Future directions include investigating more effective ways of automatically learning graph structures from text and exploiting Graph2Seq models for question generation from structured data like knowledge graphs or tables.
+
+# ACKNOWLEDGMENTS
+
+This work is supported by IBM Research AI through the IBM AI Horizons Network. We thank the human evaluators who evaluated our system. We also thank the anonymous reviewers for their constructive feedback.
+
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+
+# A DETAILS ON THE RNN DECODER
+
+At each decoding step $t$ , an attention mechanism learns to attend to the most relevant words in the input sequence, and computes a context vector $\mathbf { h } _ { t } ^ { * }$ based on the current decoding state $\mathbf { s } _ { t }$ , the current coverage vector $\mathbf { c } ^ { t }$ and the attention memory. In addition, the generation probability $p _ { \mathrm { g e n } } \in [ 0 , 1 ]$ is calculated from the context vector $\mathbf { h } _ { t } ^ { * }$ , the decoder state $\mathbf { s } _ { t }$ and the decoder input $y _ { t - 1 }$ . Next, $p _ { \mathrm { g e n } }$ is used as a soft switch to choose between generating a word from the vocabulary, or copying a word from the input sequence. We dynamically maintain an extended vocabulary which is the union of the usual vocabulary and all words appearing in a batch of source examples (i.e., passages and answers). Finally, in order to encourage the decoder to utilize the diverse components of the input sequence, a coverage mechanism is applied. At each step, we maintain a coverage vector $\mathbf { c } ^ { t }$ , which is the sum of attention distributions over all previous decoder time steps. A coverage loss is also computed to penalize repeatedly attending to the same locations of the input sequence.
+
+# B MODEL SETTINGS
+
+We keep and fix the 300-dim GloVe vectors for the most frequent 70,000 words in the training set. We compute the 1024-dim BERT embeddings on the fly for each word in text using a (trainable) weighted sum of all BERT layer outputs. The embedding sizes of case, POS and NER tags are set to 3, 12 and 8, respectively. We set the hidden state size of BiLSTM to 150 so that the concatenated state size for both directions is 300. The size of all other hidden layers is set to 300. We apply a variational dropout (Kingma et al., 2015) rate of 0.4 after word embedding layers and 0.3 after RNN layers. We set the neighborhood size to 10 for dynamic graph construction. The number of GNN hops is set to 3. During training, in each epoch, we set the initial teacher forcing probability to 0.75 and exponentially increase it to $0 . 7 5 * 0 . 9 9 9 9 ^ { i }$ where $i$ is the training step. We set $\alpha$ in the reward function to 0.1, $\gamma$ in the mixed loss function to 0.99, and the coverage loss ratio $\lambda$ to 0.4. We use Adam (Kingma & Ba, 2014) as the optimizer, and the learning rate is set to 0.001 in the pretraining stage and 0.00001 in the fine-tuning stage. We reduce the learning rate by a factor of 0.5 if the validation BLEU-4 score stops improving for three epochs. We stop the training when no improvement is seen for 10 epochs. We clip the gradient at length 10. The batch size is set to 60 and 50 on data split-1 and split-2, respectively. The beam search width is set to 5. All hyperparameters are tuned on the development set.
+
+# C SENSITIVITY ANALYSIS OF HYPERPARAMETERS
+
+![](images/d2b7a2d3a7da2a369938390729f32a3d2ba6edde823a4903bd7e887215e11b94.jpg)  
+Figure 3: Effect of the number of GNN hops.
+
+To study the effect of the number of GNN hops, we conduct experiments on the ${ \mathrm { G } } 2 { \mathrm { S } } _ { s t a }$ model on the SQuAD split-2 data. Fig. 3 shows that our model is not very sensitive to the number of GNN hops and can achieve reasonably good results with various number of hops.
+
+Table 5: Ablation study on the SQuAD split-2 test set.   
+
+<table><tr><td>Methods</td><td>BLEU-4</td><td>Methods</td><td>BLEU-4</td></tr><tr><td>G2Sdyn +BERT+RL</td><td>18.06</td><td>G2Sdyn w/o feat</td><td>16.51</td></tr><tr><td>G2S sta+BERT+RL</td><td>18.30</td><td>G2S sta w/o feat</td><td>16.65</td></tr><tr><td>G2S sta +BERT-fixed+RL</td><td>18.20</td><td>G2Sdyn W/o DAN</td><td>12.58</td></tr><tr><td>G2Sdyn+BERT</td><td>17.56</td><td>G2Ssta w/o DAN</td><td>12.62</td></tr><tr><td>G2Ssta+BERT</td><td>18.02</td><td>G2Ssta w/ DAN-word only</td><td>15.92</td></tr><tr><td>G2S sta+BERT-fixed</td><td>17.86</td><td>G2Ssta w/DAN-contextual only</td><td>16.07</td></tr><tr><td>G2Sdyn +RL</td><td>17.18</td><td>G2Ssta w/ GGNN-forward</td><td>16.53</td></tr><tr><td>G2S sta+RL</td><td>17.49</td><td>G2Ssta w/ GGNN-backward</td><td>16.75</td></tr><tr><td>G2Sdyn</td><td>16.81</td><td>G2Ssta w/o BiGGNN, w/ Seq2Seq</td><td>16.14</td></tr><tr><td>G2Ssta</td><td>16.96</td><td>G2Ssta W/o BiGGNN,w/GCN</td><td>14.47</td></tr></table>
+
+# D DETAILS ON BASELINE METHODS
+
+Transformer (Vaswani et al., 2017) We included a Transformer-based Seq2Seq model augmented with attention and copy mechanisms. We used the open source implementation3 provided by the OpenNMT (Klein et al., 2017) library and trained the model from scratch. Surprisingly, this baseline performed very poorly on the benchmarks even though we conducted moderate hyperparameter search and trained the model for a large amount of epochs. We suspect this might be partially because this method is very sensitive to hyperparameters as reported by Klein et al. (2017) and probably data-hungry on this task. We conjecture that better performance might be expected by extensively searching the hyperparameters and using a pretrained transformer model.
+
+SeqCopyNet (Zhou et al., 2018) proposed an extension to the copy mechanism which learns to copy not only single words but also sequences from the input sentence.
+
+$\mathbf { N Q G + + }$ (Zhou et al., 2017) proposed an attention-based Seq2Seq model equipped with a copy mechanism and a feature-rich encoder to encode answer position, POS and NER tag information.
+
+$\mathbf { M P Q G + R }$ (Song et al., 2017) proposed an RL-based Seq2Seq model with a multi-perspective matching encoder to incorporate answer information. Copy and coverage mechanisms are applied.
+
+AFPQA (Sun et al., 2018) consists of an answer-focused component which generates an interrogative word matching the answer type, and a position-aware component which is aware of the position of the context words when generating a question by modeling the relative distance between the context words and the answer.
+
+s2sa-at-mp-gsa (Zhao et al., 2018) proposed a model which contains a gated attention encoder and a maxout pointer decoder to tackle the challenges of processing long input sequences. For fair comparison, we report the results of the sentence-level version of their model to match with our settings.
+
+ASs2s (Kim et al., 2018) proposed an answer-separated Seq2Seq model which treats the passage and the answer separately.
+
+CGC-QG (Liu et al., 2019) proposed a multi-task learning framework to guide the model to learn the accurate boundaries between copying and generation.
+
+# E DETAILS ON HUMAN EVALUATION
+
+We conducted a small-scale (i.e., 50 random examples per system) human evaluation on the split-2 data. We asked 5 human evaluators to give feedback on the quality of questions generated by a set of anonymized competing systems. In each example, given a triple containing a source passage, a target answer and an anonymised system output, they were asked to rate the quality of the output by answering the following three questions: i) is this generated question syntactically correct? ii) is this generated question semantically correct? and iii) is this generated question relevant to the passage? For each evaluation question, the rating scale is from 1 to 5 where a higher score means better quality (i.e., 1: Poor, 2: Marginal, 3: Acceptable, 4: Good, 5: Excellent). Responses from all evaluators were collected and averaged.
+
+# F MORE RESULTS ON ABLATION STUDY
+
+We perform the comprehensive ablation study to systematically assess the impact of different model components (e.g., BERT, RL, DAN, BiGGNN, FEAT, DAN-word, and DAN-contextual) for two proposed full model variants (static vs dynamic) on the SQuAD split-2 test set. Our experimental results confirmed that every component in our proposed model makes the contribution to the overall performance.

parse/train/bJxgv5C3sYc/bJxgv5C3sYc.md

@@ -0,0 +1,315 @@
+# FEW-SHOT BAYESIAN OPTIMIZATION WITH DEEPKERNEL SURROGATES
+
+Martin Wistuba   
+IBM Research   
+Dublin, Ireland   
+martin.wistuba@ibm.com   
+Josif Grabocka   
+University of Freiburg   
+Freiburg, Germany   
+grabocka@cs.uni-freiburg.de
+
+# ABSTRACT
+
+Hyperparameter optimization (HPO) is a central pillar in the automation of machine learning solutions and is mainly performed via Bayesian optimization, where a parametric surrogate is learned to approximate the black box response function (e.g. validation error). Unfortunately, evaluating the response function is computationally intensive. As a remedy, earlier work emphasizes the need for transfer learning surrogates which learn to optimize hyperparameters for an algorithm from other tasks. In contrast to previous work, we propose to rethink HPO as a few-shot learning problem in which we train a shared deep surrogate model to quickly adapt (with few response evaluations) to the response function of a new task. We propose the use of a deep kernel network for a Gaussian process surrogate that is meta-learned in an end-to-end fashion in order to jointly approximate the response functions of a collection of training data sets. As a result, the novel few-shot optimization of our deep kernel surrogate leads to new state-of-the-art results at HPO compared to several recent methods on diverse metadata sets.
+
+# 1 INTRODUCTION
+
+Many machine learning models have very sensitive hyperparameters that must be carefully tuned for efficient use. Unfortunately, finding the right setting is a tedious trial and error process that requires expert knowledge. AutoML methods address this problem by providing tools to automate hyperparameter optimization, where Bayesian optimization has become the standard for this task (Snoek et al., 2012). It treats the problem of hyperparameter optimization as a black box optimization problem. Here the black box function is the hyperparameter response function, which maps a hyperparameter setting to the validation loss. Bayesian optimization consists of two parts. First, a surrogate model, often a Gaussian process, is used to approximate the response function. Second, an acquisition function is used that balances the trade-off between exploration and exploitation. In a sequential process, hyperparameter settings are selected and evaluated, followed by an update of the surrogate model.
+
+Recently, several attempts have been made to extend Bayesian optimization to account for a transfer learning setup. It is assumed here that historical information on machine learning algorithms is available with different hyperparameters. This can either be because this information is publicly available (e.g. OpenML) or because the algorithm is repeatedly optimized for different data sets. To this end, several transfer learning surrogates have been proposed that use this additional information to reduce the convergence time of Bayesian optimization.
+
+We propose a new paradigm for accomplishing the knowledge transfer by reconceptualizing the process as a few-shot learning task. Inspiration is drawn from the fact that there are a limited number of black box function evaluations for a new hyperparameter optimization task (i.e. few shots) but there are ample evaluations of related black box objectives (i.e. evaluated hyperparameters on other data sets). This approach has several advantages. First, a single model is learned that is trained to quickly adapt to a new task when few examples are available. This is exactly the challenge we face when optimizing hyperparameters. Second, this method can be scaled very well for any number of considered tasks. This not only enables the learning from large metadata sets but also enables the problem of label normalization to be dealt with in a new way. Finally, we present an evolutionary algorithm that can use surrogate models to get a warm start initialization for Bayesian optimization. All of our contributions are empirically compared with several competitive methods in three different problems. Two ablation studies provide information about the influence of the individual components. Concluding, our contributions in this work are:
+
+• This is the first work that, in the context of hyperparameter optimization (HPO), trains the initialization of the parameters of a probabilistic surrogate model from a collection of meta-tasks by few-shot learning and then transfers it by fine-tuning the initialized kernel parameters to a target task.   
+We are the first to consider transfer learning in HPO as a few-shot learning task.   
+We set the new state of the art in transfer learning for the HPO and provide ample evidence that we outperform strong baselines published at ICLR and NeurIPS with a statistically significant margin.   
+• We present an evolutionary algorithm that can use surrogate models to get a warm start initialization for Bayesian optimization.
+
+# 2 RELATED WORK
+
+The idea of using transfer learning to improve Bayesian optimization is being investigated in several papers. The early work suggests learning a single Gaussian process for the entire data (Bardenet et al., 2013; Swersky et al., 2013; Yogatama & Mann, 2014). Since the training of a Gaussian process is cubic in the number of training points, this idea does not scale well. This problem has recently been addressed by several proposals to use ensembles of Gaussian processes where a Gaussian process is learned for each task (Wistuba et al., 2018; Feurer et al., 2018). This idea scales linearly in the number of tasks but still cubically in the number of training points per task. Thus, the problem persists in scenarios where there is a lot of data available for each task.
+
+Bayesian neural networks are a possible more scalable way of learning with large amounts of data. For example, Schilling et al. (2015) propose to use a neural network with task embedding and variable interactions. To obtain mean and variance predictions, the authors propose using an ensemble of models. In contrast, Springenberg et al. (2016) use a Bayesian multi-task neural network. However, since training Bayesian neural networks is computationally intensive, Perrone et al. (2018) propose a more scalable approach. They suggest using a neural network that is shared by all tasks and using Bayesian linear regression for each task. The parameters are trained jointly on the entire data. While our work shares some similarities with the previous work, our algorithm has unique properties. First of all, a meta-learning algorithm is used, which is motivated by recent work on model-agnostic meta-learning for few-shot learning (Finn et al., 2017). This will allow us to integrate all task-specific parameters out such that the model does not grow with the number of tasks. As a result, our algorithm scales very well with the number of tasks. Second, while we are also using a neural network, we combine it with a Gaussian process with a nonlinear kernel in order to obtain uncertainty predictions.
+
+A simpler solution for using the transfer learning idea in Bayesian optimization is initialization (Feurer et al., 2015; Wistuba et al., 2015a). The standard Bayesian optimization routine with a simple Gaussian process is used in this case but it is warm-started by a number of hyperparameter settings that work well for related tasks. We also explore this idea in the context of this paper by proposing a simple evolutionary algorithm that can use a surrogate model to estimate a data-driven warm start initialization sequence. The use of an evolutionary algorithm is motivated by its ease of implementation and natural capability to deal with continuous and discrete hyperparameters.
+
+# 3 PRELIMINARIES
+
+# 3.1 BAYESIAN OPTIMIZATION
+
+Bayesian optimization is an optimization method for computationally intensive black box functions that consists of two main components, the surrogate model and the acquisition function. The surrogate model is a probabilistic model with mean $\mu$ and variance $\sigma ^ { 2 }$ , which tries to approximate the unknown black box function $f$ , in the following also response function. The acquisition function can provide a score for each feasible solution based on the prediction of the surrogate model. This score balances between exploration and exploitation. The following steps are carried out sequentially. First, the feasible solution that maximizes the acquisition function is evaluated. In this way a new observation $\left( \mathbf { x } , f ( \mathbf { x } ) \right)$ is obtained. Then, the surrogate model is updated based on the entirety of all observations $\mathcal { D }$ . This sequence of steps is carried out until a previously defined convergence criterion occurs (e.g. exhaustion of the time budget). In Figure 1 we provide an example how Bayesian optimization is used to maximize a sine wave.
+
+Expected Improvement (Jones et al., 1998) is one of the most commonly used acquisition functions and will also be used in all our experiments. It is defined as
+
+$$
+a ( \mathbf { x } | \mathcal { D } ) = \mathbb { E } \left[ \operatorname* { m a x } \left\{ f ( \mathbf { x } ) - y _ { \operatorname* { m a x } } , 0 \right\} \right] ,
+$$
+
+where $y _ { \mathrm { m a x } }$ is the largest observed value of $f$
+
+# 3.2 GAUSSIAN PROCESSES
+
+We have a training set $\mathcal { D }$ of $n$ observations, $\mathcal { D } = \{ ( \mathbf { x } _ { i } , y _ { i } ) | i = 1 , \dots , n \}$ , where $y _ { i }$ are noisy observations of the function values $y _ { i } = f ( \mathbf { x } _ { i } ) + \varepsilon$ . We assume that the noise $\varepsilon$ is additive independent identically distributed Gaussian with variance $\sigma _ { n } ^ { 2 }$ . For Gaussian processes (Rasmussen & Williams, 2006) we consider $y _ { i }$ to be a random variable and the joint distribution of all $y _ { i }$ is assumed to be multivariate Gaussian distributed:
+
+$$
+\mathbf { y } \sim { \mathcal { N } } \left( m \left( \mathbf { X } \right) , k \left( \mathbf { X } , \mathbf { X } \right) \right) \ .
+$$
+
+A Gaussian process is completely specified by its mean function $m$ , its covariance function $k$ , and may depend on parameters $\pmb { \theta }$ . A common choice is to set $m ( { \bf x } _ { i } ) = 0$ . At inference time for instances $\mathbf { x } _ { * }$ , the assumption is that their ground truth $\mathbf { f } _ { * }$ is jointly Gaussian with $\mathbf { y }$
+
+$$
+\begin{array}{c} \left[ \begin{array} { c } { \mathbf { y } } \\ { \mathbf { f _ { * } } } \end{array} \right] \sim \mathcal { N } \left( \mathbf { 0 } , \left( \begin{array} { c c } { \mathbf { K } _ { n } } & { \mathbf { K } _ { * } } \\ { \mathbf { K } _ { * } ^ { T } } & { \mathbf { K } _ { * * } } \end{array} \right) \right)  \end{array} ,
+$$
+
+where
+
+$$
+\mathbf { K } _ { n } = k \left( \mathbf { X } , \mathbf { X } | \pmb { \theta } \right) + \sigma _ { n } ^ { 2 } \mathbf { I } , \ \mathbf { K } _ { * } = k \left( \mathbf { X } , \mathbf { X } _ { * } | \pmb { \theta } \right) , \ \mathbf { K } _ { * * } = k \left( \mathbf { X } _ { * } , \mathbf { X } _ { * } | \pmb { \theta } \right)
+$$
+
+for brevity. Then, the posterior predictive distribution has mean and covariance
+
+$$
+\mathbb { E } \left[ \mathbf { f } _ { * } | \mathbf { X } , \mathbf { y } , \mathbf { X } _ { * } \right] = \mathbf { K } _ { * } ^ { T } \mathbf { K } _ { n } ^ { - 1 } \mathbf { y } , \thinspace \thinspace \mathrm { c o v } \left[ \mathbf { f } _ { * } | \mathbf { X } , \mathbf { X } _ { * } \right] = \mathbf { K } _ { * * } - \mathbf { K } _ { * } ^ { T } \mathbf { K } _ { n } ^ { - 1 } \mathbf { K } _ { * }
+$$
+
+Examples for covariance functions are the linear or squared exponential kernel. However, these kernel functions are designed by hand. The idea of deep kernel learning (Wilson et al., 2016) is to learn the kernel function. They propose to use a neural network $\varphi$ to transform the input $\mathbf { x }$ to a latent representation which serves as input for the kernel function.
+
+$$
+k _ { \mathrm { d e e p } } ( \mathbf { x } , \mathbf { x } ^ { \prime } | \boldsymbol { \theta } , \mathbf { w } ) = k ( \varphi ( \mathbf { x } , \mathbf { w } ) , \varphi ( \mathbf { x } ^ { \prime } , \mathbf { w } ) | \boldsymbol { \theta } )
+$$
+
+# 4 FEW-SHOT BAYESIAN OPTIMIZATION
+
+Hyperparameter optimization is traditionally either tackled as an optimization problem without prior knowledge about the response function or alternatively as a multi-task or transfer learning problem. In the former, every search basically starts from scratch. In the latter, one or multiple models are trained that attempt to reuse knowledge of the source tasks for the target task.
+
+In this work we will address the problem as a few-shot problem. Given $T$ related source tasks and very few examples of the target task, we want to make reliable predictions. For each of the source tasks we have observations $\bar { \mathcal { D } } ^ { ( t ) } = \{ ( \mathbf { x } _ { i } ^ { ( t ) } , y _ { i } ^ { ( t ) } ) \} _ { i = 1 \ldots n ^ { ( t ) } }$ , where $\mathbf { x } _ { i } ^ { ( t ) }$ is a hyperparameter setting and $y _ { i } ^ { ( t ) }$ is the noisy observations of $f ^ { ( t ) } ( \mathbf { x } _ { i } ^ { ( t ) } )$ , i.e. a validation score of a machine learning model on data set $t$ . In the following we will denote the set of all data points by $\textstyle { \mathcal { D } } : = \bigcup _ { t = 1 } ^ { T } { \mathcal { D } } ^ { ( t ) }$ . Modelagnostic meta-learning (Finn et al., 2017) has become a popular choice for few-shot learning and we propose to use an adaptation for Gaussian processes (Patacchiola et al., 2020) as a surrogate model within the Bayesian optimization framework. The idea is simple. A deep kernel $\varphi$ is used to learn parameters across tasks such that all its parameters $\pmb \theta$ and w are task-independent. All task-dependent parameters are kept separate which allows to marginalize its corresponding variable out when solely optimizing for the task-independent parameters. If we assume that the posteriors over $\pmb \theta$ and w are dominated by their respective maximum likelihood estimates $\hat { \pmb { \theta } }$ and $\hat { \mathbf { w } }$ , we can approximate the posterior predictive distribution by
+
+$$
+p \left( f _ { * } | \mathbf { x } _ { * } , \mathcal { D } \right) = \int p \left( f _ { * } | \mathbf { x } _ { * } , \pmb { \theta } , \mathbf { w } \right) p \left( \pmb { \theta } , \mathbf { w } | \mathcal { D } \right) \mathrm { d } \pmb { \theta } , \mathbf { w } \approx p \left( f _ { * } | \mathbf { x } _ { * } , \mathcal { D } , \pmb { \hat { \theta } } , \hat { \mathbf { w } } \right)
+$$
+
+We estimate $\hat { \pmb { \theta } }$ and $\hat { \mathbf { w } }$ by maximizing the log marginal likelihood for all tasks.
+
+$$
+\begin{array} { l } { \log p \left( \mathbf { y } ^ { ( 1 ) } , \dots , \mathbf { y } ^ { ( T ) } | \mathbf { X } ^ { ( 1 ) } , \dots , \mathbf { X } ^ { ( T ) } , \pmb { \theta } , \mathbf { w } \right) = \displaystyle \sum _ { t = 1 } ^ { T } \log p \left( \mathbf { y } ^ { ( t ) } | \mathbf { X } ^ { ( t ) } , \pmb { \theta } , \mathbf { w } \right) } \\ { \propto - \displaystyle \sum _ { t = 1 } ^ { T } \left( \mathbf { y } ^ { ( t ) ^ { \mathrm { T } } } \mathbf { K } _ { n } ^ { ( t ) ^ { - 1 } } \mathbf { y } ^ { ( t ) } + \log \left| \mathbf { K } _ { n } ^ { ( t ) } \right| \right) } \end{array}
+$$
+
+We maximize the marginal likelihood with stochastic gradient ascent (SGA). Each batch contains data for one task. It is important to note that this batch data does not correspond to the data in $\mathcal { D } ^ { ( t ) }$ , but rather a subset of it. With small batch sizes, this is an efficient way to train the Gaussian process, regardless of how many tasks knowledge is transferred from or how large $\mathcal { D } ^ { ( t ) }$ is. It is important to note that there is a correlation between all data per task and not just one batch. Hence the stochastic gradient is a biased estimator of the full gradient. For a long time we lacked the theoretical understanding of how SGA would behave in this situation. Fortunately, recent work (Chen et al., 2020) proved that SGA still successfully converges and restores model parameters in these cases. The results by Chen et al. (2020) guarantee a $\mathcal { O } ( 1 / K )$ optimization margin of error, where $K$ is the number of iterations.
+
+The training works as follows. In each iteration, a task $t$ is sampled uniformly at random from all $T$ tasks. Then we sample a batch of training instances uniformly at random from $\mathcal { D } ^ { ( t ) }$ . Finally, we calculate the log marginal likelihood for this batch (Equation 8) and update the kernel parameters with one step in the direction of the gradient.
+
+In hyperparameter optimization, we are only interested in the hyperparameter setting that works best according to a predefined metric of interest. Therefore, only the ranking of the hyperparameter settings is important, not the actual value of the metric. Therefore, we are interested in a surrogate model whose prediction strongly correlate with the response function and the squared error is of little to no interest for us. In practice, however, the minimum / maximum score and the range of values differ significantly between the tasks which makes it challenging to obtain a strongly correlating surrogate model. The most common way to address this problem is to normalize the labels, e.g. by $\mathbf { Z }$ -normalizing or scaling the labels to $[ 0 , 1 ]$ per data set. However, this does not fully solve the problem. The main problem is that this label normalization must also be applied to the target task. This is not possible with a satisfactory degree of accuracy, especially when only a few examples are available, since the approximate values for minimum / maximum or mean / standard deviation are insufficiently accurate.
+
+Since our proposed method scales easily to any number of tasks, we can afford to consider another option. We propose a task augmentation strategy that addresses the label normalization issue by randomly scaling the labels for each batch. Since $y _ { \mathrm { { m i n } } }$ and $y _ { \mathrm { m a x } }$ are the minimum and maximum values for all $T$ tasks, we can generate augmented tasks for each batch $B = \{ \mathbf { x } _ { i } , y _ { i } \} _ { i = 1 , \ldots , b } \sim \mathcal { D } ^ { ( t ) }$ where $b$ is the batch size, as follows. A lower and upper limit is sampled for each sample batch,
+
+$$
+l \sim \mathcal { U } ( y _ { \mathrm { m i n } } , y _ { \mathrm { m a x } } ) , u \sim \mathcal { U } ( y _ { \mathrm { m i n } } , y _ { \mathrm { m a x } } ) ~ ,
+$$
+
+such that $l < u$ holds. Then the labels for that batch are scaled to this range,
+
+$$
+\mathbf { y } \gets \frac { \mathbf { y } - l } { u - l } .
+$$
+
+No further changes are required. We summarize this procedure in Algorithm 1. The idea here is to learn a representation that is invariant with respect to various offsets and ranges so that the target task does not require normalization. This strategy is not possible for other hyperparameter optimization methods, since this either increases the training data set size even further and thus becomes
+
+# Algorithm 1: Few-Shot GP Surrogate
+
+Input: Learning rates $\alpha$ and $\beta$ , training data $\mathcal { D }$ , kernel $k$ , and neural network $\varphi$ .   
+while not done do Sample task $t \sim \mathcal { U } ( \{ 1 , \dots , T \} )$ ; Estimate $l$ and $u$ (Equation 10); for $b _ { n }$ times do Sample batch $B = \{ ( \mathbf { x } _ { i } , y _ { i } ) \} _ { i = 1 , \ldots , b } \sim \mathcal { D } ^ { ( t ) }$ and scale labels $\mathbf { y }$ using $l$ and $u$ ; Compute marginal likelihood $\mathcal { L }$ on $\boldsymbol { B }$ . (Equation 8); $\pmb { \theta }  \pmb { \theta } + \alpha \nabla _ { \pmb { \theta } } \mathcal { L }$ ; $\mathbf { w } \gets \mathbf { w } + \beta \nabla _ { \mathbf { w } } \mathcal { L } ;$ ; end   
+end
+
+![](images/eedb4f8e35c73478805b6df22db0fab343c2f801f56fc02053d1e6efa0660124.jpg)  
+Figure 1: Demonstration of five steps of Bayesian Optimization with FSBO for maximizing a sine wave (blue). One maximum is discovered within only three steps. Expected Improvement has been scaled to improve readability. In black the predictions of the surrogate model. Bottom right are examples of the source tasks. The deep kernel consists of a spectral kernel (Wilson & Adams, 2013) combined with a two-layer neural network $\mathrm { 1  6 4  6 4 }$ ).
+
+computationally impossible (Bardenet et al., 2013; Swersky et al., 2013; Yogatama & Mann, 2014) or thousands of new tasks would have to be generated, which is also is not practical (Springenberg et al., 2016; Perrone et al., 2018; Wistuba et al., 2018; Feurer et al., 2018).
+
+At test time, the posterior predictive distribution or the target task $T + 1$ is computed according to Equation 5. In practice, the target data set can be very different to the learned prior. For this reason, the deep kernel parameters are fine-tuned on the target task’s data for few epochs. Our task augmentation strategy described earlier has resulted in a model that has become invariant for different scales of y. For this reason we do not apply this strategy to the target task $T + 1$ and only use it for all source tasks 1 to $T$ . We discuss the usefulness of this step in more detail when we discuss the empirical results.
+
+# 5 A MOTIVATING EXAMPLE
+
+We would like to first motivate our method, FSBO, with an example, similar to the one described in Nichol et al. (2018). The aim is to find the argument $x$ that maximizes a one-dimensional sine wave of the form $f ^ { ( t ) } ( x ) = a ^ { ( t ) } \sin ( x + b ^ { ( t ) } )$ . It is not known that the function is a sine wave and that one only has to determine the amplitude $a$ and phase $b$ . However, information about other sine waves is available to the optimizer which are considered to be related. These source tasks $t = 1 , \dots , T$ are generated randomly with $a \sim \mathcal { U } ( 0 . 1 , 5 )$ and $b \sim \mathcal { U } ( 0 , 2 \pi )$ . 50 examples $( x _ { i } ^ { ( t ) } , f ^ { ( t ) } ( x _ { i } ^ { ( t ) } ) )$ are available for each task $t$ , whereby the points $x _ { i } ^ { ( t ) } \in [ - 5 , 5 ]$ are evenly spaced.
+
+![](images/937e452ee66004a79a138ec13aa35ab753d35161c5d4481d3dc1bed46fae866f.jpg)  
+Figure 2: Examples for the mutation and crossover operation with $I = 3$
+
+It should be noted at this point that the expected value for each $x _ { i }$ is 0. Thus, training on the data described above leads to a model predicting 0 for every $x _ { i }$ . However, the used meta-learning procedure allows for accurately reconstructing the underlying sine wave after only a few examples (see Figure 1). Provided that the response function of the original task is similar to the target task, this is a very promising technique for accelerating the search for good hyperparameter settings.
+
+# 6 A DATA-DRIVEN INITIALIZATION
+
+The proposed few-shot surrogate model requires only a few examples before reliable predictions can be made for the target data set. How do you choose the initial hyper parameter settings? The simplest idea would be to pick them at random or use Latin Hypercube Sampling (LHS) (McKay et al., 1979). Since we already have data from various tasks and a surrogate model that can impute missing values, we propose a data-driven warm start approach. If we evaluate the performance of a hyperparameter optimization algorithm with a loss function $\mathcal { L }$ , we use an evolutionary algorithm to find a set with $I$ hyperparameter settings which minimizes this loss on the source tasks.
+
+$$
+\mathbf { X } ^ { ( \mathrm { i n i t } ) } = \underset { \mathbf { X } \in \mathcal { X } ^ { I } } { \arg \operatorname* { m i n } } \sum _ { t = 1 } ^ { T } \mathcal { L } \left( f ^ { ( t ) } , \mathbf { X } \right) = \underset { \mathbf { X } \in \mathcal { X } ^ { I } } { \arg \operatorname* { m i n } } \sum _ { t = 1 } ^ { T } \underset { \mathbf { x } \in \mathbf { X } } { \operatorname* { m i n } } \tilde { f } ^ { ( t ) } ( \mathbf { x } )
+$$
+
+The loss of a set of hyperparameters depends on the response function values for each of the elements. Since these are not available for every $\mathbf { x }$ , we approximate $f ^ { ( t ) } ( \mathbf { x } )$ with the prediction of the surrogate model described in the last section whenever this is necessary.
+
+Arbitrary loss function can be considered. The specific loss function used in our experiments is the normalized regret and is defined on the right side of Equation 12, where $\tilde { f } ^ { ( t ) }$ is derived from $f ^ { ( t ) }$ by scaling it to $[ 0 , 1 ]$ range:
+
+$$
+\tilde { f } ^ { ( t ) } ( \mathbf { x } ) = \frac { f ^ { ( t ) } ( \mathbf { x } ) - f _ { \operatorname* { m i n } } ^ { ( t ) } } { f _ { \operatorname* { m a x } } ^ { ( t ) } - f _ { \operatorname* { m i n } } ^ { ( t ) } } ,
+$$
+
+where $f _ { \mathrm { m a x } } ^ { ( t ) }$ and $f _ { \operatorname* { m i n } } ^ { ( t ) }$ are the maximum and minimum of $f ^ { ( t ) }$ , respectively.
+
+The evolutionary algorithm works as follows. We initialize the population with sets containing $I$ random settings, the settings being sampled in proportion to their performance according to the predictions. Precisely, a setting $\mathbf { x }$ is sampled proportional to
+
+$$
+\exp \left( - \operatorname* { m i n } _ { t \in \{ 1 , . . . , T \} } \tilde { f } ^ { ( t ) } ( { \bf x } ) \right) .
+$$
+
+The best sets are then selected to be refined using an evolutionary algorithm. The algorithm randomly chooses either to mutate a set or to perform a crossover operation between two sets. When mutating a set, a setting is removed uniformly at random and a new setting is added proportional to its predicted performance (Figure 2, left). The crossover operation creates a new set and adds elements from the union of the parent sets until the new set has $I$ settings (Figure 2, right). The new set is added to the population. After 100,000 steps, the algorithm is stopped and the best set is used as the set of initial hyperparameter settings. In Figure 4 we compare this warm start initialization with the simple random or LHS initialization.
+
+# 7 EXPERIMENTS
+
+We used three different optimization problems to compare the different hyperparameter optimization methods: AdaBoost, GLMNet, and SVM. We created the GLMNet and SVM metadata set by downloading the 30 data sets with the most reported hyperparameter settings from OpenML for each problem. The AdaBoost data set is publicly available (Wistuba et al., 2018). The number of settings per data set varies, the number of settings across all tasks for the GLMNet problem is approximately 800,000. See the appendix for more details about the metadata sets. We compare the following list of hyperparameter optimization methods.
+
+Random Search This is a simple but strong baseline for hyperparameter optimization (Bergstra & Bengio, 2012). Hyperparameters settings are selected uniformly at random from the search space.
+
+Gaussian Process (GP) Bayesian optimization with a Gaussian process as a surrogate model is a standard and strong baseline (Snoek et al., 2012). We use a Matern 5/2 kernel with ARD and rely ´ on the scikit-optimize implementation. We compare with two variations. The first is the vanilla method, which uses Latin Hypercube Sampling (LHS) with design size of 10 to initialize. The second method uses the warm start (WS) method described in Section 6 to find an initial set of 5 settings to use the knowledge from other data sets.
+
+RGPE RGPE (Feurer et al., 2018) is one of the latest examples of methods that use GP ensembles to transfer knowledge across task (Wistuba et al., 2016; Schilling et al., 2016; Wistuba et al., 2018). In this case a GP is trained for each individual task and in a final step the predictions of the GPs are aggregated. These ensembles scale linearly in the number of tasks, but the computation time is still cubic and the space requirement is still quadratic in the number of data points per task. For this reason we can only present results for AdaBoost and not for the other optimization problems, which have significantly more data points.
+
+MetaBO Using the acquisition function instead of the surrogate model for transfer learning is another option. MetaBO is the state-of-the-art transfer acquisition function and was presented last year at ICLR (Volpp et al., 2020). We use the implementation provided by the authors.
+
+ABLR The state-of-the-art surrogate model for scalable hyperparameter transfer learning (Perrone et al., 2018). This method uses a multi-task GP which combines a linear kernel with a neural network and scales to very large data sets. Transfer learning is achieved by sharing the neural network parameters across different tasks. By ABLR (WS) we are referring to a version of ABLR that uses the same warm start as FSBO.
+
+Multi-Head GPs This closely resembles ABLR but uses the same deep kernel as our proposed method. The main difference from our proposed method is that it is using a GP for every task, only shares the neural network across tasks, and uses standard stochastic gradient ascent.
+
+Few-Shot Bayesian Optimization (FSBO) Our proposed method described in Section 4. The deep kernel is composed of a two-layer neural network $\mathrm { 1 2 8 }  \mathrm { 1 2 8 } \mathrm { , }$ ) with ReLU activations and a squared-exponential kernel. We use the Adam optimizer with learning rate $1 0 ^ { - 3 }$ and a batch size of fifty. The warm start length is five.
+
+The experiments are repeated ten times and evaluated in a leave-one-task-out cross-validation. This means that all transfer learning methods use one task as the target task and all other tasks as source tasks. For AdaBoost we use the same train/test split as used by Volpp et al. (2020) instead. We report the aggregated results for all tasks within one problem class with respect to the mean of normalized regrets. The normalized regret for a task is obtained by first scaling the response function values between 0 and 1 before calculating the regret (Wistuba et al., 2018), i.e.
+
+$$
+\tilde { r } ^ { ( t ) } = \tilde { f } ^ { ( t ) } ( \mathbf x _ { \mathrm { m i n } } ) - \tilde { f } _ { \mathrm { m i n } } ^ { ( t ) } ,
+$$
+
+<table><tr><td rowspan="2">Method</td><td colspan="3">AdaBoost</td><td colspan="3">GLMNet</td><td colspan="3">SVM</td></tr><tr><td>15</td><td>33</td><td>50</td><td>33</td><td>67</td><td>100</td><td>33</td><td>67</td><td>100</td></tr><tr><td>RANDOM</td><td>4.87</td><td>3.02</td><td>2.16</td><td>0.85</td><td>0.40</td><td>0.29</td><td>2.01</td><td>1.12</td><td>0.83</td></tr><tr><td>GP (LHS)</td><td>3.87</td><td>2.49</td><td>2.23</td><td>1.32</td><td>1.01</td><td>0.73</td><td>1.94</td><td>1.40</td><td>1.14</td></tr><tr><td>GP (WS)</td><td>3.25</td><td>1.66</td><td>1.02</td><td>0.86</td><td>0.36</td><td>0.24</td><td>1.10</td><td>0.81</td><td>0.65</td></tr><tr><td>RGPE</td><td>5.29</td><td>3.26</td><td>2.83</td><td>_1</td><td>_1</td><td>_1</td><td>_1</td><td>1</td><td>_1</td></tr><tr><td>METABO</td><td>5.27</td><td>3.52</td><td>1.96</td><td>11.02</td><td>10.97</td><td>10.96</td><td>12.39</td><td>12.39</td><td>11.93</td></tr><tr><td>ABLR</td><td>4.56</td><td>1.44</td><td>1.24</td><td>1.77</td><td>0.50</td><td>0.40</td><td>1.81</td><td>1.23</td><td>0.84</td></tr><tr><td>ABLR(WS)</td><td>3.17</td><td>1.72</td><td>1.17</td><td>0.54</td><td>0.36</td><td>0.29</td><td>1.47</td><td>1.08</td><td>0.87</td></tr><tr><td>MULTI-HEAD GPs(WS)</td><td>6.78</td><td>3.45</td><td>1.80</td><td>2.83</td><td>2.80</td><td>2.68</td><td>11.20</td><td>9.82</td><td>8.72</td></tr><tr><td>FSBO</td><td>3.10</td><td>1.13</td><td>0.80</td><td>0.42</td><td>0.22</td><td>0.16</td><td>0.79</td><td>0.51</td><td>0.36</td></tr></table>
+
+Table 1: FSBO obtains better results for all hyperparameter optimization problems. The best results are in bold. Results that are not significantly worse than the best are in italics. Used initialization in parentheses, (LHS) - Latin Hypercube Sampling, (WS) - Warm Start.
+
+here $\tilde { f } ^ { ( t ) }$ is the normalized response function (Equation 13), $\tilde { f } _ { \mathrm { m i n } } ^ { ( t ) }$ is the global minimum for task $t$ $\mathbf { x } _ { \mathrm { m i n } }$ is the best discovered hyperparameter setting by the optimizer.
+
+# 7.1 HYPERPARAMETER OPTIMIZATION
+
+We conduct experiments on three different metadata sets and report the aggregated mean of normalized regrets in Table 1. The best results are in bold. Results that are not significantly worse than the best are in italics. We determined the significance using the Wilcoxon signed rank test with a confidence level of $9 5 \%$ . Results are reported after every 33 trials of Bayesian optimization for GLMNet and SVM. Since AdaBoost has fewer test examples, we report its results in shorter intervals.
+
+Our proposed FSBO method outperforms all other transfer methods in all three tasks. Results are significantly better but in the case of AdaBoost where GP (WS) achieves similar but on average worse results. The results obtained for MetaBO are the worst. We contacted Volpp et al. (2020) and they confirmed that our results are obtained correctly. The problem is apparently that the Reinforcement Learning method does not work well for larger number of trials. We provide a longer discussion in the appendix. Also ABLR and the very related Multi-Head GP are not performing very well. As we show in the next section, one possible reason for this might be because the neural network parameters are fixed which will prohibit a possibly required adaptation to the target task. The vanilla GP and its transfer variant that uses a warm start turn out to be among the strongest baselines. This is in particular true for the warm start version which is the second best method. This simple baseline of knowledge transfer is often not taken into account when comparing transfer surrogates, although it is easy to implement. Due to the very competitive results, we recommend using it as a standard baseline to assess the usefulness of new transfer surrogates.
+
+# 7.2 COMPONENT CONTRIBUTIONS
+
+In this section we highlight the various components that make a significant difference to ABLR. The results for each setup are reported in Figure 3. The Multi-Head GP is a surrogate model that shares the neural network between tasks but uses a separate Gaussian process for each task. It closely resembles the idea of ABLR but is closer to our proposed implementation of FSBO. Starting from this configuration, we add various components that will eventually lead to our proposed model FSBO. We consider Multi-Head GP (WS), a version that additionally uses the warm start method described in Section 6 instead of a random initialization. Multi-Head GP (fine-tune) not only updates the kernel parameters when a new observation is received but also fine-tunes the parameters of the neural network. Finally, FSBO is our proposed method, which uses only one Gaussian process for all tasks. We see that all Multi-Head GP versions have a problem adapting to the target tasks efficiently. Fine-tuning the deep kernel is an important part of learning FSBO. Although FSBO outperforms all Multi-Head GP versions without fine-tuning, the additional use of it makes for a significant improvement. We analyzed the reason for this in more detail. We observed that the learned neural network provides a strong prior that leads to strong exploitation behavior. Fine-tuning prevents this and thus ensures that the method does not get stuck in local optima.
+
+![](images/eb4e6e3741711d97d4722a59d7acbb1aa450ff05dcd8da6c5ab71455aa0c9e48.jpg)  
+Figure 3: Comparison of the contribution of the various FSBO components to the final solution. Each component makes its own orthogonal contribution.
+
+![](images/198844a05885f005ffafff1b96f57361c178cfeae541a69459e23fdb9aec3908.jpg)  
+Figure 4: The warm start initialization yields the best results on GLMNet and SVM for all initialization lengths. For AdaBoost it is comparable to LHS.
+
+# 7.3 DATA-DRIVEN WARM START
+
+The use of a warm start is not the main contribution of this paper but it is interesting to explore further nonetheless. First of all, for some methods this is the only differentiating factor. We compare our suggested warm start initialization with a random and a Latin hypercube sampling (LHS) initialization in Figure 4. Considering that GP (WS) always performed better than GP (LHS) in Table 1, one would expect that the warm start itself also performs better than LHS. While this is the case for GLMNet and SVM, there are no significant differences for the AdaBoost problem. Our assumption is that in this case the warm start might not have always found good settings as part of the initialization, it still explored areas in the search space close to the optimum. This would facilitate finding a better solution in one of the subsequent steps of Bayesian optimization.
+
+# 8 CONCLUSIONS
+
+In this work, we propose to rethink hyperparameter optimization as a few-shot learning problem in which we train a shared deep surrogate model to quickly adapt (with few response evaluations) to the response function of a new task. We propose the use of a deep kernel network for a Gaussian process surrogate that is meta-learned in an end-to-end fashion in order to jointly approximate the response functions of a collection of training data sets. This few-shot surrogate model is used for two different purposes. First, we use it in combination with an evolutionary algorithm in order to estimate a data-driven warm start initialization for Bayesian optimization. Second, we use it directly for Bayesian optimization. In our empirical evaluation on three hyperparameter optimization problems, we observe significantly better results than with state-of-the-art methods that use transfer learning.
+
+# ACKNOWLEDGMENTS
+
+This work has been supported by European Union’s Horizon 2020 research and innovation programme under grant number 951911 - AI4Media. Prof. Grabocka is thankful to the Eva Mayr-Stihl Foundation for their generous research grant.
+
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+
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+
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+
+![](images/52bd45ca2cd10392be7bcf35ae3e85883edee009bb9a2fedb85da9ccb2823033.jpg)  
+Figure 5: FSBO is the best of all considered methods.
+
+# A HYPERPARAMETER OPTIMIZATION
+
+In the main paper we report the normalized regret after every 33 trials of Bayesian optimization in a table. This allows us to also report statistical significance. In Figure 5 we show results for all 100 trials. The conclusions remain unchanged. FSBO provides consistently the best results.
+
+# B CHALLENGES WITH METABO
+
+The results reported in the main paper for MetaBO seem not to align with the numbers reported by Volpp et al. (2020), in particular for AdaBoost. In order to understand this behavior, we conducted a deeper analysis for the AdaBoost metadata set which was used in the evaluation of Volpp et al. (2020) as well. Using the authors’ code2, we executed the same scripts used by the authors to report their results for $T = 1 5$ trials. Furthermore, we changed two lines in the scripts to train MetaBO for $T = 5 0$ , the number of trials considered for AdaBoost in our experiment. We report the outcome of these two experiments in Figure 6. In the two left plots we show how the learned policies for $T = 1 5$ and $T = 5 0$ perform on validation and test during training. The authors’ default setting for the total number of iterations is 2000. However, we noticed a sudden drop in regret for the setting $T = 5 0$ around 2000 iterations. For that reason we assumed that further training might help to further improve MetaBO and increased it to 5000 iterations. As can be seen in the middle plot, that was not the case. We observe that the policy learned for $T = 1 5$ has a lower regret than the one for $T = 5 0$ even though it is only trying 15 compared to 50 trials. In the right plot we show the results compared to a simple random search baseline. The results for $T = 1 5$ are in line with those reported by Volpp et al. (2020). We contacted the authors and they confirmed that we use their code correctly. Their explanation is that the increase of episode length makes this a harder problem for Reinforcement Learning. This results in worse results for $T = 5 0$ and even worse for $T = 1 0 0$ in case of the GLMNet and SVM problem. They proposed to randomly vary $T$ between 5 and 50 during the training phase. We also considered this training protocol and report the results in Figure 6. This improves over training only with $T = 5 0$ but does not improve over a simple random search.
+
+# C METADATA
+
+We created the GLMNet and SVM metadata using OpenML. We chose the 30 OpenML flows with the most observations. We also limited ourselves to the uploaded results from user with ID 2702 (OpenML Bot R), who uploaded the majority of all runs, to ensure that the metadata was generated under similar circumstances. In some cases the choice of a single hyperparameter is not indicated. In such a case, we assume that the default value was used. The GLMNet metadata has two continous hyperparameters: the elastic-net mixing parameter $\alpha \in [ 0 , 1 ]$ and the regularization parameter $\lambda \in \mathsf { [ \bar { 2 } ^ { - 1 0 } , 2 ^ { 1 0 } ] }$ . The SVM metadata has two mandatory hyperparameters and two conditional
+
+![](images/efd123033581afd4b192793e0edc0650c43c8423b0076c5eea25fd6bb98bdfd5.jpg)  
+Figure 6: Left two plots: Training MetaBO for different number of trials $T$ . Right plot: comparing the final policy against a random search. Results for $T = 1 5$ match the results reported by Volpp et al. (2020).
+
+<table><tr><td>Method</td><td colspan="3">AdaBoost</td><td colspan="3">GLMNet</td><td colspan="3">SVM</td></tr><tr><td></td><td>15</td><td>33</td><td>50</td><td>33</td><td>67</td><td>100</td><td>33</td><td>67</td><td>100</td></tr><tr><td>FSBO</td><td>3.10</td><td>1.13</td><td>0.80</td><td>0.42</td><td>0.22</td><td>0.16</td><td>0.79</td><td>0.51</td><td>0.36</td></tr><tr><td>FSBO (REPTILE)</td><td>3.80</td><td>2.13</td><td>1.21</td><td>0.36</td><td>0.19</td><td>0.14</td><td>1.48</td><td>0.89</td><td>0.72</td></tr></table>
+
+Table 2: Comparing different model-agnostic meta-learning approaches. Again, the best results are in bold and results that are not significantly worse than the best are in italics.
+
+hyperparameters. The kernel (linear, polynomial or RBF) and the continuous trade-off parameter $\dot { C } \in \mathsf { [ 2 ^ { - 1 0 } , 2 ^ { 1 0 } ] }$ must always be selected. The degree $d \in \{ 2 , 3 , 4 , 5 \}$ is only considered for the polynomial kernel and the bandwidth $\gamma \in [ 2 ^ { - 1 0 } , 2 ^ { 1 \overline { { 0 } } } ]$ is only considered for the RBF kernel.
+
+We further use the AdaBoost metadata which has been used to evaluate MetaBO (Volpp et al., 2020). According to Wistuba et al. (2015b), this metadata was created using AdaBoost (Kegl & ´ Busa-Fekete, 2009) with decision products as weak learners. The authors designed a grid over the two continuous hyperparameters: the number of iterations and the number of product terms. The number of iterations was chosen from $\{ 2 , 5 , 1 0 , 2 0 , 5 0 , 1 0 0 , 2 0 0 , 5 0 0 , 1 0 ^ { 3 } , 2 \cdot 1 0 ^ { 3 } , 5 \cdot 1 0 ^ { 3 } , 1 0 ^ { 4 } \}$ , the number of product term was chosen from $\{ 2 , 3 , 4 , 5 , 7 , 1 0 , 1 5 , 2 0 , 3 0 \}$ .
+
+The metadata was created by running a grid search on 50 different classification data sets, using classification accuracy as an objective. Compared to the other two metadata sets, this one is significantly smaller. Its use is mainly motivated by comparing to MetaBO under the same circumstances as used to evaluate MetaBO. For that reason, we do not use the leave-one-data-set-out cross-validation protocol but instead use the same fixed split created by Volpp et al. (2020).
+
+Statistics about these metadata sets are provided in Table 3.
+
+# D META-LEARNING WITH REPTILE
+
+In principle our few-shot surrogate can be combined with any model-agnostic meta-learning approach. In this section we compare our proposed meta-learning approach against Reptile (Nichol et al., 2018), a first-order approximation of MAML. We use the same hyperparameters and data augmentation as described in Algorithm 1. Reptile introduces a new hyperparameter, an outer learning rate. We set it to 0.1 and decay it linearly to 0. As summarized in Table 2, FSBO with the meta-learning approach described in Algorithm 1 is either better or not significantly worse than its variation with Reptile.
+
+Table 3: Metadata set statistics. OpenML ID refers to a task on openml.org.   
+
+<table><tr><td rowspan="2">AdaBoost</td><td colspan="2">GLMNet</td><td colspan="3">SVM</td></tr><tr><td>RUNS</td><td>OPENML ID</td><td>RUNS</td><td>OPENML ID</td><td>RUNS</td></tr><tr><td>DATA SET</td><td></td><td></td><td></td><td></td><td>2429</td></tr><tr><td>A9A W8A</td><td>108 108</td><td>3 31</td><td>12796 60721</td><td>37 3485</td><td>1000</td></tr><tr><td></td><td>108</td><td>37</td><td>17950</td><td>3492</td><td>3217</td></tr><tr><td>ABALONE</td><td></td><td>219</td><td>13963</td><td>3493</td><td>1828</td></tr><tr><td>APPENDICITIS</td><td>108 108</td><td></td><td>16968</td><td>3494</td><td>1854</td></tr><tr><td>AUSTRALIAN AUTOMOBILE</td><td>108</td><td>3492 3493</td><td>19931</td><td>3889</td><td>1951</td></tr><tr><td>BANANA</td><td>108</td><td>3889</td><td>13942</td><td>3891</td><td>3083</td></tr><tr><td>BANDS</td><td>108</td><td>3891</td><td>13917</td><td>3899</td><td>2487</td></tr><tr><td>BREAST-CANCER</td><td>108</td><td>3899</td><td>15954</td><td>3902</td><td>1105</td></tr><tr><td>BUPA</td><td>108</td><td>3903</td><td>39155</td><td>3903</td><td>1018</td></tr><tr><td>CAR</td><td>108</td><td>3913</td><td>22949</td><td>3913</td><td>1448</td></tr><tr><td>CHESS</td><td>108</td><td>3917</td><td>16960</td><td>3918</td><td>1459</td></tr><tr><td>COD-RNA</td><td>108</td><td>3918</td><td>20984</td><td>3950</td><td>1998</td></tr><tr><td>COIL2000</td><td>108</td><td>9914</td><td>37635</td><td>6566</td><td>2702</td></tr><tr><td>COLON-CANCER</td><td>108</td><td>9946</td><td>25532</td><td>9889</td><td>2485</td></tr><tr><td>CRX</td><td>108</td><td>9952</td><td>25854</td><td>9914</td><td>1498</td></tr><tr><td>DIABETES</td><td>108</td><td>9967</td><td>21994</td><td>9946</td><td>1291</td></tr><tr><td>ECOLI</td><td>108</td><td>9978</td><td>19956</td><td>9952</td><td>2483</td></tr><tr><td>GERMAN-NUMER</td><td>108</td><td>9980</td><td>31961</td><td>9967</td><td>1492</td></tr><tr><td>HABERMAN</td><td>108</td><td>9983</td><td>37134</td><td>9971</td><td>1381</td></tr><tr><td>HOUSEVOTES</td><td>108</td><td>10101</td><td>66277</td><td>9976</td><td>1473</td></tr><tr><td>IJCNN1</td><td>108</td><td>125923</td><td>35555</td><td>9978</td><td>1503</td></tr><tr><td>KR-VS-K</td><td>108</td><td>145847</td><td>29927</td><td>9980</td><td>2943</td></tr><tr><td>LED7DIGIT</td><td>108</td><td>145857</td><td>31180</td><td>9983</td><td>987</td></tr><tr><td>LETTER</td><td>108</td><td>145862</td><td>12954</td><td>10101</td><td>1871</td></tr><tr><td>LYMPHOGRAPHY</td><td>108</td><td>145872</td><td>17835</td><td>14951</td><td>1765</td></tr><tr><td>MAGIC</td><td>108</td><td>145953</td><td>47439</td><td>34536</td><td>1499</td></tr><tr><td>MONK-2</td><td>108</td><td>145972</td><td>19972</td><td>34537</td><td>1489</td></tr><tr><td>PENDIGITS</td><td>108</td><td>145979</td><td>12979</td><td>145878</td><td>1244</td></tr><tr><td>PHONEME</td><td>108</td><td>146064</td><td>43785</td><td>146064</td><td>1435</td></tr><tr><td>PIMA</td><td>108</td><td></td><td></td><td></td><td></td></tr><tr><td>RING</td><td>108</td><td></td><td></td><td></td><td></td></tr><tr><td>SAHEART</td><td>108</td><td></td><td></td><td></td><td></td></tr><tr><td>SEGMENT</td><td>108</td><td></td><td></td><td></td><td></td></tr><tr><td>SEISMIC</td><td>108</td><td></td><td></td><td></td><td></td></tr><tr><td>SHUTTLE</td><td>108</td><td></td><td></td><td></td><td></td></tr><tr><td>SONAR-SCALE</td><td>108</td><td></td><td></td><td></td><td></td></tr><tr><td>SPAMBASE</td><td>108</td><td></td><td></td><td></td><td></td></tr><tr><td>SPECTFHEART</td><td>108</td><td></td><td></td><td></td><td></td></tr><tr><td>SPLICE</td><td>108</td><td></td><td></td><td></td><td></td></tr><tr><td>TIC-TAC-TOE</td><td>108</td><td></td><td></td><td></td><td></td></tr><tr><td>TITANIC</td><td>108</td><td></td><td></td><td></td><td></td></tr><tr><td>TWONORM</td><td>108</td><td></td><td></td><td></td><td></td></tr><tr><td>USPS</td><td>108</td><td></td><td></td><td></td><td></td></tr><tr><td>VEHICLE</td><td>108</td><td></td><td></td><td></td><td></td></tr><tr><td>WDBC</td><td>108</td><td></td><td></td><td></td><td></td></tr><tr><td>WINE</td><td>108</td><td></td><td></td><td></td><td></td></tr><tr><td>WINEQUALITY-RED</td><td>108</td><td></td><td></td><td></td><td></td></tr><tr><td>WISCONSIN</td><td>108</td><td></td><td></td><td></td><td></td></tr><tr><td>YEAST</td><td>108 5400</td><td>TOTAL</td><td>804159</td><td>TOTAL</td><td></td></tr><tr><td>TOTAL</td><td></td><td></td><td></td><td></td><td>54418</td></tr></table>

parse/train/bJxgv5C3sYc/bJxgv5C3sYc_content_list.json

@@ -0,0 +1,1638 @@
+[
+    {
+        "type": "text",
+        "text": "FEW-SHOT BAYESIAN OPTIMIZATION WITH DEEPKERNEL SURROGATES",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 0
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+    {
+        "type": "text",
+        "text": "Martin Wistuba   \nIBM Research   \nDublin, Ireland   \nmartin.wistuba@ibm.com   \nJosif Grabocka   \nUniversity of Freiburg   \nFreiburg, Germany   \ngrabocka@cs.uni-freiburg.de ",
+        "bbox": [
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            483,
+            171,
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+            227
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "ABSTRACT ",
+        "text_level": 1,
+        "bbox": [
+            454,
+            262,
+            544,
+            277
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Hyperparameter optimization (HPO) is a central pillar in the automation of machine learning solutions and is mainly performed via Bayesian optimization, where a parametric surrogate is learned to approximate the black box response function (e.g. validation error). Unfortunately, evaluating the response function is computationally intensive. As a remedy, earlier work emphasizes the need for transfer learning surrogates which learn to optimize hyperparameters for an algorithm from other tasks. In contrast to previous work, we propose to rethink HPO as a few-shot learning problem in which we train a shared deep surrogate model to quickly adapt (with few response evaluations) to the response function of a new task. We propose the use of a deep kernel network for a Gaussian process surrogate that is meta-learned in an end-to-end fashion in order to jointly approximate the response functions of a collection of training data sets. As a result, the novel few-shot optimization of our deep kernel surrogate leads to new state-of-the-art results at HPO compared to several recent methods on diverse metadata sets. ",
+        "bbox": [
+            233,
+            295,
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "1 INTRODUCTION ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            516,
+            336,
+            532
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Many machine learning models have very sensitive hyperparameters that must be carefully tuned for efficient use. Unfortunately, finding the right setting is a tedious trial and error process that requires expert knowledge. AutoML methods address this problem by providing tools to automate hyperparameter optimization, where Bayesian optimization has become the standard for this task (Snoek et al., 2012). It treats the problem of hyperparameter optimization as a black box optimization problem. Here the black box function is the hyperparameter response function, which maps a hyperparameter setting to the validation loss. Bayesian optimization consists of two parts. First, a surrogate model, often a Gaussian process, is used to approximate the response function. Second, an acquisition function is used that balances the trade-off between exploration and exploitation. In a sequential process, hyperparameter settings are selected and evaluated, followed by an update of the surrogate model. ",
+        "bbox": [
+            174,
+            547,
+            825,
+            700
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Recently, several attempts have been made to extend Bayesian optimization to account for a transfer learning setup. It is assumed here that historical information on machine learning algorithms is available with different hyperparameters. This can either be because this information is publicly available (e.g. OpenML) or because the algorithm is repeatedly optimized for different data sets. To this end, several transfer learning surrogates have been proposed that use this additional information to reduce the convergence time of Bayesian optimization. ",
+        "bbox": [
+            174,
+            708,
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+            791
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "We propose a new paradigm for accomplishing the knowledge transfer by reconceptualizing the process as a few-shot learning task. Inspiration is drawn from the fact that there are a limited number of black box function evaluations for a new hyperparameter optimization task (i.e. few shots) but there are ample evaluations of related black box objectives (i.e. evaluated hyperparameters on other data sets). This approach has several advantages. First, a single model is learned that is trained to quickly adapt to a new task when few examples are available. This is exactly the challenge we face when optimizing hyperparameters. Second, this method can be scaled very well for any number of considered tasks. This not only enables the learning from large metadata sets but also enables the problem of label normalization to be dealt with in a new way. Finally, we present an evolutionary algorithm that can use surrogate models to get a warm start initialization for Bayesian optimization. All of our contributions are empirically compared with several competitive methods in three different problems. Two ablation studies provide information about the influence of the individual components. Concluding, our contributions in this work are: ",
+        "bbox": [
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "• This is the first work that, in the context of hyperparameter optimization (HPO), trains the initialization of the parameters of a probabilistic surrogate model from a collection of meta-tasks by few-shot learning and then transfers it by fine-tuning the initialized kernel parameters to a target task.   \nWe are the first to consider transfer learning in HPO as a few-shot learning task.   \nWe set the new state of the art in transfer learning for the HPO and provide ample evidence that we outperform strong baselines published at ICLR and NeurIPS with a statistically significant margin.   \n• We present an evolutionary algorithm that can use surrogate models to get a warm start initialization for Bayesian optimization. ",
+        "bbox": [
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "2 RELATED WORK ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            342,
+            344,
+            359
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "The idea of using transfer learning to improve Bayesian optimization is being investigated in several papers. The early work suggests learning a single Gaussian process for the entire data (Bardenet et al., 2013; Swersky et al., 2013; Yogatama & Mann, 2014). Since the training of a Gaussian process is cubic in the number of training points, this idea does not scale well. This problem has recently been addressed by several proposals to use ensembles of Gaussian processes where a Gaussian process is learned for each task (Wistuba et al., 2018; Feurer et al., 2018). This idea scales linearly in the number of tasks but still cubically in the number of training points per task. Thus, the problem persists in scenarios where there is a lot of data available for each task. ",
+        "bbox": [
+            174,
+            373,
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Bayesian neural networks are a possible more scalable way of learning with large amounts of data. For example, Schilling et al. (2015) propose to use a neural network with task embedding and variable interactions. To obtain mean and variance predictions, the authors propose using an ensemble of models. In contrast, Springenberg et al. (2016) use a Bayesian multi-task neural network. However, since training Bayesian neural networks is computationally intensive, Perrone et al. (2018) propose a more scalable approach. They suggest using a neural network that is shared by all tasks and using Bayesian linear regression for each task. The parameters are trained jointly on the entire data. While our work shares some similarities with the previous work, our algorithm has unique properties. First of all, a meta-learning algorithm is used, which is motivated by recent work on model-agnostic meta-learning for few-shot learning (Finn et al., 2017). This will allow us to integrate all task-specific parameters out such that the model does not grow with the number of tasks. As a result, our algorithm scales very well with the number of tasks. Second, while we are also using a neural network, we combine it with a Gaussian process with a nonlinear kernel in order to obtain uncertainty predictions. ",
+        "bbox": [
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "A simpler solution for using the transfer learning idea in Bayesian optimization is initialization (Feurer et al., 2015; Wistuba et al., 2015a). The standard Bayesian optimization routine with a simple Gaussian process is used in this case but it is warm-started by a number of hyperparameter settings that work well for related tasks. We also explore this idea in the context of this paper by proposing a simple evolutionary algorithm that can use a surrogate model to estimate a data-driven warm start initialization sequence. The use of an evolutionary algorithm is motivated by its ease of implementation and natural capability to deal with continuous and discrete hyperparameters. ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "3 PRELIMINARIES ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "3.1 BAYESIAN OPTIMIZATION ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            842,
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+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Bayesian optimization is an optimization method for computationally intensive black box functions that consists of two main components, the surrogate model and the acquisition function. The surrogate model is a probabilistic model with mean $\\mu$ and variance $\\sigma ^ { 2 }$ , which tries to approximate the unknown black box function $f$ , in the following also response function. The acquisition function can provide a score for each feasible solution based on the prediction of the surrogate model. This score balances between exploration and exploitation. The following steps are carried out sequentially. First, the feasible solution that maximizes the acquisition function is evaluated. In this way a new observation $\\left( \\mathbf { x } , f ( \\mathbf { x } ) \\right)$ is obtained. Then, the surrogate model is updated based on the entirety of all observations $\\mathcal { D }$ . This sequence of steps is carried out until a previously defined convergence criterion occurs (e.g. exhaustion of the time budget). In Figure 1 we provide an example how Bayesian optimization is used to maximize a sine wave. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Expected Improvement (Jones et al., 1998) is one of the most commonly used acquisition functions and will also be used in all our experiments. It is defined as ",
+        "bbox": [
+            173,
+            208,
+            820,
+            236
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/1fc92d39ec90d9d2393cebdc7861d150dd5bb54d8ca89270d4052efec7ef5417.jpg",
+        "text": "$$\na ( \\mathbf { x } | \\mathcal { D } ) = \\mathbb { E } \\left[ \\operatorname* { m a x } \\left\\{ f ( \\mathbf { x } ) - y _ { \\operatorname* { m a x } } , 0 \\right\\} \\right] ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            372,
+            242,
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+            260
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "where $y _ { \\mathrm { m a x } }$ is the largest observed value of $f$ ",
+        "bbox": [
+            176,
+            266,
+            470,
+            281
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "3.2 GAUSSIAN PROCESSES ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            296,
+            372,
+            311
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "We have a training set $\\mathcal { D }$ of $n$ observations, $\\mathcal { D } = \\{ ( \\mathbf { x } _ { i } , y _ { i } ) | i = 1 , \\dots , n \\}$ , where $y _ { i }$ are noisy observations of the function values $y _ { i } = f ( \\mathbf { x } _ { i } ) + \\varepsilon$ . We assume that the noise $\\varepsilon$ is additive independent identically distributed Gaussian with variance $\\sigma _ { n } ^ { 2 }$ . For Gaussian processes (Rasmussen & Williams, 2006) we consider $y _ { i }$ to be a random variable and the joint distribution of all $y _ { i }$ is assumed to be multivariate Gaussian distributed: ",
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/d7f765f0fa21db9166323d5beb04fd51da92d373e927c98f444c41cea217dbb5.jpg",
+        "text": "$$\n\\mathbf { y } \\sim { \\mathcal { N } } \\left( m \\left( \\mathbf { X } \\right) , k \\left( \\mathbf { X } , \\mathbf { X } \\right) \\right) \\ .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            400,
+            398,
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "A Gaussian process is completely specified by its mean function $m$ , its covariance function $k$ , and may depend on parameters $\\pmb { \\theta }$ . A common choice is to set $m ( { \\bf x } _ { i } ) = 0$ . At inference time for instances $\\mathbf { x } _ { * }$ , the assumption is that their ground truth $\\mathbf { f } _ { * }$ is jointly Gaussian with $\\mathbf { y }$ ",
+        "bbox": [
+            174,
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+            464
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/8089466f957996ee919aea5d034ddcefa5df6bbd099df9763302da9d543a561d.jpg",
+        "text": "$$\n\\begin{array}{c} \\left[ \\begin{array} { c } { \\mathbf { y } } \\\\ { \\mathbf { f _ { * } } } \\end{array} \\right] \\sim \\mathcal { N } \\left( \\mathbf { 0 } , \\left( \\begin{array} { c c } { \\mathbf { K } _ { n } } & { \\mathbf { K } _ { * } } \\\\ { \\mathbf { K } _ { * } ^ { T } } & { \\mathbf { K } _ { * * } } \\end{array} \\right) \\right)  \\end{array} ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            370,
+            469,
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "where ",
+        "bbox": [
+            174,
+            511,
+            217,
+            523
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/3dade55fb8b59b1edc4dcced6164a32396602a635ec3615ffb714e963b5d3879.jpg",
+        "text": "$$\n\\mathbf { K } _ { n } = k \\left( \\mathbf { X } , \\mathbf { X } | \\pmb { \\theta } \\right) + \\sigma _ { n } ^ { 2 } \\mathbf { I } , \\ \\mathbf { K } _ { * } = k \\left( \\mathbf { X } , \\mathbf { X } _ { * } | \\pmb { \\theta } \\right) , \\ \\mathbf { K } _ { * * } = k \\left( \\mathbf { X } _ { * } , \\mathbf { X } _ { * } | \\pmb { \\theta } \\right)\n$$",
+        "text_format": "latex",
+        "bbox": [
+            267,
+            522,
+            728,
+            540
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "for brevity. Then, the posterior predictive distribution has mean and covariance ",
+        "bbox": [
+            169,
+            542,
+            696,
+            558
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/2aeee13dbdad74214e8a7d66daf801896994140db2fd30fadc443c889806e197.jpg",
+        "text": "$$\n\\mathbb { E } \\left[ \\mathbf { f } _ { * } | \\mathbf { X } , \\mathbf { y } , \\mathbf { X } _ { * } \\right] = \\mathbf { K } _ { * } ^ { T } \\mathbf { K } _ { n } ^ { - 1 } \\mathbf { y } , \\thinspace \\thinspace \\mathrm { c o v } \\left[ \\mathbf { f } _ { * } | \\mathbf { X } , \\mathbf { X } _ { * } \\right] = \\mathbf { K } _ { * * } - \\mathbf { K } _ { * } ^ { T } \\mathbf { K } _ { n } ^ { - 1 } \\mathbf { K } _ { * }\n$$",
+        "text_format": "latex",
+        "bbox": [
+            272,
+            563,
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+            582
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Examples for covariance functions are the linear or squared exponential kernel. However, these kernel functions are designed by hand. The idea of deep kernel learning (Wilson et al., 2016) is to learn the kernel function. They propose to use a neural network $\\varphi$ to transform the input $\\mathbf { x }$ to a latent representation which serves as input for the kernel function. ",
+        "bbox": [
+            173,
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/cdc64024ae0e4b77ab75f6549e63b8848865f28cc3ae770e8240886865a92783.jpg",
+        "text": "$$\nk _ { \\mathrm { d e e p } } ( \\mathbf { x } , \\mathbf { x } ^ { \\prime } | \\boldsymbol { \\theta } , \\mathbf { w } ) = k ( \\varphi ( \\mathbf { x } , \\mathbf { w } ) , \\varphi ( \\mathbf { x } ^ { \\prime } , \\mathbf { w } ) | \\boldsymbol { \\theta } )\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "4 FEW-SHOT BAYESIAN OPTIMIZATION ",
+        "text_level": 1,
+        "bbox": [
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+        "type": "text",
+        "text": "Hyperparameter optimization is traditionally either tackled as an optimization problem without prior knowledge about the response function or alternatively as a multi-task or transfer learning problem. In the former, every search basically starts from scratch. In the latter, one or multiple models are trained that attempt to reuse knowledge of the source tasks for the target task. ",
+        "bbox": [
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+        "type": "text",
+        "text": "In this work we will address the problem as a few-shot problem. Given $T$ related source tasks and very few examples of the target task, we want to make reliable predictions. For each of the source tasks we have observations $\\bar { \\mathcal { D } } ^ { ( t ) } = \\{ ( \\mathbf { x } _ { i } ^ { ( t ) } , y _ { i } ^ { ( t ) } ) \\} _ { i = 1 \\ldots n ^ { ( t ) } }$ , where $\\mathbf { x } _ { i } ^ { ( t ) }$ is a hyperparameter setting and $y _ { i } ^ { ( t ) }$ is the noisy observations of $f ^ { ( t ) } ( \\mathbf { x } _ { i } ^ { ( t ) } )$ , i.e. a validation score of a machine learning model on data set $t$ . In the following we will denote the set of all data points by $\\textstyle { \\mathcal { D } } : = \\bigcup _ { t = 1 } ^ { T } { \\mathcal { D } } ^ { ( t ) }$ . Modelagnostic meta-learning (Finn et al., 2017) has become a popular choice for few-shot learning and we propose to use an adaptation for Gaussian processes (Patacchiola et al., 2020) as a surrogate model within the Bayesian optimization framework. The idea is simple. A deep kernel $\\varphi$ is used to learn parameters across tasks such that all its parameters $\\pmb \\theta$ and w are task-independent. All task-dependent parameters are kept separate which allows to marginalize its corresponding variable out when solely optimizing for the task-independent parameters. If we assume that the posteriors over $\\pmb \\theta$ and w are dominated by their respective maximum likelihood estimates $\\hat { \\pmb { \\theta } }$ and $\\hat { \\mathbf { w } }$ , we can approximate the posterior predictive distribution by ",
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+        "type": "text",
+        "text": "",
+        "bbox": [
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+        "type": "equation",
+        "img_path": "images/4ba388c34c023e6f8308413050a01620602a10653b4c770f9d2732179a08f14c.jpg",
+        "text": "$$\np \\left( f _ { * } | \\mathbf { x } _ { * } , \\mathcal { D } \\right) = \\int p \\left( f _ { * } | \\mathbf { x } _ { * } , \\pmb { \\theta } , \\mathbf { w } \\right) p \\left( \\pmb { \\theta } , \\mathbf { w } | \\mathcal { D } \\right) \\mathrm { d } \\pmb { \\theta } , \\mathbf { w } \\approx p \\left( f _ { * } | \\mathbf { x } _ { * } , \\mathcal { D } , \\pmb { \\hat { \\theta } } , \\hat { \\mathbf { w } } \\right)\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "We estimate $\\hat { \\pmb { \\theta } }$ and $\\hat { \\mathbf { w } }$ by maximizing the log marginal likelihood for all tasks. ",
+        "bbox": [
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+        "page_idx": 3
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+    {
+        "type": "equation",
+        "img_path": "images/22b548627fb16dae6b2265228805721fde0e20116ffecb2dbf7fd0e0f79bcc6e.jpg",
+        "text": "$$\n\\begin{array} { l } { \\log p \\left( \\mathbf { y } ^ { ( 1 ) } , \\dots , \\mathbf { y } ^ { ( T ) } | \\mathbf { X } ^ { ( 1 ) } , \\dots , \\mathbf { X } ^ { ( T ) } , \\pmb { \\theta } , \\mathbf { w } \\right) = \\displaystyle \\sum _ { t = 1 } ^ { T } \\log p \\left( \\mathbf { y } ^ { ( t ) } | \\mathbf { X } ^ { ( t ) } , \\pmb { \\theta } , \\mathbf { w } \\right) } \\\\ { \\propto - \\displaystyle \\sum _ { t = 1 } ^ { T } \\left( \\mathbf { y } ^ { ( t ) ^ { \\mathrm { T } } } \\mathbf { K } _ { n } ^ { ( t ) ^ { - 1 } } \\mathbf { y } ^ { ( t ) } + \\log \\left| \\mathbf { K } _ { n } ^ { ( t ) } \\right| \\right) } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "We maximize the marginal likelihood with stochastic gradient ascent (SGA). Each batch contains data for one task. It is important to note that this batch data does not correspond to the data in $\\mathcal { D } ^ { ( t ) }$ , but rather a subset of it. With small batch sizes, this is an efficient way to train the Gaussian process, regardless of how many tasks knowledge is transferred from or how large $\\mathcal { D } ^ { ( t ) }$ is. It is important to note that there is a correlation between all data per task and not just one batch. Hence the stochastic gradient is a biased estimator of the full gradient. For a long time we lacked the theoretical understanding of how SGA would behave in this situation. Fortunately, recent work (Chen et al., 2020) proved that SGA still successfully converges and restores model parameters in these cases. The results by Chen et al. (2020) guarantee a $\\mathcal { O } ( 1 / K )$ optimization margin of error, where $K$ is the number of iterations. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "The training works as follows. In each iteration, a task $t$ is sampled uniformly at random from all $T$ tasks. Then we sample a batch of training instances uniformly at random from $\\mathcal { D } ^ { ( t ) }$ . Finally, we calculate the log marginal likelihood for this batch (Equation 8) and update the kernel parameters with one step in the direction of the gradient. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "In hyperparameter optimization, we are only interested in the hyperparameter setting that works best according to a predefined metric of interest. Therefore, only the ranking of the hyperparameter settings is important, not the actual value of the metric. Therefore, we are interested in a surrogate model whose prediction strongly correlate with the response function and the squared error is of little to no interest for us. In practice, however, the minimum / maximum score and the range of values differ significantly between the tasks which makes it challenging to obtain a strongly correlating surrogate model. The most common way to address this problem is to normalize the labels, e.g. by $\\mathbf { Z }$ -normalizing or scaling the labels to $[ 0 , 1 ]$ per data set. However, this does not fully solve the problem. The main problem is that this label normalization must also be applied to the target task. This is not possible with a satisfactory degree of accuracy, especially when only a few examples are available, since the approximate values for minimum / maximum or mean / standard deviation are insufficiently accurate. ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Since our proposed method scales easily to any number of tasks, we can afford to consider another option. We propose a task augmentation strategy that addresses the label normalization issue by randomly scaling the labels for each batch. Since $y _ { \\mathrm { { m i n } } }$ and $y _ { \\mathrm { m a x } }$ are the minimum and maximum values for all $T$ tasks, we can generate augmented tasks for each batch $B = \\{ \\mathbf { x } _ { i } , y _ { i } \\} _ { i = 1 , \\ldots , b } \\sim \\mathcal { D } ^ { ( t ) }$ where $b$ is the batch size, as follows. A lower and upper limit is sampled for each sample batch, ",
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+        "type": "equation",
+        "img_path": "images/d3dfa998b7a7dec0da41d01b105d5a144d1e06de8752cb703f877ebabcda74eb.jpg",
+        "text": "$$\nl \\sim \\mathcal { U } ( y _ { \\mathrm { m i n } } , y _ { \\mathrm { m a x } } ) , u \\sim \\mathcal { U } ( y _ { \\mathrm { m i n } } , y _ { \\mathrm { m a x } } ) ~ ,\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "such that $l < u$ holds. Then the labels for that batch are scaled to this range, ",
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+        "page_idx": 3
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+    {
+        "type": "equation",
+        "img_path": "images/2063b3b0850a1fa73935f4c41dd939cbd50f8fb9633568c7bddfc73d37202ab0.jpg",
+        "text": "$$\n\\mathbf { y } \\gets \\frac { \\mathbf { y } - l } { u - l } .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "No further changes are required. We summarize this procedure in Algorithm 1. The idea here is to learn a representation that is invariant with respect to various offsets and ranges so that the target task does not require normalization. This strategy is not possible for other hyperparameter optimization methods, since this either increases the training data set size even further and thus becomes ",
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+    {
+        "type": "text",
+        "text": "Algorithm 1: Few-Shot GP Surrogate ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Input: Learning rates $\\alpha$ and $\\beta$ , training data $\\mathcal { D }$ , kernel $k$ , and neural network $\\varphi$ .   \nwhile not done do Sample task $t \\sim \\mathcal { U } ( \\{ 1 , \\dots , T \\} )$ ; Estimate $l$ and $u$ (Equation 10); for $b _ { n }$ times do Sample batch $B = \\{ ( \\mathbf { x } _ { i } , y _ { i } ) \\} _ { i = 1 , \\ldots , b } \\sim \\mathcal { D } ^ { ( t ) }$ and scale labels $\\mathbf { y }$ using $l$ and $u$ ; Compute marginal likelihood $\\mathcal { L }$ on $\\boldsymbol { B }$ . (Equation 8); $\\pmb { \\theta }  \\pmb { \\theta } + \\alpha \\nabla _ { \\pmb { \\theta } } \\mathcal { L }$ ; $\\mathbf { w } \\gets \\mathbf { w } + \\beta \\nabla _ { \\mathbf { w } } \\mathcal { L } ;$ ; end   \nend ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "image",
+        "img_path": "images/eedb4f8e35c73478805b6df22db0fab343c2f801f56fc02053d1e6efa0660124.jpg",
+        "image_caption": [
+            "Figure 1: Demonstration of five steps of Bayesian Optimization with FSBO for maximizing a sine wave (blue). One maximum is discovered within only three steps. Expected Improvement has been scaled to improve readability. In black the predictions of the surrogate model. Bottom right are examples of the source tasks. The deep kernel consists of a spectral kernel (Wilson & Adams, 2013) combined with a two-layer neural network $\\mathrm { 1  6 4  6 4 }$ ). "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "computationally impossible (Bardenet et al., 2013; Swersky et al., 2013; Yogatama & Mann, 2014) or thousands of new tasks would have to be generated, which is also is not practical (Springenberg et al., 2016; Perrone et al., 2018; Wistuba et al., 2018; Feurer et al., 2018). ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "At test time, the posterior predictive distribution or the target task $T + 1$ is computed according to Equation 5. In practice, the target data set can be very different to the learned prior. For this reason, the deep kernel parameters are fine-tuned on the target task’s data for few epochs. Our task augmentation strategy described earlier has resulted in a model that has become invariant for different scales of y. For this reason we do not apply this strategy to the target task $T + 1$ and only use it for all source tasks 1 to $T$ . We discuss the usefulness of this step in more detail when we discuss the empirical results. ",
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+    {
+        "type": "text",
+        "text": "5 A MOTIVATING EXAMPLE ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "We would like to first motivate our method, FSBO, with an example, similar to the one described in Nichol et al. (2018). The aim is to find the argument $x$ that maximizes a one-dimensional sine wave of the form $f ^ { ( t ) } ( x ) = a ^ { ( t ) } \\sin ( x + b ^ { ( t ) } )$ . It is not known that the function is a sine wave and that one only has to determine the amplitude $a$ and phase $b$ . However, information about other sine waves is available to the optimizer which are considered to be related. These source tasks $t = 1 , \\dots , T$ are generated randomly with $a \\sim \\mathcal { U } ( 0 . 1 , 5 )$ and $b \\sim \\mathcal { U } ( 0 , 2 \\pi )$ . 50 examples $( x _ { i } ^ { ( t ) } , f ^ { ( t ) } ( x _ { i } ^ { ( t ) } ) )$ are available for each task $t$ , whereby the points $x _ { i } ^ { ( t ) } \\in [ - 5 , 5 ]$ are evenly spaced. ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "image",
+        "img_path": "images/937e452ee66004a79a138ec13aa35ab753d35161c5d4481d3dc1bed46fae866f.jpg",
+        "image_caption": [
+            "Figure 2: Examples for the mutation and crossover operation with $I = 3$ "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "",
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+    {
+        "type": "text",
+        "text": "It should be noted at this point that the expected value for each $x _ { i }$ is 0. Thus, training on the data described above leads to a model predicting 0 for every $x _ { i }$ . However, the used meta-learning procedure allows for accurately reconstructing the underlying sine wave after only a few examples (see Figure 1). Provided that the response function of the original task is similar to the target task, this is a very promising technique for accelerating the search for good hyperparameter settings. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "6 A DATA-DRIVEN INITIALIZATION ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "The proposed few-shot surrogate model requires only a few examples before reliable predictions can be made for the target data set. How do you choose the initial hyper parameter settings? The simplest idea would be to pick them at random or use Latin Hypercube Sampling (LHS) (McKay et al., 1979). Since we already have data from various tasks and a surrogate model that can impute missing values, we propose a data-driven warm start approach. If we evaluate the performance of a hyperparameter optimization algorithm with a loss function $\\mathcal { L }$ , we use an evolutionary algorithm to find a set with $I$ hyperparameter settings which minimizes this loss on the source tasks. ",
+        "bbox": [
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+    },
+    {
+        "type": "equation",
+        "img_path": "images/937778cae14fe0c56d7e610cd4f25ad1fad349f73d059b61cfabd479039e8d3d.jpg",
+        "text": "$$\n\\mathbf { X } ^ { ( \\mathrm { i n i t } ) } = \\underset { \\mathbf { X } \\in \\mathcal { X } ^ { I } } { \\arg \\operatorname* { m i n } } \\sum _ { t = 1 } ^ { T } \\mathcal { L } \\left( f ^ { ( t ) } , \\mathbf { X } \\right) = \\underset { \\mathbf { X } \\in \\mathcal { X } ^ { I } } { \\arg \\operatorname* { m i n } } \\sum _ { t = 1 } ^ { T } \\underset { \\mathbf { x } \\in \\mathbf { X } } { \\operatorname* { m i n } } \\tilde { f } ^ { ( t ) } ( \\mathbf { x } )\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "The loss of a set of hyperparameters depends on the response function values for each of the elements. Since these are not available for every $\\mathbf { x }$ , we approximate $f ^ { ( t ) } ( \\mathbf { x } )$ with the prediction of the surrogate model described in the last section whenever this is necessary. ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Arbitrary loss function can be considered. The specific loss function used in our experiments is the normalized regret and is defined on the right side of Equation 12, where $\\tilde { f } ^ { ( t ) }$ is derived from $f ^ { ( t ) }$ by scaling it to $[ 0 , 1 ]$ range: ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "equation",
+        "img_path": "images/98d2c8d928b40e91be8102cad898d0e33c712437437ef83f02fb619564792372.jpg",
+        "text": "$$\n\\tilde { f } ^ { ( t ) } ( \\mathbf { x } ) = \\frac { f ^ { ( t ) } ( \\mathbf { x } ) - f _ { \\operatorname* { m i n } } ^ { ( t ) } } { f _ { \\operatorname* { m a x } } ^ { ( t ) } - f _ { \\operatorname* { m i n } } ^ { ( t ) } } ,\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "where $f _ { \\mathrm { m a x } } ^ { ( t ) }$ and $f _ { \\operatorname* { m i n } } ^ { ( t ) }$ are the maximum and minimum of $f ^ { ( t ) }$ , respectively. ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "The evolutionary algorithm works as follows. We initialize the population with sets containing $I$ random settings, the settings being sampled in proportion to their performance according to the predictions. Precisely, a setting $\\mathbf { x }$ is sampled proportional to ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "equation",
+        "img_path": "images/a3976e1b837340ccd6cda2ec391049f5f497d766915955771514431deed37c43.jpg",
+        "text": "$$\n\\exp \\left( - \\operatorname* { m i n } _ { t \\in \\{ 1 , . . . , T \\} } \\tilde { f } ^ { ( t ) } ( { \\bf x } ) \\right) .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "The best sets are then selected to be refined using an evolutionary algorithm. The algorithm randomly chooses either to mutate a set or to perform a crossover operation between two sets. When mutating a set, a setting is removed uniformly at random and a new setting is added proportional to its predicted performance (Figure 2, left). The crossover operation creates a new set and adds elements from the union of the parent sets until the new set has $I$ settings (Figure 2, right). The new set is added to the population. After 100,000 steps, the algorithm is stopped and the best set is used as the set of initial hyperparameter settings. In Figure 4 we compare this warm start initialization with the simple random or LHS initialization. ",
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+        "page_idx": 5
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+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+        "page_idx": 6
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+    {
+        "type": "text",
+        "text": "7 EXPERIMENTS ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "We used three different optimization problems to compare the different hyperparameter optimization methods: AdaBoost, GLMNet, and SVM. We created the GLMNet and SVM metadata set by downloading the 30 data sets with the most reported hyperparameter settings from OpenML for each problem. The AdaBoost data set is publicly available (Wistuba et al., 2018). The number of settings per data set varies, the number of settings across all tasks for the GLMNet problem is approximately 800,000. See the appendix for more details about the metadata sets. We compare the following list of hyperparameter optimization methods. ",
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Random Search This is a simple but strong baseline for hyperparameter optimization (Bergstra & Bengio, 2012). Hyperparameters settings are selected uniformly at random from the search space. ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Gaussian Process (GP) Bayesian optimization with a Gaussian process as a surrogate model is a standard and strong baseline (Snoek et al., 2012). We use a Matern 5/2 kernel with ARD and rely ´ on the scikit-optimize implementation. We compare with two variations. The first is the vanilla method, which uses Latin Hypercube Sampling (LHS) with design size of 10 to initialize. The second method uses the warm start (WS) method described in Section 6 to find an initial set of 5 settings to use the knowledge from other data sets. ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "RGPE RGPE (Feurer et al., 2018) is one of the latest examples of methods that use GP ensembles to transfer knowledge across task (Wistuba et al., 2016; Schilling et al., 2016; Wistuba et al., 2018). In this case a GP is trained for each individual task and in a final step the predictions of the GPs are aggregated. These ensembles scale linearly in the number of tasks, but the computation time is still cubic and the space requirement is still quadratic in the number of data points per task. For this reason we can only present results for AdaBoost and not for the other optimization problems, which have significantly more data points. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "MetaBO Using the acquisition function instead of the surrogate model for transfer learning is another option. MetaBO is the state-of-the-art transfer acquisition function and was presented last year at ICLR (Volpp et al., 2020). We use the implementation provided by the authors. ",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "ABLR The state-of-the-art surrogate model for scalable hyperparameter transfer learning (Perrone et al., 2018). This method uses a multi-task GP which combines a linear kernel with a neural network and scales to very large data sets. Transfer learning is achieved by sharing the neural network parameters across different tasks. By ABLR (WS) we are referring to a version of ABLR that uses the same warm start as FSBO. ",
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+    {
+        "type": "text",
+        "text": "Multi-Head GPs This closely resembles ABLR but uses the same deep kernel as our proposed method. The main difference from our proposed method is that it is using a GP for every task, only shares the neural network across tasks, and uses standard stochastic gradient ascent. ",
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+        "text": "Few-Shot Bayesian Optimization (FSBO) Our proposed method described in Section 4. The deep kernel is composed of a two-layer neural network $\\mathrm { 1 2 8 }  \\mathrm { 1 2 8 } \\mathrm { , }$ ) with ReLU activations and a squared-exponential kernel. We use the Adam optimizer with learning rate $1 0 ^ { - 3 }$ and a batch size of fifty. The warm start length is five. ",
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+        "text": "The experiments are repeated ten times and evaluated in a leave-one-task-out cross-validation. This means that all transfer learning methods use one task as the target task and all other tasks as source tasks. For AdaBoost we use the same train/test split as used by Volpp et al. (2020) instead. We report the aggregated results for all tasks within one problem class with respect to the mean of normalized regrets. The normalized regret for a task is obtained by first scaling the response function values between 0 and 1 before calculating the regret (Wistuba et al., 2018), i.e. ",
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+        "text": "$$\n\\tilde { r } ^ { ( t ) } = \\tilde { f } ^ { ( t ) } ( \\mathbf x _ { \\mathrm { m i n } } ) - \\tilde { f } _ { \\mathrm { m i n } } ^ { ( t ) } ,\n$$",
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+        "table_body": "<table><tr><td rowspan=\"2\">Method</td><td colspan=\"3\">AdaBoost</td><td colspan=\"3\">GLMNet</td><td colspan=\"3\">SVM</td></tr><tr><td>15</td><td>33</td><td>50</td><td>33</td><td>67</td><td>100</td><td>33</td><td>67</td><td>100</td></tr><tr><td>RANDOM</td><td>4.87</td><td>3.02</td><td>2.16</td><td>0.85</td><td>0.40</td><td>0.29</td><td>2.01</td><td>1.12</td><td>0.83</td></tr><tr><td>GP (LHS)</td><td>3.87</td><td>2.49</td><td>2.23</td><td>1.32</td><td>1.01</td><td>0.73</td><td>1.94</td><td>1.40</td><td>1.14</td></tr><tr><td>GP (WS)</td><td>3.25</td><td>1.66</td><td>1.02</td><td>0.86</td><td>0.36</td><td>0.24</td><td>1.10</td><td>0.81</td><td>0.65</td></tr><tr><td>RGPE</td><td>5.29</td><td>3.26</td><td>2.83</td><td>_1</td><td>_1</td><td>_1</td><td>_1</td><td>1</td><td>_1</td></tr><tr><td>METABO</td><td>5.27</td><td>3.52</td><td>1.96</td><td>11.02</td><td>10.97</td><td>10.96</td><td>12.39</td><td>12.39</td><td>11.93</td></tr><tr><td>ABLR</td><td>4.56</td><td>1.44</td><td>1.24</td><td>1.77</td><td>0.50</td><td>0.40</td><td>1.81</td><td>1.23</td><td>0.84</td></tr><tr><td>ABLR(WS)</td><td>3.17</td><td>1.72</td><td>1.17</td><td>0.54</td><td>0.36</td><td>0.29</td><td>1.47</td><td>1.08</td><td>0.87</td></tr><tr><td>MULTI-HEAD GPs(WS)</td><td>6.78</td><td>3.45</td><td>1.80</td><td>2.83</td><td>2.80</td><td>2.68</td><td>11.20</td><td>9.82</td><td>8.72</td></tr><tr><td>FSBO</td><td>3.10</td><td>1.13</td><td>0.80</td><td>0.42</td><td>0.22</td><td>0.16</td><td>0.79</td><td>0.51</td><td>0.36</td></tr></table>",
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+    {
+        "type": "text",
+        "text": "Table 1: FSBO obtains better results for all hyperparameter optimization problems. The best results are in bold. Results that are not significantly worse than the best are in italics. Used initialization in parentheses, (LHS) - Latin Hypercube Sampling, (WS) - Warm Start. ",
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+        "type": "text",
+        "text": "here $\\tilde { f } ^ { ( t ) }$ is the normalized response function (Equation 13), $\\tilde { f } _ { \\mathrm { m i n } } ^ { ( t ) }$ is the global minimum for task $t$ $\\mathbf { x } _ { \\mathrm { m i n } }$ is the best discovered hyperparameter setting by the optimizer. ",
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+    {
+        "type": "text",
+        "text": "7.1 HYPERPARAMETER OPTIMIZATION ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "We conduct experiments on three different metadata sets and report the aggregated mean of normalized regrets in Table 1. The best results are in bold. Results that are not significantly worse than the best are in italics. We determined the significance using the Wilcoxon signed rank test with a confidence level of $9 5 \\%$ . Results are reported after every 33 trials of Bayesian optimization for GLMNet and SVM. Since AdaBoost has fewer test examples, we report its results in shorter intervals. ",
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+    {
+        "type": "text",
+        "text": "Our proposed FSBO method outperforms all other transfer methods in all three tasks. Results are significantly better but in the case of AdaBoost where GP (WS) achieves similar but on average worse results. The results obtained for MetaBO are the worst. We contacted Volpp et al. (2020) and they confirmed that our results are obtained correctly. The problem is apparently that the Reinforcement Learning method does not work well for larger number of trials. We provide a longer discussion in the appendix. Also ABLR and the very related Multi-Head GP are not performing very well. As we show in the next section, one possible reason for this might be because the neural network parameters are fixed which will prohibit a possibly required adaptation to the target task. The vanilla GP and its transfer variant that uses a warm start turn out to be among the strongest baselines. This is in particular true for the warm start version which is the second best method. This simple baseline of knowledge transfer is often not taken into account when comparing transfer surrogates, although it is easy to implement. Due to the very competitive results, we recommend using it as a standard baseline to assess the usefulness of new transfer surrogates. ",
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+    {
+        "type": "text",
+        "text": "7.2 COMPONENT CONTRIBUTIONS ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "In this section we highlight the various components that make a significant difference to ABLR. The results for each setup are reported in Figure 3. The Multi-Head GP is a surrogate model that shares the neural network between tasks but uses a separate Gaussian process for each task. It closely resembles the idea of ABLR but is closer to our proposed implementation of FSBO. Starting from this configuration, we add various components that will eventually lead to our proposed model FSBO. We consider Multi-Head GP (WS), a version that additionally uses the warm start method described in Section 6 instead of a random initialization. Multi-Head GP (fine-tune) not only updates the kernel parameters when a new observation is received but also fine-tunes the parameters of the neural network. Finally, FSBO is our proposed method, which uses only one Gaussian process for all tasks. We see that all Multi-Head GP versions have a problem adapting to the target tasks efficiently. Fine-tuning the deep kernel is an important part of learning FSBO. Although FSBO outperforms all Multi-Head GP versions without fine-tuning, the additional use of it makes for a significant improvement. We analyzed the reason for this in more detail. We observed that the learned neural network provides a strong prior that leads to strong exploitation behavior. Fine-tuning prevents this and thus ensures that the method does not get stuck in local optima. ",
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+    {
+        "type": "image",
+        "img_path": "images/eb4e6e3741711d97d4722a59d7acbb1aa450ff05dcd8da6c5ab71455aa0c9e48.jpg",
+        "image_caption": [
+            "Figure 3: Comparison of the contribution of the various FSBO components to the final solution. Each component makes its own orthogonal contribution. "
+        ],
+        "image_footnote": [],
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+        "img_path": "images/198844a05885f005ffafff1b96f57361c178cfeae541a69459e23fdb9aec3908.jpg",
+        "image_caption": [
+            "Figure 4: The warm start initialization yields the best results on GLMNet and SVM for all initialization lengths. For AdaBoost it is comparable to LHS. "
+        ],
+        "image_footnote": [],
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+    {
+        "type": "text",
+        "text": "",
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+    {
+        "type": "text",
+        "text": "7.3 DATA-DRIVEN WARM START ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "The use of a warm start is not the main contribution of this paper but it is interesting to explore further nonetheless. First of all, for some methods this is the only differentiating factor. We compare our suggested warm start initialization with a random and a Latin hypercube sampling (LHS) initialization in Figure 4. Considering that GP (WS) always performed better than GP (LHS) in Table 1, one would expect that the warm start itself also performs better than LHS. While this is the case for GLMNet and SVM, there are no significant differences for the AdaBoost problem. Our assumption is that in this case the warm start might not have always found good settings as part of the initialization, it still explored areas in the search space close to the optimum. This would facilitate finding a better solution in one of the subsequent steps of Bayesian optimization. ",
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+    {
+        "type": "text",
+        "text": "8 CONCLUSIONS ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "In this work, we propose to rethink hyperparameter optimization as a few-shot learning problem in which we train a shared deep surrogate model to quickly adapt (with few response evaluations) to the response function of a new task. We propose the use of a deep kernel network for a Gaussian process surrogate that is meta-learned in an end-to-end fashion in order to jointly approximate the response functions of a collection of training data sets. This few-shot surrogate model is used for two different purposes. First, we use it in combination with an evolutionary algorithm in order to estimate a data-driven warm start initialization for Bayesian optimization. Second, we use it directly for Bayesian optimization. In our empirical evaluation on three hyperparameter optimization problems, we observe significantly better results than with state-of-the-art methods that use transfer learning. ",
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+    {
+        "type": "text",
+        "text": "ACKNOWLEDGMENTS ",
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+        "type": "text",
+        "text": "This work has been supported by European Union’s Horizon 2020 research and innovation programme under grant number 951911 - AI4Media. Prof. Grabocka is thankful to the Eva Mayr-Stihl Foundation for their generous research grant. ",
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+        "type": "text",
+        "text": "Martin Wistuba, Nicolas Schilling, and Lars Schmidt-Thieme. Sequential model-free hyperparameter tuning. In 2015 IEEE International Conference on Data Mining, ICDM 2015, Atlantic City, NJ, USA, November 14-17, 2015, pp. 1033–1038, 2015b. doi: 10.1109/ICDM.2015.20. URL https://doi.org/10.1109/ICDM.2015.20. ",
+        "bbox": [
+            174,
+            867,
+            823,
+            924
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Martin Wistuba, Nicolas Schilling, and Lars Schmidt-Thieme. Two-stage transfer surrogate model for automatic hyperparameter optimization. In Machine Learning and Knowledge Discovery in Databases - European Conference, ECML PKDD 2016, Riva del Garda, Italy, September 19- 23, 2016, Proceedings, Part I, pp. 199–214, 2016. doi: 10.1007/978-3-319-46128-1\\ 13. URL https://doi.org/10.1007/978-3-319-46128-1_13. ",
+        "bbox": [
+            174,
+            103,
+            825,
+            174
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Martin Wistuba, Nicolas Schilling, and Lars Schmidt-Thieme. Scalable gaussian process-based transfer surrogates for hyperparameter optimization. Mach. Learn., 107(1):43–78, 2018. doi: 10. 1007/s10994-017-5684-y. URL https://doi.org/10.1007/s10994-017-5684-y. ",
+        "bbox": [
+            174,
+            183,
+            825,
+            224
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Dani Yogatama and Gideon Mann. Efficient transfer learning method for automatic hyperparameter tuning. In Proceedings of the Seventeenth International Conference on Artificial Intelligence and Statistics, AISTATS 2014, Reykjavik, Iceland, April 22-25, 2014, pp. 1077–1085, 2014. URL http://proceedings.mlr.press/v33/yogatama14.html. ",
+        "bbox": [
+            174,
+            234,
+            825,
+            290
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "image",
+        "img_path": "images/52bd45ca2cd10392be7bcf35ae3e85883edee009bb9a2fedb85da9ccb2823033.jpg",
+        "image_caption": [
+            "Figure 5: FSBO is the best of all considered methods. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            176,
+            101,
+            823,
+            272
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "A HYPERPARAMETER OPTIMIZATION ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            332,
+            501,
+            347
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "In the main paper we report the normalized regret after every 33 trials of Bayesian optimization in a table. This allows us to also report statistical significance. In Figure 5 we show results for all 100 trials. The conclusions remain unchanged. FSBO provides consistently the best results. ",
+        "bbox": [
+            174,
+            363,
+            825,
+            405
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "B CHALLENGES WITH METABO ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            425,
+            459,
+            441
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "The results reported in the main paper for MetaBO seem not to align with the numbers reported by Volpp et al. (2020), in particular for AdaBoost. In order to understand this behavior, we conducted a deeper analysis for the AdaBoost metadata set which was used in the evaluation of Volpp et al. (2020) as well. Using the authors’ code2, we executed the same scripts used by the authors to report their results for $T = 1 5$ trials. Furthermore, we changed two lines in the scripts to train MetaBO for $T = 5 0$ , the number of trials considered for AdaBoost in our experiment. We report the outcome of these two experiments in Figure 6. In the two left plots we show how the learned policies for $T = 1 5$ and $T = 5 0$ perform on validation and test during training. The authors’ default setting for the total number of iterations is 2000. However, we noticed a sudden drop in regret for the setting $T = 5 0$ around 2000 iterations. For that reason we assumed that further training might help to further improve MetaBO and increased it to 5000 iterations. As can be seen in the middle plot, that was not the case. We observe that the policy learned for $T = 1 5$ has a lower regret than the one for $T = 5 0$ even though it is only trying 15 compared to 50 trials. In the right plot we show the results compared to a simple random search baseline. The results for $T = 1 5$ are in line with those reported by Volpp et al. (2020). We contacted the authors and they confirmed that we use their code correctly. Their explanation is that the increase of episode length makes this a harder problem for Reinforcement Learning. This results in worse results for $T = 5 0$ and even worse for $T = 1 0 0$ in case of the GLMNet and SVM problem. They proposed to randomly vary $T$ between 5 and 50 during the training phase. We also considered this training protocol and report the results in Figure 6. This improves over training only with $T = 5 0$ but does not improve over a simple random search. ",
+        "bbox": [
+            173,
+            457,
+            825,
+            750
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "C METADATA ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            768,
+            302,
+            785
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "We created the GLMNet and SVM metadata using OpenML. We chose the 30 OpenML flows with the most observations. We also limited ourselves to the uploaded results from user with ID 2702 (OpenML Bot R), who uploaded the majority of all runs, to ensure that the metadata was generated under similar circumstances. In some cases the choice of a single hyperparameter is not indicated. In such a case, we assume that the default value was used. The GLMNet metadata has two continous hyperparameters: the elastic-net mixing parameter $\\alpha \\in [ 0 , 1 ]$ and the regularization parameter $\\lambda \\in \\mathsf { [ \\bar { 2 } ^ { - 1 0 } , 2 ^ { 1 0 } ] }$ . The SVM metadata has two mandatory hyperparameters and two conditional ",
+        "bbox": [
+            174,
+            800,
+            825,
+            898
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "image",
+        "img_path": "images/efd123033581afd4b192793e0edc0650c43c8423b0076c5eea25fd6bb98bdfd5.jpg",
+        "image_caption": [
+            "Figure 6: Left two plots: Training MetaBO for different number of trials $T$ . Right plot: comparing the final policy against a random search. Results for $T = 1 5$ match the results reported by Volpp et al. (2020). "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            176,
+            99,
+            821,
+            282
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "table",
+        "img_path": "images/215dc1f312d89560bc3fef0a5849998fc04ec39164d15f8f6f3b5d97cb0f0e4f.jpg",
+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Method</td><td colspan=\"3\">AdaBoost</td><td colspan=\"3\">GLMNet</td><td colspan=\"3\">SVM</td></tr><tr><td></td><td>15</td><td>33</td><td>50</td><td>33</td><td>67</td><td>100</td><td>33</td><td>67</td><td>100</td></tr><tr><td>FSBO</td><td>3.10</td><td>1.13</td><td>0.80</td><td>0.42</td><td>0.22</td><td>0.16</td><td>0.79</td><td>0.51</td><td>0.36</td></tr><tr><td>FSBO (REPTILE)</td><td>3.80</td><td>2.13</td><td>1.21</td><td>0.36</td><td>0.19</td><td>0.14</td><td>1.48</td><td>0.89</td><td>0.72</td></tr></table>",
+        "bbox": [
+            215,
+            358,
+            779,
+            425
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "Table 2: Comparing different model-agnostic meta-learning approaches. Again, the best results are in bold and results that are not significantly worse than the best are in italics. ",
+        "bbox": [
+            173,
+            441,
+            823,
+            469
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "hyperparameters. The kernel (linear, polynomial or RBF) and the continuous trade-off parameter $\\dot { C } \\in \\mathsf { [ 2 ^ { - 1 0 } , 2 ^ { 1 0 } ] }$ must always be selected. The degree $d \\in \\{ 2 , 3 , 4 , 5 \\}$ is only considered for the polynomial kernel and the bandwidth $\\gamma \\in [ 2 ^ { - 1 0 } , 2 ^ { 1 \\overline { { 0 } } } ]$ is only considered for the RBF kernel. ",
+        "bbox": [
+            174,
+            496,
+            825,
+            539
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "We further use the AdaBoost metadata which has been used to evaluate MetaBO (Volpp et al., 2020). According to Wistuba et al. (2015b), this metadata was created using AdaBoost (Kegl & ´ Busa-Fekete, 2009) with decision products as weak learners. The authors designed a grid over the two continuous hyperparameters: the number of iterations and the number of product terms. The number of iterations was chosen from $\\{ 2 , 5 , 1 0 , 2 0 , 5 0 , 1 0 0 , 2 0 0 , 5 0 0 , 1 0 ^ { 3 } , 2 \\cdot 1 0 ^ { 3 } , 5 \\cdot 1 0 ^ { 3 } , 1 0 ^ { 4 } \\}$ , the number of product term was chosen from $\\{ 2 , 3 , 4 , 5 , 7 , 1 0 , 1 5 , 2 0 , 3 0 \\}$ . ",
+        "bbox": [
+            174,
+            545,
+            825,
+            628
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "The metadata was created by running a grid search on 50 different classification data sets, using classification accuracy as an objective. Compared to the other two metadata sets, this one is significantly smaller. Its use is mainly motivated by comparing to MetaBO under the same circumstances as used to evaluate MetaBO. For that reason, we do not use the leave-one-data-set-out cross-validation protocol but instead use the same fixed split created by Volpp et al. (2020). ",
+        "bbox": [
+            174,
+            635,
+            825,
+            705
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "Statistics about these metadata sets are provided in Table 3. ",
+        "bbox": [
+            174,
+            712,
+            562,
+            727
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "D META-LEARNING WITH REPTILE ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            746,
+            483,
+            762
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "In principle our few-shot surrogate can be combined with any model-agnostic meta-learning approach. In this section we compare our proposed meta-learning approach against Reptile (Nichol et al., 2018), a first-order approximation of MAML. We use the same hyperparameters and data augmentation as described in Algorithm 1. Reptile introduces a new hyperparameter, an outer learning rate. We set it to 0.1 and decay it linearly to 0. As summarized in Table 2, FSBO with the meta-learning approach described in Algorithm 1 is either better or not significantly worse than its variation with Reptile. ",
+        "bbox": [
+            173,
+            779,
+            825,
+            876
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "table",
+        "img_path": "images/3033dab78bcc89a32ce1b5a5d71e93b877d5e18f6f323018d2ff4be2c7bc9459.jpg",
+        "table_caption": [
+            "Table 3: Metadata set statistics. OpenML ID refers to a task on openml.org. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=\"2\">AdaBoost</td><td colspan=\"2\">GLMNet</td><td colspan=\"3\">SVM</td></tr><tr><td>RUNS</td><td>OPENML ID</td><td>RUNS</td><td>OPENML ID</td><td>RUNS</td></tr><tr><td>DATA SET</td><td></td><td></td><td></td><td></td><td>2429</td></tr><tr><td>A9A W8A</td><td>108 108</td><td>3 31</td><td>12796 60721</td><td>37 3485</td><td>1000</td></tr><tr><td></td><td>108</td><td>37</td><td>17950</td><td>3492</td><td>3217</td></tr><tr><td>ABALONE</td><td></td><td>219</td><td>13963</td><td>3493</td><td>1828</td></tr><tr><td>APPENDICITIS</td><td>108 108</td><td></td><td>16968</td><td>3494</td><td>1854</td></tr><tr><td>AUSTRALIAN AUTOMOBILE</td><td>108</td><td>3492 3493</td><td>19931</td><td>3889</td><td>1951</td></tr><tr><td>BANANA</td><td>108</td><td>3889</td><td>13942</td><td>3891</td><td>3083</td></tr><tr><td>BANDS</td><td>108</td><td>3891</td><td>13917</td><td>3899</td><td>2487</td></tr><tr><td>BREAST-CANCER</td><td>108</td><td>3899</td><td>15954</td><td>3902</td><td>1105</td></tr><tr><td>BUPA</td><td>108</td><td>3903</td><td>39155</td><td>3903</td><td>1018</td></tr><tr><td>CAR</td><td>108</td><td>3913</td><td>22949</td><td>3913</td><td>1448</td></tr><tr><td>CHESS</td><td>108</td><td>3917</td><td>16960</td><td>3918</td><td>1459</td></tr><tr><td>COD-RNA</td><td>108</td><td>3918</td><td>20984</td><td>3950</td><td>1998</td></tr><tr><td>COIL2000</td><td>108</td><td>9914</td><td>37635</td><td>6566</td><td>2702</td></tr><tr><td>COLON-CANCER</td><td>108</td><td>9946</td><td>25532</td><td>9889</td><td>2485</td></tr><tr><td>CRX</td><td>108</td><td>9952</td><td>25854</td><td>9914</td><td>1498</td></tr><tr><td>DIABETES</td><td>108</td><td>9967</td><td>21994</td><td>9946</td><td>1291</td></tr><tr><td>ECOLI</td><td>108</td><td>9978</td><td>19956</td><td>9952</td><td>2483</td></tr><tr><td>GERMAN-NUMER</td><td>108</td><td>9980</td><td>31961</td><td>9967</td><td>1492</td></tr><tr><td>HABERMAN</td><td>108</td><td>9983</td><td>37134</td><td>9971</td><td>1381</td></tr><tr><td>HOUSEVOTES</td><td>108</td><td>10101</td><td>66277</td><td>9976</td><td>1473</td></tr><tr><td>IJCNN1</td><td>108</td><td>125923</td><td>35555</td><td>9978</td><td>1503</td></tr><tr><td>KR-VS-K</td><td>108</td><td>145847</td><td>29927</td><td>9980</td><td>2943</td></tr><tr><td>LED7DIGIT</td><td>108</td><td>145857</td><td>31180</td><td>9983</td><td>987</td></tr><tr><td>LETTER</td><td>108</td><td>145862</td><td>12954</td><td>10101</td><td>1871</td></tr><tr><td>LYMPHOGRAPHY</td><td>108</td><td>145872</td><td>17835</td><td>14951</td><td>1765</td></tr><tr><td>MAGIC</td><td>108</td><td>145953</td><td>47439</td><td>34536</td><td>1499</td></tr><tr><td>MONK-2</td><td>108</td><td>145972</td><td>19972</td><td>34537</td><td>1489</td></tr><tr><td>PENDIGITS</td><td>108</td><td>145979</td><td>12979</td><td>145878</td><td>1244</td></tr><tr><td>PHONEME</td><td>108</td><td>146064</td><td>43785</td><td>146064</td><td>1435</td></tr><tr><td>PIMA</td><td>108</td><td></td><td></td><td></td><td></td></tr><tr><td>RING</td><td>108</td><td></td><td></td><td></td><td></td></tr><tr><td>SAHEART</td><td>108</td><td></td><td></td><td></td><td></td></tr><tr><td>SEGMENT</td><td>108</td><td></td><td></td><td></td><td></td></tr><tr><td>SEISMIC</td><td>108</td><td></td><td></td><td></td><td></td></tr><tr><td>SHUTTLE</td><td>108</td><td></td><td></td><td></td><td></td></tr><tr><td>SONAR-SCALE</td><td>108</td><td></td><td></td><td></td><td></td></tr><tr><td>SPAMBASE</td><td>108</td><td></td><td></td><td></td><td></td></tr><tr><td>SPECTFHEART</td><td>108</td><td></td><td></td><td></td><td></td></tr><tr><td>SPLICE</td><td>108</td><td></td><td></td><td></td><td></td></tr><tr><td>TIC-TAC-TOE</td><td>108</td><td></td><td></td><td></td><td></td></tr><tr><td>TITANIC</td><td>108</td><td></td><td></td><td></td><td></td></tr><tr><td>TWONORM</td><td>108</td><td></td><td></td><td></td><td></td></tr><tr><td>USPS</td><td>108</td><td></td><td></td><td></td><td></td></tr><tr><td>VEHICLE</td><td>108</td><td></td><td></td><td></td><td></td></tr><tr><td>WDBC</td><td>108</td><td></td><td></td><td></td><td></td></tr><tr><td>WINE</td><td>108</td><td></td><td></td><td></td><td></td></tr><tr><td>WINEQUALITY-RED</td><td>108</td><td></td><td></td><td></td><td></td></tr><tr><td>WISCONSIN</td><td>108</td><td></td><td></td><td></td><td></td></tr><tr><td>YEAST</td><td>108 5400</td><td>TOTAL</td><td>804159</td><td>TOTAL</td><td></td></tr><tr><td>TOTAL</td><td></td><td></td><td></td><td></td><td>54418</td></tr></table>",
+        "bbox": [
+            232,
+            151,
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+            854
+        ],
+        "page_idx": 14
+    }
+]

parse/train/rJeqeCEtvH/rJeqeCEtvH.md

@@ -0,0 +1,299 @@
+# SEMI-SUPERVISED GENERATIVE MODELING FOR CONTROLLABLE SPEECH SYNTHESIS
+
+Raza Habib1∗ Soroosh Mariooryad2 Matt Shannon2 Eric Battenberg2 RJ Skerry-Ryan2   
+Daisy Stanton2 David Kao2 Tom Bagby2
+
+1University College London (UCL) 2Google Research.
+
+raza.habib@cs.ucl.ac.uk {soroosh, mattshannon, ebattenberg, rjryan, daisy, davidkao, tombagby}@google.com
+
+# ABSTRACT
+
+We present a novel generative model that combines state-of-the-art neural textto-speech (TTS) with semi-supervised probabilistic latent variable models. By providing partial supervision to some of the latent variables, we are able to force them to take on consistent and interpretable purposes, which previously hasn’t been possible with purely unsupervised TTS models. We demonstrate that our model is able to reliably discover and control important but rarely labelled attributes of speech, such as affect and speaking rate, with as little as 30 minutes supervision. Even at such low supervision levels we do not observe a degradation of synthesis quality compared to a state-of-the-art baseline. Audio samples are available on the web1.
+
+# 1 INTRODUCTION
+
+The ability to reliably control high level attributes of speech, such as emotional expression (affect) or speaking rate, is often desirable in speech synthesis applications. Achieving this control however is made difficult by the necessity of acquiring a large quantity of high quality labels. In this paper we show that semi-supervised latent variable models can take us a significant step closer towards solving this problem.
+
+Combining state-of-the-art neural text-to-speech (TTS) systems with probabilistic latent variable models provides a natural framework for discovering aspects of speech that are rarely labelled or even difficult to describe. Both inferring the latent prosody and generating samples with sufficient variety requires reasoning about uncertainty and is thus a natural fit for deep generative models.
+
+There has been recent progress in applying stochastic gradient variational Bayes (SGVB) (Kingma & Welling, 2013; Rezende et al., 2014) to training probabilistic neural TTS models. Battenberg et al. (2019) and Hsu et al. (2018) have shown that it is possible to use latent variable models to discover features such as speaking style, speaking rate, arousal, gender and even the quality of the recording environment.
+
+However, these models are formally non-identifiable (Hyvarinen & Pajunen, 1999; Locatello et al., ¨ 2019) and this implies that repeated training runs will not reliably discover the same latent attributes. Even if they did, a lengthy human post-processing stage is necessary to identify what the model has learned on any given training run. In order to be of practical use for control, it is not enough for the models to discover latent attributes, they need to do so reliably and in a way that is robust to random initialization and to changes in the model. We demonstrate that the addition of even modest amounts of supervision can be sufficient to achieve this reliability.
+
+By augmenting state-of-the art neural TTS with semi-supervised deep generative models within the VAE framework (Kingma et al., 2014; Narayanaswamy et al., 2017), we show that it is possible to not only discover latent attributes of speech but to do so in a reliable and controllable manner. In particular we are able to achieve reliable control over affect, speaking rate and F0 variation (F0 is the fundamental frequency of oscillation of the vocal folds). Further, we provide demonstrations that it is possible to transfer controllability to speakers for whom we have no labels. Our core contributions are:
+
+![](images/59e6cf3a31971d9ed7b0556a2024cd69e20dde3e13fdf4de40afa3963a643163.jpg)  
+Figure 1: Schematic showing how we parameterize the conditional likelihood $p ( x | y , z _ { u } , z _ { s } )$ . Left: A block of 1-D convolutions and RNNs originally introduced by Wang et al. (2017) and described in detail in the appendix. Right: Schematic of the sequence-to-sequence network that outputs the means of our auto-regressive distribution. At each decoder time step, the network outputs the means for the next two spectrogram frames.
+
+• To combine semi-supervised latent variable models with Neural TTS systems, producing a system that can reliably discover attributes of speech we wish to control. • To demonstrate that as little as 30 minutes supervision can be sufficient to improve prosody and allow control over speaking rate, fundamental frequency (F0) variation and affect, a problem of interest to the speech community for well over two decades (Schroder, 2001). ¨ • To imbue TTS models with control over affect, F0 and speaking rate whilst still maintaining prosodic variation when sampling.
+
+# 2 GENERATIVE MODEL
+
+Our generative model, shown in figures 1 and 2a, consists of an autoregressive distribution over a sequence of acoustic features, $x _ { 1 \ldots t }$ , that are generated conditioned on a sequence of text, $y _ { 1 \ldots k }$ , and on two latent variables, $z _ { u }$ and $z _ { s }$ . The latent variables can be discrete or continuous. $z _ { s }$ represents the variations in prosody that we seek to control and is semi-supervised. $z _ { u }$ is fully unobserved and represents latent variations in prosody (intonation, rhythm, stress) that we wish to model but not explicitly control. Once trained, our model can be used to synthesize acoustic features from text. Similar to Tacotron 2 (Shen et al., 2018), we then generate waveforms by training a second network such as WaveNet (van den Oord et al., 2016) or WaveRNN (Kalchbrenner et al., 2018) to act as a vocoder. In our case we use WaveRNN.
+
+We parameterize our likelihood $p ( x _ { 1 \ldots t } | y _ { 1 \ldots k } , z _ { u } , z _ { s } , \theta )$ by a sequence-to-sequence neural network with attention (Shen et al., 2018; Graves, 2013; Bahdanau et al., 2014) that is shown schematically in figure 1. Details largely follow Tacotron (Wang et al., 2017) and are given in appendix A. At each time step we model a mel-spectrogram frame with a fixed variance isotropic Laplace distribution whose mean is output by the neural network. We condition each of the latent variables by concatenating the vectors $z _ { u }$ and $z _ { s }$ to the representation of the text-encoder, before the application of the attention mechanism. In the case of continuous $z$ we use a standard normal prior and in the case of discrete $z$ we use a uniform categorical prior with one-hot encoding.
+
+![](images/6af43fa6423e9f76d05ad6f4fd5b83fd36082ecc584f1c386911a0ead48a50ef.jpg)  
+Figure 2: Left: The graphical model showing the conditional independence assumptions between each of the stochastic variables. Centre: The structure of the variational distribution used to approximate the posterior for fully unsupervised data points and Right: supervised points.
+
+# 2.1 SEMI-SUPERVISED TRAINING
+
+Following Kingma et al. (2014); Narayanaswamy et al. (2017), we train our model via stochastic gradient variational Bayes (SGVB). That is we approximately maximize the log-likelihood of our training data by maximizing a variational lower bound using stochastic gradient ascent. Since we are training with semi-supervision we in fact need two lower bounds: one for the data points for which $z _ { s }$ is observed; one for the case where $z _ { s }$ is unobserved. In our models the fully latent variable $z _ { u }$ is always continuous but the semi-supervised latent $z _ { s }$ can be continuous or discrete. The conditional independence structure of our variational distributions is shown in figures 2b and 2c. On supervised data, the per-datapoint bound is:
+
+$$
+\begin{array} { r l } & { \log p ( x , z _ { s } | y , \theta ) = \log \int p ( x , z _ { u } , z _ { s } | y , \theta ) \mathrm { d } z _ { u } } \\ & { \qquad \geq \mathbb { E } _ { q ( z _ { u } | x , y , z _ { s } , \phi ) } \left[ \log \left( \frac { p ( x | y , z _ { u } , z _ { s } , \theta ) p ( z _ { u } ) p ( z _ { s } ) } { q ( z _ { u } | x , y , z _ { s } , \phi ) } \right) \right] } \\ & { \qquad = \mathbb { E } _ { q ( z _ { u } | x , y , z _ { s } , \phi ) } \left[ \log p ( x | y , z _ { u } , z _ { s } , \theta ) \right] + \log p ( z _ { s } ) - D _ { \mathrm { K L } } ( q ( z _ { u } | x , y , z _ { s } , \phi ) \| p ( z _ { u } ) ) } \\ & { \qquad = \mathcal { L } _ { s } ( \theta , \phi ; x , y , z _ { s } ) } \end{array}
+$$
+
+Where $q ( z _ { u } | x , y , z _ { s } , \phi )$ is a parametric variational distribution introduced to approximately marginalize $z _ { u }$ . $\theta$ are the parameters of the generative model and $\phi$ are the parameters of the variational distributions. The intractable integrals are approximated with reparameterized samples. For the cases where $z _ { s }$ is unobserved and discrete, the bound is:
+
+$$
+\begin{array} { l } { \log p ( x | y , \theta ) = \log \displaystyle \int \displaystyle \sum _ { z _ { s } } p ( x , z _ { u } , z _ { s } | y , \theta ) \mathrm { d } z _ { u } } \\ { \displaystyle \qquad \geq \sum _ { z _ { s } } q ( z _ { s } | x , y , \phi ) \mathcal { L } _ { s } ( \theta , \phi ; x , y , z _ { s } ) + H ( q ( z _ { s } | x , y , \phi ) ) } \\ { \displaystyle \qquad = \mathcal { L } _ { u } ( \theta , \phi ; x , y ) } \end{array}
+$$
+
+and when $z _ { s }$ is continuous we replace the sum above with an integral and again approximate with reparameterized samples. The variational distributions are parameterized by a neural network that takes as input the text, spectrograms and other conditioning variables and outputs the parameters of the distribution. The exact structure of this network is given in appendix A. We have implicitly assumed that $q ( z _ { u } , z _ { s } | x , y , \phi )$ may be factorized as $q ( z _ { u } , z _ { s } | x , y , \phi ) = q ( z _ { u } | x , y , z _ { s } , \phi ) q ( z _ { s } | x , y , \phi )$ with shared parameters between these two distributions (see appendix A). Optimizing the variational objective with respect to the parameters $\phi$ encourages the variational distributions to match the posterior of the generative model $p ( z _ { u } , z _ { s } | x , y , \theta )$ . Unlike previous work (Hsu et al., 2018), we do not assume that the posterior on the latents is independent of the text, as this dependence likely exists in the model due to explaining away. That is to say that although the text and the latents are independent in our prior, observing the spectrogram correlates them in the posterior because they
+
+![](images/b9b30b9a2b861960a061a036bd85ea3a7fb8c4eba78836b734e5f9854b94822c.jpg)  
+Figure 3: The circumplex model of emotion. Each possible emotion is represented in a 2 dimensional plane consisting of an arousal dimension and valence dimension. This figure is borrowed from Munoz-de Escalona & Canas (2017).
+
+both explain variation in the spectrogram. This has been shown to be significant by Battenberg et al.   
+(2019).
+
+If we define:
+
+$$
+\tilde { q } ( z _ { s } | x , y ) = \left\{ \begin{array} { l l } { q ( z _ { s } | x , y , \phi ) } & { \mathrm { i f ~ u n s u p e r v i s e d } } \\ { \gamma \delta ( z _ { s } - z _ { s _ { \mathrm { o b s e r v e d } } } ) } & { \mathrm { i f ~ s u p e r v i s e d } } \end{array} \right.
+$$
+
+then we can write the overall objective over both the supervised and unsupervised points succinctly $\mathrm { a s } ^ { 2 }$ :
+
+$$
+\mathcal { L } ( \theta , \phi ) = \mathbb { E } _ { x , y } \left[ \sum _ { z _ { s } } \tilde { q } ( z _ { s } | x , y , \phi ) \mathcal { L } _ { s } ( \theta , \phi ; x , y , z _ { s } ) + H ( \tilde { q } ( z _ { s } | x , y , \phi ) ) \right]
+$$
+
+where summation would again be replaced by integration for continuous $z _ { s }$ and $\gamma$ (shown in equation 4) is a weighting factor that pre-multiplies the loss for any supervised point. This weighting was also used in previous work such as Narayanaswamy et al. (2017), who showed it to be beneficial at very low levels of supervision.
+
+Writing the objective in this form allows an intuitive interpretation for the semi-supervised training procedure. When supervision is provided, our objective function is evaluated at the observed value of $z _ { s }$ . When supervision is not provided, we evaluate the objective function for every possible value of $z _ { s }$ and take a (potentially infinite for continuous $z _ { s }$ ) weighted average. The weighting in the average is given by $q ( \boldsymbol { z } _ { s } | \boldsymbol { x } , \boldsymbol { y } , \phi )$ , which is simultaneously trained to approximate the posterior $p ( z _ { s } | x , y , \theta )$ . In other words, on unsupervised utterances, we evaluate our objective for each possible value of the latent attribute and weight by the (approximate) posterior probability that this value of the latent was responsible for generating the utterance.
+
+As $q ( \boldsymbol { z } _ { s } | \boldsymbol { x } , \boldsymbol { y } , \phi )$ is trained to approximate $p ( z _ { s } | x , y , \theta )$ we can expect it to become a reasonable classifier/regressor for the semi-supervised latent attribute as the model improves. For example when $z _ { s }$ represents an affect label, $p ( z _ { s } | x , y , \theta )$ is the posterior probability, of the model, over affect given text and speech. By taking the most likely posterior class, this distribution can be used as an affect classifier. However, this variational distribution is only trained on unsupervised training points and so does not benefit directly from the supervised data. To overcome this problem we follow Kingma et al. (2014) and add a classification loss to our objective. The overall objective becomes:
+
+$$
+\mathcal { L } _ { \mathrm { t o t a l } } ( \theta , \phi ) = \mathcal { L } ( \theta , \phi ) + \alpha \mathbb { E } _ { x , y , z _ { s } } [ \log q ( z _ { s } | x , y , \phi ) ]
+$$
+
+Where $\alpha$ is a hyperparameter which adjusts the contribution of this term.
+
+# 3 DATA
+
+We have used a proprietary high quality labeled data-set of 40 English speakers. The training set consists of 72,405 utterances with durations of at most 5 seconds (45 hours). The validation and test sets each contain 745 utterances or roughly 30 minutes of data. We vary the amount of supervision in the experiments below. We also experimented with transferring controllability to a fully unlabeled data-set of audiobook recordings by Catherine Byers (the speaker from the 2013 Blizzard Challenge), which exhibits high variation in affect and prosody and to other speakers who were less expressive. We strongly encourage the reader to listen to the synthesized samples on our demo page3.
+
+In this work we chose to focus on learning to control affect with a discrete representation, as well as speaking rate and F0 variation with a continuous representation, as these are challenging aspects of prosody to control. Our method could be applied to other factors without modification.
+
+# 3.1 AFFECT CONTROL
+
+The best way to represent emotion is an actively researched area and many models of affect exist. In this work we chose to follow the circumplex model of emotion (Russell, 1980) which posits that most affective states can be represented in a 2 dimensional plane with one axis representing arousal and the other axis representing valence. Arousal measures the level of excitement or energy and valence measures positivity or negativity. Figure 3, shows a chart of emotions plotted in the arousalvalence plane where we can see that, for example, high arousal and high valence corresponds to joy or happiness whereas high arousal and low valence might correspond to anger or frustration.
+
+Our data-set was recorded under studio conditions with trained voice actors who were prompted to read dialogues in one of three valences:-2, -1, $+ 2$ and two arousal values:-2 (low), $+ 2$ (high). This was achieved by prompting the actors to read dialogues in either a happy, sad or angry voice at two levels of arousal. This results in 6 possible affective states which we chose to model as discrete and use as our supervision labels.
+
+# 3.2 SPEAKING RATE AND F0 VARIATION CONTROL
+
+In order to demonstrate that we can control continuous attributes we also created approximate realvalued labels for speaking rate and arousal for all of our data. We generate the approximate speaking rate as number of syllables per second in each utterance. F0, also known as the fundamental frequency, measures the frequency of vibration of the vocal folds during voiced sounds. Variation in F0 is highly correlated with arousal and roughly measures how expressive an utterance is. To create approximate arousal labels we extracted the F0 contour from each of our utterances, using the YIN algorithm (De Cheveigne & Kawahara, 2002), and measured its standard deviation. We then per- ´ formed a whitening transform on these two approximate labels in order to match it to our standard normal prior.
+
+These artificial labels would of course be cheap to obtain for the entire data-set and would not justify the use of semi-supervision in real applications. But, our objective here is to evaluate/demonstrate the efficacy of semi-supervision rather than to specifically control a particular attribute. We have chosen syllable rate and F0 standard deviation, because they both correspond to subjectively distinct variations of interest, and they are more easily quantifiable than affect and so provide strong evidence of controllability. For the continuous latents we are not only able to interpolate speaking-rates and F0 variations but also to extrapolate outside of our training data. We provide examples on our demo page of samples with significantly greater/lower speed and F0 variation than typically observed in natural speech.
+
+# 4 EXPERIMENTS AND RESULTS
+
+To evaluate the efficacy of semi-supervised latent variable models for controllable TTS we trained the model described in section 2 on the above data-sets at varying levels of supervision as well as for
+
+<table><tr><td rowspan="2" colspan="2"></td><td colspan="3">Valence</td><td rowspan="2">Arousal low vs. high</td></tr><tr><td>baseline vs.angry</td><td>baseline vs. sad</td><td>baseline vs. happy</td></tr><tr><td rowspan="3">preference score</td><td>27 min (1%)</td><td>-0.20± 0.10</td><td>-0.60± 0.08</td><td>-0.43 ± 0.09</td><td>-0.50 ± 0.09</td></tr><tr><td>135 min (5%)</td><td>-0.74 ± 0.07</td><td>-0.83 ± 0.06</td><td>-0.83 ± 0.06</td><td>-0.57± 0.08</td></tr><tr><td>270 min (10%)</td><td>-0.71 ± 0.07</td><td>-0.86 ± -0.95</td><td>-0.61 ± 0.08</td><td>-0.59 ± 0.08</td></tr></table>
+
+Table 1: Subjective metrics for affect control. Negative is a preference for the controlled model. $+ 1$ and $^ { - 1 }$ indicate a preference for samples A and B, respectively. For valence, raters are told that a sample is intended to convey a particular emotion, e.g. happy, and then presented with sample from baseline without control (A), and controlled model (B), and asked to choose between them. For arousal, raters are told to choose the sample that is more vocally aroused, and presented with controlled samples in low (A) and high (B) arousal. To avoid bias, the orders are randomly altered during rating. We show preference score and $9 5 \%$ confidence intervals at multiple supervision levels.
+
+varying settings of the hyperparameters: $\alpha$ which controls the supervision loss and $\gamma$ , which over emphasizes supervised training points. We found that a value of $\alpha = 1$ was optimal for the discrete experiments and $\alpha = 0$ for the continuous experiments, which corresponds to simply optimizing the ELBO. For each experiment we report the results for the best $\gamma$ found, and $\gamma = 1$ . $\gamma = 1$ corresponds to experiments with no over-weighting of the supervised points. All models were trained using the ADAM optimizer with learning rate of $1 0 ^ { - 3 }$ and run for 300, 000 training steps with a batch size of 256, distributed across 32 Google Cloud TPU chips. All models were implemented using tensorflow 1 (Abadi et al., 2016).
+
+Assessing the degree of control is challenging as interpreting affect is subjective. We used two objective metrics of control as well as subjective evaluation from human raters and a third objective metric of overall quality. For affect, the first objective metric we introduced was the test-set accuracy of a 6-class affect classifier trained on the ground truth training data and applied to generated samples from the model (shown in figure 4a). The classifier is a convolutional neural network whose structure mirrors the posterior network $q ( \boldsymbol { z } _ { s } | \boldsymbol { x } , \boldsymbol { y } , \phi )$ and its exact architecture is given in appendix A. We also provide subjective metrics of controllability, shown in table 1. For speaking rate control, we measure the mean syllable rate error on a held out test-set. The syllable rate error is calculated as the absolute difference in syllable rate between the desired syllable rate and that measured from the synthesized sample. We calculate an analogous error rate for F0 variation.
+
+Whilst the two metrics above measure controllability they don’t tell us if this comes at the expense of a degradation in synthesis quality. To probe quality we use three further metrics. The first was MelCepstral-Distortion-Dynamic-Time-Warping (MCD-DTW) (Kubichek, 1993) on a held out test-set, shown in figure 4d. MCD-DTW is a measure of the difference between the ground-truth spectrogram and the synthesized mel spectrogram that is known to correlate well with human perception (Kubichek, 1993). The second metric of quality was crowd sourced mean-opinion-scores (MOS). The third metric of quality is speech recognition word error rate (WER) and character error rate (CER) on audio samples. The MOS and speech recognition results are summarized in table 2.
+
+To demonstrate that semi-supervision by including unlabelled data is beneficial, we also provide MOS and speech recognition errors for fully supervised subsets of the data in table 2. These show that at least close to 5 hours of data is required to train a reasonable quality TTS model, far above the 30 minutes supervision needed to control prosodic aspects of speech.
+
+We provide further details of all of these metrics in the appendix B, and sample spectrograms are provided in appendix C.
+
+# 5 DISCUSSION
+
+The classification accuracy (see figure 4a), subjective metrics (see table 1) and error-rate results (see figure 4b-4c) provide a clear demonstration that using semi-supervised latent variables, we are able to achieve control of both continuous and discrete attributes of speech. There is not a significant degradation in the overall quality and this is evidenced by the mean opinion scores which are above the baseline, Tacotron, and also speech recognition errors (see table 2). We also include a baseline of our Tacotron model augmented only by the unsupervised latent $z _ { s }$ , to aid comparison.
+
+![](images/0fae0afd533b1326cdaeaec9b47571daf6fdeeb2657b06fa42da5fa889699f75.jpg)  
+Figure 4: Objective evaluation metrics as a function of supervision fraction. $1 0 0 \%$ supervision corresponds to 45 hours of supervised training data and $0 \%$ supervision corresponds to base tacotron. For MCD-DTW and error-rates lower is better.
+
+<table><tr><td rowspan="2" colspan="4"></td><td colspan="3">Semi-Supervised (1O% supervision)</td></tr><tr><td colspan="2">continuous latent</td><td>discrete latent</td></tr><tr><td></td><td>ground truth</td><td>baseline</td><td>baseline with zu</td><td>F0</td><td>speaking-rate</td><td>affect</td></tr><tr><td>MOS</td><td>4.52±0.07</td><td>4.09±0.09</td><td>4.24±0.08</td><td>4.28±0.07</td><td>4.16±0.08</td><td>4.17±0.09</td></tr><tr><td>WER</td><td>4.49</td><td>4.93</td><td>4.93</td><td>4.06</td><td>2.53</td><td>6.09</td></tr><tr><td>CER</td><td>2.11</td><td>2.38</td><td>2.22</td><td>1.80</td><td>1.14</td><td>3.12</td></tr></table>
+
+Table 2: Metrics of overall quality: Mean Opinion Scores (MOS) alongside $9 5 \%$ confidence intervals, speech recognition word error rate (WER), and chararacter error rate (CER). The results show no degradation in performance compared to the baseline.   
+
+<table><tr><td></td><td>27 min (1%)</td><td>54 min (2%)</td><td>108 min (4%)</td><td>135 min (5%)</td><td>270 min (10%)</td><td>45 hours (100%)</td></tr><tr><td>MOS</td><td>unintelligible</td><td>unintelligible</td><td>3.20±0.13</td><td>3.52±0.11</td><td>4.03±0.08</td><td>4.08±0.09</td></tr><tr><td>WER</td><td>91.95</td><td>96.31</td><td>19.83</td><td>7.55</td><td>5.56</td><td>4.93</td></tr><tr><td>CER</td><td>74.57</td><td>78.9</td><td>12.54</td><td>4.46</td><td>2.79</td><td>2.22</td></tr></table>
+
+Table 3: Metrics of overall quality for fully supervised data at varying data-set sizes, showing significant degradation below 270 minutes.
+
+The MCD-DTW scores for F0 variation and affect are improved at all levels of supervision (figure 4d). Whilst the MCD-DTW is degraded for speaking rate, this is likely a misleading metric when targeting changes in timing as the dynamic-time-warping component of MCD-DTW changes exactly the aspect we wish to control. For speaking rate the combination of MOS and samples is a better indication of the overall quality. We are able to reduce the supervision level to levels as low as $1 \%$ or 30 minutes and still have a significant degree of control. We show on our demo page4 that even at 3 minutes of supervision we can still achieve control of speaking rate and that we are able to extrapolate outside the range of values seen during training. On the affect data our classification accuracy doesn’t degrade significantly until we reach $1 0 \%$ (300 minutes) supervision and remains significantly above chance down to levels as low as $1 \%$ (30 minutes), see figure 4a and table 1. Obtaining 30 minutes of supervised data is likely within reach of most teams constructing TTS systems. Unlike previous work on generative modelling for control (Hsu et al., 2018; Wang et al., 2018), we do not require a post-processing stage to determine what our latent variables control and we can pre-determine what aspects we wish to control through choice of data. By separating our latent variables into those that are partially supervised and those which are fully unsupervised we retain the ability to model other latent aspects of prosody; this means that we can still draw samples of varying prosody whilst holding constant the affect, speaking rate or F0 variation.
+
+We observe the greatest degree of affect control, as measured by classifier accuracy, when $\alpha = 1$ and $\gamma = 1 0 0$ . This means that to achieve the highest controllability we needed to 1) provide extra information to our approximate posterior $q ( \boldsymbol { z } _ { s } | \boldsymbol { x } , \boldsymbol { y } , \phi )$ and 2) to over-represent the supervised data at low levels of supervision. Although both of these hyperparameters have been used in the literature before (Narayanaswamy et al., 2017; Kingma et al., 2014) and shown to be either beneficial or necessary, they aren’t strictly required by our probabilistic framework and so it is worth considering why they are needed. There are three potential sources of error in any generative model trained with SGVB: the model itself may be mis-specified such that the true data-generating distribution is not in the model class, the parametric family chosen to approximate the posterior may be overly restrictive and finally the optimization landscape may contain undesirable local minima. These problems have afflicted previous work with deep latent variable models trained with SGVB, resulting in models that don’t use their latent variables unless trained with complex annealing schedules (Bowman et al., 2015). In our case we believe that the necessity to set $\alpha$ and $\gamma$ arises from a combination of model mis-specification and local minima. If $\alpha$ is set to 0, then at the start of training $q ( \boldsymbol { z } _ { s } | \boldsymbol { x } , \boldsymbol { y } , \phi )$ is trained only to approximate $p ( z _ { s } | x , y , \theta _ { 0 } )$ , which is randomly initialized. We found empirically that in our discrete-latent experiments this resulted in $q ( \boldsymbol { z } _ { s } | \boldsymbol { x } , \boldsymbol { y } , \phi )$ collapsing early in training to a point-mass on a single class. Having ended up in this undesirable local minimum the posterior distribution never recovered, despite this being an obviously poor approximation to the model posterior later in training. The addition of the classification loss and supervision weighting were sufficient to overcome this collapse and allow $q$ to continue to model the posterior.
+
+The optimization landscape is strongly affected by the relative size of the conditional likelihood and KL terms in our objective. These are in turn strongly affected by our choice of conditional independence assumptions and output-distributions. Thus, a natural direction for further work is to increase the expressivity of the conditional likelihood $p ( x | y , z _ { s } , z _ { u } , \phi )$ to reduce model mis-specification. This could be done by learning the variance of the Laplace-distribution we currently use or by parameterizing more expressive output distributions that do not assume conditional independence across spectrogram channels. We conjecture that with more expressive output distributions, it may be possible to reduce the need for the $\alpha$ and $\gamma$ terms in the objective. In this work we chose to use quite simple unconditional diagonal Gaussian priors, as our primary goal was to demonstrate the practicality of semi-supervision. Another natural extension would be to use conditional-priors $p ( z | y )$ and to use more expressive priors such as mixtures as was done in Hsu et al. (2018).
+
+# 5.1 RELATED WORK
+
+There has been enormous recent progress in neural TTS with numerous novel models proposed in recent years to synthesize speech directly from characters or phonemes (Shen et al., 2018; Arik et al., 2017; Gibiansky et al., 2017; Ping et al., 2017; Vasquez & Lewis, 2019; Taigman et al., 2017). Differentiating factors between these models include the degree of parallelism, with some models using Transformer based architectures (Ren et al., 2019), the choice of conditional independence assumptions made (Vasquez & Lewis, 2019) or the number of separately trained components (Gibiansky et al., 2017). Our work here is largely orthogonal to the exact structure of the conditional likelihood $p ( x | y , z _ { s } , z _ { u } )$ and could be combined with all of the above methods.
+
+Much of the recent research focus has been on modeling latent aspects of prosody. Early attempts include Global Style Tokens (Wang et al., 2018) which attempted to learn a trainable set of styleembeddings. Wang et al. (2018) condition the Tacotron decoder on a linear combination of embedding vectors whose weights during training are predicted from the ground-truth spectrogram.
+
+They were able to achieve prosodic control but there is no straightforward way to sample utterances of varying prosody. More recently, attempts have also been made to combine probabilistic latent variable models trained using SGVB (Akuzawa et al., 2018; Wan et al., 2019). These models use a fully unsupervised and non-identifiable approach, which makes it difficult to disentangle or interpret their latent variables for control. Hsu et al. (2018) attempt to overcome this problem by using a Gaussian mixture as the latent prior and so perform clustering in the latent space. Battenberg et al. (2019) introduce a hierarchical latent variable model to separate the modelling of style from prosody. However, all of these methods are fully unsupervised and this results in latents that can be hard to interpret or require complex post-processing.
+
+The work most similar to ours is Wu et al. (2019) which also attempts to achieve affect control using semi-supervision with a heuristic approach based on Global Style Tokens (Wang et al., 2018). Wu et al. (2019) add a cross-entropy objective to the weightings of the style-tokens that encourages them to be one-hot on points with supervision. Similar to our method, they are able to achieve control over affect but unlike our method they do not have a principled probabilistic interpretation nor the ability to simultaneously model aspects of prosody other than emotion. The result is that their method is not able to draw samples of varying prosody for the same utterance with fixed emotion. Furthermore, whilst our method can be applied to both continuous and discrete controllable factors, its not clear how to extend the style-token based approach to handle continuous latent factors.
+
+In the wider generative modelling literature, the combination of semi-supervision and deep latent variable models was first introduced in Kingma et al. (2014) who focus on using unlabelled data to improve classification accuracy. The potential to use the same technique for controllable generation was recognized by Narayanaswamy et al. (2017) who also provided demonstrations on image synthesis tasks. Since that work, interest in learning disentangled latent variables has grown but generally pursued alternate directions such as re-weighting the ELBO (Higgins et al., 2017), augmenting the objective to encourage factorization (Kim & Mnih, 2018) or using adversarial training (Mathieu et al., 2016). The ability to transfer controllability to speakers for whom we do not have supervision is referred to as domain transfer and our model bears similarities to that introduced by Ilse et al. (2019) but they use a mixture in their latent space more similar to Hsu et al. (2018).
+
+# 5.2 ETHICAL CONSIDERATIONS
+
+As with many advances in speech synthesis, progress in controllability raises the prospect that bad actors may misuse the technology either for misinformation or to commit fraud. Improvements in data efficiency and realism increase these risks and, when publishing, a consideration has to be made as to whether the benefits of the developments outweigh the risks. It is the opinion of the authors in this case that, since the focus of this work is on improved prosody, with potential benefits to human-computer interfaces, the benefits likely outweigh the risks. We nonetheless urge the research community to take seriously the potential for misuse both of this work and broader advances in TTS.
+
+# 6 CONCLUSION
+
+We have shown that the combination of semi-supervised latent variable models with neural TTS presents a practical and principled path towards building speech synthesizers we can control. Unlike previous fully unsupervised methods, we are able to consistently and reliably learn to control predetermined aspects of prosody. Our method can be applied to any latent attribute of speech for which a modest amount of labelling can be obtained, whether it be continuous or discrete. In our experiments we found that 30 minutes of supervision was sufficient, a volume of data that is within the reach of most research teams. We are able to learn to control subtle characteristics of speech such as affect and for continuous attributes we have provided demonstrations of extrapolation to ranges never seen during training, and to speakers with no supervision. Augmenting existing stateof-the-art TTS systems with latent variables does not degrade synthesis quality and we evidence this with crowd sourced mean opinion scores. Unlike similar heuristic methods, our probabilistic formulation, allows us to draw samples of varying prosody whilst holding constant some attribute we wish to control.
+
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+H. Kim and A. Mnih. Disentangling by factorising. arXiv preprint arXiv:1802.05983, 2018.   
+D. P. Kingma and M. Welling. Auto-encoding variational Bayes. arXiv preprint arXiv:1312.6114, 2013.   
+D. P. Kingma, S. Mohamed, D. J. Rezende, and M. Welling. Semi-supervised learning with deep generative models. In Advances in neural information processing systems, pp. 3581–3589, 2014.   
+R. Kubichek. Mel-cepstral distance measure for objective speech quality assessment. In Proceedings of IEEE Pacific Rim Conference on Communications Computers and Signal Processing, 1993.   
+F. Locatello, S. Bauer, M. Lucic, G. Raetsch, S. Gelly, B. Scholkopf, and O. Bachem. Challenging ¨ common assumptions in the unsupervised learning of disentangled representations. In Proceedings of the 36th International Conference on Machine Learning, volume 97 of Proceedings of Machine Learning Research, pp. 4114–4124, Long Beach, California, USA, 2019. PMLR.   
+M. F. Mathieu, J. J. Zhao, J. Zhao, A. Ramesh, P. Sprechmann, and Y. LeCun. Disentangling factors of variation in deep representation using adversarial training. In Advances in Neural Information Processing Systems, pp. 5040–5048, 2016.   
+E. Munoz-de Escalona and J. J. Canas. Online measuring of available resources. In $H .$ -Workload 2017: The first international symposium on human mental workload. Dublin Institute of Technology, 2017.   
+S. Narayanaswamy, B. T. Paige, J. Van de Meent, A. Desmaison, N. Goodman, P. Kohli, F. Wood, and P. Torr. Learning disentangled representations with semi-supervised deep generative models. In Advances in Neural Information Processing Systems, pp. 5925–5935, 2017.   
+W. Ping, K. Peng, A. Gibiansky, S. O. Arik, A. Kannan, S. Narang, J. Raiman, and J. Miller. Deep voice 3: Scaling text-to-speech with convolutional sequence learning. arXiv preprint arXiv:1710.07654, 2017.   
+Y. Ren, Y. Ruan, X. Tan, T. Qin, S. Zhao, Z. Zhao, and T. Liu. FastSpeech: Fast, robust and controllable text-to-speech. arXiv preprint arXiv:1905.09263, 2019.   
+D. J. Rezende, S. Mohamed, and D. Wierstra. Stochastic backpropagation and approximate inference in deep generative models. In Proceedings of The International Conference on Machine Learning, 2014.   
+J. A. Russell. A circumplex model of affect. Journal of personality and social psychology, 1980.   
+T. Salimans, A. Karpathy, X. Chen, and D. P. Kingma. PixelCNN $^ { + + }$ : Improving the PixelCNN with discretized logistic mixture likelihood and other modifications. arXiv preprint arXiv:1701.05517, 2017.   
+M. Schroder. Emotional speech synthesis: A review. In ¨ Seventh European Conference on Speech Communication and Technology, 2001.   
+J. Shen, R. Pang, R. J. Weiss, M. Schuster, N. Jaitly, Z. Yang, Z. Chen, Y. Zhang, Y. Wang, and R. Skerrv-Ryan. Natural TTS synthesis by conditioning WaveNet on mel spectrogram predictions. In 2018 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), 2018.   
+R. Skerry-Ryan, E. Battenberg, Y. Xiao, Y. Wang, D. Stanton, J. Shor, R. Weiss, R. Clark, and R. A. Saurous. Towards end-to-end prosody transfer for expressive speech synthesis with Tacotron. In Proceedings of the 35th International Conference on Machine Learning, 2018.   
+Y. Taigman, L. Wolf, A. Polyak, and E. Nachmani. Voiceloop: Voice fitting and synthesis via a phonological loop. arXiv preprint arXiv:1707.06588, 2017.   
+A. van den Oord, S. Dieleman, H. Zen, K. Simonyan, O. Vinyals, A. Graves, N. Kalchbrenner, A. Senior, and K. Kavukcuoglu. WaveNet: A generative model for raw audio. arXiv preprint arXiv:1609.03499, 2016.   
+S. Vasquez and M. Lewis. MelNet: A generative model for audio in the frequency domain. arXiv preprint arXiv:1906.01083, 2019.   
+V. M. Velichko and N. G. Zagoruyko. Automatic recognition of 200 words. International Journal of Man-Machine Studies, 2(3):223–234, 1970.   
+V. Wan, C. Chan, T. Kenter, J. Vit, and R. Clark. CHiVE: Varying prosody in speech synthesis with a linguistically driven dynamic hierarchical conditional variational network. arXiv preprint arXiv:1905.07195, 2019.   
+Y. Wang, R. Skerry-Ryan, D. Stanton, Y. Wu, R. J. Weiss, N. Jaitly, Z. Yang, Y. Xiao, Z. Chen, S. Bengio, et al. Tacotron: Towards end-to-end speech synthesis. arXiv preprint arXiv:1703.10135, 2017.   
+Y. Wang, D. Stanton, Y. Zhang, R. J. Skerry-Ryan, E. Battenberg, J. Shor, Y. Xiao, F. Ren, Y. Jia, and R. A. Saurous. Style tokens: Unsupervised style modeling, control and transfer in end-to-end speech synthesis. arXiv preprint arXiv:1803.09017, 2018.   
+P. Wu, Z. Ling, L. Liu, Y. Jiang, H. Wu, and L. Dai. End-to-end emotional speech synthesis using style tokens and semi-supervised training. arXiv preprint arXiv:1906.10859, 2019.   
+H. Zen, V. Dang, R. Clark, Y. Zhang, R. J. Weiss, Y. Jia, Z. Chen, and Y. Wu. LibriTTS: A corpus derived from LibriSpeech for text-to-speech. In Interspeech 2019, 2019.
+
+A NEURAL NETWORK ARCHITECTURE   
+Table 4: Summary of the hyperparameters described below.   
+
+<table><tr><td>Module</td><td>hyperparameters</td></tr><tr><td>Input</td><td>Text normalized phonemes</td></tr><tr><td>Phoneme embedding</td><td>256-D</td></tr><tr><td>Pre-net</td><td>FC-256-Relu-Dropout(0.5)</td></tr><tr><td rowspan="4">CBHG text encoder</td><td>→FC-128-Relu-Dropout(0.5)</td></tr><tr><td>Conv1D bank:K=16,conv-k-128-Relu →Max pooling with stride=1 width=2</td></tr><tr><td>→ Conv1D projections: conv-3-128-Relu</td></tr><tr><td>→ conv-3-128-Linear →Highway net: 4 layers of FC-128-Relu</td></tr><tr><td>Attention type</td><td>→Bidirectional GRU:128 cells 5 component GMM attention w/ softplus (Graves, 2013)</td></tr><tr><td>Attention RNN</td><td>LSTM-256-Zoneout(0.1) →FC-128-tanh</td></tr><tr><td>DecoderRNN</td><td>2-layer residual-LSTM-265-zoneout(0.1)</td></tr><tr><td>Frames-per-timestep (reduction factor) 2</td><td>→FC-80-Linear</td></tr><tr><td>WaveRNN</td><td>5 layers DilatedConv1D-512</td></tr><tr><td rowspan="3">Variational posterior</td><td>-→2 layers TransposeConv + ReLu</td></tr><tr><td>→ GRU-768 conditioned on 5 previous samples →FC-768-relu</td></tr><tr><td>→3 component MoL, 24kHz sample rate Spectrogram</td></tr><tr><td></td><td>→ 6 Conv-layers 32-32-64-64-128-128 →LSTM-128 → FC-128-tanh</td></tr><tr><td>Optimizer</td><td>ADAM with learning rate 1O-³,batch-size 256</td></tr><tr><td>Speaker embedding</td><td>64-D</td></tr></table>
+
+Sequence-to-sequence model Our sequence-to-sequence network is modelled on Tacotron (Wang et al., 2018) but uses some modifications introduced in Skerry-Ryan et al. (2018). Input to the model consists of sequences of phonemes produced by a text normalization pipeline rather than character inputs. The CBHG text encoder from Wang et al. (2017) is used to convert the input phonemes into a sequence of text embeddings. The phoneme inputs are converted to learned 256-dimensional embeddings and passed through a pre-net composed of two fully connected ReLU layers (with 256 and 128 units, respectively), with dropout of 0.5 applied to the output of each layer, before being fed to the encoder. For multi-speaker models, a learned embedding for the target speaker is broadcastconcatenated to the output of the text encoder. The attention module uses a single LSTM layer with 256 units and zoneout of 0.1 followed by an MLP with 128 tanh hidden units to compute parameters for the monotonic 5-component GMM attention window. We use GMMv2b attention mechanism described in Battenberg et al. (2020). Instead of using the exponential function to compute the shift and scale parameters of the GMM components as in Graves (2013), GMMv2vb uses the softplus function, and also adds initial bias to these parameters, which we found leads to faster alignment and more stable optimization. The attention weights predicted by the attention network are used to compute a weighted sum of output of the text encoder, producing a context vector. The context vector is concatenated with the output of the attention LSTM layer before being passed to the first decoder LSTM layer. The autoregressive decoder module consists of 2 LSTM layers each with 256 units, zoneout of 0.1, and residual connections between the layers. The spectrogram output is produced using a linear layer on top of the 2 LSTM layers, and we use a reduction factor of 2, meaning we predict two spectrogram frames for each decoder step. The decoder is fed the last frame of its most recent prediction (or the previous ground truth frame during training) and the current context as computed by the attention module. Before being fed to the decoder, the previous prediction is passed through a pre-net with the same same structure used before the text encoder above but its own parameters.
+
+CBHG text encoder We reuse the CHGB text encoder introduced in Wang et al. (2018). The text encoder consists of a bank of 1-D convolutional filters, followed by highway networks and a bidirectional gated recurrent unit (GRU) recurrent neural net (RNN). The input sequence is first convolved with K sets of 1-D convolutional filters, where the $k$ -th set contains $C _ { k }$ filters of width $k$ . The convolution outputs are stacked together and further max pooled, preserving time. As in the original paper we use a stride of 1 to preserve the original time resolution. We further pass the processed sequence to a few fixed-width 1-D convolutions, whose outputs are added with the original input sequence via residual connections. Batch normalization is used for all convolutional layers. The convolution outputs are fed into a multi-layer highway network to extract high-level features. Finally, we stack a bidirectional GRU RNN on top to extract sequential features from both forward and backward context.
+
+Variational posteriors The variational distributions $q ( z _ { s } | x , y )$ and $q ( z _ { u } | x , y , z _ { s } )$ are both structured as diagonal Gaussian distributions whose mean and variance are parameterized by neural networks. For discrete supervision we replace $q ( z _ { s } | x , y )$ by a categorical distribution and use the same network to output just the mean. The input to the distribution starts from the mel spectrogram $x$ and passes it through a stack of 6 convolutional layers, each using ReLU non-linearities, $3 \mathrm { x } 3$ filters, $2 \mathbf { x } 2$ stride, and batch normalization. The 6 layers have 32, 32, 64, 64, 128, and 128 filters, respectively. The output of this convolution stack is fed into a unidirectional LSTM with 128 units. We pass the final output of this LSTM (and potentially vectors describing the text and/or speaker) through an MLP with 128 tanh hidden units to produce the parameters of the diagonal Gaussian posterior which we sample from. All but the last linear layer of these networks is shared between the two distributions $q ( z _ { s } | x , y )$ and $q ( z _ { u } | x , y , z _ { s } )$ . The resulting sample is broadcast-concatenated to the output of the text encoder. In our experiments $z _ { u }$ is always 32-dimensional and $z _ { s }$ is either a one-hot vector across 6 classes or a 1 dimensional continuous value.
+
+Conditional inputs When providing information about the text to the variational posterior, we pass the sequence of text embeddings produced by the text encoder to a unidirectional RNN with 128 units and use its final output as a fixed-length text summary that is passed to the posterior MLP. Speaker information is passed to the posterior MLP via a learned speaker embedding.
+
+WaveRNN We used a WaveRNN model similar to that described in Kalchbrenner et al. (2018) as our vocoder. Our WaveRNN uses discretized mixture of logistics output as described in Salimans et al. (2017) instead of the dual softmax from that paper, and conditions on 5 previous samples at each step instead of only 1 previous. We trained the network to map from synthesized melspectrograms to waveforms, training on 900 sample windows. A conditioning stack of dilated convolution and transpose convolutions is applied to the input spectrogram before tiling to upsample to the audio sample rate.
+
+# B EVALUATION
+
+Mel spectrograms The mel spectrograms the model predicts are computed from $2 4 ~ \mathrm { k H z }$ audio using a frame size of $5 0 \mathrm { m s }$ , a hop size of $1 2 . 5 \mathrm { m s }$ , an FFT size of 2048, and a Hann window. From the FFT energies, we compute $8 0 \mathrm { m e l }$ bins distributed between $8 0 \mathrm { H z }$ and $1 2 \mathrm { k H z }$ .
+
+MCD-DTW To compute mel cepstral distortion (MCD) (Kubichek, 1993), we use the same mel spectrogram parameters described above and take the discrete-cosine-transform to compute the first 13 MFCCs (not including the 0th coefficient). The MCD between two frames is the Euclidean distance between their MFCC vectors. Then we use the dynamic time warping (DTW) algorithm (Velichko & Zagoruyko, 1970) (with a warp penalty of 1.0) to find an alignment between two spectrograms that produces the minimum MCD cost (including the total warp penalty). We report the average per-frame MCD-DTW.
+
+Affect classifier The affect classifier has a very similar structure to the variational posterior. The input to the classifier starts from the mel spectrogram $x$ and passes it through a stack of 6 convolutional layers, each using ReLU non-linearities, 3x3 filters, $2 \mathbf { x } 2$ stride, and batch normalization. The 6 layers have 32, 32, 64, 64, 128, and 128 filters, respectively. The output of this convolution stack is fed into a unidirectional LSTM with 128 units. The final output of the LSTM is then passed through a softmax non-linearity to get logits over the training classes. We use the same data splits described in section 3 to train and evaluate the classifier. The classifier is tuned on the validation set achieving $8 4 . 3 3 \%$ classification accuracy, generalizing well to the test set with $8 3 . 9 4 \%$ accuracy.
+
+Mean opinion scores We use a human rating service similar to Amazon’s Mechanical Turk, with a large pool of English speakers to collect MOS evaluations. The MOS template is shown in figure 5. A human rater is presented with a single speech sample and is asked to rate perceived naturalness on a scale of 1 to 5, where 1 is “Bad” and 5 is “Excellent”. We have selected the utterances of one male, and one female speaker in our test set, totalling 371 utterances to evaluate. For each sample, we collect 1 rating, and no rater is used for more than 6 items in a single evaluation set. In total, 270 unique raters completed the 6 evaluation sets presented in table 2. Since raters are randomly selected for each set, some raters have assessed multiple methods. Across the 6 evaluation sets, the average and median of total number of ratings per rater was 8.24, and 6, respectively. To analyze the data from these subjective tests, we average the scores and compute $9 5 \%$ confidence intervals. Natural human speech is typically rated around 4.5. Samples used for MOS from our model were drawn using the mean of $z _ { u }$ , whilst sampling $z _ { s }$ .
+
+# Instruction
+
+IMPORTANT:   
+In this project, you willisten to audio samples.Please release this task if any of the following is true:   
+1) Youdonothave headphones   
+2) You think you do not have good listening ability   
+3)Thereisconsiderable backgroundnoise (streetoise,loudfan/airconditioner,open TV/radio,people taking,etc)   
+4)Foranyreason,youcan'thear theaudiosamples If youare not sure which type you have,please see thisWikipedia article.   
+2)Please set the volume of youraudiodevice toacomfortable level
+
+t thesentence sounded.You should notjudge the grammaror thecontent of the sentence,just howit sounds.
+
+1) Listen to each sample at least twice,with at least a one sec break between them
+
+se spc
+
+e speech sample falls between two levels.
+
+Naturalness Scale:   
+
+<table><tr><td>Score</td><td>Naturalness</td><td>Description</td><td>Reference</td></tr><tr><td>5.0</td><td>Excellent</td><td>Completely natural speech</td><td>Listen</td></tr><tr><td>4.0</td><td>Good</td><td>Mostly natural speech</td><td>Listen</td></tr><tr><td>3.0</td><td>Fair</td><td>Equally natural and unnatural speech</td><td>Listen</td></tr><tr><td>2.0</td><td>Poor</td><td>Mostly unnatural speech</td><td>Listen</td></tr><tr><td>1.0</td><td>Bad</td><td>Completely unnatural speech</td><td>Listen</td></tr></table>
+
+# How are you listening to the speech sample?
+
+$1 0$ ii and fans, etc.). Hdi
+
+playingiod.)
+
+# Speech sample (please listen at least twice)
+
+![](images/9a6c86d397f0f3c2141c88093c5d3236aff6deacaf8553e165a2715cb76f5244.jpg)  
+Figure 5: Mean opinion score (MOS) evaluation template. For each utterance, the human raters assign a 1–5 score of the perceived naturalness, with 1 being “Bad” and 5 being “Excellent”.
+
+Subjective affect control evaluation We use the same rater pool, and the same set of 371 utterances used for the MOS evaluations. The A/B template is shown in figure 6. For each utterance, the human rater is presented with a pair of utterances to choose the one that better conveys the target emotion (e.g., happy in the figure). Both utterances are generated with the same text. To evaluate the control over valence, we present baseline (i.e, no control) against utterances generated in specific valence category (angry, happy and sad). To evaluate the control over arousal, we present samples generated at low arousal against samples generated at high arousal, and ask the rater to choose the utterance that is more vocally aroused. We use mean of $z _ { u }$ to generate all the samples.
+
+![](images/5ff645d5991f5a882e6a826f33757c218f6b3c9859d7a071136e17b7b683fd35.jpg)
+
+ltis beti diference,makeaquick intuitive guess.
+
+#
+
+How are you listening to these speech samples?
+
+● Headphones, with no noise in the background Headphones,with some low-levelnoise in thebackground Oother
+
+MPORTANT:Ifyoudon'thaveheadphonespleasereleasethistasforthereason,"Idonotmeetheupfrontrequirementsforthistas
+
+#
+
+![](images/40e401510743a860818648832370982a2bca41ba89b9dc8873a957c96bd329d3.jpg)  
+Figure 6: A/B evaluation affect control evaluation template. The emotion label (Happy in the figure) varies depending on the task.
+
+# C SAMPLE SPECTROGRAMS
+
+Controlling affect Table 5 shows the effect of varying the valence and arousal latent variable on the spectrogram and F0 track. We can see that a low valence for sadness corresponds to the flattest F0 track, and high arousal manifests in higher F0 values and variations.
+
+Controlling speaking rate and pitch variations Table 6 shows the effect of varying speaking rate, and F0 variation control variables on a sample spectrogram and F0 track. When controlling speaking rate (first column), the duration reduces as we increase the input speaking rate control, while the F0 variation remains stable. When controlling the F0 variation (second column), the pitch dynamic range increases, while the duration remains constant, which demonstrates controllablity and also some degree of disentanglement.
+
+# D REPRODUCING RESULTS ON LIBRITTS PUBLIC DATASET
+
+To verify the reproduciblity of our results on a public dataset, we trained models to control speaking rate and F0 variation on clean subset of LibriTTS dataset (Zen et al., 2019). We only use the utterances below 5 seconds, which is 62 hours of data. We have done no tuning on this dataset, and directly used the hyperparameters we used for our internal dataset. Figures 7a and 7b show the errors of producing the desired speaking rate, and F0 standard deviation, which generally go down as function of supervision level, with exception of $1 0 \%$ supervision for controlling F0 variation 5. Given, this is a lower quality dataset, with many more speakers and with much smaller data per speaker and also the fact that we have done zero hyperparameter tuning on this dataset, this result look very encouraging.
+
+![](images/7cf32eb806dde73a728066853cbadebc9a267e286893e2cf4719cbbcab0529e7.jpg)  
+Table 5: Sample spectrogram and F0 track plots, generated by varying affect labels, valence in y-axis, and arousal in $\mathbf { X }$ -axis.
+
+![](images/74cb0a213e7b8f8f2451c4a4b889014a82ebbe4c1bfdc9b3df4effc481cc1dd9.jpg)  
+Figure 7: Objective controllability of speaking rate and F0 variation evaluation metrics presented at multiple supervision levels, on LibriTTS (Zen et al., 2019) datasets. $1 0 0 \%$ supervision corresponds to 62 hours of supervised data.
+
+![](images/7734fdea3db70aaf56bede90d9baec21ab384b45bc4aeb3b88637957c98254a0.jpg)  
+Table 6: Sample spectrogram and F0 track plots, generated by varying the speaking rate (first column) and F0 variation (second column). We use standard normal prior for these factors and this table demos varying the control factor from $- 5 \sigma$ to $5 \sigma$ , demonstrating the controllability, interpolation and extrapolation of conditional generation, and also disentanglement of these factors.

parse/train/rJeqeCEtvH/rJeqeCEtvH_content_list.json

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+        "text": "SEMI-SUPERVISED GENERATIVE MODELING FOR CONTROLLABLE SPEECH SYNTHESIS ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "Raza Habib1∗ Soroosh Mariooryad2 Matt Shannon2 Eric Battenberg2 RJ Skerry-Ryan2   \nDaisy Stanton2 David Kao2 Tom Bagby2 ",
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+    {
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+        "text": "1University College London (UCL) 2Google Research. ",
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+    },
+    {
+        "type": "text",
+        "text": "raza.habib@cs.ucl.ac.uk {soroosh, mattshannon, ebattenberg, rjryan, daisy, davidkao, tombagby}@google.com ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "ABSTRACT ",
+        "text_level": 1,
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+    },
+    {
+        "type": "text",
+        "text": "We present a novel generative model that combines state-of-the-art neural textto-speech (TTS) with semi-supervised probabilistic latent variable models. By providing partial supervision to some of the latent variables, we are able to force them to take on consistent and interpretable purposes, which previously hasn’t been possible with purely unsupervised TTS models. We demonstrate that our model is able to reliably discover and control important but rarely labelled attributes of speech, such as affect and speaking rate, with as little as 30 minutes supervision. Even at such low supervision levels we do not observe a degradation of synthesis quality compared to a state-of-the-art baseline. Audio samples are available on the web1. ",
+        "bbox": [
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+        "type": "text",
+        "text": "1 INTRODUCTION ",
+        "text_level": 1,
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "The ability to reliably control high level attributes of speech, such as emotional expression (affect) or speaking rate, is often desirable in speech synthesis applications. Achieving this control however is made difficult by the necessity of acquiring a large quantity of high quality labels. In this paper we show that semi-supervised latent variable models can take us a significant step closer towards solving this problem. ",
+        "bbox": [
+            174,
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+            569
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Combining state-of-the-art neural text-to-speech (TTS) systems with probabilistic latent variable models provides a natural framework for discovering aspects of speech that are rarely labelled or even difficult to describe. Both inferring the latent prosody and generating samples with sufficient variety requires reasoning about uncertainty and is thus a natural fit for deep generative models. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "There has been recent progress in applying stochastic gradient variational Bayes (SGVB) (Kingma & Welling, 2013; Rezende et al., 2014) to training probabilistic neural TTS models. Battenberg et al. (2019) and Hsu et al. (2018) have shown that it is possible to use latent variable models to discover features such as speaking style, speaking rate, arousal, gender and even the quality of the recording environment. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "However, these models are formally non-identifiable (Hyvarinen & Pajunen, 1999; Locatello et al., ¨ 2019) and this implies that repeated training runs will not reliably discover the same latent attributes. Even if they did, a lengthy human post-processing stage is necessary to identify what the model has learned on any given training run. In order to be of practical use for control, it is not enough for the models to discover latent attributes, they need to do so reliably and in a way that is robust to random initialization and to changes in the model. We demonstrate that the addition of even modest amounts of supervision can be sufficient to achieve this reliability. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "By augmenting state-of-the art neural TTS with semi-supervised deep generative models within the VAE framework (Kingma et al., 2014; Narayanaswamy et al., 2017), we show that it is possible to not only discover latent attributes of speech but to do so in a reliable and controllable manner. In particular we are able to achieve reliable control over affect, speaking rate and F0 variation (F0 is the fundamental frequency of oscillation of the vocal folds). Further, we provide demonstrations that it is possible to transfer controllability to speakers for whom we have no labels. Our core contributions are: ",
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+        "image_caption": [
+            "Figure 1: Schematic showing how we parameterize the conditional likelihood $p ( x | y , z _ { u } , z _ { s } )$ . Left: A block of 1-D convolutions and RNNs originally introduced by Wang et al. (2017) and described in detail in the appendix. Right: Schematic of the sequence-to-sequence network that outputs the means of our auto-regressive distribution. At each decoder time step, the network outputs the means for the next two spectrogram frames. "
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+        "text": "• To combine semi-supervised latent variable models with Neural TTS systems, producing a system that can reliably discover attributes of speech we wish to control. • To demonstrate that as little as 30 minutes supervision can be sufficient to improve prosody and allow control over speaking rate, fundamental frequency (F0) variation and affect, a problem of interest to the speech community for well over two decades (Schroder, 2001). ¨ • To imbue TTS models with control over affect, F0 and speaking rate whilst still maintaining prosodic variation when sampling. ",
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+        "type": "text",
+        "text": "2 GENERATIVE MODEL ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "Our generative model, shown in figures 1 and 2a, consists of an autoregressive distribution over a sequence of acoustic features, $x _ { 1 \\ldots t }$ , that are generated conditioned on a sequence of text, $y _ { 1 \\ldots k }$ , and on two latent variables, $z _ { u }$ and $z _ { s }$ . The latent variables can be discrete or continuous. $z _ { s }$ represents the variations in prosody that we seek to control and is semi-supervised. $z _ { u }$ is fully unobserved and represents latent variations in prosody (intonation, rhythm, stress) that we wish to model but not explicitly control. Once trained, our model can be used to synthesize acoustic features from text. Similar to Tacotron 2 (Shen et al., 2018), we then generate waveforms by training a second network such as WaveNet (van den Oord et al., 2016) or WaveRNN (Kalchbrenner et al., 2018) to act as a vocoder. In our case we use WaveRNN. ",
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+        "type": "text",
+        "text": "We parameterize our likelihood $p ( x _ { 1 \\ldots t } | y _ { 1 \\ldots k } , z _ { u } , z _ { s } , \\theta )$ by a sequence-to-sequence neural network with attention (Shen et al., 2018; Graves, 2013; Bahdanau et al., 2014) that is shown schematically in figure 1. Details largely follow Tacotron (Wang et al., 2017) and are given in appendix A. At each time step we model a mel-spectrogram frame with a fixed variance isotropic Laplace distribution whose mean is output by the neural network. We condition each of the latent variables by concatenating the vectors $z _ { u }$ and $z _ { s }$ to the representation of the text-encoder, before the application of the attention mechanism. In the case of continuous $z$ we use a standard normal prior and in the case of discrete $z$ we use a uniform categorical prior with one-hot encoding. ",
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+        "image_caption": [
+            "Figure 2: Left: The graphical model showing the conditional independence assumptions between each of the stochastic variables. Centre: The structure of the variational distribution used to approximate the posterior for fully unsupervised data points and Right: supervised points. "
+        ],
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+        "type": "text",
+        "text": "2.1 SEMI-SUPERVISED TRAINING ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "Following Kingma et al. (2014); Narayanaswamy et al. (2017), we train our model via stochastic gradient variational Bayes (SGVB). That is we approximately maximize the log-likelihood of our training data by maximizing a variational lower bound using stochastic gradient ascent. Since we are training with semi-supervision we in fact need two lower bounds: one for the data points for which $z _ { s }$ is observed; one for the case where $z _ { s }$ is unobserved. In our models the fully latent variable $z _ { u }$ is always continuous but the semi-supervised latent $z _ { s }$ can be continuous or discrete. The conditional independence structure of our variational distributions is shown in figures 2b and 2c. On supervised data, the per-datapoint bound is: ",
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+        "type": "equation",
+        "img_path": "images/9cd62dbcd69a893e921bc8e35d6bf8a721c3b38ad3052dd6657049b3442283bb.jpg",
+        "text": "$$\n\\begin{array} { r l } & { \\log p ( x , z _ { s } | y , \\theta ) = \\log \\int p ( x , z _ { u } , z _ { s } | y , \\theta ) \\mathrm { d } z _ { u } } \\\\ & { \\qquad \\geq \\mathbb { E } _ { q ( z _ { u } | x , y , z _ { s } , \\phi ) } \\left[ \\log \\left( \\frac { p ( x | y , z _ { u } , z _ { s } , \\theta ) p ( z _ { u } ) p ( z _ { s } ) } { q ( z _ { u } | x , y , z _ { s } , \\phi ) } \\right) \\right] } \\\\ & { \\qquad = \\mathbb { E } _ { q ( z _ { u } | x , y , z _ { s } , \\phi ) } \\left[ \\log p ( x | y , z _ { u } , z _ { s } , \\theta ) \\right] + \\log p ( z _ { s } ) - D _ { \\mathrm { K L } } ( q ( z _ { u } | x , y , z _ { s } , \\phi ) \\| p ( z _ { u } ) ) } \\\\ & { \\qquad = \\mathcal { L } _ { s } ( \\theta , \\phi ; x , y , z _ { s } ) } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Where $q ( z _ { u } | x , y , z _ { s } , \\phi )$ is a parametric variational distribution introduced to approximately marginalize $z _ { u }$ . $\\theta$ are the parameters of the generative model and $\\phi$ are the parameters of the variational distributions. The intractable integrals are approximated with reparameterized samples. For the cases where $z _ { s }$ is unobserved and discrete, the bound is: ",
+        "bbox": [
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+        "page_idx": 2
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+    {
+        "type": "equation",
+        "img_path": "images/9924683bd350dd81a986a76d260598c4a930710b948f7f89d6b792dc12673ea5.jpg",
+        "text": "$$\n\\begin{array} { l } { \\log p ( x | y , \\theta ) = \\log \\displaystyle \\int \\displaystyle \\sum _ { z _ { s } } p ( x , z _ { u } , z _ { s } | y , \\theta ) \\mathrm { d } z _ { u } } \\\\ { \\displaystyle \\qquad \\geq \\sum _ { z _ { s } } q ( z _ { s } | x , y , \\phi ) \\mathcal { L } _ { s } ( \\theta , \\phi ; x , y , z _ { s } ) + H ( q ( z _ { s } | x , y , \\phi ) ) } \\\\ { \\displaystyle \\qquad = \\mathcal { L } _ { u } ( \\theta , \\phi ; x , y ) } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "and when $z _ { s }$ is continuous we replace the sum above with an integral and again approximate with reparameterized samples. The variational distributions are parameterized by a neural network that takes as input the text, spectrograms and other conditioning variables and outputs the parameters of the distribution. The exact structure of this network is given in appendix A. We have implicitly assumed that $q ( z _ { u } , z _ { s } | x , y , \\phi )$ may be factorized as $q ( z _ { u } , z _ { s } | x , y , \\phi ) = q ( z _ { u } | x , y , z _ { s } , \\phi ) q ( z _ { s } | x , y , \\phi )$ with shared parameters between these two distributions (see appendix A). Optimizing the variational objective with respect to the parameters $\\phi$ encourages the variational distributions to match the posterior of the generative model $p ( z _ { u } , z _ { s } | x , y , \\theta )$ . Unlike previous work (Hsu et al., 2018), we do not assume that the posterior on the latents is independent of the text, as this dependence likely exists in the model due to explaining away. That is to say that although the text and the latents are independent in our prior, observing the spectrogram correlates them in the posterior because they ",
+        "bbox": [
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+        "page_idx": 2
+    },
+    {
+        "type": "image",
+        "img_path": "images/b9b30b9a2b861960a061a036bd85ea3a7fb8c4eba78836b734e5f9854b94822c.jpg",
+        "image_caption": [
+            "Figure 3: The circumplex model of emotion. Each possible emotion is represented in a 2 dimensional plane consisting of an arousal dimension and valence dimension. This figure is borrowed from Munoz-de Escalona & Canas (2017). "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        "page_idx": 3
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+    {
+        "type": "text",
+        "text": "both explain variation in the spectrogram. This has been shown to be significant by Battenberg et al.   \n(2019). ",
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+    {
+        "type": "text",
+        "text": "If we define: ",
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+    {
+        "type": "equation",
+        "img_path": "images/34699753a1f69ec6ec43b0eeecb72102e925b7777a07887bfcfa2edc5b735d73.jpg",
+        "text": "$$\n\\tilde { q } ( z _ { s } | x , y ) = \\left\\{ \\begin{array} { l l } { q ( z _ { s } | x , y , \\phi ) } & { \\mathrm { i f ~ u n s u p e r v i s e d } } \\\\ { \\gamma \\delta ( z _ { s } - z _ { s _ { \\mathrm { o b s e r v e d } } } ) } & { \\mathrm { i f ~ s u p e r v i s e d } } \\end{array} \\right.\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 3
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+    {
+        "type": "text",
+        "text": "then we can write the overall objective over both the supervised and unsupervised points succinctly $\\mathrm { a s } ^ { 2 }$ : ",
+        "bbox": [
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+    },
+    {
+        "type": "equation",
+        "img_path": "images/01d2ffafe31d8d98d5406e62d2202216fe52fd3a3b76424e0c3d3a4790b5e02b.jpg",
+        "text": "$$\n\\mathcal { L } ( \\theta , \\phi ) = \\mathbb { E } _ { x , y } \\left[ \\sum _ { z _ { s } } \\tilde { q } ( z _ { s } | x , y , \\phi ) \\mathcal { L } _ { s } ( \\theta , \\phi ; x , y , z _ { s } ) + H ( \\tilde { q } ( z _ { s } | x , y , \\phi ) ) \\right]\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "where summation would again be replaced by integration for continuous $z _ { s }$ and $\\gamma$ (shown in equation 4) is a weighting factor that pre-multiplies the loss for any supervised point. This weighting was also used in previous work such as Narayanaswamy et al. (2017), who showed it to be beneficial at very low levels of supervision. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Writing the objective in this form allows an intuitive interpretation for the semi-supervised training procedure. When supervision is provided, our objective function is evaluated at the observed value of $z _ { s }$ . When supervision is not provided, we evaluate the objective function for every possible value of $z _ { s }$ and take a (potentially infinite for continuous $z _ { s }$ ) weighted average. The weighting in the average is given by $q ( \\boldsymbol { z } _ { s } | \\boldsymbol { x } , \\boldsymbol { y } , \\phi )$ , which is simultaneously trained to approximate the posterior $p ( z _ { s } | x , y , \\theta )$ . In other words, on unsupervised utterances, we evaluate our objective for each possible value of the latent attribute and weight by the (approximate) posterior probability that this value of the latent was responsible for generating the utterance. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "As $q ( \\boldsymbol { z } _ { s } | \\boldsymbol { x } , \\boldsymbol { y } , \\phi )$ is trained to approximate $p ( z _ { s } | x , y , \\theta )$ we can expect it to become a reasonable classifier/regressor for the semi-supervised latent attribute as the model improves. For example when $z _ { s }$ represents an affect label, $p ( z _ { s } | x , y , \\theta )$ is the posterior probability, of the model, over affect given text and speech. By taking the most likely posterior class, this distribution can be used as an affect classifier. However, this variational distribution is only trained on unsupervised training points and so does not benefit directly from the supervised data. To overcome this problem we follow Kingma et al. (2014) and add a classification loss to our objective. The overall objective becomes: ",
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+    {
+        "type": "equation",
+        "img_path": "images/e7795f2949a598eff6565948844d1485e39be707f0197f2a753623eabd8a899f.jpg",
+        "text": "$$\n\\mathcal { L } _ { \\mathrm { t o t a l } } ( \\theta , \\phi ) = \\mathcal { L } ( \\theta , \\phi ) + \\alpha \\mathbb { E } _ { x , y , z _ { s } } [ \\log q ( z _ { s } | x , y , \\phi ) ]\n$$",
+        "text_format": "latex",
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+    {
+        "type": "text",
+        "text": "Where $\\alpha$ is a hyperparameter which adjusts the contribution of this term. ",
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+    {
+        "type": "text",
+        "text": "3 DATA ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "We have used a proprietary high quality labeled data-set of 40 English speakers. The training set consists of 72,405 utterances with durations of at most 5 seconds (45 hours). The validation and test sets each contain 745 utterances or roughly 30 minutes of data. We vary the amount of supervision in the experiments below. We also experimented with transferring controllability to a fully unlabeled data-set of audiobook recordings by Catherine Byers (the speaker from the 2013 Blizzard Challenge), which exhibits high variation in affect and prosody and to other speakers who were less expressive. We strongly encourage the reader to listen to the synthesized samples on our demo page3. ",
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+    {
+        "type": "text",
+        "text": "In this work we chose to focus on learning to control affect with a discrete representation, as well as speaking rate and F0 variation with a continuous representation, as these are challenging aspects of prosody to control. Our method could be applied to other factors without modification. ",
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+    {
+        "type": "text",
+        "text": "3.1 AFFECT CONTROL ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "The best way to represent emotion is an actively researched area and many models of affect exist. In this work we chose to follow the circumplex model of emotion (Russell, 1980) which posits that most affective states can be represented in a 2 dimensional plane with one axis representing arousal and the other axis representing valence. Arousal measures the level of excitement or energy and valence measures positivity or negativity. Figure 3, shows a chart of emotions plotted in the arousalvalence plane where we can see that, for example, high arousal and high valence corresponds to joy or happiness whereas high arousal and low valence might correspond to anger or frustration. ",
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+        "type": "text",
+        "text": "Our data-set was recorded under studio conditions with trained voice actors who were prompted to read dialogues in one of three valences:-2, -1, $+ 2$ and two arousal values:-2 (low), $+ 2$ (high). This was achieved by prompting the actors to read dialogues in either a happy, sad or angry voice at two levels of arousal. This results in 6 possible affective states which we chose to model as discrete and use as our supervision labels. ",
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+    {
+        "type": "text",
+        "text": "3.2 SPEAKING RATE AND F0 VARIATION CONTROL ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "In order to demonstrate that we can control continuous attributes we also created approximate realvalued labels for speaking rate and arousal for all of our data. We generate the approximate speaking rate as number of syllables per second in each utterance. F0, also known as the fundamental frequency, measures the frequency of vibration of the vocal folds during voiced sounds. Variation in F0 is highly correlated with arousal and roughly measures how expressive an utterance is. To create approximate arousal labels we extracted the F0 contour from each of our utterances, using the YIN algorithm (De Cheveigne & Kawahara, 2002), and measured its standard deviation. We then per- ´ formed a whitening transform on these two approximate labels in order to match it to our standard normal prior. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "These artificial labels would of course be cheap to obtain for the entire data-set and would not justify the use of semi-supervision in real applications. But, our objective here is to evaluate/demonstrate the efficacy of semi-supervision rather than to specifically control a particular attribute. We have chosen syllable rate and F0 standard deviation, because they both correspond to subjectively distinct variations of interest, and they are more easily quantifiable than affect and so provide strong evidence of controllability. For the continuous latents we are not only able to interpolate speaking-rates and F0 variations but also to extrapolate outside of our training data. We provide examples on our demo page of samples with significantly greater/lower speed and F0 variation than typically observed in natural speech. ",
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+    {
+        "type": "text",
+        "text": "4 EXPERIMENTS AND RESULTS ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "To evaluate the efficacy of semi-supervised latent variable models for controllable TTS we trained the model described in section 2 on the above data-sets at varying levels of supervision as well as for ",
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+    {
+        "type": "table",
+        "img_path": "images/67e3cdea42c94b11f3962a2e9acf9e9c7eefb18780046b8b52c3c70aa0c5add5.jpg",
+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=\"2\" colspan=\"2\"></td><td colspan=\"3\">Valence</td><td rowspan=\"2\">Arousal low vs. high</td></tr><tr><td>baseline vs.angry</td><td>baseline vs. sad</td><td>baseline vs. happy</td></tr><tr><td rowspan=\"3\">preference score</td><td>27 min (1%)</td><td>-0.20± 0.10</td><td>-0.60± 0.08</td><td>-0.43 ± 0.09</td><td>-0.50 ± 0.09</td></tr><tr><td>135 min (5%)</td><td>-0.74 ± 0.07</td><td>-0.83 ± 0.06</td><td>-0.83 ± 0.06</td><td>-0.57± 0.08</td></tr><tr><td>270 min (10%)</td><td>-0.71 ± 0.07</td><td>-0.86 ± -0.95</td><td>-0.61 ± 0.08</td><td>-0.59 ± 0.08</td></tr></table>",
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+    {
+        "type": "text",
+        "text": "Table 1: Subjective metrics for affect control. Negative is a preference for the controlled model. $+ 1$ and $^ { - 1 }$ indicate a preference for samples A and B, respectively. For valence, raters are told that a sample is intended to convey a particular emotion, e.g. happy, and then presented with sample from baseline without control (A), and controlled model (B), and asked to choose between them. For arousal, raters are told to choose the sample that is more vocally aroused, and presented with controlled samples in low (A) and high (B) arousal. To avoid bias, the orders are randomly altered during rating. We show preference score and $9 5 \\%$ confidence intervals at multiple supervision levels. ",
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+    {
+        "type": "text",
+        "text": "varying settings of the hyperparameters: $\\alpha$ which controls the supervision loss and $\\gamma$ , which over emphasizes supervised training points. We found that a value of $\\alpha = 1$ was optimal for the discrete experiments and $\\alpha = 0$ for the continuous experiments, which corresponds to simply optimizing the ELBO. For each experiment we report the results for the best $\\gamma$ found, and $\\gamma = 1$ . $\\gamma = 1$ corresponds to experiments with no over-weighting of the supervised points. All models were trained using the ADAM optimizer with learning rate of $1 0 ^ { - 3 }$ and run for 300, 000 training steps with a batch size of 256, distributed across 32 Google Cloud TPU chips. All models were implemented using tensorflow 1 (Abadi et al., 2016). ",
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+    {
+        "type": "text",
+        "text": "Assessing the degree of control is challenging as interpreting affect is subjective. We used two objective metrics of control as well as subjective evaluation from human raters and a third objective metric of overall quality. For affect, the first objective metric we introduced was the test-set accuracy of a 6-class affect classifier trained on the ground truth training data and applied to generated samples from the model (shown in figure 4a). The classifier is a convolutional neural network whose structure mirrors the posterior network $q ( \\boldsymbol { z } _ { s } | \\boldsymbol { x } , \\boldsymbol { y } , \\phi )$ and its exact architecture is given in appendix A. We also provide subjective metrics of controllability, shown in table 1. For speaking rate control, we measure the mean syllable rate error on a held out test-set. The syllable rate error is calculated as the absolute difference in syllable rate between the desired syllable rate and that measured from the synthesized sample. We calculate an analogous error rate for F0 variation. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Whilst the two metrics above measure controllability they don’t tell us if this comes at the expense of a degradation in synthesis quality. To probe quality we use three further metrics. The first was MelCepstral-Distortion-Dynamic-Time-Warping (MCD-DTW) (Kubichek, 1993) on a held out test-set, shown in figure 4d. MCD-DTW is a measure of the difference between the ground-truth spectrogram and the synthesized mel spectrogram that is known to correlate well with human perception (Kubichek, 1993). The second metric of quality was crowd sourced mean-opinion-scores (MOS). The third metric of quality is speech recognition word error rate (WER) and character error rate (CER) on audio samples. The MOS and speech recognition results are summarized in table 2. ",
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+        "type": "text",
+        "text": "To demonstrate that semi-supervision by including unlabelled data is beneficial, we also provide MOS and speech recognition errors for fully supervised subsets of the data in table 2. These show that at least close to 5 hours of data is required to train a reasonable quality TTS model, far above the 30 minutes supervision needed to control prosodic aspects of speech. ",
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+        "text": "We provide further details of all of these metrics in the appendix B, and sample spectrograms are provided in appendix C. ",
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+        "type": "text",
+        "text": "5 DISCUSSION ",
+        "text_level": 1,
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+        "text": "The classification accuracy (see figure 4a), subjective metrics (see table 1) and error-rate results (see figure 4b-4c) provide a clear demonstration that using semi-supervised latent variables, we are able to achieve control of both continuous and discrete attributes of speech. There is not a significant degradation in the overall quality and this is evidenced by the mean opinion scores which are above the baseline, Tacotron, and also speech recognition errors (see table 2). We also include a baseline of our Tacotron model augmented only by the unsupervised latent $z _ { s }$ , to aid comparison. ",
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+        "image_caption": [
+            "Figure 4: Objective evaluation metrics as a function of supervision fraction. $1 0 0 \\%$ supervision corresponds to 45 hours of supervised training data and $0 \\%$ supervision corresponds to base tacotron. For MCD-DTW and error-rates lower is better. "
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+        "table_body": "<table><tr><td rowspan=\"2\" colspan=\"4\"></td><td colspan=\"3\">Semi-Supervised (1O% supervision)</td></tr><tr><td colspan=\"2\">continuous latent</td><td>discrete latent</td></tr><tr><td></td><td>ground truth</td><td>baseline</td><td>baseline with zu</td><td>F0</td><td>speaking-rate</td><td>affect</td></tr><tr><td>MOS</td><td>4.52±0.07</td><td>4.09±0.09</td><td>4.24±0.08</td><td>4.28±0.07</td><td>4.16±0.08</td><td>4.17±0.09</td></tr><tr><td>WER</td><td>4.49</td><td>4.93</td><td>4.93</td><td>4.06</td><td>2.53</td><td>6.09</td></tr><tr><td>CER</td><td>2.11</td><td>2.38</td><td>2.22</td><td>1.80</td><td>1.14</td><td>3.12</td></tr></table>",
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+            "Table 2: Metrics of overall quality: Mean Opinion Scores (MOS) alongside $9 5 \\%$ confidence intervals, speech recognition word error rate (WER), and chararacter error rate (CER). The results show no degradation in performance compared to the baseline. "
+        ],
+        "table_footnote": [
+            "Table 3: Metrics of overall quality for fully supervised data at varying data-set sizes, showing significant degradation below 270 minutes. "
+        ],
+        "table_body": "<table><tr><td></td><td>27 min (1%)</td><td>54 min (2%)</td><td>108 min (4%)</td><td>135 min (5%)</td><td>270 min (10%)</td><td>45 hours (100%)</td></tr><tr><td>MOS</td><td>unintelligible</td><td>unintelligible</td><td>3.20±0.13</td><td>3.52±0.11</td><td>4.03±0.08</td><td>4.08±0.09</td></tr><tr><td>WER</td><td>91.95</td><td>96.31</td><td>19.83</td><td>7.55</td><td>5.56</td><td>4.93</td></tr><tr><td>CER</td><td>74.57</td><td>78.9</td><td>12.54</td><td>4.46</td><td>2.79</td><td>2.22</td></tr></table>",
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+        "text": "The MCD-DTW scores for F0 variation and affect are improved at all levels of supervision (figure 4d). Whilst the MCD-DTW is degraded for speaking rate, this is likely a misleading metric when targeting changes in timing as the dynamic-time-warping component of MCD-DTW changes exactly the aspect we wish to control. For speaking rate the combination of MOS and samples is a better indication of the overall quality. We are able to reduce the supervision level to levels as low as $1 \\%$ or 30 minutes and still have a significant degree of control. We show on our demo page4 that even at 3 minutes of supervision we can still achieve control of speaking rate and that we are able to extrapolate outside the range of values seen during training. On the affect data our classification accuracy doesn’t degrade significantly until we reach $1 0 \\%$ (300 minutes) supervision and remains significantly above chance down to levels as low as $1 \\%$ (30 minutes), see figure 4a and table 1. Obtaining 30 minutes of supervised data is likely within reach of most teams constructing TTS systems. Unlike previous work on generative modelling for control (Hsu et al., 2018; Wang et al., 2018), we do not require a post-processing stage to determine what our latent variables control and we can pre-determine what aspects we wish to control through choice of data. By separating our latent variables into those that are partially supervised and those which are fully unsupervised we retain the ability to model other latent aspects of prosody; this means that we can still draw samples of varying prosody whilst holding constant the affect, speaking rate or F0 variation. ",
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+        "text": "We observe the greatest degree of affect control, as measured by classifier accuracy, when $\\alpha = 1$ and $\\gamma = 1 0 0$ . This means that to achieve the highest controllability we needed to 1) provide extra information to our approximate posterior $q ( \\boldsymbol { z } _ { s } | \\boldsymbol { x } , \\boldsymbol { y } , \\phi )$ and 2) to over-represent the supervised data at low levels of supervision. Although both of these hyperparameters have been used in the literature before (Narayanaswamy et al., 2017; Kingma et al., 2014) and shown to be either beneficial or necessary, they aren’t strictly required by our probabilistic framework and so it is worth considering why they are needed. There are three potential sources of error in any generative model trained with SGVB: the model itself may be mis-specified such that the true data-generating distribution is not in the model class, the parametric family chosen to approximate the posterior may be overly restrictive and finally the optimization landscape may contain undesirable local minima. These problems have afflicted previous work with deep latent variable models trained with SGVB, resulting in models that don’t use their latent variables unless trained with complex annealing schedules (Bowman et al., 2015). In our case we believe that the necessity to set $\\alpha$ and $\\gamma$ arises from a combination of model mis-specification and local minima. If $\\alpha$ is set to 0, then at the start of training $q ( \\boldsymbol { z } _ { s } | \\boldsymbol { x } , \\boldsymbol { y } , \\phi )$ is trained only to approximate $p ( z _ { s } | x , y , \\theta _ { 0 } )$ , which is randomly initialized. We found empirically that in our discrete-latent experiments this resulted in $q ( \\boldsymbol { z } _ { s } | \\boldsymbol { x } , \\boldsymbol { y } , \\phi )$ collapsing early in training to a point-mass on a single class. Having ended up in this undesirable local minimum the posterior distribution never recovered, despite this being an obviously poor approximation to the model posterior later in training. The addition of the classification loss and supervision weighting were sufficient to overcome this collapse and allow $q$ to continue to model the posterior. ",
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+        "text": "The optimization landscape is strongly affected by the relative size of the conditional likelihood and KL terms in our objective. These are in turn strongly affected by our choice of conditional independence assumptions and output-distributions. Thus, a natural direction for further work is to increase the expressivity of the conditional likelihood $p ( x | y , z _ { s } , z _ { u } , \\phi )$ to reduce model mis-specification. This could be done by learning the variance of the Laplace-distribution we currently use or by parameterizing more expressive output distributions that do not assume conditional independence across spectrogram channels. We conjecture that with more expressive output distributions, it may be possible to reduce the need for the $\\alpha$ and $\\gamma$ terms in the objective. In this work we chose to use quite simple unconditional diagonal Gaussian priors, as our primary goal was to demonstrate the practicality of semi-supervision. Another natural extension would be to use conditional-priors $p ( z | y )$ and to use more expressive priors such as mixtures as was done in Hsu et al. (2018). ",
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+        "text": "5.1 RELATED WORK ",
+        "text_level": 1,
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+        "text": "There has been enormous recent progress in neural TTS with numerous novel models proposed in recent years to synthesize speech directly from characters or phonemes (Shen et al., 2018; Arik et al., 2017; Gibiansky et al., 2017; Ping et al., 2017; Vasquez & Lewis, 2019; Taigman et al., 2017). Differentiating factors between these models include the degree of parallelism, with some models using Transformer based architectures (Ren et al., 2019), the choice of conditional independence assumptions made (Vasquez & Lewis, 2019) or the number of separately trained components (Gibiansky et al., 2017). Our work here is largely orthogonal to the exact structure of the conditional likelihood $p ( x | y , z _ { s } , z _ { u } )$ and could be combined with all of the above methods. ",
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+        "text": "Much of the recent research focus has been on modeling latent aspects of prosody. Early attempts include Global Style Tokens (Wang et al., 2018) which attempted to learn a trainable set of styleembeddings. Wang et al. (2018) condition the Tacotron decoder on a linear combination of embedding vectors whose weights during training are predicted from the ground-truth spectrogram. ",
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+        "text": "They were able to achieve prosodic control but there is no straightforward way to sample utterances of varying prosody. More recently, attempts have also been made to combine probabilistic latent variable models trained using SGVB (Akuzawa et al., 2018; Wan et al., 2019). These models use a fully unsupervised and non-identifiable approach, which makes it difficult to disentangle or interpret their latent variables for control. Hsu et al. (2018) attempt to overcome this problem by using a Gaussian mixture as the latent prior and so perform clustering in the latent space. Battenberg et al. (2019) introduce a hierarchical latent variable model to separate the modelling of style from prosody. However, all of these methods are fully unsupervised and this results in latents that can be hard to interpret or require complex post-processing. ",
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+        "type": "text",
+        "text": "The work most similar to ours is Wu et al. (2019) which also attempts to achieve affect control using semi-supervision with a heuristic approach based on Global Style Tokens (Wang et al., 2018). Wu et al. (2019) add a cross-entropy objective to the weightings of the style-tokens that encourages them to be one-hot on points with supervision. Similar to our method, they are able to achieve control over affect but unlike our method they do not have a principled probabilistic interpretation nor the ability to simultaneously model aspects of prosody other than emotion. The result is that their method is not able to draw samples of varying prosody for the same utterance with fixed emotion. Furthermore, whilst our method can be applied to both continuous and discrete controllable factors, its not clear how to extend the style-token based approach to handle continuous latent factors. ",
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+    {
+        "type": "text",
+        "text": "In the wider generative modelling literature, the combination of semi-supervision and deep latent variable models was first introduced in Kingma et al. (2014) who focus on using unlabelled data to improve classification accuracy. The potential to use the same technique for controllable generation was recognized by Narayanaswamy et al. (2017) who also provided demonstrations on image synthesis tasks. Since that work, interest in learning disentangled latent variables has grown but generally pursued alternate directions such as re-weighting the ELBO (Higgins et al., 2017), augmenting the objective to encourage factorization (Kim & Mnih, 2018) or using adversarial training (Mathieu et al., 2016). The ability to transfer controllability to speakers for whom we do not have supervision is referred to as domain transfer and our model bears similarities to that introduced by Ilse et al. (2019) but they use a mixture in their latent space more similar to Hsu et al. (2018). ",
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+        "type": "text",
+        "text": "5.2 ETHICAL CONSIDERATIONS ",
+        "text_level": 1,
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+        "text": "As with many advances in speech synthesis, progress in controllability raises the prospect that bad actors may misuse the technology either for misinformation or to commit fraud. Improvements in data efficiency and realism increase these risks and, when publishing, a consideration has to be made as to whether the benefits of the developments outweigh the risks. It is the opinion of the authors in this case that, since the focus of this work is on improved prosody, with potential benefits to human-computer interfaces, the benefits likely outweigh the risks. We nonetheless urge the research community to take seriously the potential for misuse both of this work and broader advances in TTS. ",
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+        "type": "text",
+        "text": "6 CONCLUSION ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "We have shown that the combination of semi-supervised latent variable models with neural TTS presents a practical and principled path towards building speech synthesizers we can control. Unlike previous fully unsupervised methods, we are able to consistently and reliably learn to control predetermined aspects of prosody. Our method can be applied to any latent attribute of speech for which a modest amount of labelling can be obtained, whether it be continuous or discrete. In our experiments we found that 30 minutes of supervision was sufficient, a volume of data that is within the reach of most research teams. We are able to learn to control subtle characteristics of speech such as affect and for continuous attributes we have provided demonstrations of extrapolation to ranges never seen during training, and to speakers with no supervision. Augmenting existing stateof-the-art TTS systems with latent variables does not degrade synthesis quality and we evidence this with crowd sourced mean opinion scores. Unlike similar heuristic methods, our probabilistic formulation, allows us to draw samples of varying prosody whilst holding constant some attribute we wish to control. ",
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Velichko and N. G. Zagoruyko. Automatic recognition of 200 words. International Journal of Man-Machine Studies, 2(3):223–234, 1970.   \nV. Wan, C. Chan, T. Kenter, J. Vit, and R. Clark. CHiVE: Varying prosody in speech synthesis with a linguistically driven dynamic hierarchical conditional variational network. arXiv preprint arXiv:1905.07195, 2019.   \nY. Wang, R. Skerry-Ryan, D. Stanton, Y. Wu, R. J. Weiss, N. Jaitly, Z. Yang, Y. Xiao, Z. Chen, S. Bengio, et al. Tacotron: Towards end-to-end speech synthesis. arXiv preprint arXiv:1703.10135, 2017.   \nY. Wang, D. Stanton, Y. Zhang, R. J. Skerry-Ryan, E. Battenberg, J. Shor, Y. Xiao, F. Ren, Y. Jia, and R. A. Saurous. Style tokens: Unsupervised style modeling, control and transfer in end-to-end speech synthesis. arXiv preprint arXiv:1803.09017, 2018.   \nP. Wu, Z. Ling, L. Liu, Y. Jiang, H. Wu, and L. Dai. End-to-end emotional speech synthesis using style tokens and semi-supervised training. arXiv preprint arXiv:1906.10859, 2019.   \nH. Zen, V. Dang, R. Clark, Y. Zhang, R. J. Weiss, Y. Jia, Z. Chen, and Y. Wu. LibriTTS: A corpus derived from LibriSpeech for text-to-speech. In Interspeech 2019, 2019. ",
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+        "table_caption": [
+            "A NEURAL NETWORK ARCHITECTURE ",
+            "Table 4: Summary of the hyperparameters described below. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Module</td><td>hyperparameters</td></tr><tr><td>Input</td><td>Text normalized phonemes</td></tr><tr><td>Phoneme embedding</td><td>256-D</td></tr><tr><td>Pre-net</td><td>FC-256-Relu-Dropout(0.5)</td></tr><tr><td rowspan=\"4\">CBHG text encoder</td><td>→FC-128-Relu-Dropout(0.5)</td></tr><tr><td>Conv1D bank:K=16,conv-k-128-Relu →Max pooling with stride=1 width=2</td></tr><tr><td>→ Conv1D projections: conv-3-128-Relu</td></tr><tr><td>→ conv-3-128-Linear →Highway net: 4 layers of FC-128-Relu</td></tr><tr><td>Attention type</td><td>→Bidirectional GRU:128 cells 5 component GMM attention w/ softplus (Graves, 2013)</td></tr><tr><td>Attention RNN</td><td>LSTM-256-Zoneout(0.1) →FC-128-tanh</td></tr><tr><td>DecoderRNN</td><td>2-layer residual-LSTM-265-zoneout(0.1)</td></tr><tr><td>Frames-per-timestep (reduction factor) 2</td><td>→FC-80-Linear</td></tr><tr><td>WaveRNN</td><td>5 layers DilatedConv1D-512</td></tr><tr><td rowspan=\"3\">Variational posterior</td><td>-→2 layers TransposeConv + ReLu</td></tr><tr><td>→ GRU-768 conditioned on 5 previous samples →FC-768-relu</td></tr><tr><td>→3 component MoL, 24kHz sample rate Spectrogram</td></tr><tr><td></td><td>→ 6 Conv-layers 32-32-64-64-128-128 →LSTM-128 → FC-128-tanh</td></tr><tr><td>Optimizer</td><td>ADAM with learning rate 1O-³,batch-size 256</td></tr><tr><td>Speaker embedding</td><td>64-D</td></tr></table>",
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+    },
+    {
+        "type": "text",
+        "text": "Sequence-to-sequence model Our sequence-to-sequence network is modelled on Tacotron (Wang et al., 2018) but uses some modifications introduced in Skerry-Ryan et al. (2018). Input to the model consists of sequences of phonemes produced by a text normalization pipeline rather than character inputs. The CBHG text encoder from Wang et al. (2017) is used to convert the input phonemes into a sequence of text embeddings. The phoneme inputs are converted to learned 256-dimensional embeddings and passed through a pre-net composed of two fully connected ReLU layers (with 256 and 128 units, respectively), with dropout of 0.5 applied to the output of each layer, before being fed to the encoder. For multi-speaker models, a learned embedding for the target speaker is broadcastconcatenated to the output of the text encoder. The attention module uses a single LSTM layer with 256 units and zoneout of 0.1 followed by an MLP with 128 tanh hidden units to compute parameters for the monotonic 5-component GMM attention window. We use GMMv2b attention mechanism described in Battenberg et al. (2020). Instead of using the exponential function to compute the shift and scale parameters of the GMM components as in Graves (2013), GMMv2vb uses the softplus function, and also adds initial bias to these parameters, which we found leads to faster alignment and more stable optimization. The attention weights predicted by the attention network are used to compute a weighted sum of output of the text encoder, producing a context vector. The context vector is concatenated with the output of the attention LSTM layer before being passed to the first decoder LSTM layer. The autoregressive decoder module consists of 2 LSTM layers each with 256 units, zoneout of 0.1, and residual connections between the layers. The spectrogram output is produced using a linear layer on top of the 2 LSTM layers, and we use a reduction factor of 2, meaning we predict two spectrogram frames for each decoder step. The decoder is fed the last frame of its most recent prediction (or the previous ground truth frame during training) and the current context as computed by the attention module. Before being fed to the decoder, the previous prediction is passed through a pre-net with the same same structure used before the text encoder above but its own parameters. ",
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+        "type": "text",
+        "text": "CBHG text encoder We reuse the CHGB text encoder introduced in Wang et al. (2018). The text encoder consists of a bank of 1-D convolutional filters, followed by highway networks and a bidirectional gated recurrent unit (GRU) recurrent neural net (RNN). The input sequence is first convolved with K sets of 1-D convolutional filters, where the $k$ -th set contains $C _ { k }$ filters of width $k$ . The convolution outputs are stacked together and further max pooled, preserving time. As in the original paper we use a stride of 1 to preserve the original time resolution. We further pass the processed sequence to a few fixed-width 1-D convolutions, whose outputs are added with the original input sequence via residual connections. Batch normalization is used for all convolutional layers. The convolution outputs are fed into a multi-layer highway network to extract high-level features. Finally, we stack a bidirectional GRU RNN on top to extract sequential features from both forward and backward context. ",
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+    {
+        "type": "text",
+        "text": "Variational posteriors The variational distributions $q ( z _ { s } | x , y )$ and $q ( z _ { u } | x , y , z _ { s } )$ are both structured as diagonal Gaussian distributions whose mean and variance are parameterized by neural networks. For discrete supervision we replace $q ( z _ { s } | x , y )$ by a categorical distribution and use the same network to output just the mean. The input to the distribution starts from the mel spectrogram $x$ and passes it through a stack of 6 convolutional layers, each using ReLU non-linearities, $3 \\mathrm { x } 3$ filters, $2 \\mathbf { x } 2$ stride, and batch normalization. The 6 layers have 32, 32, 64, 64, 128, and 128 filters, respectively. The output of this convolution stack is fed into a unidirectional LSTM with 128 units. We pass the final output of this LSTM (and potentially vectors describing the text and/or speaker) through an MLP with 128 tanh hidden units to produce the parameters of the diagonal Gaussian posterior which we sample from. All but the last linear layer of these networks is shared between the two distributions $q ( z _ { s } | x , y )$ and $q ( z _ { u } | x , y , z _ { s } )$ . The resulting sample is broadcast-concatenated to the output of the text encoder. In our experiments $z _ { u }$ is always 32-dimensional and $z _ { s }$ is either a one-hot vector across 6 classes or a 1 dimensional continuous value. ",
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+        "type": "text",
+        "text": "Conditional inputs When providing information about the text to the variational posterior, we pass the sequence of text embeddings produced by the text encoder to a unidirectional RNN with 128 units and use its final output as a fixed-length text summary that is passed to the posterior MLP. Speaker information is passed to the posterior MLP via a learned speaker embedding. ",
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+    {
+        "type": "text",
+        "text": "WaveRNN We used a WaveRNN model similar to that described in Kalchbrenner et al. (2018) as our vocoder. Our WaveRNN uses discretized mixture of logistics output as described in Salimans et al. (2017) instead of the dual softmax from that paper, and conditions on 5 previous samples at each step instead of only 1 previous. We trained the network to map from synthesized melspectrograms to waveforms, training on 900 sample windows. A conditioning stack of dilated convolution and transpose convolutions is applied to the input spectrogram before tiling to upsample to the audio sample rate. ",
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+    {
+        "type": "text",
+        "text": "B EVALUATION ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "Mel spectrograms The mel spectrograms the model predicts are computed from $2 4 ~ \\mathrm { k H z }$ audio using a frame size of $5 0 \\mathrm { m s }$ , a hop size of $1 2 . 5 \\mathrm { m s }$ , an FFT size of 2048, and a Hann window. From the FFT energies, we compute $8 0 \\mathrm { m e l }$ bins distributed between $8 0 \\mathrm { H z }$ and $1 2 \\mathrm { k H z }$ . ",
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+    {
+        "type": "text",
+        "text": "MCD-DTW To compute mel cepstral distortion (MCD) (Kubichek, 1993), we use the same mel spectrogram parameters described above and take the discrete-cosine-transform to compute the first 13 MFCCs (not including the 0th coefficient). The MCD between two frames is the Euclidean distance between their MFCC vectors. Then we use the dynamic time warping (DTW) algorithm (Velichko & Zagoruyko, 1970) (with a warp penalty of 1.0) to find an alignment between two spectrograms that produces the minimum MCD cost (including the total warp penalty). We report the average per-frame MCD-DTW. ",
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+    {
+        "type": "text",
+        "text": "Affect classifier The affect classifier has a very similar structure to the variational posterior. The input to the classifier starts from the mel spectrogram $x$ and passes it through a stack of 6 convolutional layers, each using ReLU non-linearities, 3x3 filters, $2 \\mathbf { x } 2$ stride, and batch normalization. The 6 layers have 32, 32, 64, 64, 128, and 128 filters, respectively. The output of this convolution stack is fed into a unidirectional LSTM with 128 units. The final output of the LSTM is then passed through a softmax non-linearity to get logits over the training classes. We use the same data splits described in section 3 to train and evaluate the classifier. The classifier is tuned on the validation set achieving $8 4 . 3 3 \\%$ classification accuracy, generalizing well to the test set with $8 3 . 9 4 \\%$ accuracy. ",
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+    {
+        "type": "text",
+        "text": "",
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+    {
+        "type": "text",
+        "text": "Mean opinion scores We use a human rating service similar to Amazon’s Mechanical Turk, with a large pool of English speakers to collect MOS evaluations. The MOS template is shown in figure 5. A human rater is presented with a single speech sample and is asked to rate perceived naturalness on a scale of 1 to 5, where 1 is “Bad” and 5 is “Excellent”. We have selected the utterances of one male, and one female speaker in our test set, totalling 371 utterances to evaluate. For each sample, we collect 1 rating, and no rater is used for more than 6 items in a single evaluation set. In total, 270 unique raters completed the 6 evaluation sets presented in table 2. Since raters are randomly selected for each set, some raters have assessed multiple methods. Across the 6 evaluation sets, the average and median of total number of ratings per rater was 8.24, and 6, respectively. To analyze the data from these subjective tests, we average the scores and compute $9 5 \\%$ confidence intervals. Natural human speech is typically rated around 4.5. Samples used for MOS from our model were drawn using the mean of $z _ { u }$ , whilst sampling $z _ { s }$ . ",
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+    {
+        "type": "text",
+        "text": "Instruction ",
+        "text_level": 1,
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+    },
+    {
+        "type": "text",
+        "text": "IMPORTANT:   \nIn this project, you willisten to audio samples.Please release this task if any of the following is true:   \n1) Youdonothave headphones   \n2) You think you do not have good listening ability   \n3)Thereisconsiderable backgroundnoise (streetoise,loudfan/airconditioner,open TV/radio,people taking,etc)   \n4)Foranyreason,youcan'thear theaudiosamples If youare not sure which type you have,please see thisWikipedia article.   \n2)Please set the volume of youraudiodevice toacomfortable level ",
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+    },
+    {
+        "type": "text",
+        "text": "",
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+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "t thesentence sounded.You should notjudge the grammaror thecontent of the sentence,just howit sounds. ",
+        "bbox": [
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+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "1) Listen to each sample at least twice,with at least a one sec break between them ",
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+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "se spc ",
+        "bbox": [
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+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "e speech sample falls between two levels. ",
+        "bbox": [
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+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "table",
+        "img_path": "images/877eec3934373c8f0c0f81454bd0f881428f949887804828dc2f85b16e5f34ca.jpg",
+        "table_caption": [
+            "Naturalness Scale: "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Score</td><td>Naturalness</td><td>Description</td><td>Reference</td></tr><tr><td>5.0</td><td>Excellent</td><td>Completely natural speech</td><td>Listen</td></tr><tr><td>4.0</td><td>Good</td><td>Mostly natural speech</td><td>Listen</td></tr><tr><td>3.0</td><td>Fair</td><td>Equally natural and unnatural speech</td><td>Listen</td></tr><tr><td>2.0</td><td>Poor</td><td>Mostly unnatural speech</td><td>Listen</td></tr><tr><td>1.0</td><td>Bad</td><td>Completely unnatural speech</td><td>Listen</td></tr></table>",
+        "bbox": [
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+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "How are you listening to the speech sample? ",
+        "text_level": 1,
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+    },
+    {
+        "type": "text",
+        "text": "$1 0$ ii and fans, etc.). Hdi ",
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+    },
+    {
+        "type": "text",
+        "text": "playingiod.) ",
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+    },
+    {
+        "type": "text",
+        "text": "Speech sample (please listen at least twice) ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "image",
+        "img_path": "images/9a6c86d397f0f3c2141c88093c5d3236aff6deacaf8553e165a2715cb76f5244.jpg",
+        "image_caption": [
+            "Figure 5: Mean opinion score (MOS) evaluation template. For each utterance, the human raters assign a 1–5 score of the perceived naturalness, with 1 being “Bad” and 5 being “Excellent”. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Subjective affect control evaluation We use the same rater pool, and the same set of 371 utterances used for the MOS evaluations. The A/B template is shown in figure 6. For each utterance, the human rater is presented with a pair of utterances to choose the one that better conveys the target emotion (e.g., happy in the figure). Both utterances are generated with the same text. To evaluate the control over valence, we present baseline (i.e, no control) against utterances generated in specific valence category (angry, happy and sad). To evaluate the control over arousal, we present samples generated at low arousal against samples generated at high arousal, and ask the rater to choose the utterance that is more vocally aroused. We use mean of $z _ { u }$ to generate all the samples. ",
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+    {
+        "type": "text",
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+        "page_idx": 15
+    },
+    {
+        "type": "image",
+        "img_path": "images/5ff645d5991f5a882e6a826f33757c218f6b3c9859d7a071136e17b7b683fd35.jpg",
+        "image_caption": [],
+        "image_footnote": [],
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+    },
+    {
+        "type": "text",
+        "text": "ltis beti diference,makeaquick intuitive guess. ",
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+        "page_idx": 15
+    },
+    {
+        "type": "text",
+        "text": "",
+        "text_level": 1,
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+    },
+    {
+        "type": "text",
+        "text": "How are you listening to these speech samples? ",
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+        ],
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+    },
+    {
+        "type": "text",
+        "text": "● Headphones, with no noise in the background Headphones,with some low-levelnoise in thebackground Oother ",
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+    },
+    {
+        "type": "text",
+        "text": "MPORTANT:Ifyoudon'thaveheadphonespleasereleasethistasforthereason,\"Idonotmeetheupfrontrequirementsforthistas ",
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+        "page_idx": 15
+    },
+    {
+        "type": "image",
+        "img_path": "images/40e401510743a860818648832370982a2bca41ba89b9dc8873a957c96bd329d3.jpg",
+        "image_caption": [
+            "Figure 6: A/B evaluation affect control evaluation template. The emotion label (Happy in the figure) varies depending on the task. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "C SAMPLE SPECTROGRAMS ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "text",
+        "text": "Controlling affect Table 5 shows the effect of varying the valence and arousal latent variable on the spectrogram and F0 track. We can see that a low valence for sadness corresponds to the flattest F0 track, and high arousal manifests in higher F0 values and variations. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Controlling speaking rate and pitch variations Table 6 shows the effect of varying speaking rate, and F0 variation control variables on a sample spectrogram and F0 track. When controlling speaking rate (first column), the duration reduces as we increase the input speaking rate control, while the F0 variation remains stable. When controlling the F0 variation (second column), the pitch dynamic range increases, while the duration remains constant, which demonstrates controllablity and also some degree of disentanglement. ",
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+    {
+        "type": "text",
+        "text": "D REPRODUCING RESULTS ON LIBRITTS PUBLIC DATASET ",
+        "text_level": 1,
+        "bbox": [
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+        ],
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+    {
+        "type": "text",
+        "text": "To verify the reproduciblity of our results on a public dataset, we trained models to control speaking rate and F0 variation on clean subset of LibriTTS dataset (Zen et al., 2019). We only use the utterances below 5 seconds, which is 62 hours of data. We have done no tuning on this dataset, and directly used the hyperparameters we used for our internal dataset. Figures 7a and 7b show the errors of producing the desired speaking rate, and F0 standard deviation, which generally go down as function of supervision level, with exception of $1 0 \\%$ supervision for controlling F0 variation 5. Given, this is a lower quality dataset, with many more speakers and with much smaller data per speaker and also the fact that we have done zero hyperparameter tuning on this dataset, this result look very encouraging. ",
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+    },
+    {
+        "type": "image",
+        "img_path": "images/7cf32eb806dde73a728066853cbadebc9a267e286893e2cf4719cbbcab0529e7.jpg",
+        "image_caption": [
+            "Table 5: Sample spectrogram and F0 track plots, generated by varying affect labels, valence in y-axis, and arousal in $\\mathbf { X }$ -axis. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        ],
+        "page_idx": 16
+    },
+    {
+        "type": "image",
+        "img_path": "images/74cb0a213e7b8f8f2451c4a4b889014a82ebbe4c1bfdc9b3df4effc481cc1dd9.jpg",
+        "image_caption": [
+            "Figure 7: Objective controllability of speaking rate and F0 variation evaluation metrics presented at multiple supervision levels, on LibriTTS (Zen et al., 2019) datasets. $1 0 0 \\%$ supervision corresponds to 62 hours of supervised data. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        ],
+        "page_idx": 16
+    },
+    {
+        "type": "text",
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+    {
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+        "img_path": "images/7734fdea3db70aaf56bede90d9baec21ab384b45bc4aeb3b88637957c98254a0.jpg",
+        "image_caption": [
+            "Table 6: Sample spectrogram and F0 track plots, generated by varying the speaking rate (first column) and F0 variation (second column). We use standard normal prior for these factors and this table demos varying the control factor from $- 5 \\sigma$ to $5 \\sigma$ , demonstrating the controllability, interpolation and extrapolation of conditional generation, and also disentanglement of these factors. "
+        ],
+        "image_footnote": [],
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+    }
+]