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+# Parallel Bayesian Optimization of Multiple Noisy Objectives with Expected Hypervolume Improvement
+
+Samuel Daulton Facebook, University of Oxford sdaulton@fb.com
+
+Maximilian Balandat Facebook balandat@fb.com
+
+Eytan Bakshy Facebook ebakshy@fb.com
+
+# Abstract
+
+Optimizing multiple competing black-box objectives is a challenging problem in many fields, including science, engineering, and machine learning. Multi-objective Bayesian optimization (MOBO) is a sample-efficient approach for identifying the optimal trade-offs between the objectives. However, many existing methods perform poorly when the observations are corrupted by noise. We propose a novel acquisition function, NEHVI, that overcomes this important practical limitation by applying a Bayesian treatment to the popular expected hypervolume improvement (EHVI) criterion and integrating over this uncertainty in the Pareto frontier. We argue that, even in the noiseless setting, generating multiple candidates in parallel is an incarnation of EHVI with uncertainty in the Pareto frontier and therefore can be addressed using the same underlying technique. Through this lens, we derive a natural parallel variant, $q \mathrm { N E H V I }$ , that reduces computational complexity of parallel EHVI from exponential to polynomial with respect to the batch size. $q \mathrm { N E H V I }$ is one-step Bayes-optimal for hypervolume maximization in both noisy and noiseless environments, and we show that it can be optimized effectively with gradient-based methods via sample average approximation. Empirically, we demonstrate not only that $q \mathrm { N E H V I }$ is substantially more robust to observation noise than existing MOBO approaches, but also that it achieves state-of-the-art optimization performance and competitive wall-times in large-batch environments.
+
+# 1 Introduction
+
+Black-box optimization problems that involve multiple competing noisy objectives are ubiquitous in science and engineering. For example, a real-time communications service may be interested in tuning the parameters of a control policy to adapt video quality in real time in order to maximize video quality and minimize latency [10, 17]. In robotics, scientists may seek to design hardware components that maximize locomotive speed and minimize energy expended [8, 38]. In agriculture, development agencies may seek to balance crop yield and environmental impact [28]. For such multi-objective optimization (MOO) problems, there typically is no single solution that is best with respect to all objectives. Rather, the goal is to identify the Pareto frontier: a set of optimal trade-offs such that improving one objective means deteriorating another. In many cases, the objectives are expensive to evaluate. For instance, randomized trials used in agriculture and the internet industry may take weeks or months to conduct and incur opportunity costs, and manufacturing and testing hardware is both costly and time-consuming. Therefore, it is imperative to be able to identify good trade-offs with as few objective evaluations as possible.
+
+Bayesian optimization (BO), a method for efficient global black-box optimization, is often used to tackle such problems. BO employs a probabilistic surrogate model in conjunction with an acquisition function to navigate the trade-off between exploration (evaluating designs with high uncertainty) and exploitation (evaluating designs that are believed to be optimal). Although a significant number of works have explored multi-objective Bayesian optimization (MOBO), most available methods [3, 39, 51, 60] do not take into account the fact that, in practice, observations are often subject to noise. For example, results of an A/B test are highly variable due to heterogeneity in the underlying user population and other factors. Agricultural trials are affected by the stochastic nature of plant growth and environmental factors such as soil composition or wind currents. In robotics, devices are subject to manufacturing tolerances, and observations of quantities such as locomotive speed and efficiency may be corrupted by measurement error from noisy sensors and environmental factors such as temperature or surface friction. While previous work has shown that a principled treatment of noisy observations can significantly improve optimization performance in the single-objective case [24, 37], this issue is understudied in the multi-objective setting. Furthermore, many applications in which evaluations take a long time require evaluating large batches of candidates in parallel in order to achieve reasonable throughput. For example, when firms optimize systems via A/B tests, it may take several weeks to test any particular configuration. Because of this, large batches of candidate policies are tested simultaneously [36]. In biochemistry and materials design, dozens of tests can be conducted parallel on a single microplate [63]. Even in sophisticated high throughput chemistry settings, these batches may take several hours or days to set up and evaluate [42]. Most existing MOBO methods, however, are either designed for purely sequential optimization [3, 51] or do not scale well to large batch sizes [11].
+
+Contributions: In this work, we propose a novel MOBO algorithm, based on expected hypervolume improvement (EHVI), that scales to highly parallel evaluations of noisy objectives. Our approach is made possible by a general-purpose, differentiable, cached box decomposition (CBD) implementation that dramatically speeds up critical computations needed to account for uncertainty introduced by noisy observations and generate new candidate points for highly parallel batch or asynchronous evaluation. In particular, our CBD-based approach solves the fundamental problem of scaling parallel EHVI-based methods to large batch sizes, reducing time and space complexity from exponential to polynomial. Our proposed algorithm, noisy expected hypervolume improvement (NEHVI), is the onestep Bayes-optimal policy for hypervolume improvement and provides state-of-the-art performance across a variety of benchmarks. To our knowledge, our work provides the most extensive evaluation of noisy parallel MOBO to date. A high-quality implementation of $q \mathrm { N E H V I }$ , as well as many of the baselines considered here, will be made available as open-source software upon publication.
+
+# 2 Preliminaries
+
+Our goal is to find the set of optimal designs $_ { \textbf { \em x } }$ over a bounded set $\mathcal { X } \subset \mathbb { R } ^ { d }$ that maximize one or more objectives $\pmb { f } ( \pmb { x } ) \in \mathbb { R } ^ { M }$ , with no known analytical expression nor gradient information of $f$ .
+
+Multi-Objective Optimization (MOO) aims to identify the set of Pareto optimal objective tradeoffs. We say a solution ${ \pmb f } ( { \pmb x } ) = \left[ f ^ { ( 1 ) } ( { \pmb x } ) , . . . , f ^ { ( M ) } ( { \pmb x } ) \right]$ dominates another solution $\pmb { f } ( \pmb { x } ) \succ \pmb { f } ( \pmb { x } ^ { \prime } )$ if $f ^ { ( m ) } ( { \pmb x } ) \geq f ^ { ( m ) } ( { \pmb x } ^ { \prime } )$ for $m = 1 , . . . , M$ and $\exists m \in \{ 1 , . . . , M \}$ s.t. $f ^ { ( m ) } ( { \pmb x } ) > f ^ { ( m ) } ( { \pmb x } ^ { \prime } )$ . We define the Pareto frontier as ${ \mathcal { P } } ^ { * } = \{ { \pmb { f } } ( { \pmb { x } } ) : { \pmb { x } } \in { \pmb { \chi } }$ , $\nexists \mathbf { \boldsymbol { x } } ^ { \prime } \in \mathcal { X }$ s.t. $f ( { \pmb x } ^ { \prime } ) \succ f ( { \pmb x } ) \}$ , and denote the set of Pareto optimal designs as $\mathcal { X } ^ { \ast } = \{ \pmb { x } : \pmb { f } ( \pmb { x } ) \in \mathcal { P } ^ { * } \}$ . Since the Pareto frontier (PF) is often an infinite set of points, MOO algorithms usually aim to identify a finite approximate PF $\mathcal { P }$ . A natural measure of the quality of a PF is the hypervolume of the region of objective space that is dominated by the PF and bounded from below by a reference point. Provided with the approximate PF, the decision-maker can select a particular Pareto optimal trade-off according to their preferences.
+
+Bayesian Optimization (BO) is a sample-efficient optimization method that leverages a probabilistic surrogate model to make principled decisions to balance exploration and exploitation [19, 50]. Typically, the surrogate is a Gaussian Process (GP), a flexible, non-parametric model known for its well-calibrated predictive uncertainty [47]. To decide which points to evaluate next, BO employs an acquisition function $\alpha ( \cdot )$ that specifies the value of evaluating a set of new points $_ { \textbf { \em x } }$ based on the surrogate’s predictive distribution at . While evaluating the true black-box function $f$ is timeconsuming or costly, evaluating the surrogate is cheap and relatively fast; therefore, numerical optimization can be used to find the maximizer of the acquisition function $\pmb { x } ^ { * } = \arg \operatorname* { m a x } _ { \pmb { x } \in \mathcal { X } } \alpha ( \pmb { x } )$ to evaluate next on the black-box function. BO sequentially selects new points to evaluate and updates the model to incorporate the new observations.
+
+Evolutionary algorithms (EAs) such as NSGA-II [12] are a popular choice for solving MOO problems (see Zitzler et al. [67] for a review of various other approaches). However, EAs generally suffer from high sample complexity, rendering them infeasible for optimizing expensive-to-evaluate black-box functions. Multi-objective Bayesian optimization (MOBO), which combines a Bayesian surrogate with an acquisition function designed for MOO, provides a much more sample-efficient alternative.
+
+# 3 Related Work
+
+Methods based on hypervolume improvement (HVI) seek to expand the volume of the objective space dominated by the Pareto frontier. Expected hypervolume improvement (EHVI) [16] is a natural extension of the popular expected improvement (EI) [29] acquisition function to the MOO setting. Recent work has led to efficient computational paradigms using box decomposition algorithms [59] and practical enhancements such as support for parallel candidate generation and gradient-based acquisition optimization [11, 58]. However, EHVI still suffers from some limitations, including (i) the assumption that observations are noise-free, and (ii) the exponential scaling of its batch variant, $q \mathrm { E H V I }$ , in the batch size $q$ , which precludes large-batch optimization. DGEMO [39] is a recent method for parallel MOBO that greedily maximizes HVI while balancing the diversity of the design points being sampled. Although DGEMO scales well to large batch sizes, it does not account for noisy observations. TSEMO [5] is a Thompson sampling (TS) heuristic that can acquire batches of points by optimizing a random fourier feature (RFF) [46] approximation of a GP surrogate using NSGA-II and selecting a subset of points from the EA’s population to sequentially greedily maximize HVI. This heuristic approach for maximizing HVI currently has no theoretical guarantees and relies on zeroth-order optimization methods, which tend to be slower and exhibit worse optimization performance than gradient-based approaches.
+
+Entropy-based methods such as PESMO [25], MESMO [3], and PFES [51] are an alternative to EHVI. Of these three methods, PESMO is the only one that accounts for observation noise. However, PESMO involves intractable entropy computations and therefore relies on complex approximations, as well as challenging and time-consuming numerical optimization procedures [25]. Garrido-Merchán & Hernández-Lobato [21] recently proposed an extension to PESMO that supports parallel candidate generation. However, the authors of this work provide limited evaluation and have not provided code to reproduce their results.1
+
+MOO can also be cast into a single-objective problem by applying a random scalarization of the objectives. ParEGO maximizes the expected improvement using random augmented Chebyshev scalarizations [32]. MOEA/D-EGO [64] extends ParEGO to the batch setting using multiple random scalarizations and the genetic algorithm MOEA/D [65] to optimize these scalarizations in parallel. Recently, $q \mathrm { P a r E G O }$ , another batch variant of ParEGO was proposed that uses compositional Monte Carlo objectives and sequential greedy candidate selection [11]. Additionally, the authors proposed a noisy variant, qNParEGO, but the empirical evaluation of that variant was limited. TS-TCH [45] combines random Chebyshev scalarizations with Thompson sampling [54], which is naturally robust to noise when the objective is scalarized. Golovin & Zhang [23] propose to use a hypervolume scalarization with the property that the expected value of the scalarization over a specific distribution of weights is equivalent to the hypervolume indicator. The authors propose a upper confidence bound algorithm using randomly sampled weights, but provide a very limited empirical evaluation.
+
+Many prior attempts by the simulation community to handle MOO with noisy observations found that accounting for the noise did not improve optimization performance: Horn et al. [26] suggest that the best approach is to ignore noise, and Koch et al. [33] concluded that further research was needed to determine if modeling techniques such as re-interpolation could improve BO performance with noisy observations. In contrast, we find that accounting for noise does substantially improve performance in noisy settings.
+
+Lastly, previous works have considered methods for quantifying and monitoring uncertainty in the Pareto frontiers during the optimization [4, 7]. In contrast, we provide a solution to performing MOBO in noisy settings, rather than purely reasoning about the uncertainty in the Pareto frontier.
+
+# 4 Background on Expected Hypervolume Improvement
+
+In this section, we review hypervolume, hypervolume improvement, and expected hypervolume improvement as well as efficient methods for computing these metrics using box decompositions.
+
+Definition 1. The hypervolume indicator (HV) of a finite approximate Pareto frontier $\mathcal { P }$ is the $M$ -dimensional Lebesgue measure $\lambda _ { M }$ of the space dominated by $\mathcal { P }$ and bounded from below by $a$ reference point. $\begin{array} { r } { \pmb { r } \in \mathbb { R } ^ { M } \colon \mathrm { H V } ( \mathcal { P } | \pmb { r } ) = \lambda _ { M } \big ( \bigcup _ { \pmb { v } \in \mathcal { P } } [ \pmb { r } , \pmb { v } ] \big ) } \end{array}$ , where $[ r , v ]$ denotes the hyper-rectangle bounded by vertices $\mathbfit { \Delta } \mathbf { r }$ and $\textbf {  { v } }$ .
+
+As in previous work, we assume that the reference point $\pmb { r }$ is known and specified by the decision maker [58].
+
+Definition 2. The hypervolume improvement (HVI) of a set of points ${ \mathcal { P } } ^ { \prime } w . r . t .$ . an existing approximate Pareto frontier $\mathcal { P }$ and reference point $\pmb { r }$ is defined $a s ^ { 2 } \operatorname { H V I } ( \mathcal { P } ^ { \prime } | \mathcal { P } , \boldsymbol { r } ) = \operatorname { H V } ( \mathcal { P } \cup \mathcal { P } ^ { \prime } | \boldsymbol { r } ) - \operatorname { H V } ( \mathcal { P } | \boldsymbol { r } ) .$
+
+Computing HV requires calculating the volume of a typically non-rectangular polytope and is known to have time complexity that is super-polynomial in the number of objectives [59]. An efficient approach for computing HV is to (i) decompose the region that is dominated by the Pareto frontier $\mathcal { P }$ and bounded from below by the reference point $\mathbfit { \Delta } \mathbf { r }$ into disjoint axis-aligned hyperrectangles [34], (ii) compute the volume of each hyperrectangle in the decomposition, and (iii) sum over all hyperrectangles. So-called box decomposition algorithms have also been applied to partition the region that is not dominated by the Pareto frontier $\mathcal { P }$ , which can be used to compute the HVI from a set of new points [15, 59]. See Appendix $\mathbf { B }$ for further details.
+
+Expected Hypervolume Improvement: Since function values at unobserved points are unknown in black-box optimization, so is the HVI of an out-of-sample point. However, in BO the probabilistic surrogate model provides a posterior distribution $p ( \pmb { f } ( \pmb { x } ) | \bar { \mathcal { D } } )$ over the function values for each $_ { \textbf { \em x } }$ , which can be used to compute the expected hypervolume improvement (EHVI) acquisition function: $\alpha _ { \mathrm { E H V I } } ( \pmb { x } | \mathcal { P } ) = \mathbb { E } \big [ \mathrm { H V I } ( \mathbf { \dot { f } } ( \pmb { x } ) | \mathcal { P } ) \big ]$ . Although $\alpha _ { \mathrm { E H V I } }$ can be expressed analytically when (i) the objectives are assumed to be conditionally independent given $_ { \textbf { \em x } }$ and (ii) the candidates are generated and evaluated sequentially [58], Monte Carlo (MC) integration is commonly used since it does not require either assumption [16]. The more general parallel variant using MC integration is given by
+
+$$
+\alpha _ { q \mathrm { E H V I } } \big ( \mathcal { X } _ { \mathrm { c a n d } } \big | \mathcal { P } \big ) \approx \hat { \alpha } _ { q \mathrm { E H V I } } \big ( \mathcal { X } _ { \mathrm { c a n d } } \big | \mathcal { P } \big ) = \frac { 1 } { N } \sum _ { t = 1 } ^ { N } \mathrm { H V I } \big ( \tilde { f } _ { t } \big ( \mathcal { X } _ { \mathrm { c a n d } } \big ) \big | \mathcal { P } \big ) ,
+$$
+
+where $\tilde { f } _ { t } \sim p ( f | \mathcal { D } )$ for $t = 1 , . . . , N$ and $\mathcal { X } _ { \mathrm { c a n d } } = \{ x _ { i } \} _ { i = 1 } ^ { q }$ [11]. The same box decomposition algorithms used to compute HVI can be used to compute EHVI (either analytic or via MC) using piece-wise integration. EHVI computation is agnostic to the choice of box decomposition algorithm (and can also use approximate methods [9]). Similar to EI in the single-objective case, EHVI is a one-step Bayes-optimal algorithm for maximizing hypervolume in the MOO setting under the following assumptions: (i) only a single design will be generated and evaluated, (ii) the observations are noise-free, (iii) the final approximate Pareto frontier (and final design that will be deployed) will be drawn from the set of observed points [19].
+
+# 5 Expected Hypervolume Improvement with Noisy Observations
+
+We consider the case that frequently arises in practice where we only receive noisy observations ${ \pmb y } _ { i } = f ( { \pmb x } _ { i } ) + \epsilon _ { i }$ , $\epsilon _ { i } \sim \mathcal { N } ( 0 , \mathbf { \bar { \Sigma } } _ { i } )$ , where $\Sigma _ { i }$ is the noise covariance. In this setting, EHVI is no longer (one-step) Bayes-optimal. This is because we can no longer compute the true Pareto frontier ${ \mathcal { P } } _ { n } { \overset { \cdot } { = } } \{ f ( { \pmb x } ) \mid { \overset { \cdot } { \pmb x } } \in { \dot { X } } _ { n }$ , $\bar { \nexists } \boldsymbol { x ^ { \prime } } \in X _ { n }$ s.t. $f ( { \pmb x } ^ { \prime } ) \succ f ( { \pmb x } ) \}$ over the previously evaluated points $X _ { n } =$ $\{ { \pmb x } _ { i } \} _ { i = 1 } ^ { n }$ . Simply using the observed Pareto frontier, $\mathcal { V } _ { n } = \{ \pmb { y } \vert \pmb { y } \in Y _ { n }$ , $\bar { \nexists } \ : y ^ { \prime } \in Y _ { n }$ s.t. $y ^ { \prime } \succ y , y \}$ where $\bar { Y } _ { n } = \{ { \pmb y } _ { i } \} _ { i = 1 } ^ { n }$ , can have strong detrimental effects on optimization performance. This is illustrated in Figure 1, which shows how EHVI is misled by noisy observations that appear to be Pareto optimal. EHVI proceeds to spend its evaluation budget trying to optimize noise, resulting in a clumped Pareto frontier that lacks diversity. Although the posterior mean could serve as a "plug-in" estimate of the true function values at the observed points and provide some regularization [61], we find that this heuristic also leads to clustered Pareto frontiers (EHVI-PM in Fig. 1). Similar patterns emerge with DGEMO (which does not account for noise), and other baselines that utilize the posterior mean rather than the observed values when computing hypervolume improvement (see Appendix H). To our knowledge, all previous work on EHVI assumes that observations are noiseless [16, 58] or imputes the unknown true function values with the posterior mean.
+
+![](images/956f97dc2146646d797257b5483737aca4c5f1c6a8ae6460f9c3c6c2a9f3f493.jpg)  
+Figure 1: An illustration of the effect of noisy observations on the true noiseless Pareto frontiers identified by NEHVI (our proposed algorithm), EHVI, and EHVI-PM, which uses the modeled posterior mean as point estimate of the true in-sample function values. All algorithms are tested on a BraninCurrin synthetic problem, where observations are corrupted with zero-mean, additive Gaussian noise with a standard deviation of $5 \%$ of the range of respective objective. All methods use sequential $( q = 1 )$ ) optimization. See Appendix G for details.
+
+# 5.1 A Bayes-optimal algorithm for hypervolume maximization in noisy environments
+
+In contrast with EHVI(-PM), we instead approach the problem of hypervolume maximization under noisy observations from a Bayesian perspective and derive a novel one-step Bayes-optimal expected hypervolume improvement criterion that iterates the expectation over the posterior distribution $p \big ( \bar { f } ( X _ { n } ) | \mathcal { D } _ { n } \big )$ of the function values at the previously evaluated points $X _ { n }$ given noisy observations $\mathcal { D } _ { n } = \{ \mathbf { { \boldsymbol { x } } } _ { i } , \mathbf { { \boldsymbol { y } } } _ { i } , ( \Sigma _ { i } ) \} _ { i = 1 } ^ { n }$ . Our acquisition function, noisy expected hypervolume improvement (NEHVI), is defined as
+
+$$
+\alpha _ { \mathrm { N E H V I } } ( { \pmb x } ) = \int \alpha _ { \mathrm { E H V I } } ( { \pmb x } | \mathcal { P } _ { n } ) p ( { \pmb f } | \mathcal { D } _ { n } ) d { \pmb f }
+$$
+
+where $P _ { n }$ denotes the Pareto frontier over $f ( X _ { n } )$ .
+
+By integrating over the uncertainty in the function values at the observed points, NEHVI retains one-step Bayes-optimality in noisy environments (in noiseless environments, NEHVI is equivalent to EHVI). Empirically, Figure 1 shows that NEHVI is robust to noise and identifies a well-distributed Pareto frontier with no signs of clumping, even under very noisy observations.3
+
+The integral in (2) is analytically intractable, but can easily be approximated using MC integration. Let $\tilde { \pmb { f } } _ { t } \sim p ( \pmb { f } | D _ { n } )$ for $t = 1 , . . . N$ be samples from the posterior, and let $\mathcal { P } _ { t } ~ { = } ~ \{ \tilde { f } _ { t } ( { \pmb x } ) \} ~ \mathbf { { \sigma } } | ~ { \pmb x } ~ \in$ $X _ { n } , \tilde { { f } } _ { t } ( { \pmb x } ) \succ \tilde { { f } } _ { t } ( { \pmb x } ^ { \prime } ) \forall { \pmb x } ^ { \prime } \in X _ { n } \}$ be the Pareto frontier over the previously evaluated points under the sampled function $\tilde { \pmb { f } } _ { t }$ . Then, $\begin{array} { r } { \alpha _ { \mathrm { N E H V I } } ( \pmb { x } ) \approx \frac { 1 } { N } \sum _ { t = 1 } ^ { N } \alpha _ { \mathrm { E H V I } } ( \pmb { x } | \mathcal { P } _ { t } ) } \end{array}$ . Using MC integration, we can compute the inner expectation in $\alpha _ { \mathrm { E H V I } }$ simultaneously using samples from the joint posterior $\tilde { \pmb f _ { t } } ( X _ { n } , \pmb x ) \sim p ( \pmb f ( X _ { n } , \pmb x ) | \mathcal D _ { n } )$ over $_ { \textbf { \em x } }$ and $X _ { n }$ :
+
+$$
+\hat { \alpha } _ { \mathrm { N E H V I } } ( \pmb { x } ) = \frac { 1 } { N } \sum _ { t = 1 } ^ { N } \mathrm { H V I } ( \tilde { \pmb { f } _ { t } } ( \pmb { x } ) | \mathcal { P } _ { t } ) .
+$$
+
+See Appendix B for details on computing (3) using box decompositions. Note that this “full-MC” variant of NEHVI does not require objectives to be modeled independently, and supports multi-task covariance functions across correlated objectives.
+
+# 5.2 Parallel Noisy Expected Hypervolume Improvement
+
+Generating and evaluating batches of candidates is imperative to achieving adequate throughput in many real-world scenarios. qNEHVI can naturally be extended to the parallel (asynchronous or batch) setting by evaluating HVI with respect to a batch of $q$ points $\mathcal { X } _ { \mathrm { c a n d } } = \{ \pmb { x } _ { i } \} _ { i = 1 } ^ { q }$
+
+$$
+\varepsilon _ { q \mathrm { N E H V I } } ( \mathcal { X } _ { \mathrm { c a n d } } ) = \int \alpha _ { q \mathrm { E H V I } } ( \mathcal { X } _ { \mathrm { c a n d } } | \mathcal { P } _ { n } ) p ( f | \mathcal { D } _ { n } ) d f \approx \hat { \alpha } _ { q \mathrm { N E H V I } } ( \mathcal { X } _ { \mathrm { c a n d } } ) = \frac { 1 } { N } \sum _ { t = 1 } ^ { N } \mathrm { H V I } ( \tilde { f } _ { t } ( \mathcal { X } _ { \mathrm { c a n d } } ) | \mathcal { P } _ { t } )
+$$
+
+Since optimizing $q$ candidates jointly is a difficult numerical optimization problem over a qddimensional domain, we use a sequential greedy approximation in the parallel setting and solve a sequence of $q$ simpler optimization problems with $d$ dimensions, which been shown empirically to improve optimization performance [57]. While selecting candidates according to a “sequential greedy” policy does not guarantee that the selected batch of candidates is a maximizer of the $\alpha _ { q \mathrm { N E H V I } }$ the submodularity of $\alpha _ { q \mathrm { N E H V I } }$ allows us to bound the regret of this approximation to be no more than $\frac { 1 } { e } \alpha _ { q \mathrm { N E H V I } } ^ { * }$ , where $\begin{array} { r } { \alpha _ { q \mathrm { N E H V I } } ^ { * } = \operatorname* { m a x } _ { \mathcal { X } _ { \mathrm { c a n d } } \in \mathcal { X } } \alpha _ { q \mathrm { N E H V I } } ( \mathcal { X } _ { \mathrm { c a n d } } ) } \end{array}$ (see Appendix F).
+
+3This noise level is ${ 5 } \mathbf { x }$ greater than the ones considered by previous works that evaluate noisy MOBO [25].
+
+# 6 Efficient Evaluation with Cached Box Decompositions
+
+Although $\hat { \alpha } _ { \mathrm { N E H V I } } ( \pmb { x } )$ in (3) has a concise mathematical form, computing it requires determining the Pareto frontier $\mathcal { P } _ { t }$ under each sample $\tilde { f } _ { t }$ for $t = 1 , . . . , N$ and then partitioning the region that is not dominated by $\mathcal { P } _ { t }$ into disjoint hyperrectangles $\{ S _ { k _ { t } } \} _ { k _ { t } = 1 } ^ { K _ { t } }$ . Optimizing the unbiased MC estimator of $\alpha _ { \mathrm { { N E H V I } } }$ would require re-sampling $\{ \tilde { f } _ { t } \} _ { t = 1 } ^ { N }$ at each evaluation of $\alpha _ { \mathrm { { N E H V I } } }$ . However, computing the Pareto frontier and performing a box decomposition under each of the $N$ samples during every evaluation of $\alpha _ { \mathrm { N E H V I } }$ in the inner optimization loop $\begin{array} { r } { ( \pmb { x } ^ { * } = \arg \operatorname* { m a x } _ { \pmb { x } } \alpha _ { \mathrm { N E H V I } } ( \pmb { x } | \mathcal { D } _ { n } ) ) } \end{array}$ would be prohibitively expensive. This is because box decomposition algorithms have super-polynomial time complexity in the number of objectives [59]. We instead propose an efficient alternative computational technique for repeated evaluations of EHVI with uncertain Pareto frontiers.
+
+Cached Box Decompositions: For repeated evaluations of the integral in (2), we use a set of fixed samples $\{ \tilde { f } _ { t } ( X _ { n } ) \} _ { t = 1 } ^ { N }$ , which allows us to compute the Pareto frontiers and box decompositions once, and cache them for the entirety of the acquisition function optimization, thereby making those two computationally intensive operations a one-time cost per BO iteration.4 We refer to this approach as using cached box decompositions (CBD). The method of optimizing over fixed random samples is known as sample average approximation (SAA) [2].
+
+Conditional Posterior Sampling: Under the CBD formulation, computing $\hat { \alpha } _ { \mathrm { N E H V I } } ( \pmb { x } )$ with joint samples from $\tilde { \pmb f _ { t } } ( X _ { n } , \pmb x ) \sim p ( \pmb f ( X _ { n } , \pmb x ) | \mathcal D _ { n } )$ requires sampling from the conditional distributions
+
+$$
+\tilde { { f } } _ { t } ( x ) \sim p \big ( f ( x ) | f ( X _ { n } ) = \tilde { f } _ { t } ( X _ { n } ) , \mathcal { D } _ { n } \big ) ,
+$$
+
+where $t = 1 , . . . , N$ and $\{ \tilde { f } _ { t } ( X _ { n } ) \} _ { t = 1 } ^ { N }$ are the realized samples at the previously evaluated points. For multivariate Gaussian posteriors (as is the case with GP surrogates), we can sample from $p ( f ( X _ { n } ) | \mathcal { D } _ { n } )$ via the reparameterization trick [30] by evaluating $\begin{array} { r } { \tilde { \pmb f } _ { t } ( { \pmb x } ) = { \pmb \mu } _ { n } + L _ { n } ^ { T } \zeta _ { n , t } } \end{array}$ , where $\zeta _ { n , t } \sim \mathcal { N } ( \mathbf { 0 } , I _ { n M } )$ , $\pmb { \mu _ { n } } \in \mathbb { R } ^ { n M }$ is the posterior mean, and $L _ { n } \in \mathbb { R } ^ { n M \times n M }$ is a lower triangular root decomposition of the posterior covariance matrix, typically a Cholesky decomposition. Given $L _ { n }$ , we can obtain a root decomposition $L _ { n } ^ { \prime }$ of the covariance matrix of the joint posterior $p ( f ( X _ { n } , \pmb { x } ) | \mathcal { D } _ { n } )$ by performing efficient low-rank updates [44]. Given $L _ { n } ^ { \prime }$ and the posterior mean of $p ( f ( X _ { n } , \pmb { x } ) | \mathcal { D } _ { n } )$ , we can sample from (5) via the reparameterization trick by augmenting the existing base samples $\zeta _ { n , t }$ with $M$ new base samples for the new point.
+
+# 6.1 Efficient Sequential Greedy Batch Selection using CBD
+
+The CBD technique addresses the general problem of inefficient repeated evaluations of EHVI with uncertain Pareto frontiers. In this section, we show that sequential greedy batch selection (with both $q \mathrm { E H V I }$ and $q \mathrm { N E H V I }$ is an incarnation of EHVI with uncertain Pareto frontiers.
+
+The original formulation of parallel EHVI in Daulton et al. [11] uses the inclusion-exclusion principle (IEP), which involves computing the volume jointly dominated by each of the $2 ^ { q } - 1$ nonempty subsets of points in $\mathcal { X } _ { \mathrm { c a n d } }$ . However, using large batch sizes is not computationally feasible under this formulation because time and space complexity are exponential in $q$ and multiplicative in the number of hyperrectangles in the box decomposition [11] (see Appendix D for a complexity analysis). Although $q \mathrm { E H V I }$ is optimized using sequential greedy batch selection, the IEP is used over all candidates $\pmb { x } _ { 1 } , . . . , \pmb { x } _ { i }$ when selecting candidate $i$ . Although the IEP could similarly be used to compute $q \mathrm { N E H V I }$ , we instead leverage CBD, which yields a sequential greedy approximation of the joint (noisy) EHVI that is mathematically equivalent to the IEP formulation, but significantly reduces computational overhead. That is, the IEP and CBD approaches produce exactly the same acquisition value for a given set of points $\mathcal { X } _ { \mathrm { c a n d } }$ , but the IEP and the CBD approaches have exponential and polynomial time complexities in $q$ , respectively.
+
+When selecting $\mathbf { \Delta } _ { \mathbf { \mathcal { X } } _ { i } }$ for $i \in \{ 2 , \ldots , q \}$ , all $\mathbf { \Delta } _ { \mathbf { \mathcal { X } } _ { j } }$ for which $j < i$ have already been selected and are therefore held constant. Thus, we can decompose $q \mathrm { N E H V I }$ into the $q \mathrm { N E H V I }$ from the previously selected candidates $\pmb { x } _ { 1 } , \ldots , \pmb { x } _ { i - 1 }$ and NEHVI from $\mathbf { \boldsymbol { x } } _ { i }$ given the previously selected candidates
+
+![](images/a468f1bb1946757538c10a82e4cba5957f0190cedcd03a0cdcef5e345becbb5b.jpg)  
+Figure 2: Acquisition optimization wall time under a sequential greedy approximation using L-BFGS-B. CBD enables scaling to much larger batch sizes $q$ than using the IEP and avoids running out-of-memory (OOM) on a GPU. Independent GPs are used for each outcome. The Pareto frontier of of the 2-objective, 6- dimensional DTLZ2 problem [13] is initialized with 20 points. Wall times were measured on a Tesla V100 SXM2 GPU (16GB RAM) and a $2 \mathbf { x }$ Intel Xeon 6138 CPU $@$ 2GHz (251GB RAM). See Appendix H.2 for results with more objectives.
+
+$$
+\tilde { \tau } _ { q \mathrm { N E H V I } } ( \{ x _ { j } \} _ { j = 1 } ^ { i } ) = \frac { 1 } { N } \sum _ { t = 1 } ^ { N } \mathrm { H V I } \big ( \{ \tilde { f } _ { t } ( x _ { j } ) \} _ { j = 1 } ^ { i - 1 } \big \} | \mathcal { P } _ { t } \big ) + \frac { 1 } { N } \sum _ { t = 1 } ^ { N } \mathrm { H V I } \big ( \tilde { f } _ { t } ( x _ { i } ) | \mathcal { P } _ { t } \cup \{ \tilde { f } _ { t } ( x _ { j } ) \} _ { j = 1 } ^ { i - 1 } \big \} \big )
+$$
+
+Note that the first term on the right hand side is constant, since $\{ { \pmb x } _ { j } \} _ { j = 1 } ^ { i - 1 }$ and $\{ \tilde { f } _ { t } ( \pmb { x } _ { j } ) \} _ { j = 1 } ^ { i - 1 }$ are fixed for all $t = 1 , . . . , N$ . The second term is $\hat { \alpha } _ { \mathrm { N E H V I } } ( \pmb { x } _ { i } )$ , where the NEHVI is taken with respect to the Pareto frontier across $\pmb { f } ( X _ { n } , \pmb { x } _ { 1 } , . . . , \pmb { x } _ { i - 1 } )$ and computed using the fixed samples $\{ \tilde { f } _ { t } ( X _ { n } , \pmb { x } _ { 1 } , . . . \pmb { x } _ { i - 1 } ) \} _ { t = 1 } ^ { N }$ . To compute the second term when selecting candidate $\mathbf { \Delta } _ { \mathbf { \mathcal { X } } _ { i } }$ , the $N$ Pareto frontiers and CBDs are updated to include $\{ \tilde { f } _ { t } ( X _ { n } , \pmb { x } _ { 1 } , . . . \pmb { x } _ { i - 1 } ) \} _ { t = 1 } ^ { N }$ . As in the sequential $q = 1$ setting, the box decompositions are only computed and cached while selecting each candidate point. See Appendix C.2 for a derivation of (6). Although we have focused on $q \mathrm { N E H V I }$ in the above, the CBD formulation for $q \mathrm { E H V I }$ is obtained by simply replacing $\mathcal { P } _ { t }$ with the Pareto frontier over the observed values ${ \mathcal { V } } _ { n }$ .
+
+Despite computing $N$ box decompositions when selecting each candidate $\mathbf { \Delta } _ { \mathbf { \mathcal { X } } _ { i } }$ for $i = 2 , . . . , q$ , the CBD approach reduces the time and space complexity from exponential (under the IEP) to polynomial in $q$ (see Appendix $\mathrm { D }$ for details on time and space complexity). Figure 2 shows the total acquisition optimization time (including box decompositions) for various batch sizes and demonstrates that using CBD allows to scale to batch sizes that are completely infeasible when using IEP.
+
+# 7 Optimizing NEHVI
+
+Differentiability: Importantly, $\hat { \alpha } _ { \mathrm { N E H V I } } ( \pmb { x } )$ is differentiable w.r.t. $_ { \textbf { \em x } }$ . Although determining the Pareto frontier and computing the box decompositions are non-differentiable operations, these operations do not involve $_ { \textbf { \em x } }$ , even when re-sampling from the joint posterior $p ( f ( X _ { n } , \pmb { x } ) | \mathcal { D } _ { n } )$ . Exact sample-path gradients of $\nabla _ { \pmb { x } } \hat { \alpha } _ { \mathrm { N E H V I } } ( \pmb { x } )$ can easily be computed using auto-differentiation in modern computational frameworks. This enables efficient gradient-based optimization of $q \mathrm { N E H V I }$ .5
+
+SAA Convergence Results: In addition to approximating the outer expectation over $f ( X _ { n } )$ with fixed posterior samples, we can similarly fix the base samples used for the new candidate point $_ { \textbf { \em x } }$ This approach yields a deterministic acquisition function, which enables using (quasi-) higher-order optimization methods to obtain fast convergence rates for acquisition optimization [2]. Importantly, we prove that the theoretical convergence guarantees on acquisition optimization under the SAA approach proposed by Balandat et al. [2] also hold for NEHVI.
+
+Theorem 1. Suppose $\mathcal { X }$ is compact and $f$ has a multi-output $G P$ prior with continuously differentiable mean and covariance functions. Let $X _ { n } = \{ \pmb { x } _ { i } \} _ { i = 1 } ^ { n }$ denote the previously evaluated points and $\{ \zeta \} _ { t = 1 } ^ { N }$ be base samples $\zeta \sim \mathcal { N } ( \mathbf { 0 } , I _ { ( n + 1 ) M } )$ . Let $\hat { \alpha }$ NEHVI denote the deterministic acquisition function computed using $\{ \zeta \} _ { t = 1 } ^ { N }$ as $\hat { \alpha } _ { \mathrm { N E H V I } } ^ { N }$ and define $S ^ { * } : = \arg \operatorname* { m a x } _ { { \pmb x } \in { \pmb X } }$ $\alpha _ { \mathrm { N E H V I } } ( \pmb { x } )$ to be the set of maximizers of $\alpha _ { \mathrm { N E H V I } } ( \pmb { x } )$ over $\mathcal { X }$ . Suppose $\hat { \pmb { x } } _ { N } ^ { * } \in \arg \operatorname* { m a x } _ { { \pmb { x } } \in { \mathcal { X } } } \hat { \alpha } _ { \mathrm { N E H V I } } ^ { N } ( { \pmb { x } } )$ . Then $( l )$ $\hat { \alpha } _ { \mathrm { N E H V I } } ^ { N } ( \hat { \pmb { x } } _ { N } ^ { * } )  \alpha _ { \mathrm { N E H V I } } ( \pmb { x } _ { N } ^ { * } )$ almost surely, and (2) $\mathrm { d i s t } ( \hat { \pmb { x } } _ { N } ^ { * } , S ^ { * } )  0$ , where d $l i s t ( \hat { \pmb { x } } _ { N } ^ { * } , S ^ { * } ) : =$ $\mathrm { i n f } _ { { \pmb x } \in S ^ { * } } \| \hat { \pmb x } _ { N } ^ { * } - { \pmb x } \|$ is the Euclidean distance between $\hat { \pmb { x } } _ { N } ^ { * }$ and the set $S ^ { * }$ .
+
+Theorem 1 also holds in the parallel setting, so $q \mathrm { N E H V I }$ enjoys the same convergence guarantees as NEHVI on acquisition optimization under the SAA. See Appendix $\mathrm { E }$ for further details and proof.
+
+# 8 Approximation of $q$ NEHVI using Approximate GP Sample Paths
+
+Although CBD yields polynomial complexity of $q \mathrm { N E H V I }$ with respect to $q$ (rather than exponential complexity with the IEP), it still requires computing $N$ box decompositions and repeatedly evaluating the joint posterior over ${ \pmb f } ( X _ { n } , \{ { \pmb x } _ { j } \} _ { j = 1 } ^ { i - 1 } )$ for selecting each candidate $\mathbf { \Delta } _ { \mathbf { \mathcal { X } } _ { i } }$ for $i = 1 , . . . , q$ . A cheaper alternative is to approximate the integral in (4) using a single approximate GP sample path $\tilde { \pmb { f } } _ { i }$ using RFFs when optimizing candidate $\mathbf { \Delta } _ { \mathbf { \mathcal { X } } _ { i } }$ . A single-sample approximation of $q \mathrm { N E H V I }$ , which we refer to as $q \mathrm { N E H V I - 1 }$ , can be computed by using $\tilde { \pmb { f } } _ { i }$ as the sampled GP in (6). Since the RFF is a deterministic model, it is much less computationally expensive to evaluate than the GP posterior on out-of-sample points, and exact gradients of $q \mathrm { N E H V I - 1 }$ with respect to current candidate $\mathbf { \Delta } _ { \mathbf { \mathcal { X } } _ { i } }$ can be computed and used for efficient multi-start optimization of $q \mathrm { N E H V I - 1 }$ using second-order gradient methods. $q \mathrm { N E H V I - 1 }$ requires CBD for efficient sequential greedy batch selection and gradient-based optimization, but does not use a sample average approximation for optimizing a new candidate $\mathbf { \Delta } _ { \mathbf { \mathcal { X } } _ { i } }$ ; instead, it uses an approximate sample path. See Rahimi & Recht [46] for details on RFFs.
+
+$q$ NEHVI-1 is related to TSEMO in that both use sequential greedy batch selection using HVI based on RFF samples. However, TSEMO does not directly maximize HVI when selecting candidate $\mathbf { \Delta } _ { \mathbf { \mathcal { X } } _ { i } }$ , where $i = 1 , . . . , q$ ; rather, it relies on a heuristic approach of running NSGA-II on an RFF sample of each objective to create a discrete population of candidates and then selecting the point from the discrete population that maximizes HVI under the RFF sample. In contrast, $q \mathrm { N E H V I - 1 }$ directly optimizes HVI under the RFF using exact sample-path gradients, which leads to improved optimization performance (see Appendix H). Furthermore, we find that $q \mathrm { N E H V I - 1 }$ is significantly faster than TSEMO, because rather than using NSGA-II it uses second order gradient methods to optimize HVI (see Appendix H). Gradient-based optimization is only possible because CBD enables scalable, differentiable HVI computation. While the primary goal of this work is to develop a principled, scalable method for parallel EHVI in noisy environments, we include empirical comparisons with qNEHVI-1 throughout the appendix to demonstrate the generalizablility of the CBD approach and practical performance of the $q \mathrm { N E H V I - 1 }$ approximation. $q \mathrm { N E H V I - 1 }$ achieves the fastest batch selection timesof any method tested on a GPU on every problem; in many cases, this is an order of magnitude speed-up over $q \mathrm { N E H V I }$ . Moreover, qNEHVI-1 has a remarkable ability to scale to large batch sizes when the dimensionality of optimization problem is modest. Further investigation of $q \mathrm { N E H V I - 1 }$ is needed, but we hope that the readers can recognize the ways in which $q \mathrm { N E H V I }$ can create broader opportunities for research into hypervolume improvement based acquisition functions.
+
+# 9 Experiments
+
+We empirically evaluate $q \mathrm { N E H V I }$ on a set of synthetic and real-world benchmark problems. We compare it against the following recently proposed methods from the literature: PESMO, MESMO (which we extend to the handle noisy observations using the noisy information gain from Takeno et al. [52]), PFES, DGEMO, MOEA/D-EGO, TSEMO, TS-TCH, $q \mathrm { E H V I }$ (and qEHVI-PM-CBD, which uses the posterior mean as a plug-in estimate for the function values at the in-sample points, along with CBD to scale to large batch sizes), and qNParEGO. We optimize all methods using multi-start L-BFGS-B with exact gradients (except for PFES, which uses gradients approximated via finite differences), including TS-TCH where we optimize approximate function samples using RFFs with 500 basis functions. We model each outcome with an independent GP with a Matérn 5/2 ARD kernel and infer the GP hyperparameters via maximum a posteriori (MAP) estimation. For all problems, we assume that the noise variances are observed (except ABR, where we infer the noise level). See Appendix G for more details on the experiments and acquisition function implementations.
+
+We evaluate all methods using the logarithm of the difference in hypervolume between the true Pareto frontier and the approximate Pareto frontier recovered by the algorithm. Since evaluations are noisy, we compute the hypervolume dominated by the noiseless Pareto frontier across the observed points for each method.
+
+Synthetic Problems: We consider a noisy variants of the BraninCurrin problem $\quad M = 2 , d = 2 ,$ and the DTLZ2 problem $( M = 2 , d = 6 )$ ) [13], in which observations are corrupted with zero-mean additive Gaussian noise with standard deviation of $5 \%$ of the range of each objective for BraninCurrin and $10 \%$ for DTLZ2.
+
+Adaptive Bitrate (ABR) Control Policy Optimization: ABR controllers are used for real-time communication and media streaming applications. Policies for these controllers must be tuned to deliver a high quality of experience with respect to multiple objectives [40]. In industry settings, A/B tests with dozens of policies are tested simultaneously since each policy may take days or weeks to evaluate, producing noisy measurements across multiple objectives. In this experiment, we tune policies to maximize video quality (bitrate) and minimize stall time. The policy has $d = 4$ parameters, which are detailed in Appendix G. We use the Park simulator [41] and sample a random set of 100 traces to obtain noisy measurements of the objectives under a given policy. For comparing the performance of different methods, we estimate the true noiseless objective using mean objectives across 300 traces. We infer a homoskedastic noise level jointly with the GP hyperparameters via MAP estimation.
+
+![](images/6af06c98f410630170c917e8cab50477e1aa6be60e3cfa4697976649ab5238e6.jpg)  
+Figure 3: Sequential optimization performance. The shaded region indicates two standard errors of the mean over 100 replications (only 20 replications were feasible for PESMO due to large runtimes).
+
+Vehicle Design Optimization: Optimizing the design of the frame an automobile is important to maximizing passenger safety, vehicle durability and fuel efficiency. Evaluating a vehicle design is time-consuming, since either a vehicle must manufactured and crashed, or a nonlinear finite element-based crash analysis must be run to simulate a collision (which can take over 20 hours per run) [62]. Hence, evaluating many designs in parallel is critical for reducing end-to-end optimization time. Observations are often noisy due to manufacturing imperfections, measurement error, or non-deterministic simulations. In this experiment, we tune the $d = 5$ widths of various components of a vehicle’s frame to minimize proxy metrics for (1) fuel consumption, (2) passenger trauma in a full frontal collison, and (3) vehicle fragility [53]. See Appendix G for details. For this demonstration, we add zero-mean Gaussian noise with a standard deviation of $1 \%$ of the objective range, which roughly corresponds to the manufacturing noise level used in previous work [62].
+
+# 9.1 Summary of Results:
+
+We find that $q \mathrm { N E H V I }$ and $q \mathrm { N E H V I - 1 }$ outperform all other methods on the noisy benchmarks, both in the sequential and parallel setting. In the sequential setting (Fig 3), $q \mathrm { N E H V I }$ and $q \mathrm { N E H V I - 1 }$ are followed closely by $q \mathrm { E H V I - P M }$ , and in some cases, even $q \mathrm { E H V I }$ . TS-TCH is firmly in the middle of the pack, while information-theoretic acquisition functions appear to perform the worst. This is consistent across noise levels; for experiments where we add noise to the objectives, we consider noise levels ranging from $1 \%$ to $10 \%$ of the range of each objective (these are magnitudes of the noise often seen in practice). Previous works have only evaluated MOBO algorithms with noise levels of $1 \%$ [25]. In Appendix H, we perform a study showing that $q \mathrm { N E H V I }$ consistently performs best with increasing noise levels up to $30 \%$ of the range of each objective.
+
+While parallel evaluation can provide optimization speedups on order of the batch size $q$ , these evaluations do affect the overall sample complexity of the algorithm, since less information is available within the synchronous batch setting compared with fully sequential optimization. We find that, by and large, $q \mathrm { N E H V I }$ achieves the greatest hyper-volume for increasingly large batch sizes, and scales more elegantly relative to TS-TCH and the ParEGO variants (Fig 4). qNEHVI also consistently outperforms $q \mathrm { E H V I }$ -PM-CBD. In Appendix H, we observe that $q \mathrm { N E H V I }$ and $q \mathrm { N E H V I - 1 }$ provides excellent anytime performance all values of $q$ that we tested. We provide results on 4 additional test problems in Appendix H.3, and in Appendix H.8, we demonstrate that leveraging CBD and a single sample path approximation, $q$ NEHVI-1 enables scaling to 5-objective problems, which is a first for an HVI-based method, to our knowledge.
+
+![](images/871673288d167bcbd541079e5997b278cca429d8ca64fd48b785eb05c1db1a12.jpg)  
+Figure 4: The quality of the final Pareto frontier identified by each method with increasing batch sizes $q$ given a budget of 224 function evaluations. $q \mathrm { E H V I }$ is only included for $q = 1$ and $q = 8$ because the IEP scales exponential with $q$ . DGEMO is omitted on the ABR problem because it was prohibitively slow with time-consuming ABR simulations and on the VehicleSafety problem because DGEMO consistently crashed in the graph cutting algorithm.
+
+In our experiments, we find that $q \mathrm { N E H V I - 1 }$ is among the top performers on relatively lowdimensional problems. Given the strong performance of $q \mathrm { N E H V I - 1 }$ , we examine its performance as the dimensionality of the search space increases in Appendix H.5. We find that $q \mathrm { N E H V I }$ is more robust than $q \mathrm { N E H V I - 1 }$ in higher-dimensional search spaces, but further investigation is needed into how the number of the Fourier basis functions affects the performance of qNEHVI-1 in highdimensional search spaces.
+
+Optimization wall time: Across all experiments, we observe competitive wall times for optimizing $q \mathrm { N E H V I }$ and $q \mathrm { N E H V I - 1 }$ (all wall time comparisons are provided in Appendix H). On a GPU, optimizing qNEHVI-1 incurs the lowest wall time of any method that we tested on every single problem and optimizing $q \mathrm { N E H V I }$ is faster than optimizing information-theoretic methods on all problems. Using efficient low-rank Cholesky updates, qNEHVI is often faster than the qNParEGO implementation in BoTorch on a GPU.
+
+# 10 Discussion
+
+We proposed NEHVI, a novel acquisition function that provides a principled approach to parallel and noisy multi-objective Bayesian optimization. NEHVI is a one-step Bayes-optimal policy for maximizing the hypervolume dominated by the Pareto frontier in noisy and noise-free settings. NEHVI is made feasible by a new approach to computing joint hypervolumes (CBD), and we demonstrated that CBD enables scalable, parallel candidate generation with both noiseless $q \mathrm { E H V I }$ and qNEHVI. We provide theoretical results on optimizing a MC estimator of $q \mathrm { N E H V I }$ using sample average approximation and demonstrate significant improvements in optimization performance over state-of-the-art MOBO algorithms.
+
+Yet, our work has some limitations. While the information-theoretic acquisition functions tested here perform poorly on our benchmarks, they do allow for decoupled evaluations of different objectives in cases where querying one objective may be more resource-intensive than querying other objectives. Optimizing such acquisition functions is a non-trivial task, and it is possible that with improved procedures, such acquisition functions could yield improved performance and provide a principled approach to selecting evaluation sources on a budget. Although practically fast enough for most Bayesian optimization tasks, exact hypervolume computation has super-polynomial complexity in the number of objectives. Combining $q \mathrm { N E H V I }$ with differentiable approximate methods for computing hypervolume (e.g. Couckuyt et al. [9], Golovin & Zhang [23]) could lead to further speed-ups.
+
+We hope that the core ideas presented in this work, including the CBD approach, can provide a framework to support the development of new computationally efficient MOBO methods.
+
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+        "type": "text",
+        "text": "Parallel Bayesian Optimization of Multiple Noisy Objectives with Expected Hypervolume Improvement ",
+        "text_level": 1,
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+        "text": "Samuel Daulton Facebook, University of Oxford sdaulton@fb.com ",
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+        "text": "Maximilian Balandat Facebook balandat@fb.com ",
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+        "text": "Eytan Bakshy Facebook ebakshy@fb.com ",
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+        "type": "text",
+        "text": "Abstract ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "Optimizing multiple competing black-box objectives is a challenging problem in many fields, including science, engineering, and machine learning. Multi-objective Bayesian optimization (MOBO) is a sample-efficient approach for identifying the optimal trade-offs between the objectives. However, many existing methods perform poorly when the observations are corrupted by noise. We propose a novel acquisition function, NEHVI, that overcomes this important practical limitation by applying a Bayesian treatment to the popular expected hypervolume improvement (EHVI) criterion and integrating over this uncertainty in the Pareto frontier. We argue that, even in the noiseless setting, generating multiple candidates in parallel is an incarnation of EHVI with uncertainty in the Pareto frontier and therefore can be addressed using the same underlying technique. Through this lens, we derive a natural parallel variant, $q \\mathrm { N E H V I }$ , that reduces computational complexity of parallel EHVI from exponential to polynomial with respect to the batch size. $q \\mathrm { N E H V I }$ is one-step Bayes-optimal for hypervolume maximization in both noisy and noiseless environments, and we show that it can be optimized effectively with gradient-based methods via sample average approximation. Empirically, we demonstrate not only that $q \\mathrm { N E H V I }$ is substantially more robust to observation noise than existing MOBO approaches, but also that it achieves state-of-the-art optimization performance and competitive wall-times in large-batch environments. ",
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+        "text": "1 Introduction ",
+        "text_level": 1,
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+        "text": "Black-box optimization problems that involve multiple competing noisy objectives are ubiquitous in science and engineering. For example, a real-time communications service may be interested in tuning the parameters of a control policy to adapt video quality in real time in order to maximize video quality and minimize latency [10, 17]. In robotics, scientists may seek to design hardware components that maximize locomotive speed and minimize energy expended [8, 38]. In agriculture, development agencies may seek to balance crop yield and environmental impact [28]. For such multi-objective optimization (MOO) problems, there typically is no single solution that is best with respect to all objectives. Rather, the goal is to identify the Pareto frontier: a set of optimal trade-offs such that improving one objective means deteriorating another. In many cases, the objectives are expensive to evaluate. For instance, randomized trials used in agriculture and the internet industry may take weeks or months to conduct and incur opportunity costs, and manufacturing and testing hardware is both costly and time-consuming. Therefore, it is imperative to be able to identify good trade-offs with as few objective evaluations as possible. ",
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+    {
+        "type": "text",
+        "text": "Bayesian optimization (BO), a method for efficient global black-box optimization, is often used to tackle such problems. BO employs a probabilistic surrogate model in conjunction with an acquisition function to navigate the trade-off between exploration (evaluating designs with high uncertainty) and exploitation (evaluating designs that are believed to be optimal). Although a significant number of works have explored multi-objective Bayesian optimization (MOBO), most available methods [3, 39, 51, 60] do not take into account the fact that, in practice, observations are often subject to noise. For example, results of an A/B test are highly variable due to heterogeneity in the underlying user population and other factors. Agricultural trials are affected by the stochastic nature of plant growth and environmental factors such as soil composition or wind currents. In robotics, devices are subject to manufacturing tolerances, and observations of quantities such as locomotive speed and efficiency may be corrupted by measurement error from noisy sensors and environmental factors such as temperature or surface friction. While previous work has shown that a principled treatment of noisy observations can significantly improve optimization performance in the single-objective case [24, 37], this issue is understudied in the multi-objective setting. Furthermore, many applications in which evaluations take a long time require evaluating large batches of candidates in parallel in order to achieve reasonable throughput. For example, when firms optimize systems via A/B tests, it may take several weeks to test any particular configuration. Because of this, large batches of candidate policies are tested simultaneously [36]. In biochemistry and materials design, dozens of tests can be conducted parallel on a single microplate [63]. Even in sophisticated high throughput chemistry settings, these batches may take several hours or days to set up and evaluate [42]. Most existing MOBO methods, however, are either designed for purely sequential optimization [3, 51] or do not scale well to large batch sizes [11]. ",
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+        "text": "Contributions: In this work, we propose a novel MOBO algorithm, based on expected hypervolume improvement (EHVI), that scales to highly parallel evaluations of noisy objectives. Our approach is made possible by a general-purpose, differentiable, cached box decomposition (CBD) implementation that dramatically speeds up critical computations needed to account for uncertainty introduced by noisy observations and generate new candidate points for highly parallel batch or asynchronous evaluation. In particular, our CBD-based approach solves the fundamental problem of scaling parallel EHVI-based methods to large batch sizes, reducing time and space complexity from exponential to polynomial. Our proposed algorithm, noisy expected hypervolume improvement (NEHVI), is the onestep Bayes-optimal policy for hypervolume improvement and provides state-of-the-art performance across a variety of benchmarks. To our knowledge, our work provides the most extensive evaluation of noisy parallel MOBO to date. A high-quality implementation of $q \\mathrm { N E H V I }$ , as well as many of the baselines considered here, will be made available as open-source software upon publication. ",
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+    {
+        "type": "text",
+        "text": "2 Preliminaries ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 1
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+    {
+        "type": "text",
+        "text": "Our goal is to find the set of optimal designs $_ { \\textbf { \\em x } }$ over a bounded set $\\mathcal { X } \\subset \\mathbb { R } ^ { d }$ that maximize one or more objectives $\\pmb { f } ( \\pmb { x } ) \\in \\mathbb { R } ^ { M }$ , with no known analytical expression nor gradient information of $f$ . ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Multi-Objective Optimization (MOO) aims to identify the set of Pareto optimal objective tradeoffs. We say a solution ${ \\pmb f } ( { \\pmb x } ) = \\left[ f ^ { ( 1 ) } ( { \\pmb x } ) , . . . , f ^ { ( M ) } ( { \\pmb x } ) \\right]$ dominates another solution $\\pmb { f } ( \\pmb { x } ) \\succ \\pmb { f } ( \\pmb { x } ^ { \\prime } )$ if $f ^ { ( m ) } ( { \\pmb x } ) \\geq f ^ { ( m ) } ( { \\pmb x } ^ { \\prime } )$ for $m = 1 , . . . , M$ and $\\exists m \\in \\{ 1 , . . . , M \\}$ s.t. $f ^ { ( m ) } ( { \\pmb x } ) > f ^ { ( m ) } ( { \\pmb x } ^ { \\prime } )$ . We define the Pareto frontier as ${ \\mathcal { P } } ^ { * } = \\{ { \\pmb { f } } ( { \\pmb { x } } ) : { \\pmb { x } } \\in { \\pmb { \\chi } }$ , $\\nexists \\mathbf { \\boldsymbol { x } } ^ { \\prime } \\in \\mathcal { X }$ s.t. $f ( { \\pmb x } ^ { \\prime } ) \\succ f ( { \\pmb x } ) \\}$ , and denote the set of Pareto optimal designs as $\\mathcal { X } ^ { \\ast } = \\{ \\pmb { x } : \\pmb { f } ( \\pmb { x } ) \\in \\mathcal { P } ^ { * } \\}$ . Since the Pareto frontier (PF) is often an infinite set of points, MOO algorithms usually aim to identify a finite approximate PF $\\mathcal { P }$ . A natural measure of the quality of a PF is the hypervolume of the region of objective space that is dominated by the PF and bounded from below by a reference point. Provided with the approximate PF, the decision-maker can select a particular Pareto optimal trade-off according to their preferences. ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Bayesian Optimization (BO) is a sample-efficient optimization method that leverages a probabilistic surrogate model to make principled decisions to balance exploration and exploitation [19, 50]. Typically, the surrogate is a Gaussian Process (GP), a flexible, non-parametric model known for its well-calibrated predictive uncertainty [47]. To decide which points to evaluate next, BO employs an acquisition function $\\alpha ( \\cdot )$ that specifies the value of evaluating a set of new points $_ { \\textbf { \\em x } }$ based on the surrogate’s predictive distribution at . While evaluating the true black-box function $f$ is timeconsuming or costly, evaluating the surrogate is cheap and relatively fast; therefore, numerical optimization can be used to find the maximizer of the acquisition function $\\pmb { x } ^ { * } = \\arg \\operatorname* { m a x } _ { \\pmb { x } \\in \\mathcal { X } } \\alpha ( \\pmb { x } )$ to evaluate next on the black-box function. BO sequentially selects new points to evaluate and updates the model to incorporate the new observations. ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Evolutionary algorithms (EAs) such as NSGA-II [12] are a popular choice for solving MOO problems (see Zitzler et al. [67] for a review of various other approaches). However, EAs generally suffer from high sample complexity, rendering them infeasible for optimizing expensive-to-evaluate black-box functions. Multi-objective Bayesian optimization (MOBO), which combines a Bayesian surrogate with an acquisition function designed for MOO, provides a much more sample-efficient alternative. ",
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+        "type": "text",
+        "text": "",
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+    {
+        "type": "text",
+        "text": "3 Related Work ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            136,
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+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Methods based on hypervolume improvement (HVI) seek to expand the volume of the objective space dominated by the Pareto frontier. Expected hypervolume improvement (EHVI) [16] is a natural extension of the popular expected improvement (EI) [29] acquisition function to the MOO setting. Recent work has led to efficient computational paradigms using box decomposition algorithms [59] and practical enhancements such as support for parallel candidate generation and gradient-based acquisition optimization [11, 58]. However, EHVI still suffers from some limitations, including (i) the assumption that observations are noise-free, and (ii) the exponential scaling of its batch variant, $q \\mathrm { E H V I }$ , in the batch size $q$ , which precludes large-batch optimization. DGEMO [39] is a recent method for parallel MOBO that greedily maximizes HVI while balancing the diversity of the design points being sampled. Although DGEMO scales well to large batch sizes, it does not account for noisy observations. TSEMO [5] is a Thompson sampling (TS) heuristic that can acquire batches of points by optimizing a random fourier feature (RFF) [46] approximation of a GP surrogate using NSGA-II and selecting a subset of points from the EA’s population to sequentially greedily maximize HVI. This heuristic approach for maximizing HVI currently has no theoretical guarantees and relies on zeroth-order optimization methods, which tend to be slower and exhibit worse optimization performance than gradient-based approaches. ",
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Entropy-based methods such as PESMO [25], MESMO [3], and PFES [51] are an alternative to EHVI. Of these three methods, PESMO is the only one that accounts for observation noise. However, PESMO involves intractable entropy computations and therefore relies on complex approximations, as well as challenging and time-consuming numerical optimization procedures [25]. Garrido-Merchán & Hernández-Lobato [21] recently proposed an extension to PESMO that supports parallel candidate generation. However, the authors of this work provide limited evaluation and have not provided code to reproduce their results.1 ",
+        "bbox": [
+            174,
+            391,
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "MOO can also be cast into a single-objective problem by applying a random scalarization of the objectives. ParEGO maximizes the expected improvement using random augmented Chebyshev scalarizations [32]. MOEA/D-EGO [64] extends ParEGO to the batch setting using multiple random scalarizations and the genetic algorithm MOEA/D [65] to optimize these scalarizations in parallel. Recently, $q \\mathrm { P a r E G O }$ , another batch variant of ParEGO was proposed that uses compositional Monte Carlo objectives and sequential greedy candidate selection [11]. Additionally, the authors proposed a noisy variant, qNParEGO, but the empirical evaluation of that variant was limited. TS-TCH [45] combines random Chebyshev scalarizations with Thompson sampling [54], which is naturally robust to noise when the objective is scalarized. Golovin & Zhang [23] propose to use a hypervolume scalarization with the property that the expected value of the scalarization over a specific distribution of weights is equivalent to the hypervolume indicator. The authors propose a upper confidence bound algorithm using randomly sampled weights, but provide a very limited empirical evaluation. ",
+        "bbox": [
+            174,
+            494,
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Many prior attempts by the simulation community to handle MOO with noisy observations found that accounting for the noise did not improve optimization performance: Horn et al. [26] suggest that the best approach is to ignore noise, and Koch et al. [33] concluded that further research was needed to determine if modeling techniques such as re-interpolation could improve BO performance with noisy observations. In contrast, we find that accounting for noise does substantially improve performance in noisy settings. ",
+        "bbox": [
+            174,
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Lastly, previous works have considered methods for quantifying and monitoring uncertainty in the Pareto frontiers during the optimization [4, 7]. In contrast, we provide a solution to performing MOBO in noisy settings, rather than purely reasoning about the uncertainty in the Pareto frontier. ",
+        "bbox": [
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "4 Background on Expected Hypervolume Improvement ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            820,
+            651,
+            838
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "In this section, we review hypervolume, hypervolume improvement, and expected hypervolume improvement as well as efficient methods for computing these metrics using box decompositions. ",
+        "bbox": [
+            176,
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+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "Definition 1. The hypervolume indicator (HV) of a finite approximate Pareto frontier $\\mathcal { P }$ is the $M$ -dimensional Lebesgue measure $\\lambda _ { M }$ of the space dominated by $\\mathcal { P }$ and bounded from below by $a$ reference point. $\\begin{array} { r } { \\pmb { r } \\in \\mathbb { R } ^ { M } \\colon \\mathrm { H V } ( \\mathcal { P } | \\pmb { r } ) = \\lambda _ { M } \\big ( \\bigcup _ { \\pmb { v } \\in \\mathcal { P } } [ \\pmb { r } , \\pmb { v } ] \\big ) } \\end{array}$ , where $[ r , v ]$ denotes the hyper-rectangle bounded by vertices $\\mathbfit { \\Delta } \\mathbf { r }$ and $\\textbf {  { v } }$ . ",
+        "bbox": [
+            174,
+            90,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "As in previous work, we assume that the reference point $\\pmb { r }$ is known and specified by the decision maker [58]. ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Definition 2. The hypervolume improvement (HVI) of a set of points ${ \\mathcal { P } } ^ { \\prime } w . r . t .$ . an existing approximate Pareto frontier $\\mathcal { P }$ and reference point $\\pmb { r }$ is defined $a s ^ { 2 } \\operatorname { H V I } ( \\mathcal { P } ^ { \\prime } | \\mathcal { P } , \\boldsymbol { r } ) = \\operatorname { H V } ( \\mathcal { P } \\cup \\mathcal { P } ^ { \\prime } | \\boldsymbol { r } ) - \\operatorname { H V } ( \\mathcal { P } | \\boldsymbol { r } ) .$ ",
+        "bbox": [
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+        "page_idx": 3
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+    {
+        "type": "text",
+        "text": "Computing HV requires calculating the volume of a typically non-rectangular polytope and is known to have time complexity that is super-polynomial in the number of objectives [59]. An efficient approach for computing HV is to (i) decompose the region that is dominated by the Pareto frontier $\\mathcal { P }$ and bounded from below by the reference point $\\mathbfit { \\Delta } \\mathbf { r }$ into disjoint axis-aligned hyperrectangles [34], (ii) compute the volume of each hyperrectangle in the decomposition, and (iii) sum over all hyperrectangles. So-called box decomposition algorithms have also been applied to partition the region that is not dominated by the Pareto frontier $\\mathcal { P }$ , which can be used to compute the HVI from a set of new points [15, 59]. See Appendix $\\mathbf { B }$ for further details. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Expected Hypervolume Improvement: Since function values at unobserved points are unknown in black-box optimization, so is the HVI of an out-of-sample point. However, in BO the probabilistic surrogate model provides a posterior distribution $p ( \\pmb { f } ( \\pmb { x } ) | \\bar { \\mathcal { D } } )$ over the function values for each $_ { \\textbf { \\em x } }$ , which can be used to compute the expected hypervolume improvement (EHVI) acquisition function: $\\alpha _ { \\mathrm { E H V I } } ( \\pmb { x } | \\mathcal { P } ) = \\mathbb { E } \\big [ \\mathrm { H V I } ( \\mathbf { \\dot { f } } ( \\pmb { x } ) | \\mathcal { P } ) \\big ]$ . Although $\\alpha _ { \\mathrm { E H V I } }$ can be expressed analytically when (i) the objectives are assumed to be conditionally independent given $_ { \\textbf { \\em x } }$ and (ii) the candidates are generated and evaluated sequentially [58], Monte Carlo (MC) integration is commonly used since it does not require either assumption [16]. The more general parallel variant using MC integration is given by ",
+        "bbox": [
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+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/57a8b36e4658a261dfb590691de231137f36c566012ab250bf1024a961ec2b90.jpg",
+        "text": "$$\n\\alpha _ { q \\mathrm { E H V I } } \\big ( \\mathcal { X } _ { \\mathrm { c a n d } } \\big | \\mathcal { P } \\big ) \\approx \\hat { \\alpha } _ { q \\mathrm { E H V I } } \\big ( \\mathcal { X } _ { \\mathrm { c a n d } } \\big | \\mathcal { P } \\big ) = \\frac { 1 } { N } \\sum _ { t = 1 } ^ { N } \\mathrm { H V I } \\big ( \\tilde { f } _ { t } \\big ( \\mathcal { X } _ { \\mathrm { c a n d } } \\big ) \\big | \\mathcal { P } \\big ) ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            276,
+            462,
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+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "where $\\tilde { f } _ { t } \\sim p ( f | \\mathcal { D } )$ for $t = 1 , . . . , N$ and $\\mathcal { X } _ { \\mathrm { c a n d } } = \\{ x _ { i } \\} _ { i = 1 } ^ { q }$ [11]. The same box decomposition algorithms used to compute HVI can be used to compute EHVI (either analytic or via MC) using piece-wise integration. EHVI computation is agnostic to the choice of box decomposition algorithm (and can also use approximate methods [9]). Similar to EI in the single-objective case, EHVI is a one-step Bayes-optimal algorithm for maximizing hypervolume in the MOO setting under the following assumptions: (i) only a single design will be generated and evaluated, (ii) the observations are noise-free, (iii) the final approximate Pareto frontier (and final design that will be deployed) will be drawn from the set of observed points [19]. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "5 Expected Hypervolume Improvement with Noisy Observations ",
+        "text_level": 1,
+        "bbox": [
+            174,
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+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "We consider the case that frequently arises in practice where we only receive noisy observations ${ \\pmb y } _ { i } = f ( { \\pmb x } _ { i } ) + \\epsilon _ { i }$ , $\\epsilon _ { i } \\sim \\mathcal { N } ( 0 , \\mathbf { \\bar { \\Sigma } } _ { i } )$ , where $\\Sigma _ { i }$ is the noise covariance. In this setting, EHVI is no longer (one-step) Bayes-optimal. This is because we can no longer compute the true Pareto frontier ${ \\mathcal { P } } _ { n } { \\overset { \\cdot } { = } } \\{ f ( { \\pmb x } ) \\mid { \\overset { \\cdot } { \\pmb x } } \\in { \\dot { X } } _ { n }$ , $\\bar { \\nexists } \\boldsymbol { x ^ { \\prime } } \\in X _ { n }$ s.t. $f ( { \\pmb x } ^ { \\prime } ) \\succ f ( { \\pmb x } ) \\}$ over the previously evaluated points $X _ { n } =$ $\\{ { \\pmb x } _ { i } \\} _ { i = 1 } ^ { n }$ . Simply using the observed Pareto frontier, $\\mathcal { V } _ { n } = \\{ \\pmb { y } \\vert \\pmb { y } \\in Y _ { n }$ , $\\bar { \\nexists } \\ : y ^ { \\prime } \\in Y _ { n }$ s.t. $y ^ { \\prime } \\succ y , y \\}$ where $\\bar { Y } _ { n } = \\{ { \\pmb y } _ { i } \\} _ { i = 1 } ^ { n }$ , can have strong detrimental effects on optimization performance. This is illustrated in Figure 1, which shows how EHVI is misled by noisy observations that appear to be Pareto optimal. EHVI proceeds to spend its evaluation budget trying to optimize noise, resulting in a clumped Pareto frontier that lacks diversity. Although the posterior mean could serve as a \"plug-in\" estimate of the true function values at the observed points and provide some regularization [61], we find that this heuristic also leads to clustered Pareto frontiers (EHVI-PM in Fig. 1). Similar patterns emerge with DGEMO (which does not account for noise), and other baselines that utilize the posterior mean rather than the observed values when computing hypervolume improvement (see Appendix H). To our knowledge, all previous work on EHVI assumes that observations are noiseless [16, 58] or imputes the unknown true function values with the posterior mean. ",
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+    {
+        "type": "image",
+        "img_path": "images/956f97dc2146646d797257b5483737aca4c5f1c6a8ae6460f9c3c6c2a9f3f493.jpg",
+        "image_caption": [
+            "Figure 1: An illustration of the effect of noisy observations on the true noiseless Pareto frontiers identified by NEHVI (our proposed algorithm), EHVI, and EHVI-PM, which uses the modeled posterior mean as point estimate of the true in-sample function values. All algorithms are tested on a BraninCurrin synthetic problem, where observations are corrupted with zero-mean, additive Gaussian noise with a standard deviation of $5 \\%$ of the range of respective objective. All methods use sequential $( q = 1 )$ ) optimization. See Appendix G for details. "
+        ],
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+    {
+        "type": "text",
+        "text": "5.1 A Bayes-optimal algorithm for hypervolume maximization in noisy environments ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "In contrast with EHVI(-PM), we instead approach the problem of hypervolume maximization under noisy observations from a Bayesian perspective and derive a novel one-step Bayes-optimal expected hypervolume improvement criterion that iterates the expectation over the posterior distribution $p \\big ( \\bar { f } ( X _ { n } ) | \\mathcal { D } _ { n } \\big )$ of the function values at the previously evaluated points $X _ { n }$ given noisy observations $\\mathcal { D } _ { n } = \\{ \\mathbf { { \\boldsymbol { x } } } _ { i } , \\mathbf { { \\boldsymbol { y } } } _ { i } , ( \\Sigma _ { i } ) \\} _ { i = 1 } ^ { n }$ . Our acquisition function, noisy expected hypervolume improvement (NEHVI), is defined as ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/6ba0f65fb5a621c0d95553507d10f2e27c01940d6043cdf28397d76590409b64.jpg",
+        "text": "$$\n\\alpha _ { \\mathrm { N E H V I } } ( { \\pmb x } ) = \\int \\alpha _ { \\mathrm { E H V I } } ( { \\pmb x } | \\mathcal { P } _ { n } ) p ( { \\pmb f } | \\mathcal { D } _ { n } ) d { \\pmb f }\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "where $P _ { n }$ denotes the Pareto frontier over $f ( X _ { n } )$ . ",
+        "bbox": [
+            174,
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "By integrating over the uncertainty in the function values at the observed points, NEHVI retains one-step Bayes-optimality in noisy environments (in noiseless environments, NEHVI is equivalent to EHVI). Empirically, Figure 1 shows that NEHVI is robust to noise and identifies a well-distributed Pareto frontier with no signs of clumping, even under very noisy observations.3 ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "The integral in (2) is analytically intractable, but can easily be approximated using MC integration. Let $\\tilde { \\pmb { f } } _ { t } \\sim p ( \\pmb { f } | D _ { n } )$ for $t = 1 , . . . N$ be samples from the posterior, and let $\\mathcal { P } _ { t } ~ { = } ~ \\{ \\tilde { f } _ { t } ( { \\pmb x } ) \\} ~ \\mathbf { { \\sigma } } | ~ { \\pmb x } ~ \\in$ $X _ { n } , \\tilde { { f } } _ { t } ( { \\pmb x } ) \\succ \\tilde { { f } } _ { t } ( { \\pmb x } ^ { \\prime } ) \\forall { \\pmb x } ^ { \\prime } \\in X _ { n } \\}$ be the Pareto frontier over the previously evaluated points under the sampled function $\\tilde { \\pmb { f } } _ { t }$ . Then, $\\begin{array} { r } { \\alpha _ { \\mathrm { N E H V I } } ( \\pmb { x } ) \\approx \\frac { 1 } { N } \\sum _ { t = 1 } ^ { N } \\alpha _ { \\mathrm { E H V I } } ( \\pmb { x } | \\mathcal { P } _ { t } ) } \\end{array}$ . Using MC integration, we can compute the inner expectation in $\\alpha _ { \\mathrm { E H V I } }$ simultaneously using samples from the joint posterior $\\tilde { \\pmb f _ { t } } ( X _ { n } , \\pmb x ) \\sim p ( \\pmb f ( X _ { n } , \\pmb x ) | \\mathcal D _ { n } )$ over $_ { \\textbf { \\em x } }$ and $X _ { n }$ : ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/17af84fd9f604ba2f997624ee6c90f575596714ce9e4803e6164c51b50f0351f.jpg",
+        "text": "$$\n\\hat { \\alpha } _ { \\mathrm { N E H V I } } ( \\pmb { x } ) = \\frac { 1 } { N } \\sum _ { t = 1 } ^ { N } \\mathrm { H V I } ( \\tilde { \\pmb { f } _ { t } } ( \\pmb { x } ) | \\mathcal { P } _ { t } ) .\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 4
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+    {
+        "type": "text",
+        "text": "See Appendix B for details on computing (3) using box decompositions. Note that this “full-MC” variant of NEHVI does not require objectives to be modeled independently, and supports multi-task covariance functions across correlated objectives. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "5.2 Parallel Noisy Expected Hypervolume Improvement ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 4
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+    {
+        "type": "text",
+        "text": "Generating and evaluating batches of candidates is imperative to achieving adequate throughput in many real-world scenarios. qNEHVI can naturally be extended to the parallel (asynchronous or batch) setting by evaluating HVI with respect to a batch of $q$ points $\\mathcal { X } _ { \\mathrm { c a n d } } = \\{ \\pmb { x } _ { i } \\} _ { i = 1 } ^ { q }$ ",
+        "bbox": [
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+    {
+        "type": "equation",
+        "img_path": "images/536d2591f15be89f8ff1bf26d90cf75e55d832a5c4d603a591c841bcb51445e5.jpg",
+        "text": "$$\n\\varepsilon _ { q \\mathrm { N E H V I } } ( \\mathcal { X } _ { \\mathrm { c a n d } } ) = \\int \\alpha _ { q \\mathrm { E H V I } } ( \\mathcal { X } _ { \\mathrm { c a n d } } | \\mathcal { P } _ { n } ) p ( f | \\mathcal { D } _ { n } ) d f \\approx \\hat { \\alpha } _ { q \\mathrm { N E H V I } } ( \\mathcal { X } _ { \\mathrm { c a n d } } ) = \\frac { 1 } { N } \\sum _ { t = 1 } ^ { N } \\mathrm { H V I } ( \\tilde { f } _ { t } ( \\mathcal { X } _ { \\mathrm { c a n d } } ) | \\mathcal { P } _ { t } )\n$$",
+        "text_format": "latex",
+        "bbox": [
+            181,
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Since optimizing $q$ candidates jointly is a difficult numerical optimization problem over a qddimensional domain, we use a sequential greedy approximation in the parallel setting and solve a sequence of $q$ simpler optimization problems with $d$ dimensions, which been shown empirically to improve optimization performance [57]. While selecting candidates according to a “sequential greedy” policy does not guarantee that the selected batch of candidates is a maximizer of the $\\alpha _ { q \\mathrm { N E H V I } }$ the submodularity of $\\alpha _ { q \\mathrm { N E H V I } }$ allows us to bound the regret of this approximation to be no more than $\\frac { 1 } { e } \\alpha _ { q \\mathrm { N E H V I } } ^ { * }$ , where $\\begin{array} { r } { \\alpha _ { q \\mathrm { N E H V I } } ^ { * } = \\operatorname* { m a x } _ { \\mathcal { X } _ { \\mathrm { c a n d } } \\in \\mathcal { X } } \\alpha _ { q \\mathrm { N E H V I } } ( \\mathcal { X } _ { \\mathrm { c a n d } } ) } \\end{array}$ (see Appendix F). ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "3This noise level is ${ 5 } \\mathbf { x }$ greater than the ones considered by previous works that evaluate noisy MOBO [25]. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "6 Efficient Evaluation with Cached Box Decompositions ",
+        "text_level": 1,
+        "bbox": [
+            173,
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Although $\\hat { \\alpha } _ { \\mathrm { N E H V I } } ( \\pmb { x } )$ in (3) has a concise mathematical form, computing it requires determining the Pareto frontier $\\mathcal { P } _ { t }$ under each sample $\\tilde { f } _ { t }$ for $t = 1 , . . . , N$ and then partitioning the region that is not dominated by $\\mathcal { P } _ { t }$ into disjoint hyperrectangles $\\{ S _ { k _ { t } } \\} _ { k _ { t } = 1 } ^ { K _ { t } }$ . Optimizing the unbiased MC estimator of $\\alpha _ { \\mathrm { { N E H V I } } }$ would require re-sampling $\\{ \\tilde { f } _ { t } \\} _ { t = 1 } ^ { N }$ at each evaluation of $\\alpha _ { \\mathrm { { N E H V I } } }$ . However, computing the Pareto frontier and performing a box decomposition under each of the $N$ samples during every evaluation of $\\alpha _ { \\mathrm { N E H V I } }$ in the inner optimization loop $\\begin{array} { r } { ( \\pmb { x } ^ { * } = \\arg \\operatorname* { m a x } _ { \\pmb { x } } \\alpha _ { \\mathrm { N E H V I } } ( \\pmb { x } | \\mathcal { D } _ { n } ) ) } \\end{array}$ would be prohibitively expensive. This is because box decomposition algorithms have super-polynomial time complexity in the number of objectives [59]. We instead propose an efficient alternative computational technique for repeated evaluations of EHVI with uncertain Pareto frontiers. ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Cached Box Decompositions: For repeated evaluations of the integral in (2), we use a set of fixed samples $\\{ \\tilde { f } _ { t } ( X _ { n } ) \\} _ { t = 1 } ^ { N }$ , which allows us to compute the Pareto frontiers and box decompositions once, and cache them for the entirety of the acquisition function optimization, thereby making those two computationally intensive operations a one-time cost per BO iteration.4 We refer to this approach as using cached box decompositions (CBD). The method of optimizing over fixed random samples is known as sample average approximation (SAA) [2]. ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Conditional Posterior Sampling: Under the CBD formulation, computing $\\hat { \\alpha } _ { \\mathrm { N E H V I } } ( \\pmb { x } )$ with joint samples from $\\tilde { \\pmb f _ { t } } ( X _ { n } , \\pmb x ) \\sim p ( \\pmb f ( X _ { n } , \\pmb x ) | \\mathcal D _ { n } )$ requires sampling from the conditional distributions ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "equation",
+        "img_path": "images/436b1c7ee6aa211b3ef781943de1fd6602816335ca173c3ebbe3c25afe9a1439.jpg",
+        "text": "$$\n\\tilde { { f } } _ { t } ( x ) \\sim p \\big ( f ( x ) | f ( X _ { n } ) = \\tilde { f } _ { t } ( X _ { n } ) , \\mathcal { D } _ { n } \\big ) ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            359,
+            395,
+            635,
+            416
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "where $t = 1 , . . . , N$ and $\\{ \\tilde { f } _ { t } ( X _ { n } ) \\} _ { t = 1 } ^ { N }$ are the realized samples at the previously evaluated points. For multivariate Gaussian posteriors (as is the case with GP surrogates), we can sample from $p ( f ( X _ { n } ) | \\mathcal { D } _ { n } )$ via the reparameterization trick [30] by evaluating $\\begin{array} { r } { \\tilde { \\pmb f } _ { t } ( { \\pmb x } ) = { \\pmb \\mu } _ { n } + L _ { n } ^ { T } \\zeta _ { n , t } } \\end{array}$ , where $\\zeta _ { n , t } \\sim \\mathcal { N } ( \\mathbf { 0 } , I _ { n M } )$ , $\\pmb { \\mu _ { n } } \\in \\mathbb { R } ^ { n M }$ is the posterior mean, and $L _ { n } \\in \\mathbb { R } ^ { n M \\times n M }$ is a lower triangular root decomposition of the posterior covariance matrix, typically a Cholesky decomposition. Given $L _ { n }$ , we can obtain a root decomposition $L _ { n } ^ { \\prime }$ of the covariance matrix of the joint posterior $p ( f ( X _ { n } , \\pmb { x } ) | \\mathcal { D } _ { n } )$ by performing efficient low-rank updates [44]. Given $L _ { n } ^ { \\prime }$ and the posterior mean of $p ( f ( X _ { n } , \\pmb { x } ) | \\mathcal { D } _ { n } )$ , we can sample from (5) via the reparameterization trick by augmenting the existing base samples $\\zeta _ { n , t }$ with $M$ new base samples for the new point. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "6.1 Efficient Sequential Greedy Batch Selection using CBD ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "The CBD technique addresses the general problem of inefficient repeated evaluations of EHVI with uncertain Pareto frontiers. In this section, we show that sequential greedy batch selection (with both $q \\mathrm { E H V I }$ and $q \\mathrm { N E H V I }$ is an incarnation of EHVI with uncertain Pareto frontiers. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "The original formulation of parallel EHVI in Daulton et al. [11] uses the inclusion-exclusion principle (IEP), which involves computing the volume jointly dominated by each of the $2 ^ { q } - 1$ nonempty subsets of points in $\\mathcal { X } _ { \\mathrm { c a n d } }$ . However, using large batch sizes is not computationally feasible under this formulation because time and space complexity are exponential in $q$ and multiplicative in the number of hyperrectangles in the box decomposition [11] (see Appendix D for a complexity analysis). Although $q \\mathrm { E H V I }$ is optimized using sequential greedy batch selection, the IEP is used over all candidates $\\pmb { x } _ { 1 } , . . . , \\pmb { x } _ { i }$ when selecting candidate $i$ . Although the IEP could similarly be used to compute $q \\mathrm { N E H V I }$ , we instead leverage CBD, which yields a sequential greedy approximation of the joint (noisy) EHVI that is mathematically equivalent to the IEP formulation, but significantly reduces computational overhead. That is, the IEP and CBD approaches produce exactly the same acquisition value for a given set of points $\\mathcal { X } _ { \\mathrm { c a n d } }$ , but the IEP and the CBD approaches have exponential and polynomial time complexities in $q$ , respectively. ",
+        "bbox": [
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+        "page_idx": 5
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+    {
+        "type": "text",
+        "text": "When selecting $\\mathbf { \\Delta } _ { \\mathbf { \\mathcal { X } } _ { i } }$ for $i \\in \\{ 2 , \\ldots , q \\}$ , all $\\mathbf { \\Delta } _ { \\mathbf { \\mathcal { X } } _ { j } }$ for which $j < i$ have already been selected and are therefore held constant. Thus, we can decompose $q \\mathrm { N E H V I }$ into the $q \\mathrm { N E H V I }$ from the previously selected candidates $\\pmb { x } _ { 1 } , \\ldots , \\pmb { x } _ { i - 1 }$ and NEHVI from $\\mathbf { \\boldsymbol { x } } _ { i }$ given the previously selected candidates ",
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+    {
+        "type": "image",
+        "img_path": "images/a468f1bb1946757538c10a82e4cba5957f0190cedcd03a0cdcef5e345becbb5b.jpg",
+        "image_caption": [
+            "Figure 2: Acquisition optimization wall time under a sequential greedy approximation using L-BFGS-B. CBD enables scaling to much larger batch sizes $q$ than using the IEP and avoids running out-of-memory (OOM) on a GPU. Independent GPs are used for each outcome. The Pareto frontier of of the 2-objective, 6- dimensional DTLZ2 problem [13] is initialized with 20 points. Wall times were measured on a Tesla V100 SXM2 GPU (16GB RAM) and a $2 \\mathbf { x }$ Intel Xeon 6138 CPU $@$ 2GHz (251GB RAM). See Appendix H.2 for results with more objectives. "
+        ],
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+        "type": "equation",
+        "img_path": "images/bbf0b59144e8d6846c1fc593ed08bd606e371673e597d9ebfbeb2703730f9ed1.jpg",
+        "text": "$$\n\\tilde { \\tau } _ { q \\mathrm { N E H V I } } ( \\{ x _ { j } \\} _ { j = 1 } ^ { i } ) = \\frac { 1 } { N } \\sum _ { t = 1 } ^ { N } \\mathrm { H V I } \\big ( \\{ \\tilde { f } _ { t } ( x _ { j } ) \\} _ { j = 1 } ^ { i - 1 } \\big \\} | \\mathcal { P } _ { t } \\big ) + \\frac { 1 } { N } \\sum _ { t = 1 } ^ { N } \\mathrm { H V I } \\big ( \\tilde { f } _ { t } ( x _ { i } ) | \\mathcal { P } _ { t } \\cup \\{ \\tilde { f } _ { t } ( x _ { j } ) \\} _ { j = 1 } ^ { i - 1 } \\big \\} \\big )\n$$",
+        "text_format": "latex",
+        "bbox": [
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+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Note that the first term on the right hand side is constant, since $\\{ { \\pmb x } _ { j } \\} _ { j = 1 } ^ { i - 1 }$ and $\\{ \\tilde { f } _ { t } ( \\pmb { x } _ { j } ) \\} _ { j = 1 } ^ { i - 1 }$ are fixed for all $t = 1 , . . . , N$ . The second term is $\\hat { \\alpha } _ { \\mathrm { N E H V I } } ( \\pmb { x } _ { i } )$ , where the NEHVI is taken with respect to the Pareto frontier across $\\pmb { f } ( X _ { n } , \\pmb { x } _ { 1 } , . . . , \\pmb { x } _ { i - 1 } )$ and computed using the fixed samples $\\{ \\tilde { f } _ { t } ( X _ { n } , \\pmb { x } _ { 1 } , . . . \\pmb { x } _ { i - 1 } ) \\} _ { t = 1 } ^ { N }$ . To compute the second term when selecting candidate $\\mathbf { \\Delta } _ { \\mathbf { \\mathcal { X } } _ { i } }$ , the $N$ Pareto frontiers and CBDs are updated to include $\\{ \\tilde { f } _ { t } ( X _ { n } , \\pmb { x } _ { 1 } , . . . \\pmb { x } _ { i - 1 } ) \\} _ { t = 1 } ^ { N }$ . As in the sequential $q = 1$ setting, the box decompositions are only computed and cached while selecting each candidate point. See Appendix C.2 for a derivation of (6). Although we have focused on $q \\mathrm { N E H V I }$ in the above, the CBD formulation for $q \\mathrm { E H V I }$ is obtained by simply replacing $\\mathcal { P } _ { t }$ with the Pareto frontier over the observed values ${ \\mathcal { V } } _ { n }$ . ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Despite computing $N$ box decompositions when selecting each candidate $\\mathbf { \\Delta } _ { \\mathbf { \\mathcal { X } } _ { i } }$ for $i = 2 , . . . , q$ , the CBD approach reduces the time and space complexity from exponential (under the IEP) to polynomial in $q$ (see Appendix $\\mathrm { D }$ for details on time and space complexity). Figure 2 shows the total acquisition optimization time (including box decompositions) for various batch sizes and demonstrates that using CBD allows to scale to batch sizes that are completely infeasible when using IEP. ",
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+    {
+        "type": "text",
+        "text": "7 Optimizing NEHVI ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Differentiability: Importantly, $\\hat { \\alpha } _ { \\mathrm { N E H V I } } ( \\pmb { x } )$ is differentiable w.r.t. $_ { \\textbf { \\em x } }$ . Although determining the Pareto frontier and computing the box decompositions are non-differentiable operations, these operations do not involve $_ { \\textbf { \\em x } }$ , even when re-sampling from the joint posterior $p ( f ( X _ { n } , \\pmb { x } ) | \\mathcal { D } _ { n } )$ . Exact sample-path gradients of $\\nabla _ { \\pmb { x } } \\hat { \\alpha } _ { \\mathrm { N E H V I } } ( \\pmb { x } )$ can easily be computed using auto-differentiation in modern computational frameworks. This enables efficient gradient-based optimization of $q \\mathrm { N E H V I }$ .5 ",
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+    {
+        "type": "text",
+        "text": "SAA Convergence Results: In addition to approximating the outer expectation over $f ( X _ { n } )$ with fixed posterior samples, we can similarly fix the base samples used for the new candidate point $_ { \\textbf { \\em x } }$ This approach yields a deterministic acquisition function, which enables using (quasi-) higher-order optimization methods to obtain fast convergence rates for acquisition optimization [2]. Importantly, we prove that the theoretical convergence guarantees on acquisition optimization under the SAA approach proposed by Balandat et al. [2] also hold for NEHVI. ",
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+        "text": "Theorem 1. Suppose $\\mathcal { X }$ is compact and $f$ has a multi-output $G P$ prior with continuously differentiable mean and covariance functions. Let $X _ { n } = \\{ \\pmb { x } _ { i } \\} _ { i = 1 } ^ { n }$ denote the previously evaluated points and $\\{ \\zeta \\} _ { t = 1 } ^ { N }$ be base samples $\\zeta \\sim \\mathcal { N } ( \\mathbf { 0 } , I _ { ( n + 1 ) M } )$ . Let $\\hat { \\alpha }$ NEHVI denote the deterministic acquisition function computed using $\\{ \\zeta \\} _ { t = 1 } ^ { N }$ as $\\hat { \\alpha } _ { \\mathrm { N E H V I } } ^ { N }$ and define $S ^ { * } : = \\arg \\operatorname* { m a x } _ { { \\pmb x } \\in { \\pmb X } }$ $\\alpha _ { \\mathrm { N E H V I } } ( \\pmb { x } )$ to be the set of maximizers of $\\alpha _ { \\mathrm { N E H V I } } ( \\pmb { x } )$ over $\\mathcal { X }$ . Suppose $\\hat { \\pmb { x } } _ { N } ^ { * } \\in \\arg \\operatorname* { m a x } _ { { \\pmb { x } } \\in { \\mathcal { X } } } \\hat { \\alpha } _ { \\mathrm { N E H V I } } ^ { N } ( { \\pmb { x } } )$ . Then $( l )$ $\\hat { \\alpha } _ { \\mathrm { N E H V I } } ^ { N } ( \\hat { \\pmb { x } } _ { N } ^ { * } )  \\alpha _ { \\mathrm { N E H V I } } ( \\pmb { x } _ { N } ^ { * } )$ almost surely, and (2) $\\mathrm { d i s t } ( \\hat { \\pmb { x } } _ { N } ^ { * } , S ^ { * } )  0$ , where d $l i s t ( \\hat { \\pmb { x } } _ { N } ^ { * } , S ^ { * } ) : =$ $\\mathrm { i n f } _ { { \\pmb x } \\in S ^ { * } } \\| \\hat { \\pmb x } _ { N } ^ { * } - { \\pmb x } \\|$ is the Euclidean distance between $\\hat { \\pmb { x } } _ { N } ^ { * }$ and the set $S ^ { * }$ . ",
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+        "type": "text",
+        "text": "Theorem 1 also holds in the parallel setting, so $q \\mathrm { N E H V I }$ enjoys the same convergence guarantees as NEHVI on acquisition optimization under the SAA. See Appendix $\\mathrm { E }$ for further details and proof. ",
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+    {
+        "type": "text",
+        "text": "8 Approximation of $q$ NEHVI using Approximate GP Sample Paths ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Although CBD yields polynomial complexity of $q \\mathrm { N E H V I }$ with respect to $q$ (rather than exponential complexity with the IEP), it still requires computing $N$ box decompositions and repeatedly evaluating the joint posterior over ${ \\pmb f } ( X _ { n } , \\{ { \\pmb x } _ { j } \\} _ { j = 1 } ^ { i - 1 } )$ for selecting each candidate $\\mathbf { \\Delta } _ { \\mathbf { \\mathcal { X } } _ { i } }$ for $i = 1 , . . . , q$ . A cheaper alternative is to approximate the integral in (4) using a single approximate GP sample path $\\tilde { \\pmb { f } } _ { i }$ using RFFs when optimizing candidate $\\mathbf { \\Delta } _ { \\mathbf { \\mathcal { X } } _ { i } }$ . A single-sample approximation of $q \\mathrm { N E H V I }$ , which we refer to as $q \\mathrm { N E H V I - 1 }$ , can be computed by using $\\tilde { \\pmb { f } } _ { i }$ as the sampled GP in (6). Since the RFF is a deterministic model, it is much less computationally expensive to evaluate than the GP posterior on out-of-sample points, and exact gradients of $q \\mathrm { N E H V I - 1 }$ with respect to current candidate $\\mathbf { \\Delta } _ { \\mathbf { \\mathcal { X } } _ { i } }$ can be computed and used for efficient multi-start optimization of $q \\mathrm { N E H V I - 1 }$ using second-order gradient methods. $q \\mathrm { N E H V I - 1 }$ requires CBD for efficient sequential greedy batch selection and gradient-based optimization, but does not use a sample average approximation for optimizing a new candidate $\\mathbf { \\Delta } _ { \\mathbf { \\mathcal { X } } _ { i } }$ ; instead, it uses an approximate sample path. See Rahimi & Recht [46] for details on RFFs. ",
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+    {
+        "type": "text",
+        "text": "$q$ NEHVI-1 is related to TSEMO in that both use sequential greedy batch selection using HVI based on RFF samples. However, TSEMO does not directly maximize HVI when selecting candidate $\\mathbf { \\Delta } _ { \\mathbf { \\mathcal { X } } _ { i } }$ , where $i = 1 , . . . , q$ ; rather, it relies on a heuristic approach of running NSGA-II on an RFF sample of each objective to create a discrete population of candidates and then selecting the point from the discrete population that maximizes HVI under the RFF sample. In contrast, $q \\mathrm { N E H V I - 1 }$ directly optimizes HVI under the RFF using exact sample-path gradients, which leads to improved optimization performance (see Appendix H). Furthermore, we find that $q \\mathrm { N E H V I - 1 }$ is significantly faster than TSEMO, because rather than using NSGA-II it uses second order gradient methods to optimize HVI (see Appendix H). Gradient-based optimization is only possible because CBD enables scalable, differentiable HVI computation. While the primary goal of this work is to develop a principled, scalable method for parallel EHVI in noisy environments, we include empirical comparisons with qNEHVI-1 throughout the appendix to demonstrate the generalizablility of the CBD approach and practical performance of the $q \\mathrm { N E H V I - 1 }$ approximation. $q \\mathrm { N E H V I - 1 }$ achieves the fastest batch selection timesof any method tested on a GPU on every problem; in many cases, this is an order of magnitude speed-up over $q \\mathrm { N E H V I }$ . Moreover, qNEHVI-1 has a remarkable ability to scale to large batch sizes when the dimensionality of optimization problem is modest. Further investigation of $q \\mathrm { N E H V I - 1 }$ is needed, but we hope that the readers can recognize the ways in which $q \\mathrm { N E H V I }$ can create broader opportunities for research into hypervolume improvement based acquisition functions. ",
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+        "type": "text",
+        "text": "9 Experiments ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "We empirically evaluate $q \\mathrm { N E H V I }$ on a set of synthetic and real-world benchmark problems. We compare it against the following recently proposed methods from the literature: PESMO, MESMO (which we extend to the handle noisy observations using the noisy information gain from Takeno et al. [52]), PFES, DGEMO, MOEA/D-EGO, TSEMO, TS-TCH, $q \\mathrm { E H V I }$ (and qEHVI-PM-CBD, which uses the posterior mean as a plug-in estimate for the function values at the in-sample points, along with CBD to scale to large batch sizes), and qNParEGO. We optimize all methods using multi-start L-BFGS-B with exact gradients (except for PFES, which uses gradients approximated via finite differences), including TS-TCH where we optimize approximate function samples using RFFs with 500 basis functions. We model each outcome with an independent GP with a Matérn 5/2 ARD kernel and infer the GP hyperparameters via maximum a posteriori (MAP) estimation. For all problems, we assume that the noise variances are observed (except ABR, where we infer the noise level). See Appendix G for more details on the experiments and acquisition function implementations. ",
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+    {
+        "type": "text",
+        "text": "We evaluate all methods using the logarithm of the difference in hypervolume between the true Pareto frontier and the approximate Pareto frontier recovered by the algorithm. Since evaluations are noisy, we compute the hypervolume dominated by the noiseless Pareto frontier across the observed points for each method. ",
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+        "type": "text",
+        "text": "Synthetic Problems: We consider a noisy variants of the BraninCurrin problem $\\quad M = 2 , d = 2 ,$ and the DTLZ2 problem $( M = 2 , d = 6 )$ ) [13], in which observations are corrupted with zero-mean additive Gaussian noise with standard deviation of $5 \\%$ of the range of each objective for BraninCurrin and $10 \\%$ for DTLZ2. ",
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+    {
+        "type": "text",
+        "text": "Adaptive Bitrate (ABR) Control Policy Optimization: ABR controllers are used for real-time communication and media streaming applications. Policies for these controllers must be tuned to deliver a high quality of experience with respect to multiple objectives [40]. In industry settings, A/B tests with dozens of policies are tested simultaneously since each policy may take days or weeks to evaluate, producing noisy measurements across multiple objectives. In this experiment, we tune policies to maximize video quality (bitrate) and minimize stall time. The policy has $d = 4$ parameters, which are detailed in Appendix G. We use the Park simulator [41] and sample a random set of 100 traces to obtain noisy measurements of the objectives under a given policy. For comparing the performance of different methods, we estimate the true noiseless objective using mean objectives across 300 traces. We infer a homoskedastic noise level jointly with the GP hyperparameters via MAP estimation. ",
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+    {
+        "type": "image",
+        "img_path": "images/6af06c98f410630170c917e8cab50477e1aa6be60e3cfa4697976649ab5238e6.jpg",
+        "image_caption": [
+            "Figure 3: Sequential optimization performance. The shaded region indicates two standard errors of the mean over 100 replications (only 20 replications were feasible for PESMO due to large runtimes). "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+        "page_idx": 8
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+    {
+        "type": "text",
+        "text": "Vehicle Design Optimization: Optimizing the design of the frame an automobile is important to maximizing passenger safety, vehicle durability and fuel efficiency. Evaluating a vehicle design is time-consuming, since either a vehicle must manufactured and crashed, or a nonlinear finite element-based crash analysis must be run to simulate a collision (which can take over 20 hours per run) [62]. Hence, evaluating many designs in parallel is critical for reducing end-to-end optimization time. Observations are often noisy due to manufacturing imperfections, measurement error, or non-deterministic simulations. In this experiment, we tune the $d = 5$ widths of various components of a vehicle’s frame to minimize proxy metrics for (1) fuel consumption, (2) passenger trauma in a full frontal collison, and (3) vehicle fragility [53]. See Appendix G for details. For this demonstration, we add zero-mean Gaussian noise with a standard deviation of $1 \\%$ of the objective range, which roughly corresponds to the manufacturing noise level used in previous work [62]. ",
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+    {
+        "type": "text",
+        "text": "9.1 Summary of Results: ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "We find that $q \\mathrm { N E H V I }$ and $q \\mathrm { N E H V I - 1 }$ outperform all other methods on the noisy benchmarks, both in the sequential and parallel setting. In the sequential setting (Fig 3), $q \\mathrm { N E H V I }$ and $q \\mathrm { N E H V I - 1 }$ are followed closely by $q \\mathrm { E H V I - P M }$ , and in some cases, even $q \\mathrm { E H V I }$ . TS-TCH is firmly in the middle of the pack, while information-theoretic acquisition functions appear to perform the worst. This is consistent across noise levels; for experiments where we add noise to the objectives, we consider noise levels ranging from $1 \\%$ to $10 \\%$ of the range of each objective (these are magnitudes of the noise often seen in practice). Previous works have only evaluated MOBO algorithms with noise levels of $1 \\%$ [25]. In Appendix H, we perform a study showing that $q \\mathrm { N E H V I }$ consistently performs best with increasing noise levels up to $30 \\%$ of the range of each objective. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "While parallel evaluation can provide optimization speedups on order of the batch size $q$ , these evaluations do affect the overall sample complexity of the algorithm, since less information is available within the synchronous batch setting compared with fully sequential optimization. We find that, by and large, $q \\mathrm { N E H V I }$ achieves the greatest hyper-volume for increasingly large batch sizes, and scales more elegantly relative to TS-TCH and the ParEGO variants (Fig 4). qNEHVI also consistently outperforms $q \\mathrm { E H V I }$ -PM-CBD. In Appendix H, we observe that $q \\mathrm { N E H V I }$ and $q \\mathrm { N E H V I - 1 }$ provides excellent anytime performance all values of $q$ that we tested. We provide results on 4 additional test problems in Appendix H.3, and in Appendix H.8, we demonstrate that leveraging CBD and a single sample path approximation, $q$ NEHVI-1 enables scaling to 5-objective problems, which is a first for an HVI-based method, to our knowledge. ",
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+    {
+        "type": "image",
+        "img_path": "images/871673288d167bcbd541079e5997b278cca429d8ca64fd48b785eb05c1db1a12.jpg",
+        "image_caption": [
+            "Figure 4: The quality of the final Pareto frontier identified by each method with increasing batch sizes $q$ given a budget of 224 function evaluations. $q \\mathrm { E H V I }$ is only included for $q = 1$ and $q = 8$ because the IEP scales exponential with $q$ . DGEMO is omitted on the ABR problem because it was prohibitively slow with time-consuming ABR simulations and on the VehicleSafety problem because DGEMO consistently crashed in the graph cutting algorithm. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+        "page_idx": 9
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+    {
+        "type": "text",
+        "text": "In our experiments, we find that $q \\mathrm { N E H V I - 1 }$ is among the top performers on relatively lowdimensional problems. Given the strong performance of $q \\mathrm { N E H V I - 1 }$ , we examine its performance as the dimensionality of the search space increases in Appendix H.5. We find that $q \\mathrm { N E H V I }$ is more robust than $q \\mathrm { N E H V I - 1 }$ in higher-dimensional search spaces, but further investigation is needed into how the number of the Fourier basis functions affects the performance of qNEHVI-1 in highdimensional search spaces. ",
+        "bbox": [
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+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Optimization wall time: Across all experiments, we observe competitive wall times for optimizing $q \\mathrm { N E H V I }$ and $q \\mathrm { N E H V I - 1 }$ (all wall time comparisons are provided in Appendix H). On a GPU, optimizing qNEHVI-1 incurs the lowest wall time of any method that we tested on every single problem and optimizing $q \\mathrm { N E H V I }$ is faster than optimizing information-theoretic methods on all problems. Using efficient low-rank Cholesky updates, qNEHVI is often faster than the qNParEGO implementation in BoTorch on a GPU. ",
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+    {
+        "type": "text",
+        "text": "10 Discussion ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "We proposed NEHVI, a novel acquisition function that provides a principled approach to parallel and noisy multi-objective Bayesian optimization. NEHVI is a one-step Bayes-optimal policy for maximizing the hypervolume dominated by the Pareto frontier in noisy and noise-free settings. NEHVI is made feasible by a new approach to computing joint hypervolumes (CBD), and we demonstrated that CBD enables scalable, parallel candidate generation with both noiseless $q \\mathrm { E H V I }$ and qNEHVI. We provide theoretical results on optimizing a MC estimator of $q \\mathrm { N E H V I }$ using sample average approximation and demonstrate significant improvements in optimization performance over state-of-the-art MOBO algorithms. ",
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+    {
+        "type": "text",
+        "text": "Yet, our work has some limitations. While the information-theoretic acquisition functions tested here perform poorly on our benchmarks, they do allow for decoupled evaluations of different objectives in cases where querying one objective may be more resource-intensive than querying other objectives. Optimizing such acquisition functions is a non-trivial task, and it is possible that with improved procedures, such acquisition functions could yield improved performance and provide a principled approach to selecting evaluation sources on a budget. Although practically fast enough for most Bayesian optimization tasks, exact hypervolume computation has super-polynomial complexity in the number of objectives. Combining $q \\mathrm { N E H V I }$ with differentiable approximate methods for computing hypervolume (e.g. Couckuyt et al. [9], Golovin & Zhang [23]) could lead to further speed-ups. ",
+        "bbox": [
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+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "We hope that the core ideas presented in this work, including the CBD approach, can provide a framework to support the development of new computationally efficient MOBO methods. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "References ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            90,
+            266,
+            106
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
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parse/train/ByMVTsR5KQ/ByMVTsR5KQ.md

@@ -0,0 +1,338 @@
+# ADVERSARIAL AUDIO SYNTHESIS
+
+Chris Donahue Department of Music UC San Diego cdonahue@ucsd.edu
+
+Julian McAuley   
+Department of Computer Science UC San Diego   
+jmcauley@eng.ucsd.edu
+
+Miller Puckette Department of Music UC San Diego msp@ucsd.edu
+
+# ABSTRACT
+
+Audio signals are sampled at high temporal resolutions, and learning to synthesize audio requires capturing structure across a range of timescales. Generative adversarial networks (GANs) have seen wide success at generating images that are both locally and globally coherent, but they have seen little application to audio generation. In this paper we introduce WaveGAN, a first attempt at applying GANs to unsupervised synthesis of raw-waveform audio. WaveGAN is capable of synthesizing one second slices of audio waveforms with global coherence, suitable for sound effect generation. Our experiments demonstrate that—without labels—WaveGAN learns to produce intelligible words when trained on a smallvocabulary speech dataset, and can also synthesize audio from other domains such as drums, bird vocalizations, and piano. We compare WaveGAN to a method which applies GANs designed for image generation on image-like audio feature representations, finding both approaches to be promising.
+
+# 1 INTRODUCTION
+
+Synthesizing audio for specific domains has many practical applications in creative sound design for music and film. Musicians and Foley artists scour large databases of sound effects to find particular audio recordings suitable for specific scenarios. This strategy is painstaking and may result in a negative outcome if the ideal sound effect does not exist in the library. A better approach might allow a sound artist to explore a compact latent space of audio, taking broad steps to find the types of sounds they are looking for (e.g. footsteps) and making small adjustments to latent variables to finetune (e.g. a large boot lands on a gravel path). However, audio signals have high temporal resolution, and strategies that learn such a representation must perform effectively in high dimensions.
+
+Generative Adversarial Networks (GANs) (Goodfellow et al., 2014) are one such unsupervised strategy for mapping low-dimensional latent vectors to high-dimensional data. The potential advantages of GAN-based approaches to audio synthesis are numerous. Firstly, GANs could be useful for data augmentation (Shrivastava et al., 2017) in data-hungry speech recognition systems. Secondly, GANs could enable rapid and straightforward sampling of large amounts of audio. Furthermore, while the usefulness of generating static images with GANs is arguable, there are many applications (e.g. Foley) for which generating sound effects is immediately useful. But despite their increasing fidelity at synthesizing images (Radford et al., 2016; Berthelot et al., 2017; Karras et al., 2018), GANs have yet to be demonstrated capable of synthesizing audio in an unsupervised setting.
+
+A na¨ıve solution for applying image-generating GANs to audio would be to operate them on imagelike spectrograms, i.e., time-frequency representations of audio. This practice of bootstrapping image recognition algorithms for audio tasks is commonplace in the discriminative setting (Hershey et al., 2017). In the generative setting however, this approach is problematic as the most perceptually-informed spectrograms are non-invertible, and hence cannot be listened to without lossy estimations (Griffin & Lim, 1984) or learned inversion models (Shen et al., 2018).
+
+Recent work (van den Oord et al., 2016; Mehri et al., 2017) has shown that neural networks can be trained with autoregression to operate on raw audio. Such approaches are attractive as they dispense with engineered feature representations. However, unlike with GANs, the autoregressive setting results in slow generation as output audio samples must be fed back into the model one at a time.
+
+In this work, we investigate both waveform and spectrogram strategies for generating one-second slices of audio with GANs.1 For our spectrogram approach (SpecGAN), we first design a spectrogram representation that allows for approximate inversion, and bootstrap the two-dimensional deep convolutional GAN (DCGAN) method (Radford et al., 2016) to operate on these spectrograms. In WaveGAN, our waveform approach, we flatten the DCGAN architecture to operate in one dimension, resulting in a model with the same number of parameters and numerical operations as its twodimensional analog. With WaveGAN, we provide both a starting point for practical audio synthesis with GANs and a recipe for modifying other image generation methods to operate on waveforms.
+
+We primarily envisage our method being applied to the generation of short sound effects suitable for use in music and film. For example, we trained a WaveGAN on drums, resulting in a procedural drum machine designed to assist electronic musicians (demo chrisdonahue.com/wavegan). However, human evaluation for such domain-specific tasks would require expert listeners. Therefore, we also consider a speech benchmark, facilitating straightforward assessment by human annotators. Specifically, we explore a task where success can easily be judged by any English speaker: generating examples of spoken digits “zero” through “nine”.
+
+Though our evaluation focuses on a speech generation task, we note that it is not our goal to develop a text-to-speech synthesizer. Instead, our investigation concerns whether unsupervised strategies can learn global structure (e.g. words in speech data) implicit in high-dimensional audio signals without conditioning. Our experiments on speech demonstrate that both WaveGAN and SpecGAN can generate spoken digits that are intelligible to humans. On criteria of sound quality and speaker diversity, human judges indicate a preference for the audio generated by WaveGAN compared to that from SpecGAN.
+
+# 2 GAN PRELIMINARIES
+
+GANs learn mappings from low-dimensional latent vectors $z \in { \mathcal { Z } }$ , i.i.d. samples from known prior $P _ { Z }$ , to points in the space of natural data $\mathcal { X }$ . In their original formulation (Goodfellow et al., 2014), a generator $G : { \mathcal { Z } } \mapsto { \mathcal { X } }$ is pitted against a discriminator $D : \mathcal { X } \mapsto [ 0 , 1 ]$ in a two-player minimax game. $G$ is trained to minimize the following value function, while $D$ is trained to maximize it:
+
+$$
+V ( D , G ) = \mathbb { E } _ { \pmb { x } \sim P _ { X } } [ \log D ( \pmb { x } ) ] + \mathbb { E } _ { \pmb { z } \sim P _ { Z } } [ \log ( 1 - D ( G ( \pmb { z } ) ) ) ] .
+$$
+
+In other words, $D$ is trained to determine if an example is real or fake, and $G$ is trained to fool the discriminator into thinking its output is real. Goodfellow et al. (2014) demonstrate that their proposed training algorithm for Equation 1 equates to minimizing the Jensen-Shannon divergence between $P _ { X }$ , the data distribution, and $P _ { G }$ , the implicit distribution of the generator when $z \sim$ $P _ { Z }$ . In this original formulation, GANs are notoriously difficult to train, and prone to catastrophic failure cases. Instead of Jensen-Shannon divergence, Arjovsky et al. (2017) suggest minimizing the smoother Wasserstein-1 distance between generated and data distributions
+
+$$
+W ( P _ { X } , P _ { G } ) = \operatorname* { s u p } _ { \| f \| _ { L } \leq 1 } \mathbb { E } _ { x \sim P _ { X } } [ f ( x ) ] - \mathbb { E } _ { x \sim P _ { G } } [ f ( x ) ]
+$$
+
+where $\| f \| _ { L } \leq 1 : \mathcal { X } \mapsto \mathbb { R }$ is the family of functions that are 1-Lipschitz.
+
+To minimize Wasserstein distance, they suggest a GAN training algorithm (WGAN), similar to that of Goodfellow et al. (2014), for the following value function:
+
+$$
+V _ { \mathrm { W G A N } } ( D _ { w } , G ) = \mathbb { E } _ { { \pmb { x } } \sim P _ { X } } [ D _ { w } ( { \pmb x } ) ] - \mathbb { E } _ { { \pmb z } \sim P _ { Z } } [ D _ { w } ( G ( { \pmb z } ) ) ] .
+$$
+
+With this formulation, $D _ { w } : \mathcal { X } \mapsto \mathbb { R }$ is not trained to identify examples as real or fake, but instead is trained as a function that assists in computing the Wasserstein distance. Arjovsky et al. (2017) suggest weight clipping as a means of enforcing that $D _ { w }$ is 1-Lipschitz. As an alternative strategy, Gulrajani et al. (2017) replace weight clipping with a gradient penalty (WGAN-GP) that also enforces the constraint. They demonstrate that their WGAN-GP strategy can successfully train a variety of model configurations where other GAN losses fail.
+
+![](images/97650d4794e4590db58a92f72cb57f49c85fef289d9d13475d57aaaedc1d8259.jpg)
+
+![](images/a3b51d30ba4da138a3efe74960c70a5d651173f0a00848c1759dfae46f250693.jpg)  
+Figure 1: First eight principal components for 5x5 patches from natural images (left) versus those of length-25 audio slices from speech (right). Periodic patterns are unusual in natural images but a fundamental structure in audio.   
+Figure 2: Depiction of the transposed convolution operation for the first layers of the DCGAN (Radford et al., 2016) (left) and WaveGAN (right) generators. DCGAN uses small (5x5), twodimensional filters while WaveGAN uses longer (length-25), one-dimensional filters and a larger upsampling factor. Both strategies have the same number of parameters and numerical operations.
+
+# 3 WAVEGAN
+
+We motivate our design choices for WaveGAN by first highlighting the different types of structure found in audio versus images.
+
+# 3.1 INTRINSIC DIFFERENCES BETWEEN AUDIO AND IMAGES
+
+One way to illustrate the differences between audio and images is by examining the axes along which these types of data vary most substantially, i.e. by principal component analysis. In Figure 1, we show the first eight principal components for patches from natural images and slices from speech. While the principal components of images generally capture intensity, gradient, and edge characteristics, those from audio form a periodic basis that decompose the audio into constituent frequency bands. In general, natural audio signals are more likely to exhibit periodicity than natural images.
+
+As a consequence, correlations across large windows are commonplace in audio. For example, in a waveform sampled at $1 6 \mathrm { k H z }$ , a $4 4 0 \mathrm { H z }$ sinusoid (the musical note A4) takes over 36 samples to complete a single cycle. This suggests that filters with larger receptive fields are needed to process raw audio. This same intuition motivated van den Oord et al. (2016) in their design of WaveNet, which uses dilated convolutions to exponentially increase the model’s effective receptive field with linear increase in layer depth.
+
+# 3.2 WAVEGAN ARCHITECTURE
+
+We base our WaveGAN architecture off of DCGAN (Radford et al., 2016) which popularized usage of GANs for image synthesis. The DCGAN generator uses the transposed convolution operation (Figure 2) to iteratively upsample low-resolution feature maps into a high-resolution image. Motivated by our above discussion, we modify this transposed convolution operation to widen its receptive field. Specifically, we use longer one-dimensional filters of length 25 instead of two-dimensional filters of size 5x5, and we upsample by a factor of 4 instead of 2 at each layer (Figure 2). We modify the discriminator in a similar way, using length-25 filters in one dimension and increasing stride from 2 to 4. These changes result in WaveGAN having the same number of parameters, numerical operations, and output dimensionality as DCGAN.
+
+Because DCGAN outputs 64x64 pixel images — equivalent to just 4096 audio samples — we add one additional layer to the model resulting in 16384 samples, slightly more than one second of audio at $1 6 \mathrm { k H z }$ . This length is already sufficient for certain sound domains (e.g. sound effects, voice commands), and future work adapting megapixel image generation techniques (Karras et al., 2018) could expand the output length to more than a minute. We requantize the real data from its 16- bit integer representation (linear pulse code modulation) to 32-bit floating point, and our generator similarly outputs floating point waveforms. A complete description of our model is in Appendix D.
+
+In summary, we outline our modifications to the DCGAN (Radford et al., 2016) method which result in WaveGAN. This straightforward recipe already produces reasonable audio, and further contributions outlined below and in Appendix A serve to refine results.
+
+1. Flatten 2D convolutions into 1D (e.g. 5x5 2D convolution becomes length-25 1D).   
+2. Increase the stride factor for all convolutions (e.g. stride $2 \mathrm { x 2 }$ becomes stride 4).   
+3. Remove batch normalization from the generator and discriminator.   
+4. Train using the WGAN-GP (Gulrajani et al., 2017) strategy.
+
+# 3.3 PHASE SHUFFLE
+
+Generative image models that upsample by transposed convolution (such as DCGAN) are known to produce characteristic “checkerboard” artifacts in images (Odena et al., 2016). Periodic patterns are less common in images (Section 3.1), and thus the discriminator can learn to reject images that contain them. For audio, analogous artifacts are perceived as pitched noise which may overlap with frequencies commonplace in the real data, making the discriminator’s objective more challenging. However, the artifact frequencies will always occur at a particular phase, allowing the discriminator to learn a trivial policy to reject generated examples. This may inhibit the overall optimization problem.
+
+To prevent the discriminator from learning such a solution, we propose the phase shuffle operation with hyperparameter $n$ . Phase shuffle randomly perturbs the phase of each layer’s activations by $- n$ to $n$ samples before input to the next layer (Figure 3). We apply phase shuffle only to the discriminator, as the latent vector already provides the generator a mechanism to manipulate the phase of a resultant waveform. Intuitively speaking, phase shuffle makes the discriminator’s job more challenging by requiring invariance to the phase of the input waveform.
+
+![](images/757554e8081d9f0db8bb7bd31943ae5c91e45cbdbbfaabbff1a9a214fa80aad7.jpg)  
+Figure 3: At each layer of the WaveGAN discriminator, the phase shuffle operation perturbs the phase of each feature map by Uniform $\sim ~ [ - n , n ]$ samples, filling in the missing samples (dashed outlines) by reflection. Here we depict all possible outcomes for a layer with four feature maps $( n = 1 )$ ).
+
+# 4 SPECGAN: GENERATING SEMI-INVERTIBLE SPECTROGRAMS
+
+While a minority of recent research in discriminative audio classification tasks has used raw audio input (Sainath et al., 2015; Lee et al., 2017), most of these approaches operate on spectrogram representations of audio. A generative model may also benefit from operating in such a time-frequency space. However, commonly-used representations in the discriminative setting are uninvertible.
+
+With SpecGAN, our frequency-domain audio generation model, we design a spectrogram representation that is both well-suited to GANs designed for image generation and can be approximately inverted. Additionally, to facilitate direct comparison, our representation is designed to use the same dimensionality per unit of time as WaveGAN (16384 samples yield a 128x128 spectrogram).
+
+To process audio into suitable spectrograms, we first perform the short-time Fourier transform with $1 6 \mathrm { m s }$ windows and $8 \mathrm { m s }$ stride, resulting in 128 frequency bins2 linearly spaced from 0 to $8 \mathrm { k H z }$ . We take the magnitude of the resultant spectra and scale amplitude values logarithmically to better-align with human perception. We then normalize each frequency bin to have zero mean and unit variance. This type of preprocessing is commonplace in audio classification, but produce spectrograms with unbounded values—a departure from image representations. We therefore clip the spectra to 3 standard deviations and rescale to [−1, 1]. Through an informal listening test, we determined that this clipping strategy did not produce an audible difference during inversion.
+
+![](images/c6d4e0d8442f77e8e8c02fbc2975b0da3dda75c387e24bdc9707590ef92e64b8.jpg)  
+Figure 4: Top: Random samples from each of the five datasets used in this study, illustrating the wide variety of spectral characteristics. Middle: Random samples generated by WaveGAN for each domain. WaveGAN operates in the time domain but results are displayed here in the frequency domain for visual comparison. Bottom: Random samples generated by SpecGAN for each domain.
+
+Once our dataset has been processed into this format, we operate the DCGAN (Radford et al., 2016) algorithm on the resultant spectra. To render the resultant generated spectrograms as waveforms, we first invert the steps of spectrogram preprocessing described above, resulting in linear-amplitude magnitude spectra. We then employ the iterative Griffin-Lim algorithm (Griffin & Lim, 1984) with 16 iterations to estimate phase and produce 16384 audio samples.
+
+# 5 EXPERIMENTAL PROTOCOL
+
+To facilitate human evaluation, our experimentation focuses on the Speech Commands Dataset (Warden, 2018). This dataset consists of many speakers recording individual words in uncontrolled recording conditions. We explore a subset consisting of the spoken digits “zero” through “nine” and refer to this subset as the Speech Commands Zero Through Nine (SC09) dataset. While this dataset is intentionally reminiscent of the popular MNIST dataset of written digits, we note that examples from SC09 are much higher dimensional $( \mathbb { R } ^ { 1 6 0 0 0 } )$ than examples from MNIST $( \mathbb { R } ^ { 2 8 \times 2 8 = 7 8 4 }$ ).
+
+These ten words encompass many phonemes and two consist of multiple syllables. Each recording is one second in length, and we do not attempt to align the words in time. There are 1850 utterances of each word in the training set, resulting in 5.3 hours of speech. The heterogeneity of alignments, speakers, and recording conditions make this a challenging dataset for generative modeling.
+
+Our baseline configuration for WaveGAN excludes phase shuffle. We compare this to the performance of WaveGAN with phase shuffle $( n \in \{ 2 , 4 \} )$ and a variant of WaveGAN which uses nearest-neighbor upsampling rather than transposed convolution (Odena et al., 2016). Hoping to reduce noisy artifacts, we also experiment with adding a wide (length-512) post-processing filter to the output of the generator and learning its parameters with the rest of the generator variables (details in Appendix A.1). We use the WGAN-GP (Gulrajani et al., 2017) algorithm for all experiments, finding it to produce reasonable results where others (Radford et al., 2016; Mao et al., 2017; Arjovsky et al., 2017) failed. We compare the performance of these configurations to that of SpecGAN.
+
+We also perform experiments on four other datasets with different characteristics (Figure 4):
+
+1. Drum sound effects (0.7 hours): Drum samples for kicks, snares, toms, and cymbals   
+2. Bird vocalizations (12.2 hours): In-the-wild recordings of many species (Boesman, 2018)   
+3. Piano (0.3 hours): Professional performer playing a variety of Bach compositions   
+4. Large vocab speech (TIMIT) (2.4 hours): Multiple speakers, clean (Garofolo et al., 1993)
+
+We train our networks using batches of size 64 on a single NVIDIA P100 GPU. During our quantitative evaluation of SC09 (discussed below), our WaveGAN networks converge by their early stopping criteria (inception score) within four days (200k iterations, around 3500 epochs), and produce speech-like audio within the first hour of training. Our SpecGAN networks converge more quickly, within two days (around 1750 epochs). On the other four datasets, we train WaveGAN for $2 0 0 \mathrm { k }$ iterations representing nearly 1500 epochs for the largest dataset. Unlike with autoregressive methods (van den Oord et al., 2016; Mehri et al., 2017), generation with WaveGAN is fully parallel and can produce an hour of audio in less than two seconds. We list all hyperparameters in Appendix E.
+
+# 6 EVALUATION METHODOLOGY
+
+Evaluation of generative models is a fraught topic. Theis et al. (2016) demonstrate that quantitative measures of sample quality are poorly correlated with each other and human judgement. Accordingly, we use several quantitative evaluation metrics for hyperparameter validation and discussion, and also evaluate our most promising models with human judges.
+
+# 6.1 INCEPTION SCORE
+
+Salimans et al. (2016) propose the inception score, which uses a pre-trained Inception classifier (Szegedy et al., 2016) to measure both the diversity and semantic discriminability of generated images, finding that the measure correlates well with human judgement.
+
+Given model scores $P ( \pmb { y } \mid \pmb { x } )$ with marginal $P ( \pmb { y } )$ , the inception score is defined as $\exp ( \mathbb { E } _ { \pmb { x } } D _ { \mathrm { K L } } ( P ( \pmb { y } \mid \pmb { x } ) | | P ( \pmb { y } ) ) )$ , and is estimated over a large number of samples (e.g. 50k). For $n$ classes, this measure ranges from 1 to $n$ , and is maximized when the model is completely confident about each prediction and predicts each label equally often. We will use this measure as our primary quantitative evaluation method and early stopping criteria.
+
+To measure inception score, we train an audio classifier on SC09. Our classifier first computes a short-time Fourier transform of the input audio with $6 4 \mathrm { m s }$ windows and 8 ms stride. This representation is projected to 128 frequency bins equally spaced on the Mel scale (Stevens et al., 1937) from $4 0 \mathrm { { H z } }$ to $7 8 0 0 \mathrm { H z }$ . Amplitudes are scaled logarithmically and normalized so that each bin has zero mean and unit variance. We process this perceptually-informed representation with four layers of convolution and pooling, projecting the result to a softmax layer with 10 classes. We perform early stopping on the minimum negative log-likelihood of the validation set; the resultant model achieves $9 3 \%$ accuracy on the test set. Because this classifier observes spectrograms, our spectrogramgenerating models may have a representational advantage over our waveform-generating models.
+
+# 6.2 NEAREST NEIGHBOR COMPARISONS
+
+Inception score has two trivial failure cases in which a poor generative model can achieve a high score. Firstly, a generative model that outputs a single example of each class with uniform probability will be assigned a high score. Secondly, a generative model that overfits the training data will achieve a high score simply by outputting examples on which the classifier was trained.
+
+We use two indicators metrics to determine if a high inception score has been caused by either of these two undesirable cases. Our first indicator, $| D | _ { \mathrm { s e l f } }$ , measures the average Euclidean distance of a set of 1k examples to their nearest neighbor within the set (other than itself). A higher $| D | _ { \mathrm { s e l f } }$ indicates higher diversity amongst samples. Because measuring Euclidean distance in time-domain
+
+Table 1: Quantitative and qualitative (human study) results for SC09 experiments comparing real and generated data. A higher inception score suggests that semantic modes of the real data distribution have been captured. $| D | _ { \mathrm { s e l f } }$ indicates the intra-dataset diversity relative to that of the real test data. $| D | _ { \mathrm { t r a i n } }$ indicates the distance between the dataset and the training set relative to that of the test data; a low value indicates a generative model that is overfit to the training data. Acc. is the overall accuracy of humans on the task of labeling class-balanced digits (random chance is 0.1). Sound quality, ease of intelligibility and speaker diversity are mean opinion scores (1-5); higher is better.
+
+<table><tr><td></td><td colspan="4">Quantitative</td><td colspan="4">Qualitative (human judges)</td></tr><tr><td>Experiment</td><td>Inception score</td><td>|D|self</td><td>|D|train</td><td>Acc.</td><td>Quality</td><td>Ease</td><td></td><td>Diversity</td></tr><tr><td>Real (train)</td><td>9.18 ± 0.04</td><td>1.1</td><td>0.0</td><td rowspan="3">0.95</td><td rowspan="3"></td><td></td><td></td><td></td></tr><tr><td>Real (test)</td><td>8.01±0.24</td><td>1.0</td><td>1.0</td><td>3.9 ±0.8</td><td>3.9 ± 1.1</td><td>3.5 ± 1.0</td></tr><tr><td>Parametric</td><td>5.02 ± 0.06</td><td>0.7</td><td>1.1</td><td></td><td></td><td></td></tr><tr><td>WaveGAN</td><td>4.12 ± 0.03</td><td>1.4</td><td>2.0</td><td rowspan="6"></td><td></td><td></td><td></td><td></td></tr><tr><td>+ Phase shuffle n = 2</td><td>4.67 ± 0.01</td><td>0.8</td><td>2.3</td><td>0.58</td><td>2.3± 0.9</td><td>2.8±0.9</td><td>3.2 ±0.9</td></tr><tr><td>+ Phase shuffle n = 4</td><td>4.54± 0.03</td><td>1.0</td><td>2.3</td><td></td><td></td><td></td><td></td></tr><tr><td>+ Nearest neighbor</td><td>3.77± 0.02</td><td>1.8</td><td>2.6</td><td></td><td></td><td></td><td></td></tr><tr><td>+ Post-processing</td><td>3.92 ± 0.03</td><td>1.4</td><td>2.9</td><td></td><td></td><td></td><td></td></tr><tr><td>+ Dropout</td><td>3.93 ± 0.03</td><td>1.0</td><td>2.6</td><td></td><td></td><td></td><td></td></tr><tr><td>SpecGAN</td><td>6.03 ± 0.04</td><td>1.1</td><td>1.4</td><td>0.66</td><td>1.9 ±0.8</td><td>2.8 ±0.9</td><td></td><td>2.6±1.0</td></tr><tr><td>+Phase shuffle n =1</td><td>3.71± 0.03</td><td>0.8</td><td>1.6</td><td></td><td></td><td></td><td></td><td></td></tr></table>
+
+audio poorly represents human perception, we evaluate distances in the same frequency-domain representation as our classifier from Section 6.1.
+
+Our second indicator, $| D | _ { \mathrm { t r a i n } }$ , measures the average Euclidean distance of 1k examples to their nearest neighbor in the real training data. If the generative model simply produces examples from the training set, this measure will be 0. We report $| D | _ { \mathrm { t r a i n } }$ and $| D | _ { \mathrm { s e l f } }$ relative to those of the test set.
+
+# 6.3 QUALITATIVE HUMAN JUDGEMENTS
+
+While inception score is a useful metric for hyperparameter validation, our ultimate goal is to produce examples that are intelligible to humans. To this end, we measure the ability of human annotators on Amazon Mechanical Turk to label the generated audio. Using our best WaveGAN and SpecGAN models as measured by inception score, we generate random examples until we have 300 for each digit (as labeled by our classifier from Section 6.1)—3000 total. In batches of ten random examples, we ask annotators to label which digit they perceive in each example, and compute their accuracy with respect to the classifier’s labels (random accuracy would be $1 \bar { 0 } \%$ ). After each batch, annotators assign subjective values of 1–5 for criteria of sound quality, ease of intelligibility, and speaker diversity. We report accuracy $n = 3 0 0 0$ ) and mean opinion scores $\mathit { n } = 3 0 0 $ ) in Table 1.
+
+# 7 RESULTS AND DISCUSSION
+
+Results for our evaluation appear in Table 1. We also evaluate our metrics on the real training data, the real test data, and a version of SC09 generated by a parametric speech synthesizer (Buchner, 2017). We also compare to SampleRNN (Mehri et al., 2017) and two public implementations of WaveNet (van den Oord et al., 2016), but neither method produced competitive results (details in Appendix B), and we excluded them from further evaluation. These autoregressive models have not previously been examined on small-vocabulary speech data, and their success at generating full words has only been demonstrated when conditioning on rich linguistic features. Sound examples for all experiments can be found at chrisdonahue.com/wavegan_examples.
+
+While the maximum inception score for SC09 is 10, any score higher than the test set score of 8 should be seen as evidence that a generative model has overfit. Our best WaveGAN model uses phase shuffle with $n = 2$ and achieves an inception score of 4.7. To compare the effect of phase shuffle to other common regularizers, we also tried using $5 0 \%$ dropout in the discriminator’s activations, which resulted in a lower score. Phase shuffle decreased the inception score of SpecGAN, possibly because the operation has an exaggerated effect when applied to the compact temporal axis of spectrograms.
+
+Most experiments produced $| D | _ { \mathrm { s e l f } }$ (diversity) values higher than that of the test data, and all experiments produced $| D | _ { \mathrm { t r a i n } }$ (distance from training data) values higher than that of the test data. While these measures indicate that our generative models produce examples with statistics that deviate from those of the real data, neither metric indicates that the models achieve high inception scores by the trivial solutions outlined in Section 6.2.
+
+Compared to examples from WaveGAN, examples from SpecGAN achieve higher inception score (6.0 vs. 4.7) and are labeled more accurately by humans ( $6 6 \%$ vs. $5 8 \%$ ). However, on subjective criteria of sound quality and speaker diversity, humans indicate a preference for examples from WaveGAN. It appears that SpecGAN might better capture the variance in the underlying data compared to WaveGAN, but its success is compromised by sound quality issues when its spectrograms are inverted to audio. It is possible that the poor qualitative ratings for examples from SpecGAN are primarily caused by the lossy Griffin-Lim inversion (Griffin & Lim, 1984) and not the generative procedure itself. We see promise in both waveform and spectrogram audio generation with GANs; our study does not suggest a decisive winner. For a more thorough investigation of spectrogram generation methods, we point to follow-up work (Engel et al., 2019).
+
+Finally, we train WaveGAN and SpecGAN models on the four other domains listed in Section 5. Somewhat surprisingly, we find that the frequency-domain spectra produced by WaveGAN (a timedomain method) are visually more consistent with the training data (e.g. in terms of sharpness) than those produced by SpecGAN (Figure 4). For drum sound effects, WaveGAN captures semantic modes such as kick and snare drums. On bird vocalizations, WaveGAN generates a variety of distinct bird sounds. On piano, WaveGAN produces musically-consonant motifs that, as with the training data, represent a variety of key signatures and rhythmic patterns. For TIMIT, a large-vocabulary speech dataset with many speakers, WaveGAN produces speech-like babbling similar to results from unconditional autoregressive models (van den Oord et al., 2016).
+
+# 8 RELATED WORK
+
+Much of the work within generative modeling of audio is within the context of text-to-speech. Textto-speech systems are primarily either concatenative or parametric. In concatenative systems, audio is generated by sequencing small, prerecorded portions of speech from a phonetically-indexed dictionary (Moulines & Charpentier, 1990; Hunt & Black, 1996). Parametric systems map text to salient parameters of speech, which are then synthesized by a vocoder (Dudley, 1939); see (Zen et al., 2009) for a comprehensive review. Some of these systems use learning-based approaches such as a hidden Markov models (Yoshimura, 2002; Tokuda et al., 2013), and separately-trained neural networks pipelines (Ling et al., 2015) to estimate speech parameters.
+
+Recently, several researchers have investigated parametric speech synthesis with end-to-end neural network approaches that learn to produce vocoder features directly from text or phonetic embeddings (Arik et al., 2017; Ping et al., 2018; Sotelo et al., 2017; Wang et al., 2017; Shen et al., 2018). These vocoder features are synthesized to raw audio using off-the-shelf methods such as WORLD (Morise et al., 2016) and Griffin-Lim (Griffin & Lim, 1984), or trained neural vocoders (Sotelo et al., 2017; Shen et al., 2018; Ping et al., 2018). All of these methods are supervised: they are trained to map linguistic features to audio outputs.
+
+Several approaches have explored unsupervised generation of raw audio. van den Oord et al. (2016) propose WaveNet, a convolutional model which learns to predict raw audio samples by autoregressive modeling. WaveNets conditioned on rich linguistic features have widely been deployed in textto-speech systems, though they have not been demonstrated capable of generating cohesive words in the unconditional setting. Engel et al. (2017) pose WaveNet as an autoencoder to generate musical instrument sounds. Chung et al. (2014); Mehri et al. (2017) both train recurrent autoregressive models which learn to predict raw audio samples. While autoregressive methods generally produce higher audio fidelity than WaveGAN, synthesis with WaveGAN is orders of magnitude faster.
+
+The application of GANs (Goodfellow et al., 2014) to audio has so far been limited to supervised learning problems in combination with traditional loss functions. Pascual et al. (2017) apply GANs to raw audio speech enhancement. Their encoder-decoder approach combines the GAN objective with an $L _ { 2 }$ loss. Fan et al. (2017); Michelsanti & Tan (2017); Donahue et al. (2018) all use GANs in combination with unstructured losses to map spectrograms in one domain to spectrograms in another. Chen et al. (2017) use GANs to map musical performance images into spectrograms.
+
+# 9 CONCLUSION
+
+We present WaveGAN, the first application of GANs to unsupervised audio generation. WaveGAN is fully parallelizable and can generate hours of audio in only a few seconds. In its current form, WaveGAN can be used for creative sound design in multimedia production. In our future work we plan to extend WaveGAN to operate on variable-length audio and also explore a variety of label conditioning strategies. By providing a template for modifying image generation models to operate on audio, we hope that this work catalyzes future investigation of GANs for audio synthesis.
+
+# ACKNOWLEDGMENTS
+
+The authors would like to thank Peter Boesman and Colin Raffel for providing training data for this work. This work was supported by the Unity Global Graduate Fellowship program and the UC San Diego Department of Computer Science. GPUs used for this work were provided by the HPC $@$ UC program and donations from NVIDIA.
+
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+Aaron van den Oord, Sander Dieleman, Heiga Zen, Karen Simonyan, Oriol Vinyals, Alex Graves, ¨ Nal Kalchbrenner, Andrew Senior, and Koray Kavukcuoglu. WaveNet: A generative model for raw audio. arXiv:1609.03499, 2016.
+
+Yuxuan Wang, RJ Skerry-Ryan, Daisy Stanton, Yonghui Wu, Ron J Weiss, Navdeep Jaitly, Zongheng Yang, Ying Xiao, Zhifeng Chen, Samy Bengio, et al. Tacotron: Towards end-to-end speech synthesis. arXiv:1703.10135, 2017.
+
+Pete Warden. Speech commands: A dataset for limited-vocabulary speech recognition. arXiv:1804.03209, 2018.
+
+Takayoshi Yoshimura. Simultaneous modeling of phonetic and prosodic parameters, and characteristic conversion for hmm-based text-to-speech systems. 2002.
+
+Heiga Zen, Keiichi Tokuda, and Alan W Black. Statistical parametric speech synthesis. Speech Communication, 2009.
+
+![](images/ec0cfc4ea50d6ec4046a3bf70aba991b6d617d7fe7a11b6d37d11eee20c254fa.jpg)  
+Figure 5: (Top): Average impulse response for 1000 random initializations of the WaveGAN generator. (Bottom): Response of learned post-processing filters for speech and bird vocalizations. Post-processing filters reject frequencies corresponding to noise byproducts created by the generative procedure (top). The filter for speech boosts signal in prominent speech bands, while the filter for bird vocalizations (which are more uniformly-distributed in frequency) simply reduces noise presence.
+
+# A UNDERSTANDING AND MITIGATING ARTIFACTS IN GENERATED AUDIO
+
+Generative models that upsample by transposed convolution are known to produce characteristic “checkerboard” artifacts in images (Odena et al., 2016), artifacts with particular spatial periodicities. The discriminator of image-generating GANs can learn to reject images with these artifacts because they are uncommon in real data (as discussed in Section 3.1). However, in the audio domain, the discriminator might not have such luxury as these artifacts correspond to frequencies which might rightfully appear in the real data.
+
+While checkerboard artifacts are an annoyance in image generation, they can be devastating to audio generation results. While our eye may perceive these types of periodic distortions as an intrusive texture, our ear perceives them as an abrasive tone. To characterize these artifacts in WaveGAN, we measure its impulse response by randomly initializing it 1000 times and passing unit impulses to its first convolutional layer. In Figure 5, we plot the average of these responses in the frequency domain. The response has sharp peaks at linear multiples of the sample rates of each convolutional layer $2 5 0 \mathrm { H z }$ , $1 \mathrm { k H z }$ , $4 \mathrm { k H z }$ , etc.). This is in agreement with our informal observation of results from WaveGAN, which often have a pitched noise close to the musical note B $( 2 4 7 \times 2 ^ { n } \mathrm { H z } )$ ).
+
+Below, we will discuss strategies we designed to mitigate these artifacts in WaveGAN.
+
+# A.1 LEARNED POST-PROCESSING FILTERS
+
+We experiment with adding a post-processing filter to the generator, giving WaveGAN a simple mechanism to filter out undesirable frequencies created by the generative process. This filter has a long window (512 samples) allowing it to represent intricate transfer functions, and the weights of the filter are learned as part of the generator’s parameters. In Figure 5, we compare the postprocessing filters that WaveGAN learns for human speech and bird vocalizations. The filters boost signal in regions of the frequency spectrum that are most prominent in the real data domain, and introduce notches at bands that are artifacts of the generative procedure as discussed in the previous section.
+
+![](images/4f414414f814f3c214d3823672dd585c77e1af9cfc6dfe9cad2ef4f56848d5cf.jpg)  
+Figure 6: Depiction of the upsampling strategy used by transposed convolution (zero insertion) and other strategies which mitigate aliasing: nearest neighbor, linear and cubic interpolation.
+
+# A.2 UPSAMPLING PROCEDURE
+
+Transposed convolution upsamples signals by inserting zeros in between samples and applying a learned filterbank. This operation introduces aliased frequencies, copies of pre-existing frequencies shifted by multiples of the new Nyquist rate, into the upsampled signal. While aliased frequencies are usually seen as undesirable artifacts of a bad upsampling procedure, in the generative setting their existence may be crucial for producing fine-grained details in the output.
+
+We experiment with three other upsampling strategies in WaveGAN: nearest-neighbor, linear and cubic interpolation, all of which attenuate aliased frequencies. In Figure 6, we compare these strategies visually. While nearest neighbor upsampling resulted in similar audio output to transposed convolution, linear and cubic interpolation strategies resulted in qualitatively poor audio output (sound examples: chrisdonahue.com/wavegan_examples). We hypothesize that the aliased frequencies produced by upsampling convolutions may be more critical to audio generation than image generation.
+
+# B EXPERIMENTS WITH AUTOREGRESSIVE WAVEFORM MODELS
+
+We developed our WaveGAN and SpecGAN models primarily to address the task of steerable sound effect generation. This is an inherently different task than text to speech (TTS), however autoregressive waveform models (e.g. WaveNet (van den Oord et al., 2016) and SampleRNN (Mehri et al., 2017)) that were developed for TTS can also be used to model and generate waveforms unconditionally. Hence, a comparison to these models for our task is reasonable. One upside of autoregressive models for our task is that they have the potential to produce high-quality audio. Potential downsides are 1) these models take several orders of magnitude longer to generate waveforms, and 2) they do not learn a compact latent space of waveforms, causing useful sound generation tasks like continuous exploration and interpolation to be impossible.
+
+We attempt to train two public implementations of WaveNet (ImplA3 and $\mathrm { I m p l B ^ { 4 } }$ ) and SampleRNN5 on our SC09 digit generation task. We use default parameters for these libraries; the only modifications we make are to reduce the training example size to one second. To our ears, all three libraries failed to produce cohesive words (you can judge for yourself from our sound examples at the bottom chrisdonahue.com/wavegan_examples). This poor subjective performance is echoed by weak inception scores (weaker than any in Table 1): $1 . 0 7 \pm 0 . 0 5$ , $1 . 2 9 \pm 0 . 0 3$ , $2 . 2 8 \pm 0 . 1 9$ for WaveNet ImplA, WaveNet ImplB, and SampleRNN respectively. Note that these inception scores were calculated on far fewer examples $( < 1 k )$ than all of the scores listed in Table 1 (which were computed on $5 0 k$ examples). This is because it took over 24 hours to produce even a thousand one-second examples with these methods (whereas our methods produce $5 0 k$ examples in a few seconds).
+
+Autoregressive methods have not been demonstrated capable of learning to synthesize coherent words without conditioning on rich linguistic features. We are not claiming that these methods cannot learn to synthesize full words, merely that three open-source implementations were unable to do so with default parameters. We want to be clear that our intent is not to disparage autoregressive waveform methods as these methods were developed for a different task, and hence we excluded these poor scores from our results table to avoid sending the wrong message. Instead, we hope to highlight that these implementations produced results that were noncompetitive for our problem domain, and less useful (due to slowness and lack of a latent space) for creative generation of sound effects.
+
+# C FELINE “TURING TEST”
+
+![](images/68f22cd7b27c7439f9717bb54e59dae169763e841d232a0f2017c28f22d8f3f0.jpg)  
+Figure 7: Compared to resting state, this cat’s level of alertness increased when presented bird vocalizations synthesized by WaveGAN and SpecGAN.
+
+As our results improved throughout the course of this research, our cats became quite intrigued by the synthetic bird vocalizations produced by WaveGAN (Figure 7). While this was of course not a formal experiment, we did find this to be encouraging evidence that our method might be capable of producing audio that could additionally convince non-human animals.
+
+Table 2: WaveGAN generator architecture   
+
+<table><tr><td>Operation</td><td>Kernel Size</td><td>Output Shape</td></tr><tr><td>Input z ~ Uniform(-1,1) Dense 1 Reshape ReLU Trans Conv1D (Stride=4)</td><td>(100,256d) (25,16d,8d)</td><td>(n,100) (n,256d) (n,16,16d) (n,16,16d) (n,64,8d) (n,16384,c)</td></tr><tr><td>ReLU Trans Conv1D (Stride=4) ReLU</td><td>(25,8d,4d)</td><td>(n, 64,8d) (n,256,4d) (n,256,4d)</td></tr><tr><td>Trans Conv1D (Stride=4) ReLU Trans Conv1D (Stride=4)</td><td>(25,4d,2d)</td><td>(n,1024,2d) (n,1024,2d)</td></tr><tr><td>ReLU Trans Conv1D (Stride=4) Tanh</td><td>(25,2d,d) (25,d,c)</td><td>(n,4096,d) (n,4096,d)</td></tr></table>
+
+Table 3: WaveGAN discriminator architecture   
+
+<table><tr><td>Operation Input x or G(z)</td><td>Kernel Size</td><td>Output Shape</td></tr><tr><td rowspan="3">Conv1D (Stride=4) LReLU (α = 0.2) Phase Shuffle (n = 2) Conv1D (Stride=4) LReLU (α = 0.2) Phase Shuffle (n = 2) Conv1D (Stride=4) LReLU (α = 0.2) Phase Shuffle (n = 2) Conv1D (Stride=4) LReLU (α = 0.2) Phase Shuffle (n = 2) Conv1D (Stride=4)</td><td>(25,c,d)</td><td>(n,16384,c) (n,4096,d) (n,4096,d) (n,4096,d)</td></tr><tr><td>(25,d,2d) (25,2d,4d)</td><td>(n,1024,2d) (n,1024,2d) (n,1024,2d)</td></tr><tr><td>(25,4d,8d) (25,8d,16d) (256d,1)</td><td>(n,256,4d) (n,256,4d) (n,256,4d) (n,64,8d) (n, 64,8d) (n,64,8d) (n,16,16d) (n,16,16d) (n,256d) (n,1)</td></tr></table>
+
+# D ARCHITECTURE DESCRIPTION
+
+In Tables 2 and 3, we list the full architectures for our WaveGAN generator and discriminator respectively. In Tables 4 and 5, we list the same for SpecGAN. In these tables, $n$ is the batch size, $d$ modifies model size, and $c$ is the number of channels in the examples. In all of our experiments in this paper, $c = 1$ . All dense and convolutional layers include biases. No batch normalization is used in WaveGAN or SpecGAN.
+
+# E TRAINING HYPERPARAMETERS
+
+In Table 6, we list the values of these and all other hyperparameters for our experiments, which constitute our out-of-the-box recommendations for WaveGAN and SpecGAN.
+
+Table 4: SpecGAN generator architecture   
+
+<table><tr><td rowspan=1 colspan=4>Operation</td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>Kernel Size</td></tr><tr><td rowspan=1 colspan=2>Inputz~ Unif</td><td rowspan=1 colspan=2>form(-1,1)</td><td rowspan=2 colspan=1>(100,256d)(5,5,16d,8d)(5,5,8d,4d)(5,5,4d,2d)(5,5,2d,d)(5,5,d,c)</td><td rowspan=2 colspan=1>(n,100)(n,256d)(n,4,4,16d)(n,4,4,16d)(n,8,8,8d)(n,8,8,8d)(n,16,16,4d)(n,16,16,4d)(n,32,32,2d)(n,32,32,2d)(n,64,64,d)(n,64,64,d)(n,128,128,c)(n,128,128,c)</td></tr><tr><td rowspan=1 colspan=4>Dense 1ReshapeReLUTrans Conv2D (Stride=2)ReLUTrans Conv2D (Stride=2)ReLUTrans Conv2D (Stride=2)ReLUTrans Conv2D (Stride=2)ReLUTrans Conv2D (Stride=2)Tanh</td><td rowspan=1 colspan=1>Dense 1</td></tr></table>
+
+Table 5: SpecGAN discriminator architecture   
+
+<table><tr><td rowspan=1 colspan=1>Operation</td><td rowspan=1 colspan=1>Kernel Size</td><td rowspan=1 colspan=1>Output Shape</td></tr><tr><td rowspan=1 colspan=1>Input x or G(z)Conv2D (Stride=2)LReLU (α = 0.2)Conv2D (Stride=2)LReLU (α = 0.2)Conv2D (Stride=2)LReLU (α = 0.2)Conv2D (Stride=2)LReLU (α = 0.2)Conv2D (Stride=2)LReLU (α= 0.2)ReshapeDense</td><td rowspan=1 colspan=1>(5,5,c,d)(5,5,d,2d)(5,5,2d,4d)(5,5,4d,8d)(5,5,8d,16d)(256d,1)</td><td rowspan=1 colspan=1>(n,128,128,c)(n,64,64,d)(n,64,64,d)(n,32,32,2d)(n,32,32,2d)(n,16,16,4d)(n,16,16,4d)(n,8,8,8d)(n,8,8,8d)(n,4,4,16d)(n,4,4,16d)(n,256d)(n,1)</td></tr></table>
+
+Table 6: WaveGAN hyperparameters   
+
+<table><tr><td>Name</td><td>Value</td></tr><tr><td>Input data type Model data type</td><td>16-bit PCM (requantized to 32-bit float) 32-bit floating point</td></tr><tr><td>Num channels (c)</td><td>1</td></tr><tr><td>Batch size (b)</td><td>64</td></tr><tr><td>Model dimensionality (d)</td><td>64</td></tr><tr><td>Phase shuffle (WaveGAN)</td><td>2</td></tr><tr><td>Phase shuffle (SpecGAN)</td><td>0</td></tr><tr><td>Loss</td><td>WGAN-GP (Gulrajani et al., 2017)</td></tr><tr><td>WGAN-GP入</td><td>10</td></tr><tr><td>D updates per G update</td><td></td></tr><tr><td></td><td>5</td></tr><tr><td>Optimizer</td><td>Adam (α=1e-4,βi= 0.5,β2=0.9)</td></tr></table>

parse/train/ByMVTsR5KQ/ByMVTsR5KQ_content_list.json

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+[
+    {
+        "type": "text",
+        "text": "ADVERSARIAL AUDIO SYNTHESIS ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Chris Donahue Department of Music UC San Diego cdonahue@ucsd.edu ",
+        "bbox": [
+            183,
+            145,
+            352,
+            202
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Julian McAuley   \nDepartment of Computer Science UC San Diego   \njmcauley@eng.ucsd.edu ",
+        "bbox": [
+            390,
+            145,
+            612,
+            202
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Miller Puckette Department of Music UC San Diego msp@ucsd.edu ",
+        "bbox": [
+            650,
+            145,
+            792,
+            202
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "ABSTRACT ",
+        "text_level": 1,
+        "bbox": [
+            454,
+            238,
+            544,
+            252
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Audio signals are sampled at high temporal resolutions, and learning to synthesize audio requires capturing structure across a range of timescales. Generative adversarial networks (GANs) have seen wide success at generating images that are both locally and globally coherent, but they have seen little application to audio generation. In this paper we introduce WaveGAN, a first attempt at applying GANs to unsupervised synthesis of raw-waveform audio. WaveGAN is capable of synthesizing one second slices of audio waveforms with global coherence, suitable for sound effect generation. Our experiments demonstrate that—without labels—WaveGAN learns to produce intelligible words when trained on a smallvocabulary speech dataset, and can also synthesize audio from other domains such as drums, bird vocalizations, and piano. We compare WaveGAN to a method which applies GANs designed for image generation on image-like audio feature representations, finding both approaches to be promising. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "1 INTRODUCTION ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            491,
+            336,
+            507
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Synthesizing audio for specific domains has many practical applications in creative sound design for music and film. Musicians and Foley artists scour large databases of sound effects to find particular audio recordings suitable for specific scenarios. This strategy is painstaking and may result in a negative outcome if the ideal sound effect does not exist in the library. A better approach might allow a sound artist to explore a compact latent space of audio, taking broad steps to find the types of sounds they are looking for (e.g. footsteps) and making small adjustments to latent variables to finetune (e.g. a large boot lands on a gravel path). However, audio signals have high temporal resolution, and strategies that learn such a representation must perform effectively in high dimensions. ",
+        "bbox": [
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Generative Adversarial Networks (GANs) (Goodfellow et al., 2014) are one such unsupervised strategy for mapping low-dimensional latent vectors to high-dimensional data. The potential advantages of GAN-based approaches to audio synthesis are numerous. Firstly, GANs could be useful for data augmentation (Shrivastava et al., 2017) in data-hungry speech recognition systems. Secondly, GANs could enable rapid and straightforward sampling of large amounts of audio. Furthermore, while the usefulness of generating static images with GANs is arguable, there are many applications (e.g. Foley) for which generating sound effects is immediately useful. But despite their increasing fidelity at synthesizing images (Radford et al., 2016; Berthelot et al., 2017; Karras et al., 2018), GANs have yet to be demonstrated capable of synthesizing audio in an unsupervised setting. ",
+        "bbox": [
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "A na¨ıve solution for applying image-generating GANs to audio would be to operate them on imagelike spectrograms, i.e., time-frequency representations of audio. This practice of bootstrapping image recognition algorithms for audio tasks is commonplace in the discriminative setting (Hershey et al., 2017). In the generative setting however, this approach is problematic as the most perceptually-informed spectrograms are non-invertible, and hence cannot be listened to without lossy estimations (Griffin & Lim, 1984) or learned inversion models (Shen et al., 2018). ",
+        "bbox": [
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Recent work (van den Oord et al., 2016; Mehri et al., 2017) has shown that neural networks can be trained with autoregression to operate on raw audio. Such approaches are attractive as they dispense with engineered feature representations. However, unlike with GANs, the autoregressive setting results in slow generation as output audio samples must be fed back into the model one at a time. ",
+        "bbox": [
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+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "In this work, we investigate both waveform and spectrogram strategies for generating one-second slices of audio with GANs.1 For our spectrogram approach (SpecGAN), we first design a spectrogram representation that allows for approximate inversion, and bootstrap the two-dimensional deep convolutional GAN (DCGAN) method (Radford et al., 2016) to operate on these spectrograms. In WaveGAN, our waveform approach, we flatten the DCGAN architecture to operate in one dimension, resulting in a model with the same number of parameters and numerical operations as its twodimensional analog. With WaveGAN, we provide both a starting point for practical audio synthesis with GANs and a recipe for modifying other image generation methods to operate on waveforms. ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "We primarily envisage our method being applied to the generation of short sound effects suitable for use in music and film. For example, we trained a WaveGAN on drums, resulting in a procedural drum machine designed to assist electronic musicians (demo chrisdonahue.com/wavegan). However, human evaluation for such domain-specific tasks would require expert listeners. Therefore, we also consider a speech benchmark, facilitating straightforward assessment by human annotators. Specifically, we explore a task where success can easily be judged by any English speaker: generating examples of spoken digits “zero” through “nine”. ",
+        "bbox": [
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+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "Though our evaluation focuses on a speech generation task, we note that it is not our goal to develop a text-to-speech synthesizer. Instead, our investigation concerns whether unsupervised strategies can learn global structure (e.g. words in speech data) implicit in high-dimensional audio signals without conditioning. Our experiments on speech demonstrate that both WaveGAN and SpecGAN can generate spoken digits that are intelligible to humans. On criteria of sound quality and speaker diversity, human judges indicate a preference for the audio generated by WaveGAN compared to that from SpecGAN. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "2 GAN PRELIMINARIES ",
+        "text_level": 1,
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+        "text": "GANs learn mappings from low-dimensional latent vectors $z \\in { \\mathcal { Z } }$ , i.i.d. samples from known prior $P _ { Z }$ , to points in the space of natural data $\\mathcal { X }$ . In their original formulation (Goodfellow et al., 2014), a generator $G : { \\mathcal { Z } } \\mapsto { \\mathcal { X } }$ is pitted against a discriminator $D : \\mathcal { X } \\mapsto [ 0 , 1 ]$ in a two-player minimax game. $G$ is trained to minimize the following value function, while $D$ is trained to maximize it: ",
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+        "type": "equation",
+        "img_path": "images/3ba1361858b61c5d5a68df6810e187cc57d1c8cb786814a159ea952de9346b28.jpg",
+        "text": "$$\nV ( D , G ) = \\mathbb { E } _ { \\pmb { x } \\sim P _ { X } } [ \\log D ( \\pmb { x } ) ] + \\mathbb { E } _ { \\pmb { z } \\sim P _ { Z } } [ \\log ( 1 - D ( G ( \\pmb { z } ) ) ) ] .\n$$",
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+        "type": "text",
+        "text": "In other words, $D$ is trained to determine if an example is real or fake, and $G$ is trained to fool the discriminator into thinking its output is real. Goodfellow et al. (2014) demonstrate that their proposed training algorithm for Equation 1 equates to minimizing the Jensen-Shannon divergence between $P _ { X }$ , the data distribution, and $P _ { G }$ , the implicit distribution of the generator when $z \\sim$ $P _ { Z }$ . In this original formulation, GANs are notoriously difficult to train, and prone to catastrophic failure cases. Instead of Jensen-Shannon divergence, Arjovsky et al. (2017) suggest minimizing the smoother Wasserstein-1 distance between generated and data distributions ",
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+        "img_path": "images/7a92f1ab3b98ea33e3d4906a01343171b8782b2d28a08687be46825296b47dce.jpg",
+        "text": "$$\nW ( P _ { X } , P _ { G } ) = \\operatorname* { s u p } _ { \\| f \\| _ { L } \\leq 1 } \\mathbb { E } _ { x \\sim P _ { X } } [ f ( x ) ] - \\mathbb { E } _ { x \\sim P _ { G } } [ f ( x ) ]\n$$",
+        "text_format": "latex",
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+        "type": "text",
+        "text": "where $\\| f \\| _ { L } \\leq 1 : \\mathcal { X } \\mapsto \\mathbb { R }$ is the family of functions that are 1-Lipschitz. ",
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+        "type": "text",
+        "text": "To minimize Wasserstein distance, they suggest a GAN training algorithm (WGAN), similar to that of Goodfellow et al. (2014), for the following value function: ",
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+        "img_path": "images/fd9fba312bd88c071c747afe9ab6275a341dd9624420e688329dde77fb0739a3.jpg",
+        "text": "$$\nV _ { \\mathrm { W G A N } } ( D _ { w } , G ) = \\mathbb { E } _ { { \\pmb { x } } \\sim P _ { X } } [ D _ { w } ( { \\pmb x } ) ] - \\mathbb { E } _ { { \\pmb z } \\sim P _ { Z } } [ D _ { w } ( G ( { \\pmb z } ) ) ] .\n$$",
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+        "text": "With this formulation, $D _ { w } : \\mathcal { X } \\mapsto \\mathbb { R }$ is not trained to identify examples as real or fake, but instead is trained as a function that assists in computing the Wasserstein distance. Arjovsky et al. (2017) suggest weight clipping as a means of enforcing that $D _ { w }$ is 1-Lipschitz. As an alternative strategy, Gulrajani et al. (2017) replace weight clipping with a gradient penalty (WGAN-GP) that also enforces the constraint. They demonstrate that their WGAN-GP strategy can successfully train a variety of model configurations where other GAN losses fail. ",
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+        "type": "image",
+        "img_path": "images/a3b51d30ba4da138a3efe74960c70a5d651173f0a00848c1759dfae46f250693.jpg",
+        "image_caption": [
+            "Figure 1: First eight principal components for 5x5 patches from natural images (left) versus those of length-25 audio slices from speech (right). Periodic patterns are unusual in natural images but a fundamental structure in audio. ",
+            "Figure 2: Depiction of the transposed convolution operation for the first layers of the DCGAN (Radford et al., 2016) (left) and WaveGAN (right) generators. DCGAN uses small (5x5), twodimensional filters while WaveGAN uses longer (length-25), one-dimensional filters and a larger upsampling factor. Both strategies have the same number of parameters and numerical operations. "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "3 WAVEGAN ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "We motivate our design choices for WaveGAN by first highlighting the different types of structure found in audio versus images. ",
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+    {
+        "type": "text",
+        "text": "3.1 INTRINSIC DIFFERENCES BETWEEN AUDIO AND IMAGES ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "One way to illustrate the differences between audio and images is by examining the axes along which these types of data vary most substantially, i.e. by principal component analysis. In Figure 1, we show the first eight principal components for patches from natural images and slices from speech. While the principal components of images generally capture intensity, gradient, and edge characteristics, those from audio form a periodic basis that decompose the audio into constituent frequency bands. In general, natural audio signals are more likely to exhibit periodicity than natural images. ",
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+    {
+        "type": "text",
+        "text": "As a consequence, correlations across large windows are commonplace in audio. For example, in a waveform sampled at $1 6 \\mathrm { k H z }$ , a $4 4 0 \\mathrm { H z }$ sinusoid (the musical note A4) takes over 36 samples to complete a single cycle. This suggests that filters with larger receptive fields are needed to process raw audio. This same intuition motivated van den Oord et al. (2016) in their design of WaveNet, which uses dilated convolutions to exponentially increase the model’s effective receptive field with linear increase in layer depth. ",
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+    {
+        "type": "text",
+        "text": "3.2 WAVEGAN ARCHITECTURE ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "We base our WaveGAN architecture off of DCGAN (Radford et al., 2016) which popularized usage of GANs for image synthesis. The DCGAN generator uses the transposed convolution operation (Figure 2) to iteratively upsample low-resolution feature maps into a high-resolution image. Motivated by our above discussion, we modify this transposed convolution operation to widen its receptive field. Specifically, we use longer one-dimensional filters of length 25 instead of two-dimensional filters of size 5x5, and we upsample by a factor of 4 instead of 2 at each layer (Figure 2). We modify the discriminator in a similar way, using length-25 filters in one dimension and increasing stride from 2 to 4. These changes result in WaveGAN having the same number of parameters, numerical operations, and output dimensionality as DCGAN. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Because DCGAN outputs 64x64 pixel images — equivalent to just 4096 audio samples — we add one additional layer to the model resulting in 16384 samples, slightly more than one second of audio at $1 6 \\mathrm { k H z }$ . This length is already sufficient for certain sound domains (e.g. sound effects, voice commands), and future work adapting megapixel image generation techniques (Karras et al., 2018) could expand the output length to more than a minute. We requantize the real data from its 16- bit integer representation (linear pulse code modulation) to 32-bit floating point, and our generator similarly outputs floating point waveforms. A complete description of our model is in Appendix D. ",
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+        "type": "text",
+        "text": "",
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+    {
+        "type": "text",
+        "text": "In summary, we outline our modifications to the DCGAN (Radford et al., 2016) method which result in WaveGAN. This straightforward recipe already produces reasonable audio, and further contributions outlined below and in Appendix A serve to refine results. ",
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+    {
+        "type": "text",
+        "text": "1. Flatten 2D convolutions into 1D (e.g. 5x5 2D convolution becomes length-25 1D).   \n2. Increase the stride factor for all convolutions (e.g. stride $2 \\mathrm { x 2 }$ becomes stride 4).   \n3. Remove batch normalization from the generator and discriminator.   \n4. Train using the WGAN-GP (Gulrajani et al., 2017) strategy. ",
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+    },
+    {
+        "type": "text",
+        "text": "3.3 PHASE SHUFFLE ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "Generative image models that upsample by transposed convolution (such as DCGAN) are known to produce characteristic “checkerboard” artifacts in images (Odena et al., 2016). Periodic patterns are less common in images (Section 3.1), and thus the discriminator can learn to reject images that contain them. For audio, analogous artifacts are perceived as pitched noise which may overlap with frequencies commonplace in the real data, making the discriminator’s objective more challenging. However, the artifact frequencies will always occur at a particular phase, allowing the discriminator to learn a trivial policy to reject generated examples. This may inhibit the overall optimization problem. ",
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+    {
+        "type": "text",
+        "text": "To prevent the discriminator from learning such a solution, we propose the phase shuffle operation with hyperparameter $n$ . Phase shuffle randomly perturbs the phase of each layer’s activations by $- n$ to $n$ samples before input to the next layer (Figure 3). We apply phase shuffle only to the discriminator, as the latent vector already provides the generator a mechanism to manipulate the phase of a resultant waveform. Intuitively speaking, phase shuffle makes the discriminator’s job more challenging by requiring invariance to the phase of the input waveform. ",
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+    {
+        "type": "image",
+        "img_path": "images/757554e8081d9f0db8bb7bd31943ae5c91e45cbdbbfaabbff1a9a214fa80aad7.jpg",
+        "image_caption": [
+            "Figure 3: At each layer of the WaveGAN discriminator, the phase shuffle operation perturbs the phase of each feature map by Uniform $\\sim ~ [ - n , n ]$ samples, filling in the missing samples (dashed outlines) by reflection. Here we depict all possible outcomes for a layer with four feature maps $( n = 1 )$ ). "
+        ],
+        "image_footnote": [],
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "4 SPECGAN: GENERATING SEMI-INVERTIBLE SPECTROGRAMS ",
+        "text_level": 1,
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "While a minority of recent research in discriminative audio classification tasks has used raw audio input (Sainath et al., 2015; Lee et al., 2017), most of these approaches operate on spectrogram representations of audio. A generative model may also benefit from operating in such a time-frequency space. However, commonly-used representations in the discriminative setting are uninvertible. ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "With SpecGAN, our frequency-domain audio generation model, we design a spectrogram representation that is both well-suited to GANs designed for image generation and can be approximately inverted. Additionally, to facilitate direct comparison, our representation is designed to use the same dimensionality per unit of time as WaveGAN (16384 samples yield a 128x128 spectrogram). ",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "To process audio into suitable spectrograms, we first perform the short-time Fourier transform with $1 6 \\mathrm { m s }$ windows and $8 \\mathrm { m s }$ stride, resulting in 128 frequency bins2 linearly spaced from 0 to $8 \\mathrm { k H z }$ . We take the magnitude of the resultant spectra and scale amplitude values logarithmically to better-align with human perception. We then normalize each frequency bin to have zero mean and unit variance. This type of preprocessing is commonplace in audio classification, but produce spectrograms with unbounded values—a departure from image representations. We therefore clip the spectra to 3 standard deviations and rescale to [−1, 1]. Through an informal listening test, we determined that this clipping strategy did not produce an audible difference during inversion. ",
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+    {
+        "type": "image",
+        "img_path": "images/c6d4e0d8442f77e8e8c02fbc2975b0da3dda75c387e24bdc9707590ef92e64b8.jpg",
+        "image_caption": [
+            "Figure 4: Top: Random samples from each of the five datasets used in this study, illustrating the wide variety of spectral characteristics. Middle: Random samples generated by WaveGAN for each domain. WaveGAN operates in the time domain but results are displayed here in the frequency domain for visual comparison. Bottom: Random samples generated by SpecGAN for each domain. "
+        ],
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+    {
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+    },
+    {
+        "type": "text",
+        "text": "Once our dataset has been processed into this format, we operate the DCGAN (Radford et al., 2016) algorithm on the resultant spectra. To render the resultant generated spectrograms as waveforms, we first invert the steps of spectrogram preprocessing described above, resulting in linear-amplitude magnitude spectra. We then employ the iterative Griffin-Lim algorithm (Griffin & Lim, 1984) with 16 iterations to estimate phase and produce 16384 audio samples. ",
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+    },
+    {
+        "type": "text",
+        "text": "5 EXPERIMENTAL PROTOCOL ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "To facilitate human evaluation, our experimentation focuses on the Speech Commands Dataset (Warden, 2018). This dataset consists of many speakers recording individual words in uncontrolled recording conditions. We explore a subset consisting of the spoken digits “zero” through “nine” and refer to this subset as the Speech Commands Zero Through Nine (SC09) dataset. While this dataset is intentionally reminiscent of the popular MNIST dataset of written digits, we note that examples from SC09 are much higher dimensional $( \\mathbb { R } ^ { 1 6 0 0 0 } )$ than examples from MNIST $( \\mathbb { R } ^ { 2 8 \\times 2 8 = 7 8 4 }$ ). ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "These ten words encompass many phonemes and two consist of multiple syllables. Each recording is one second in length, and we do not attempt to align the words in time. There are 1850 utterances of each word in the training set, resulting in 5.3 hours of speech. The heterogeneity of alignments, speakers, and recording conditions make this a challenging dataset for generative modeling. ",
+        "bbox": [
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+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Our baseline configuration for WaveGAN excludes phase shuffle. We compare this to the performance of WaveGAN with phase shuffle $( n \\in \\{ 2 , 4 \\} )$ and a variant of WaveGAN which uses nearest-neighbor upsampling rather than transposed convolution (Odena et al., 2016). Hoping to reduce noisy artifacts, we also experiment with adding a wide (length-512) post-processing filter to the output of the generator and learning its parameters with the rest of the generator variables (details in Appendix A.1). We use the WGAN-GP (Gulrajani et al., 2017) algorithm for all experiments, finding it to produce reasonable results where others (Radford et al., 2016; Mao et al., 2017; Arjovsky et al., 2017) failed. We compare the performance of these configurations to that of SpecGAN. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "We also perform experiments on four other datasets with different characteristics (Figure 4): ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "1. Drum sound effects (0.7 hours): Drum samples for kicks, snares, toms, and cymbals   \n2. Bird vocalizations (12.2 hours): In-the-wild recordings of many species (Boesman, 2018)   \n3. Piano (0.3 hours): Professional performer playing a variety of Bach compositions   \n4. Large vocab speech (TIMIT) (2.4 hours): Multiple speakers, clean (Garofolo et al., 1993) ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "We train our networks using batches of size 64 on a single NVIDIA P100 GPU. During our quantitative evaluation of SC09 (discussed below), our WaveGAN networks converge by their early stopping criteria (inception score) within four days (200k iterations, around 3500 epochs), and produce speech-like audio within the first hour of training. Our SpecGAN networks converge more quickly, within two days (around 1750 epochs). On the other four datasets, we train WaveGAN for $2 0 0 \\mathrm { k }$ iterations representing nearly 1500 epochs for the largest dataset. Unlike with autoregressive methods (van den Oord et al., 2016; Mehri et al., 2017), generation with WaveGAN is fully parallel and can produce an hour of audio in less than two seconds. We list all hyperparameters in Appendix E. ",
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+    {
+        "type": "text",
+        "text": "6 EVALUATION METHODOLOGY ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Evaluation of generative models is a fraught topic. Theis et al. (2016) demonstrate that quantitative measures of sample quality are poorly correlated with each other and human judgement. Accordingly, we use several quantitative evaluation metrics for hyperparameter validation and discussion, and also evaluate our most promising models with human judges. ",
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+    {
+        "type": "text",
+        "text": "6.1 INCEPTION SCORE ",
+        "text_level": 1,
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+    },
+    {
+        "type": "text",
+        "text": "Salimans et al. (2016) propose the inception score, which uses a pre-trained Inception classifier (Szegedy et al., 2016) to measure both the diversity and semantic discriminability of generated images, finding that the measure correlates well with human judgement. ",
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+    {
+        "type": "text",
+        "text": "Given model scores $P ( \\pmb { y } \\mid \\pmb { x } )$ with marginal $P ( \\pmb { y } )$ , the inception score is defined as $\\exp ( \\mathbb { E } _ { \\pmb { x } } D _ { \\mathrm { K L } } ( P ( \\pmb { y } \\mid \\pmb { x } ) | | P ( \\pmb { y } ) ) )$ , and is estimated over a large number of samples (e.g. 50k). For $n$ classes, this measure ranges from 1 to $n$ , and is maximized when the model is completely confident about each prediction and predicts each label equally often. We will use this measure as our primary quantitative evaluation method and early stopping criteria. ",
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+    },
+    {
+        "type": "text",
+        "text": "To measure inception score, we train an audio classifier on SC09. Our classifier first computes a short-time Fourier transform of the input audio with $6 4 \\mathrm { m s }$ windows and 8 ms stride. This representation is projected to 128 frequency bins equally spaced on the Mel scale (Stevens et al., 1937) from $4 0 \\mathrm { { H z } }$ to $7 8 0 0 \\mathrm { H z }$ . Amplitudes are scaled logarithmically and normalized so that each bin has zero mean and unit variance. We process this perceptually-informed representation with four layers of convolution and pooling, projecting the result to a softmax layer with 10 classes. We perform early stopping on the minimum negative log-likelihood of the validation set; the resultant model achieves $9 3 \\%$ accuracy on the test set. Because this classifier observes spectrograms, our spectrogramgenerating models may have a representational advantage over our waveform-generating models. ",
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+    {
+        "type": "text",
+        "text": "6.2 NEAREST NEIGHBOR COMPARISONS ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Inception score has two trivial failure cases in which a poor generative model can achieve a high score. Firstly, a generative model that outputs a single example of each class with uniform probability will be assigned a high score. Secondly, a generative model that overfits the training data will achieve a high score simply by outputting examples on which the classifier was trained. ",
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+    {
+        "type": "text",
+        "text": "We use two indicators metrics to determine if a high inception score has been caused by either of these two undesirable cases. Our first indicator, $| D | _ { \\mathrm { s e l f } }$ , measures the average Euclidean distance of a set of 1k examples to their nearest neighbor within the set (other than itself). A higher $| D | _ { \\mathrm { s e l f } }$ indicates higher diversity amongst samples. Because measuring Euclidean distance in time-domain ",
+        "bbox": [
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+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Table 1: Quantitative and qualitative (human study) results for SC09 experiments comparing real and generated data. A higher inception score suggests that semantic modes of the real data distribution have been captured. $| D | _ { \\mathrm { s e l f } }$ indicates the intra-dataset diversity relative to that of the real test data. $| D | _ { \\mathrm { t r a i n } }$ indicates the distance between the dataset and the training set relative to that of the test data; a low value indicates a generative model that is overfit to the training data. Acc. is the overall accuracy of humans on the task of labeling class-balanced digits (random chance is 0.1). Sound quality, ease of intelligibility and speaker diversity are mean opinion scores (1-5); higher is better. ",
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+    },
+    {
+        "type": "table",
+        "img_path": "images/2f85e9850d9b1902eab781d7ce0232d40046b16ffa9761c3095f301a4dd41b49.jpg",
+        "table_caption": [],
+        "table_footnote": [],
+        "table_body": "<table><tr><td></td><td colspan=\"4\">Quantitative</td><td colspan=\"4\">Qualitative (human judges)</td></tr><tr><td>Experiment</td><td>Inception score</td><td>|D|self</td><td>|D|train</td><td>Acc.</td><td>Quality</td><td>Ease</td><td></td><td>Diversity</td></tr><tr><td>Real (train)</td><td>9.18 ± 0.04</td><td>1.1</td><td>0.0</td><td rowspan=\"3\">0.95</td><td rowspan=\"3\"></td><td></td><td></td><td></td></tr><tr><td>Real (test)</td><td>8.01±0.24</td><td>1.0</td><td>1.0</td><td>3.9 ±0.8</td><td>3.9 ± 1.1</td><td>3.5 ± 1.0</td></tr><tr><td>Parametric</td><td>5.02 ± 0.06</td><td>0.7</td><td>1.1</td><td></td><td></td><td></td></tr><tr><td>WaveGAN</td><td>4.12 ± 0.03</td><td>1.4</td><td>2.0</td><td rowspan=\"6\"></td><td></td><td></td><td></td><td></td></tr><tr><td>+ Phase shuffle n = 2</td><td>4.67 ± 0.01</td><td>0.8</td><td>2.3</td><td>0.58</td><td>2.3± 0.9</td><td>2.8±0.9</td><td>3.2 ±0.9</td></tr><tr><td>+ Phase shuffle n = 4</td><td>4.54± 0.03</td><td>1.0</td><td>2.3</td><td></td><td></td><td></td><td></td></tr><tr><td>+ Nearest neighbor</td><td>3.77± 0.02</td><td>1.8</td><td>2.6</td><td></td><td></td><td></td><td></td></tr><tr><td>+ Post-processing</td><td>3.92 ± 0.03</td><td>1.4</td><td>2.9</td><td></td><td></td><td></td><td></td></tr><tr><td>+ Dropout</td><td>3.93 ± 0.03</td><td>1.0</td><td>2.6</td><td></td><td></td><td></td><td></td></tr><tr><td>SpecGAN</td><td>6.03 ± 0.04</td><td>1.1</td><td>1.4</td><td>0.66</td><td>1.9 ±0.8</td><td>2.8 ±0.9</td><td></td><td>2.6±1.0</td></tr><tr><td>+Phase shuffle n =1</td><td>3.71± 0.03</td><td>0.8</td><td>1.6</td><td></td><td></td><td></td><td></td><td></td></tr></table>",
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+    {
+        "type": "text",
+        "text": "audio poorly represents human perception, we evaluate distances in the same frequency-domain representation as our classifier from Section 6.1. ",
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+    {
+        "type": "text",
+        "text": "Our second indicator, $| D | _ { \\mathrm { t r a i n } }$ , measures the average Euclidean distance of 1k examples to their nearest neighbor in the real training data. If the generative model simply produces examples from the training set, this measure will be 0. We report $| D | _ { \\mathrm { t r a i n } }$ and $| D | _ { \\mathrm { s e l f } }$ relative to those of the test set. ",
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+    {
+        "type": "text",
+        "text": "6.3 QUALITATIVE HUMAN JUDGEMENTS ",
+        "text_level": 1,
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+    {
+        "type": "text",
+        "text": "While inception score is a useful metric for hyperparameter validation, our ultimate goal is to produce examples that are intelligible to humans. To this end, we measure the ability of human annotators on Amazon Mechanical Turk to label the generated audio. Using our best WaveGAN and SpecGAN models as measured by inception score, we generate random examples until we have 300 for each digit (as labeled by our classifier from Section 6.1)—3000 total. In batches of ten random examples, we ask annotators to label which digit they perceive in each example, and compute their accuracy with respect to the classifier’s labels (random accuracy would be $1 \\bar { 0 } \\%$ ). After each batch, annotators assign subjective values of 1–5 for criteria of sound quality, ease of intelligibility, and speaker diversity. We report accuracy $n = 3 0 0 0$ ) and mean opinion scores $\\mathit { n } = 3 0 0 $ ) in Table 1. ",
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+    {
+        "type": "text",
+        "text": "7 RESULTS AND DISCUSSION ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "Results for our evaluation appear in Table 1. We also evaluate our metrics on the real training data, the real test data, and a version of SC09 generated by a parametric speech synthesizer (Buchner, 2017). We also compare to SampleRNN (Mehri et al., 2017) and two public implementations of WaveNet (van den Oord et al., 2016), but neither method produced competitive results (details in Appendix B), and we excluded them from further evaluation. These autoregressive models have not previously been examined on small-vocabulary speech data, and their success at generating full words has only been demonstrated when conditioning on rich linguistic features. Sound examples for all experiments can be found at chrisdonahue.com/wavegan_examples. ",
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+        "type": "text",
+        "text": "While the maximum inception score for SC09 is 10, any score higher than the test set score of 8 should be seen as evidence that a generative model has overfit. Our best WaveGAN model uses phase shuffle with $n = 2$ and achieves an inception score of 4.7. To compare the effect of phase shuffle to other common regularizers, we also tried using $5 0 \\%$ dropout in the discriminator’s activations, which resulted in a lower score. Phase shuffle decreased the inception score of SpecGAN, possibly because the operation has an exaggerated effect when applied to the compact temporal axis of spectrograms. ",
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+        "type": "text",
+        "text": "",
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+    {
+        "type": "text",
+        "text": "Most experiments produced $| D | _ { \\mathrm { s e l f } }$ (diversity) values higher than that of the test data, and all experiments produced $| D | _ { \\mathrm { t r a i n } }$ (distance from training data) values higher than that of the test data. While these measures indicate that our generative models produce examples with statistics that deviate from those of the real data, neither metric indicates that the models achieve high inception scores by the trivial solutions outlined in Section 6.2. ",
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+    },
+    {
+        "type": "text",
+        "text": "Compared to examples from WaveGAN, examples from SpecGAN achieve higher inception score (6.0 vs. 4.7) and are labeled more accurately by humans ( $6 6 \\%$ vs. $5 8 \\%$ ). However, on subjective criteria of sound quality and speaker diversity, humans indicate a preference for examples from WaveGAN. It appears that SpecGAN might better capture the variance in the underlying data compared to WaveGAN, but its success is compromised by sound quality issues when its spectrograms are inverted to audio. It is possible that the poor qualitative ratings for examples from SpecGAN are primarily caused by the lossy Griffin-Lim inversion (Griffin & Lim, 1984) and not the generative procedure itself. We see promise in both waveform and spectrogram audio generation with GANs; our study does not suggest a decisive winner. For a more thorough investigation of spectrogram generation methods, we point to follow-up work (Engel et al., 2019). ",
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+    {
+        "type": "text",
+        "text": "Finally, we train WaveGAN and SpecGAN models on the four other domains listed in Section 5. Somewhat surprisingly, we find that the frequency-domain spectra produced by WaveGAN (a timedomain method) are visually more consistent with the training data (e.g. in terms of sharpness) than those produced by SpecGAN (Figure 4). For drum sound effects, WaveGAN captures semantic modes such as kick and snare drums. On bird vocalizations, WaveGAN generates a variety of distinct bird sounds. On piano, WaveGAN produces musically-consonant motifs that, as with the training data, represent a variety of key signatures and rhythmic patterns. For TIMIT, a large-vocabulary speech dataset with many speakers, WaveGAN produces speech-like babbling similar to results from unconditional autoregressive models (van den Oord et al., 2016). ",
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+    {
+        "type": "text",
+        "text": "8 RELATED WORK ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "Much of the work within generative modeling of audio is within the context of text-to-speech. Textto-speech systems are primarily either concatenative or parametric. In concatenative systems, audio is generated by sequencing small, prerecorded portions of speech from a phonetically-indexed dictionary (Moulines & Charpentier, 1990; Hunt & Black, 1996). Parametric systems map text to salient parameters of speech, which are then synthesized by a vocoder (Dudley, 1939); see (Zen et al., 2009) for a comprehensive review. Some of these systems use learning-based approaches such as a hidden Markov models (Yoshimura, 2002; Tokuda et al., 2013), and separately-trained neural networks pipelines (Ling et al., 2015) to estimate speech parameters. ",
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+    {
+        "type": "text",
+        "text": "Recently, several researchers have investigated parametric speech synthesis with end-to-end neural network approaches that learn to produce vocoder features directly from text or phonetic embeddings (Arik et al., 2017; Ping et al., 2018; Sotelo et al., 2017; Wang et al., 2017; Shen et al., 2018). These vocoder features are synthesized to raw audio using off-the-shelf methods such as WORLD (Morise et al., 2016) and Griffin-Lim (Griffin & Lim, 1984), or trained neural vocoders (Sotelo et al., 2017; Shen et al., 2018; Ping et al., 2018). All of these methods are supervised: they are trained to map linguistic features to audio outputs. ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Several approaches have explored unsupervised generation of raw audio. van den Oord et al. (2016) propose WaveNet, a convolutional model which learns to predict raw audio samples by autoregressive modeling. WaveNets conditioned on rich linguistic features have widely been deployed in textto-speech systems, though they have not been demonstrated capable of generating cohesive words in the unconditional setting. Engel et al. (2017) pose WaveNet as an autoencoder to generate musical instrument sounds. Chung et al. (2014); Mehri et al. (2017) both train recurrent autoregressive models which learn to predict raw audio samples. While autoregressive methods generally produce higher audio fidelity than WaveGAN, synthesis with WaveGAN is orders of magnitude faster. ",
+        "bbox": [
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+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "The application of GANs (Goodfellow et al., 2014) to audio has so far been limited to supervised learning problems in combination with traditional loss functions. Pascual et al. (2017) apply GANs to raw audio speech enhancement. Their encoder-decoder approach combines the GAN objective with an $L _ { 2 }$ loss. Fan et al. (2017); Michelsanti & Tan (2017); Donahue et al. (2018) all use GANs in combination with unstructured losses to map spectrograms in one domain to spectrograms in another. Chen et al. (2017) use GANs to map musical performance images into spectrograms. ",
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+    },
+    {
+        "type": "text",
+        "text": "",
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+    },
+    {
+        "type": "text",
+        "text": "9 CONCLUSION ",
+        "text_level": 1,
+        "bbox": [
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+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "We present WaveGAN, the first application of GANs to unsupervised audio generation. WaveGAN is fully parallelizable and can generate hours of audio in only a few seconds. In its current form, WaveGAN can be used for creative sound design in multimedia production. In our future work we plan to extend WaveGAN to operate on variable-length audio and also explore a variety of label conditioning strategies. By providing a template for modifying image generation models to operate on audio, we hope that this work catalyzes future investigation of GANs for audio synthesis. ",
+        "bbox": [
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+    {
+        "type": "text",
+        "text": "ACKNOWLEDGMENTS ",
+        "text_level": 1,
+        "bbox": [
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+    },
+    {
+        "type": "text",
+        "text": "The authors would like to thank Peter Boesman and Colin Raffel for providing training data for this work. This work was supported by the Unity Global Graduate Fellowship program and the UC San Diego Department of Computer Science. GPUs used for this work were provided by the HPC $@$ UC program and donations from NVIDIA. ",
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+        "text": "Aaron van den Oord, Sander Dieleman, Heiga Zen, Karen Simonyan, Oriol Vinyals, Alex Graves, ¨ Nal Kalchbrenner, Andrew Senior, and Koray Kavukcuoglu. WaveNet: A generative model for raw audio. arXiv:1609.03499, 2016. ",
+        "bbox": [
+            176,
+            381,
+            821,
+            424
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Yuxuan Wang, RJ Skerry-Ryan, Daisy Stanton, Yonghui Wu, Ron J Weiss, Navdeep Jaitly, Zongheng Yang, Ying Xiao, Zhifeng Chen, Samy Bengio, et al. Tacotron: Towards end-to-end speech synthesis. arXiv:1703.10135, 2017. ",
+        "bbox": [
+            176,
+            433,
+            823,
+            476
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Pete Warden. Speech commands: A dataset for limited-vocabulary speech recognition. arXiv:1804.03209, 2018. ",
+        "bbox": [
+            174,
+            484,
+            820,
+            513
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Takayoshi Yoshimura. Simultaneous modeling of phonetic and prosodic parameters, and characteristic conversion for hmm-based text-to-speech systems. 2002. ",
+        "bbox": [
+            173,
+            522,
+            821,
+            551
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "Heiga Zen, Keiichi Tokuda, and Alan W Black. Statistical parametric speech synthesis. Speech Communication, 2009. ",
+        "bbox": [
+            173,
+            560,
+            823,
+            589
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "image",
+        "img_path": "images/ec0cfc4ea50d6ec4046a3bf70aba991b6d617d7fe7a11b6d37d11eee20c254fa.jpg",
+        "image_caption": [
+            "Figure 5: (Top): Average impulse response for 1000 random initializations of the WaveGAN generator. (Bottom): Response of learned post-processing filters for speech and bird vocalizations. Post-processing filters reject frequencies corresponding to noise byproducts created by the generative procedure (top). The filter for speech boosts signal in prominent speech bands, while the filter for bird vocalizations (which are more uniformly-distributed in frequency) simply reduces noise presence. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            272,
+            103,
+            720,
+            443
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "A UNDERSTANDING AND MITIGATING ARTIFACTS IN GENERATED AUDIO ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            579,
+            790,
+            594
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Generative models that upsample by transposed convolution are known to produce characteristic “checkerboard” artifacts in images (Odena et al., 2016), artifacts with particular spatial periodicities. The discriminator of image-generating GANs can learn to reject images with these artifacts because they are uncommon in real data (as discussed in Section 3.1). However, in the audio domain, the discriminator might not have such luxury as these artifacts correspond to frequencies which might rightfully appear in the real data. ",
+        "bbox": [
+            174,
+            611,
+            825,
+            695
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "While checkerboard artifacts are an annoyance in image generation, they can be devastating to audio generation results. While our eye may perceive these types of periodic distortions as an intrusive texture, our ear perceives them as an abrasive tone. To characterize these artifacts in WaveGAN, we measure its impulse response by randomly initializing it 1000 times and passing unit impulses to its first convolutional layer. In Figure 5, we plot the average of these responses in the frequency domain. The response has sharp peaks at linear multiples of the sample rates of each convolutional layer $2 5 0 \\mathrm { H z }$ , $1 \\mathrm { k H z }$ , $4 \\mathrm { k H z }$ , etc.). This is in agreement with our informal observation of results from WaveGAN, which often have a pitched noise close to the musical note B $( 2 4 7 \\times 2 ^ { n } \\mathrm { H z } )$ ). ",
+        "bbox": [
+            174,
+            702,
+            825,
+            814
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "Below, we will discuss strategies we designed to mitigate these artifacts in WaveGAN. ",
+        "bbox": [
+            176,
+            820,
+            738,
+            835
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "A.1 LEARNED POST-PROCESSING FILTERS ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            854,
+            477,
+            868
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "We experiment with adding a post-processing filter to the generator, giving WaveGAN a simple mechanism to filter out undesirable frequencies created by the generative process. This filter has a long window (512 samples) allowing it to represent intricate transfer functions, and the weights of the filter are learned as part of the generator’s parameters. In Figure 5, we compare the postprocessing filters that WaveGAN learns for human speech and bird vocalizations. The filters boost signal in regions of the frequency spectrum that are most prominent in the real data domain, and introduce notches at bands that are artifacts of the generative procedure as discussed in the previous section. ",
+        "bbox": [
+            176,
+            882,
+            823,
+            924
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "image",
+        "img_path": "images/4f414414f814f3c214d3823672dd585c77e1af9cfc6dfe9cad2ef4f56848d5cf.jpg",
+        "image_caption": [
+            "Figure 6: Depiction of the upsampling strategy used by transposed convolution (zero insertion) and other strategies which mitigate aliasing: nearest neighbor, linear and cubic interpolation. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            236,
+            101,
+            758,
+            296
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            174,
+            358,
+            825,
+            429
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "A.2 UPSAMPLING PROCEDURE ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            445,
+            400,
+            460
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "Transposed convolution upsamples signals by inserting zeros in between samples and applying a learned filterbank. This operation introduces aliased frequencies, copies of pre-existing frequencies shifted by multiples of the new Nyquist rate, into the upsampled signal. While aliased frequencies are usually seen as undesirable artifacts of a bad upsampling procedure, in the generative setting their existence may be crucial for producing fine-grained details in the output. ",
+        "bbox": [
+            174,
+            472,
+            825,
+            541
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "We experiment with three other upsampling strategies in WaveGAN: nearest-neighbor, linear and cubic interpolation, all of which attenuate aliased frequencies. In Figure 6, we compare these strategies visually. While nearest neighbor upsampling resulted in similar audio output to transposed convolution, linear and cubic interpolation strategies resulted in qualitatively poor audio output (sound examples: chrisdonahue.com/wavegan_examples). We hypothesize that the aliased frequencies produced by upsampling convolutions may be more critical to audio generation than image generation. ",
+        "bbox": [
+            174,
+            549,
+            825,
+            646
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "B EXPERIMENTS WITH AUTOREGRESSIVE WAVEFORM MODELS ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            667,
+            709,
+            683
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "We developed our WaveGAN and SpecGAN models primarily to address the task of steerable sound effect generation. This is an inherently different task than text to speech (TTS), however autoregressive waveform models (e.g. WaveNet (van den Oord et al., 2016) and SampleRNN (Mehri et al., 2017)) that were developed for TTS can also be used to model and generate waveforms unconditionally. Hence, a comparison to these models for our task is reasonable. One upside of autoregressive models for our task is that they have the potential to produce high-quality audio. Potential downsides are 1) these models take several orders of magnitude longer to generate waveforms, and 2) they do not learn a compact latent space of waveforms, causing useful sound generation tasks like continuous exploration and interpolation to be impossible. ",
+        "bbox": [
+            174,
+            696,
+            825,
+            823
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "We attempt to train two public implementations of WaveNet (ImplA3 and $\\mathrm { I m p l B ^ { 4 } }$ ) and SampleRNN5 on our SC09 digit generation task. We use default parameters for these libraries; the only modifications we make are to reduce the training example size to one second. To our ears, all three libraries failed to produce cohesive words (you can judge for yourself from our sound examples at the bottom chrisdonahue.com/wavegan_examples). This poor subjective performance is echoed by weak inception scores (weaker than any in Table 1): $1 . 0 7 \\pm 0 . 0 5$ , $1 . 2 9 \\pm 0 . 0 3$ , $2 . 2 8 \\pm 0 . 1 9$ for WaveNet ImplA, WaveNet ImplB, and SampleRNN respectively. Note that these inception scores were calculated on far fewer examples $( < 1 k )$ than all of the scores listed in Table 1 (which were computed on $5 0 k$ examples). This is because it took over 24 hours to produce even a thousand one-second examples with these methods (whereas our methods produce $5 0 k$ examples in a few seconds). ",
+        "bbox": [
+            178,
+            830,
+            821,
+            872
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            174,
+            103,
+            825,
+            214
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "Autoregressive methods have not been demonstrated capable of learning to synthesize coherent words without conditioning on rich linguistic features. We are not claiming that these methods cannot learn to synthesize full words, merely that three open-source implementations were unable to do so with default parameters. We want to be clear that our intent is not to disparage autoregressive waveform methods as these methods were developed for a different task, and hence we excluded these poor scores from our results table to avoid sending the wrong message. Instead, we hope to highlight that these implementations produced results that were noncompetitive for our problem domain, and less useful (due to slowness and lack of a latent space) for creative generation of sound effects. ",
+        "bbox": [
+            173,
+            222,
+            825,
+            347
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "C FELINE “TURING TEST” ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            367,
+            408,
+            383
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "image",
+        "img_path": "images/68f22cd7b27c7439f9717bb54e59dae169763e841d232a0f2017c28f22d8f3f0.jpg",
+        "image_caption": [
+            "Figure 7: Compared to resting state, this cat’s level of alertness increased when presented bird vocalizations synthesized by WaveGAN and SpecGAN. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            179,
+            404,
+            818,
+            627
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "As our results improved throughout the course of this research, our cats became quite intrigued by the synthetic bird vocalizations produced by WaveGAN (Figure 7). While this was of course not a formal experiment, we did find this to be encouraging evidence that our method might be capable of producing audio that could additionally convince non-human animals. ",
+        "bbox": [
+            174,
+            691,
+            825,
+            747
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "table",
+        "img_path": "images/74697c5edf8307d2ea96f98c0b0a4870e002be522a9643f8403499ac4413fe21.jpg",
+        "table_caption": [
+            "Table 2: WaveGAN generator architecture "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Operation</td><td>Kernel Size</td><td>Output Shape</td></tr><tr><td>Input z ~ Uniform(-1,1) Dense 1 Reshape ReLU Trans Conv1D (Stride=4)</td><td>(100,256d) (25,16d,8d)</td><td>(n,100) (n,256d) (n,16,16d) (n,16,16d) (n,64,8d) (n,16384,c)</td></tr><tr><td>ReLU Trans Conv1D (Stride=4) ReLU</td><td>(25,8d,4d)</td><td>(n, 64,8d) (n,256,4d) (n,256,4d)</td></tr><tr><td>Trans Conv1D (Stride=4) ReLU Trans Conv1D (Stride=4)</td><td>(25,4d,2d)</td><td>(n,1024,2d) (n,1024,2d)</td></tr><tr><td>ReLU Trans Conv1D (Stride=4) Tanh</td><td>(25,2d,d) (25,d,c)</td><td>(n,4096,d) (n,4096,d)</td></tr></table>",
+        "bbox": [
+            310,
+            126,
+            686,
+            319
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "table",
+        "img_path": "images/b493444eef5745986b22bc9cdce720a6a99e795447b27e75784d741e4d5c9289.jpg",
+        "table_caption": [
+            "Table 3: WaveGAN discriminator architecture "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Operation Input x or G(z)</td><td>Kernel Size</td><td>Output Shape</td></tr><tr><td rowspan=\"3\">Conv1D (Stride=4) LReLU (α = 0.2) Phase Shuffle (n = 2) Conv1D (Stride=4) LReLU (α = 0.2) Phase Shuffle (n = 2) Conv1D (Stride=4) LReLU (α = 0.2) Phase Shuffle (n = 2) Conv1D (Stride=4) LReLU (α = 0.2) Phase Shuffle (n = 2) Conv1D (Stride=4)</td><td>(25,c,d)</td><td>(n,16384,c) (n,4096,d) (n,4096,d) (n,4096,d)</td></tr><tr><td>(25,d,2d) (25,2d,4d)</td><td>(n,1024,2d) (n,1024,2d) (n,1024,2d)</td></tr><tr><td>(25,4d,8d) (25,8d,16d) (256d,1)</td><td>(n,256,4d) (n,256,4d) (n,256,4d) (n,64,8d) (n, 64,8d) (n,64,8d) (n,16,16d) (n,16,16d) (n,256d) (n,1)</td></tr></table>",
+        "bbox": [
+            321,
+            353,
+            678,
+            583
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "D ARCHITECTURE DESCRIPTION ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            607,
+            460,
+            622
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "In Tables 2 and 3, we list the full architectures for our WaveGAN generator and discriminator respectively. In Tables 4 and 5, we list the same for SpecGAN. In these tables, $n$ is the batch size, $d$ modifies model size, and $c$ is the number of channels in the examples. In all of our experiments in this paper, $c = 1$ . All dense and convolutional layers include biases. No batch normalization is used in WaveGAN or SpecGAN. ",
+        "bbox": [
+            174,
+            637,
+            825,
+            707
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "E TRAINING HYPERPARAMETERS ",
+        "text_level": 1,
+        "bbox": [
+            176,
+            728,
+            465,
+            744
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "text",
+        "text": "In Table 6, we list the values of these and all other hyperparameters for our experiments, which constitute our out-of-the-box recommendations for WaveGAN and SpecGAN. ",
+        "bbox": [
+            171,
+            760,
+            825,
+            787
+        ],
+        "page_idx": 14
+    },
+    {
+        "type": "table",
+        "img_path": "images/acd4b0cc10c9bc78c9f2246e61be2ec0f1c73fda0f3beb002b5a15c653ae8519.jpg",
+        "table_caption": [
+            "Table 4: SpecGAN generator architecture "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=1 colspan=4>Operation</td><td rowspan=1 colspan=1></td><td rowspan=1 colspan=1>Kernel Size</td></tr><tr><td rowspan=1 colspan=2>Inputz~ Unif</td><td rowspan=1 colspan=2>form(-1,1)</td><td rowspan=2 colspan=1>(100,256d)(5,5,16d,8d)(5,5,8d,4d)(5,5,4d,2d)(5,5,2d,d)(5,5,d,c)</td><td rowspan=2 colspan=1>(n,100)(n,256d)(n,4,4,16d)(n,4,4,16d)(n,8,8,8d)(n,8,8,8d)(n,16,16,4d)(n,16,16,4d)(n,32,32,2d)(n,32,32,2d)(n,64,64,d)(n,64,64,d)(n,128,128,c)(n,128,128,c)</td></tr><tr><td rowspan=1 colspan=4>Dense 1ReshapeReLUTrans Conv2D (Stride=2)ReLUTrans Conv2D (Stride=2)ReLUTrans Conv2D (Stride=2)ReLUTrans Conv2D (Stride=2)ReLUTrans Conv2D (Stride=2)Tanh</td><td rowspan=1 colspan=1>Dense 1</td></tr></table>",
+        "bbox": [
+            299,
+            161,
+            696,
+            352
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "table",
+        "img_path": "images/7f29f442639f816f8bdab7225fc89a7b87145c6047e50c000a9fd9583b9bc496.jpg",
+        "table_caption": [
+            "Table 5: SpecGAN discriminator architecture "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=1 colspan=1>Operation</td><td rowspan=1 colspan=1>Kernel Size</td><td rowspan=1 colspan=1>Output Shape</td></tr><tr><td rowspan=1 colspan=1>Input x or G(z)Conv2D (Stride=2)LReLU (α = 0.2)Conv2D (Stride=2)LReLU (α = 0.2)Conv2D (Stride=2)LReLU (α = 0.2)Conv2D (Stride=2)LReLU (α = 0.2)Conv2D (Stride=2)LReLU (α= 0.2)ReshapeDense</td><td rowspan=1 colspan=1>(5,5,c,d)(5,5,d,2d)(5,5,2d,4d)(5,5,4d,8d)(5,5,8d,16d)(256d,1)</td><td rowspan=1 colspan=1>(n,128,128,c)(n,64,64,d)(n,64,64,d)(n,32,32,2d)(n,32,32,2d)(n,16,16,4d)(n,16,16,4d)(n,8,8,8d)(n,8,8,8d)(n,4,4,16d)(n,4,4,16d)(n,256d)(n,1)</td></tr></table>",
+        "bbox": [
+            323,
+            454,
+            676,
+            633
+        ],
+        "page_idx": 15
+    },
+    {
+        "type": "table",
+        "img_path": "images/5392982202b76c75a6696a8fa5f2fa32c4045fdc54f5a8864b5fa1d0a8f8ecd7.jpg",
+        "table_caption": [
+            "Table 6: WaveGAN hyperparameters "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Name</td><td>Value</td></tr><tr><td>Input data type Model data type</td><td>16-bit PCM (requantized to 32-bit float) 32-bit floating point</td></tr><tr><td>Num channels (c)</td><td>1</td></tr><tr><td>Batch size (b)</td><td>64</td></tr><tr><td>Model dimensionality (d)</td><td>64</td></tr><tr><td>Phase shuffle (WaveGAN)</td><td>2</td></tr><tr><td>Phase shuffle (SpecGAN)</td><td>0</td></tr><tr><td>Loss</td><td>WGAN-GP (Gulrajani et al., 2017)</td></tr><tr><td>WGAN-GP入</td><td>10</td></tr><tr><td>D updates per G update</td><td></td></tr><tr><td></td><td>5</td></tr><tr><td>Optimizer</td><td>Adam (α=1e-4,βi= 0.5,β2=0.9)</td></tr></table>",
+        "bbox": [
+            282,
+            733,
+            709,
+            888
+        ],
+        "page_idx": 15
+    }
+]

parse/train/blRJEZfyem/blRJEZfyem.md

@@ -0,0 +1,451 @@
+# Automation for Interpretable Machine Learning Through a Comparison of Loss Functions to Regularisers
+
+Anonymous Author(s)   
+Affiliation   
+Address   
+email
+
+# Abstract
+
+1 To increase the ubiquity of machine learning it needs to be automated. Automation   
+2 is cost-effective as it allows experts to spend less time tuning the approach, which   
+3 leads to shorter development times. However, while this automation produces   
+4 highly accurate architectures, they can be uninterpretable, acting as ‘black-boxes’   
+5 which produce low conventional errors but fail to model the underlying input-output   
+6 relationships—the ground truth. This paper explores the use of the Fit to Median   
+7 Error measure in machine learning regression automation, using evolutionary   
+8 computation in order to improve the approximation of the ground truth. When used   
+9 alongside conventional error measures it improves interpretability by regularising   
+10 learnt input-output relationships to the conditional median. It is compared to   
+11 traditional regularisers to illustrate that the use of the Fit to Median Error produces   
+12 regression neural networks which model more consistent input-output relationships.   
+13 The problem considered is ship power prediction using a fuel-saving air lubrication   
+14 system, which is highly stochastic in nature. The networks optimised for their   
+15 Fit to Median Error are shown to approximate the ground truth more consistently,   
+16 without sacrificing conventional Minkowski-r error values.
+
+# 17 1 Development of Interpretable Machine Learning
+
+18 Machine learning regression models are increasingly being used in industrial and engineering con  
+19 texts for high-stakes decision making, automation and control. These methods often produce low   
+20 conventional error values, yet are known to produce physically inconsistent results which cannot   
+21 generalise off test set and so cannot be relied upon to model the ground truth of the system. The   
+22 models are designed to be accurate, not interpretable, and so a human cannot understand how changes   
+23 in the inputs change the prediction. In real-world applications, where the output can have a direct   
+24 effect on human life or the environment, model accuracy alone is not sufficient.   
+25 Trust is increased if a trained model approximates the true input-output relationships, performing   
+26 accurately within the bounds of the training data set and beyond it. It has been demonstrated that for   
+27 many applications minimising traditional error measures cannot guarantee an accurate approximation   
+28 of the ground truth (Willard et al. 2020). This is due to a poor inductive bias, the inherent prioritisation   
+29 of one solution over another (Battaglia et al. 2018), produced by conventional error measures which   
+30 are based on Minkowski-r metrics (Hanson & Burr 1987).   
+31 This trust can be increased by manually tuning to remove overfitting or to provide a solution that   
+32 makes more sense to the user. However, the expert knowledge and domain experience required to   
+33 properly tune a machine learning method manually are not always available in industry. Genetic   
+34 algorithms are therefore increasingly used to search a method’s hyperparameter space more efficiently   
+35 (Yang et al. 2021) (Kumar et al. 2021); which minimise conventional error measures on a test set,   
+36 often combined with lowering the complexity of the network. This automation exacerbates the   
+37 lack of interpretability, as models have a large flexibility, and prediction accuracy is prioritised,   
+38 a low conventional error is achieved without certainty that the method has modelled the correct   
+39 internal functions. Regularisation hyperparameters can be optimised alongside other neural network   
+40 parameters (Tani et al. 2021) (Luketina et al. 2016), which increases the search space and creates   
+41 more flexibility for methods to produce ‘accurate’ predictions and avoid overfitting.   
+42 Common regression regularisation methods are l1 and l2 regularisation and dropout. For l1 and l2   
+43 regularisation, large network weights are penalised in the loss function (Nowlan & Hinton 1992). The   
+44 absolute value of the weights is penalised in l1 regularisation and the squared value in l2, meaning   
+45 l1 encourages weights towards zero and l2 encourages weights to be small but non-zero. The l1, l2   
+46 and elastic net (l1+l2) regularisers improve a networks generality, increasing the applications where   
+47 the trained methods can be applied, by penalising complexity. Dropout, where a randomly selected   
+48 subset of weights are optimised at each epoch rather than the full set, improve the generality of the   
+49 trained models by preventing co-adaption of weight values (Srivastava et al. 2014). Dropout has been   
+50 shown to be equivalent to l2 regularisation after scaling by Fisher information (Wager et al. 2013),   
+51 suggesting that the two should not be used in unison. The neural network regularisation methods   
+52 discussed above aim to improve generality, reducing overfitting by simplifying the relationships   
+53 modelled by the networks.   
+54 Regularisers improve the modelling of the ground truth in scenarios adhering to the assumptions in the   
+55 proof in Bishop (1995), under which minimum Minkowski-r error values approximate the conditional   
+56 average of the dataset. This is because the inductive bias from the loss function guides the input  
+57 output relationships towards the conditional average, while the regularisation stops overfitting by   
+58 simplfying the input-output relationships being modelled. However, these assumptions are restrictive   
+59 and it is noted that few regression applications adhere to them. For example, one assumption is that   
+60 the dataset is homoscedastic. In scenarios not adhering to these assumptions, network regularisation   
+61 simplifies the relationships being modelled but this does not necessarily improve the generality, or   
+62 model the ground truth.   
+63 The Fit to Median Error measure (Parkes et al. 2021) produces more interpretable regression, when   
+64 used in conjunction with conventional error measures. This is achieved by regularising the learnt   
+65 input-output relationships to the conditional median of the training dataset: the median output value,   
+66 conditioned on each isolated input variable in turn (Bishop 1995). For many regression applications   
+67 the conditional medians are a good approximation of the ground truth input-output relationships but   
+68 as yet it has not been explored as part of an automated approach.   
+69 A challenging regression problem is ship power prediction for a vessel using air lubrication to reduce   
+70 fuel consumption. It is chosen to be used in this study as it violates the assumptions in Bishop (1995),   
+71 where the noise in the output space is non-Gaussian and heteroscedastic. In this situation, correctly   
+72 modelling the ground truth and accurate prediction is required but there is limited understanding   
+73 of that ground truth (Parkes et al. 2018). The literature shows that shaft powering of a vessel can   
+74 be predicted with average accuracies of between $1 . 5 \mathrm { - } 5 \%$ error with the use of a regression neural   
+75 network trained with high frequency data from the vessel (Pedersen & Larsen 2009), (Petersen et al.   
+76 2012), (Le et al. 2020), (Jeon et al. 2018), (Liang et al. 2019). All neural network applications to   
+77 ship power prediction in the literature use a combination of local searches and domain knowledge to   
+78 identify hyperparameter values. The addition of an air lubrication device increases the complexity of   
+79 the regression problem, as the system interacts with a number of interrelated input variables.   
+80 This paper explores the automation of neural network training to a new problem, with a focus   
+81 on producing a network which accurately models the ground truth. It compares the ground truth   
+82 representation of a neural network when a genetic algorithm optimises the network’s hyperparameters   
+83 to reduce the Mean Fit to Median Error measure and compares it to standard regularization using l1,   
+84 l2 and dropout, and to a network optimised to minimise the Maximum Absolute Error. It is illustrated   
+85 that neural network regularisation methods (l1, l2 and dropout) can be replaced by the use of the   
+86 Mean Fit to Median performance measure as an objective in the genetic algorithm, reducing the   
+87 complexity of the search space and producing networks which more consistently model the ground   
+88 truth.   
+90 Previous applications of neural networks to ship power prediction use between 1 and 3 hidden layers   
+91 (Leifsson et al. 2008) (Parkes et al. 2019), and between 5 and 300 neurons in each hidden layer (Jeon   
+92 et al. 2018). To provide a sufficiently large search space to allow verification, or otherwise, of these   
+93 parameters a maximum of 4 hidden layers and 1000 neurons in each layer are used. The majority   
+94 of the literature treats the problem as time-invariant and use feed-forward networks, so no recurrent   
+95 parameters are optimised. As the optimiser or activation functions are rarely documented in the   
+96 literature, the state-of-the-art optimisers and activation functions available in the Keras framework   
+97 (Chollet et al. 2015) are used in the optimisation, Table 1.   
+98 The number of epochs and early stopping procedure are not optimised, as there was a need for   
+99 predictable compute requirements and allowing the optimisation of these parameters leads to unpre  
+100 dictable run times. The number of epochs to train each network increases for increasing generation   
+101 number in the genetic algorithm, from 1 epoch in the first 15 generations to 20 in the final 15. This   
+102 was also implemented to reduce compute and it was validated that when more than 20 epochs were   
+103 allowed, that the early stopping, with a patience of 5, stopped the training within 20 epochs for the   
+104 majority of networks. The loss function is similarly not optimised, the Mean Absolute Error is used,   
+105 as the conditional medians are closer to the ground truth input-output relationships in these datasets   
+106 than the conditional means.   
+107 The performance measures, or the genetic algorithm’s fitness functions, are the Mean Absolute   
+108 Relative Error, the Maximum Absolute Relative Error and the Mean Fit to Median Error. Different   
+109 combinations of these, alongside the use of regularisation parameters in the search space are compared   
+110 to illustrate the effect of different types of regularisation.
+
+Table 1: Selected Neural Network Hyperparameters   
+
+<table><tr><td rowspan=1 colspan=1>Hyperparameter</td><td rowspan=1 colspan=1>Value or set</td></tr><tr><td rowspan=1 colspan=1>Layers</td><td rowspan=1 colspan=1>[1,4]</td></tr><tr><td rowspan=1 colspan=1>Neurons in each layer</td><td rowspan=1 colspan=1>[1,1000]</td></tr><tr><td rowspan=1 colspan=1>Epochs</td><td rowspan=1 colspan=1>Increasing from 1-2O for increasing generations</td></tr><tr><td rowspan=1 colspan=1>Early stopping patience</td><td rowspan=1 colspan=1>5</td></tr><tr><td rowspan=1 colspan=1>Loss function</td><td rowspan=1 colspan=1>Mean Absolute Error</td></tr><tr><td rowspan=1 colspan=1>Performance measures</td><td rowspan=1 colspan=1>Mean Absolute Relative Error,Maximum Absolute Relative Error,Mean Fit to Median Error</td></tr><tr><td rowspan=1 colspan=1>Optimiser</td><td rowspan=1 colspan=1>SGD, Adam (Kingma &amp; Ba 2014), Nadam (Dozat 2016),RMSprop (Hinton et al. 2012), Adagrad (Duchi et al. 2011),Adadelta (Zeiler 2012),Adamax (Kingma &amp; Ba 2014)</td></tr><tr><td rowspan=1 colspan=1>Activation function</td><td rowspan=1 colspan=1>ReLU, sigmoid, softmax,softplus, softsign, tanh, selu, elu</td></tr><tr><td rowspan=1 colspan=1>11 &amp; 12 Rates</td><td rowspan=1 colspan=1>0,0.01,0.001,0.0001,0.00001</td></tr><tr><td rowspan=1 colspan=1>Dropout</td><td rowspan=1 colspan=1>[0,0.9)</td></tr><tr><td rowspan=1 colspan=1>Initialiser</td><td rowspan=1 colspan=1>Random Normal (μ = 0,σ = 0.1)</td></tr></table>
+
+# 111 3 cMLSGA Parameters
+
+Table 2: Selected cMLSGA Hyperparameters   
+
+<table><tr><td rowspan=1 colspan=1>Hyperparameter</td><td rowspan=1 colspan=1>Value or set</td></tr><tr><td rowspan=1 colspan=1>Algorithm at Individual Level</td><td rowspan=1 colspan=1>HEIA,IBEA</td></tr><tr><td rowspan=1 colspan=1>Crossover Type&amp;Rate</td><td rowspan=1 colspan=1>SBX&amp;DE,1</td></tr><tr><td rowspan=1 colspan=1>Mutation Type&amp; Rate</td><td rowspan=1 colspan=1>Polynomial, 0.08</td></tr><tr><td rowspan=1 colspan=1>Number of eliminated collectives</td><td rowspan=1 colspan=1>1</td></tr><tr><td rowspan=1 colspan=1>Generations between elimination</td><td rowspan=1 colspan=1>10</td></tr><tr><td rowspan=1 colspan=1>Population size</td><td rowspan=1 colspan=1>1000</td></tr><tr><td rowspan=1 colspan=1>Generations</td><td rowspan=1 colspan=1>300</td></tr><tr><td rowspan=1 colspan=1>Proportion elite</td><td rowspan=1 colspan=1>10%</td></tr></table>
+
+112 In this study cMLSGA1 is selected as it shows the top performance on a range of evolutionary   
+113 benchmarking problems (Grudniewski & Sobey 2021) and practical problems (Grudniewski & Sobey   
+114 2019). Genetic algorithms are increasing used to tune neural network hyperparameters including   
+115 regularisation parameters for use on new problems (Jin et al. 2004). Many approaches have multiple   
+116 genetic algorithm objectives, although these all minimise an error measure and a measure of network   
+117 complexity (Wang et al. 2019) and (Smith & Jin 2014). The use of multiple different performance   
+118 measures as objectives is yet to be explored in the literature.   
+119 Four approaches are investigated in this study, summarised in Table 3, for approach (GAi) and   
+120 (GAii) the genetic algorithm cMLSGA optimises all variables in Table 2, including the l1 and l2   
+121 regularisation rate and the dropout rate of the networks. Although it is advised that l2 regularisation   
+122 and dropout are not used in the same network the genetic algorithms are provided with zero options   
+123 for all regularisation parameters, to identify if one is preferable in this scenario.   
+124 Approach (GAi) is a single objective genetic algorithm optimising the Mean Absolute Error which is   
+125 compared to a multi-objective formulation where the (GAii) approach optimises both Mean Absolute   
+126 Error and Maximum Absolute Error. For approaches (GAiii) and (GAiv) no network regularisation   
+127 parameters are optimised: l1, l2 and dropout rates are all set permanently to zero. They avoid   
+128 producing networks that have overfitted by the use of two performance metrics as multi-objectives,   
+129 (GAiii) uses the Mean Fit to Median and Mean Absolute Errors to be minimised and (GAiv) uses the   
+130 Maximum Absolute and Mean Absolute. All approaches use 40 CPUs with $2 . 0 \mathrm { G H z }$ Intel Skylake   
+131 processors and 192 GB of DDR4 memory, and take less than 3 days, this setup may not be feasible   
+132 for widespread industrial application, although it is suggested it is within reach of some industries.
+
+Table 3: Genetic Algorithm Approaches   
+
+<table><tr><td rowspan=1 colspan=1>Approach</td><td rowspan=1 colspan=1>Objective(s)</td><td rowspan=1 colspan=1>Network Regularisation</td></tr><tr><td rowspan=1 colspan=1>GAi</td><td rowspan=1 colspan=1>Mean Absolute Error</td><td rowspan=1 colspan=1>11,12 and dropout</td></tr><tr><td rowspan=1 colspan=1>GAii</td><td rowspan=1 colspan=1>MeanAbsoluteErrorMaximum Absolute Error</td><td rowspan=1 colspan=1>11,12 and dropout</td></tr><tr><td rowspan=1 colspan=1>GAiii</td><td rowspan=1 colspan=1>MeanFitto MedianErrorMean Absolute Error</td><td rowspan=1 colspan=1>None</td></tr><tr><td rowspan=1 colspan=1>GAiv</td><td rowspan=1 colspan=1>Mean Absolute ErrorMaximum Absolute Error</td><td rowspan=1 colspan=1>None</td></tr></table>
+
+# 33 4 Data
+
+134 The data used in this study are from a large vessel equipped with the Silverstream $\textsuperscript { \textregistered }$ Air Lubrication   
+135 System. The air lubrication system works through use of fluid sheering to create an air microbubble   
+136 carpet directly captured within the boundary layer on the ship hull bottom. The bubble carpet reduces   
+137 the frictional resistance thereby increasing the speed and reducing the shaft power. Compressors   
+138 provide a constant supply of air to the hull bottom to maintain a uniform bubble carpet operated at   
+139 the optimal compressor power that maximises the energy balance. The study is performed on both   
+140 system on and system off datasets, however for brevity only results for system off are presented as   
+141 they show similar performance.This prediction is required for a baseline determination of how the   
+142 system is working, but the relationships between the power, weather, ocean and operating conditions   
+143 are complex and difficult to model.
+
+The variables considered in this study are the shaft power, speed through water, relative wind speed and direction, draught and trim, with shaft power the target variable. These are selected based on a detailed study into variable selection for shaft power prediction (Parkes et al. 2019). The speed through water is selected over the speed over ground, for use as an input variable, as it is more hydrodymanically relevant and its accuracy is validated by comparison to the speed over ground. The dataset is cleaned by removing rows with missing or non-physical values and all datapoints below 0.05 normalised shaft power are removed. The dataset is split into two using the air lubrication system status: system on and system off, where system on is defined as air lubrication system power greater than zero. The system on dataset contains 352,690 datapoints and system off contains 237,962. The data is split into training, testing and validation sets of $70 \%$ , $15 \%$ and $15 \%$ respectively. Each
+
+![](images/d407bf3bf5a570e9cd59fb59f2673801938222541b2875943708ca3c20facb8c.jpg)  
+Figure 1: The distribution of the observed shaft powers for half knot bins of speed through the water for dataset where the system is off. In the box and whisker plots the boxes contain $50 \%$ of the distribution and the whiskers extend to the datum which is at 1.5 times the interquartile range.
+
+154 network in the genetic algorithm trains on a randomly sampled 35,000 datapoints from the training   
+155 set and uses randomly sampled sets of size 7,500 from validation and testing sets for validation during   
+156 training, and testing to produce the fitness of the network for the genetic algorithm. The errors stated   
+157 in the paper are from networks on the Pareto fronts of each approach, which are validated on the full   
+158 testing set.   
+159 The datasets contain large regions of sparse data in all input variable domains, this is exemplified   
+160 by the ship speed domain where each half-knot interval below 16 knots contains less than $0 . 8 \%$ of   
+161 the data, which accounts for more than half the speed domain, Figure 1. In addition, the boxplot   
+162 ranges and outliers show high heteroscedicity with idiosyncratic noise caused by situations where the   
+163 angle of the propeller blades is varied to achieve the required speed. This highlights the complexity   
+164 in developing models of the powering of this vessel, as the dataset also contains the effects from other   
+165 latent variables, such as piloting behaviour and route taken.
+
+# 66 5 Optimisation including regularisation parameters: (GAi) and (GAii)
+
+167 Previous studies predicting ship powering using neural networks report that l1, l2 and elastic net   
+168 increase both test set and off-test set errors and that optimal values for both l1 and l2 are zero.   
+169 Therefore the genetic algorithm setup is biased towards low and zero values of regularisation rates by   
+170 using a set of exponentially decreasing values and an explicit zero option.   
+171 The single objective (GAi) fails to identify that zero regularisation rates produce the lowest errors,   
+172 favouring networks with the highest possible rate of l2 (0.01), Figure 2b. (GAi) produces networks   
+173 with the highest Mean Absolute Relative Errors of all the approaches, $( 5 . 1 9 \pm 0 . 0 0 ) \%$ from Figure   
+174 4a. In contrast, (GAii) favours lower l1 and l2 rates of 0 or 0.00001, Figures 2c and 2d, which results   
+175 in networks with the lowest Mean Absolute Relative Errors of all four approaches, on average, with a   
+176 value of $( 2 . 8 7 \pm 0 . 4 5 ) \%$ , shown in Figure 4a. This is around $0 . 5 \%$ higher than the lowest documented   
+177 error for ship power prediction.   
+178 It is posited that the high error for the single objective problem is directly related to the use of a   
+179 large l2 regularisation rate, as noted in previous studies for ship power prediction. It is possible   
+180 that the use of a multi-objective search algorithm for a single-objective problem means that the   
+181 optimal hyperparameters can’t be found, resulting in large errors. The implementation also requires   
+182 restrictions in the number of epochs used for training in the initial generations, it is possible this   
+183 biases (GAi) towards certain size networks, where higher l2 rates are preferable. This hypothesis is   
+184 supported by the fact that $7 4 . 4 \%$ of networks in the first 15 generations of (GAi) have 1 hidden layer,   
+185 and that over $9 9 . 8 \%$ of the networks in the final 15 generations have 1 hidden layer, with $8 8 0 \pm 1 7$   
+186 neurons in this layer, Figure 3b. This is significantly more neurons than those in the hidden layer of   
+187 networks in the final 15 generations of (GAii) which range from 3-952 with a median value of 709,   
+188 Figure 3b. The added objective of minimising Maximum Absolute Error in (GAii) may cause these   
+189 slightly smaller networks to be more attractive as they are in a sense regularised by their size, as they   
+190 have reduced modelling flexibility therefore are less likely to overfit and produce high Maximum   
+191 Absolute Errors.   
+192 Another explanation for the difference in l2 rates chosen by (GAi) and (GAii) is the equivalence   
+193 of l2 and dropout. Since l2 and dropout are equivalent up to a Fisher transformation, their use in   
+194 conjunction is not recommended. The evidence for this is that (GAi) favours the highest l2 rate and   
+195 has a median dropout rate in the final 15 generations of 0.116, whereas (GAii) favours the zero l2 rate   
+196 and has a median dropout rate of 0.624, Figure 3a. This illustrates that the genetic algorithms will   
+197 chose either l2 or dropout to minimise the Mean Absolute Relative Error. The l1 rates also support   
+198 this hypothesis, as chosen rates for l1 regularisation in the final 15 generations are more comparable   
+199 for (GAi) and (GAii).   
+201 For approaches (GAiii) and (GAiv) all neural network regularisation parameters are set to zero.   
+202 The regularisation is performed by minimising different network performance measures, the Mean   
+203 Absolute and Mean Fit to Median for (GAiii), and the Mean Absolute and Maximum Absolute for   
+204 (GAiv). The trade-off between the two objectives produces regularised neural networks, without   
+205 explicitly changing the architecture or loss function. The Mean Fit to Median is chosen as it indicates   
+206 how close the relationships modelled by a network are to the conditional averages of the dataset, in   
+207 many regression examples this is akin to the ground truth input-output relationships (Parkes et al.   
+208 2021). The Maximum Absolute is chosen as for many industrial applications of machine learning the   
+209 maximum prediction error is more pertinent than the mean error. The Mean Absolute Error is used   
+210 instead of the Mean Squared Error in both approaches, as the conditional medians are closer to the   
+211 ground truth input-output relationships in these datasets than the conditional means.   
+212 Differently shaped networks are favoured by (GAiii) and (GAiv), compared to (GAi) and (GAii),   
+213 focusing on networks with 3 hidden layers and on average less than 400 neurons in each layer, Figure   
+214 3b. These networks have 51 times the number of connections than the networks chosen in (GAi) and   
+215 (GAii). Apart from (GAi), (GAiii) has the most consistently sized networks in the final 15 generations,   
+216 with an interquartile range of 46 neurons, compared to (GAiv) which have an interquartile range of   
+217 131 neurons. It is suggested that as the Mean Fit to Median Error biases networks towards specific   
+218 input-output relationships, there is a smaller range of potential network architectures which habitually   
+219 model these relationships. Whereas networks which minimise the Maximum Absolute Error are less   
+220 restricted and can model a wider range of input and output relationships.   
+221 The Mean Absolute Relative Errors from networks in the Pareto fronts are $( 2 . 9 7 \pm 0 . 2 5 ) \%$ for (GAiii)   
+222 and $( 3 . 1 0 \pm 0 . 2 8 ) \%$ for (GAiv). It is expected that (GAiv) would produce higher Mean Absolute   
+223 Relative Errors as discussed above, minimising the Maximum Absolute Error should bias predictions   
+224 towards the midpoint of the conditional output distributions, whereas minimising the Mean Absolute   
+225 Error should bias predictions towards the median of these distributions. As it is established that noise   
+226 in the output distribution is non-Gaussian, Figure 1, these values will not align so some sacrifice   
+227 in Mean Absolute Error is expected from (GAiv). Both (GAiii) and (GAiv) produce comparable   
+228 Maximum Absolute Errors, of $( 4 9 . 5 \pm 1 . 1 ) \%$ and $( 5 1 . 9 \pm 4 . 5 ) \%$ . It is suggested that this is because,   
+229 although the conditional median output value and conditional midpoint output value do not align   
+230 for the majority of the input domain, they are sufficiently close to produce comparable Maximum   
+231 Absolute Errors.   
+232 Across all four approaches, the genetic algorithm producing networks with the highest Mean Absolute   
+233 Error is the approach which does not provide extra weighting to sparse areas of data. The approaches   
+234 minimising Maximum Absolute Error are implicitly biased away from networks which predict the   
+235 majority of the testing datapoints correctly, but predict one datapoint poorly, favouring networks   
+236 which predict all testing datapoints to a moderate degree of error. Approach (GAiii) more explicitly   
+237 weights prediction in sparse areas of data by favouring networks which model the conditional median   
+238 of the dataset across all input domains, irrespective of the quantity of data across each input domain.   
+239 The regression problem of ship power prediction is chosen in part because of it’s irregular data   
+240 distribution; more than $9 \%$ of the dataset lies in less than a 0.5 knot interval of ship speed, Figure 1.   
+241 This provides an explanation for the high testing errors from (GAi), where only the Mean Absolute   
+242 Error is minimised, there is little incentive for the genetic algorithm to produce networks which   
+243 generalise across the full range of the input domain well.
+
+![](images/afe28a7b829a455984b0f37805b0861128d88f6efde3154d4742534acb32a5f3.jpg)  
+Figure 2: Distribution of regularisation rates for networks in the last 15 generations of (GAi) cMLSGA with multi-objectives of minimising Maximum and Mean Absolute Error for (a) l1 and (b) l2 and (GAii) cMLSGA with the single objective of minimising Mean Absolute Error for (c) l1 and (d) l2
+
+![](images/6488971327470fa6c2cd02a09f904e1b0625e70d1cdd852951e21f9b87bd1862.jpg)  
+Figure 3: (a) Dropout rate for networks in the last 15 generations of cMLSGA with (GAi) the single objective of minimising Mean Absolute Error and (GAii) multi-objectives of minimising Maximum and Mean Absolute Error and (b) the number of neurons in each layer for networks in the last 15 generations of cMLSGA with (GAi), (GAii),(GAiii) and (GAiv).
+
+![](images/28f5a9b914cd8a34d5d8e5e440d05285f51600ee12d573441f4e55b3ae160cc3.jpg)  
+Figure 4: Mean Relative Absolute Error (a) and Maximum Absolute Error (b) from cMLSGA with (GAi) the single objective of minimising Mean Absolute Error and (GAii) multi-objectives of minimising Maximum and Mean Absolute Error, both optimising the parameters for l1, l2 regularisation and dropout in the networks, and (GAiii) and (GAiv) which do not use network regularisation but minimise Mean Fit to Median and Maximum Absolute Error respectively, alongside Mean Absolute Error
+
+# 244 7 Comparison of the interpretability
+
+245 To asses the interpretability of the networks selected by the four different approaches the learnt   
+246 relationship between an input, the ship speed, and the output, shaft power, for the networks in   
+247 the Pareto front of each approach are visualised, Figure 5. These are extracted with the following   
+248 procedure: set all but one input variable to be constant at the mode; cycle the remaining variable from   
+249 its minimum to its maximum recorded values with 150 points evenly spaced along the domain and   
+250 run the new dataset through the trained network.   
+251 The approach which produces the most consistent speed-power relationships is (GAi), with an average   
+252 variation of $1 . 8 \% ^ { 2 }$ , Figure 5a. However, the relationship modelled by the 5 networks with the lowest   
+253 Mean Absolute Relative Error in (GAi) all approximate a piece-wise linear relationship which clearly   
+254 underfits the dataset in Figure 1. The expected trend between ship speed through the water and shaft   
+255 power is a cubic polynomial, therefore as well as producing the highest Mean Absolute Relative   
+256 Errors, networks chosen by (GAi) model the ground truth input-output relationships the worst out of   
+257 the four approaches. Both (GAii) and (GAiv) produce 5 fairly consistent speed-power curves, with   
+258 average variations of $5 . 9 \%$ and $10 \%$ respectively, Figures 5b and 5d. Both approaches approximate   
+259 smooth polynomial curves, although the degrees of the polynomials might differ, as multiple curves   
+260 intersect at various points along the speed axis. The spread of learnt relationships is greater at the   
+261 highest and lowest speeds for (GAii), with a decrease in spread for speeds of around 15 knots, where   
+262 many of the curves intersect. The curves from (GAiv) show equal spread across the speed domain.   
+263 The approach with both accurate and consistent learnt speed-power curves is (GAiii), with limited   
+264 intersections of curves and an average spread of $3 . 0 \%$ . It is suggested that the reason using the   
+265 Mean Fit to Median Error as an objective in a multi-objective genetic algorithm produces more   
+266 interpretable results, or more consistent learnt relationships, is because instead of encouraging the   
+267 networks to model more simple relationships. It encourages the networks to model the conditional   
+268 median functions of the dataset, supported by the increase in network connections. Whereas the other   
+269 approaches leave room for networks to fail to model the conditional averages, especially in irregularly   
+270 distributed and non-normally distributed datasets. The Mean Absolute Error values from networks   
+271 selected by (GAiii) are on average $0 . 1 \%$ higher than those from (GAii), and the Maximum Absolute   
+272 Error values are $1 . 6 \%$ higher.   
+273 A limitation of the approach is that the Fit to Median Error measure will likely perform best at   
+274 improving interpretability on datasets which violate the assumptions in Bishop (1995); the ship   
+275 powering example is chosen to illustrate this as it provides a clearly heteroscedastic dataset. For   
+276 applications where noise profiles are Gaussian, and there is no effect from latent or interrelated input   
+277 variables, the Fit to Median Error will not improve interpretability, but will perform the same as   
+278 conventional Minkowski-r metrics, either Mean Squared or Mean Absolute Error depending on the   
+279 convexity of input-output relationships.   
+280 Interestingly, the approach producing the lowest Mean Absolute and Maximum Absolute Errors   
+281 does not model the ground truth the most accurately. This creates a potential for negative societal   
+282 impacts, as the standard performance metrics for regression neural networks do not provide a full   
+283 picture of performance or expected behaviour. Interpretability of trained methods is essential for   
+284 safe application of machine learning in the real world, especially when automated methods are   
+285 used to replace experienced professionals. (GAii) demonstrates the same accuracy of approach as   
+286 those with standard network regularisation, but with a better fit to the ground truth. This approach   
+287 bypasses the need to use, and therefore to optimise the parameters of the regularisation methods.   
+288 If evolutionary computation is already being used to optimise network parameters, then compute   
+289 is saved by removing the network regularisation parameters l1, l2 and dropout. (GAiii) completed   
+290 300 generations in 46hours whereas (GAii) required 12 hours more computation to complete 300   
+291 generations.
+
+![](images/cd83a8f2500890996747ee92b12173858cc6ee31f7cc8b3bca0dd6b66e9a5c18.jpg)  
+Figure 5: The learnt speed-power curves from 5 networks on the Pareto fronts of (GAii), (GAiii), (GAiv) and the 5 networks producing lowest Mean Absolute Relative Error from (GAi).
+
+# 92 8 Conclusion
+
+293   
+294   
+295   
+296   
+297   
+298   
+299   
+300   
+301   
+302   
+303
+
+Interpretable and accurate methods are required for widespread application of machine learning to real-world regression problems. To automate the training of neural networks so that the result is interpretable, three different genetic algorithm approaches are compared: one to minimise the Maximum Absolute Error of the networks, which includes standard regularization using l1, l2 and dropout; and two which do not use any network regularisation, one minimising the Mean Fit to Median and one to minimise the Maximum Absolute Error. The results show that all three approaches give similar Mean Absolute Errors from networks on their Pareto fronts, from $2 . 9 \%$ for the approach with regularisation to $3 . 1 \%$ for the approach minimising Maximum Absolute Error. However, the Mean Fit to the Median approach shows a considerably better interpretability, or fit to the ground truth, with a spread in predicted input-output curves of $3 \%$ compared to a spread of $6 \%$ for the approach using regularisation and $10 \%$ when minimising the Maximum Absolute Error.
+
+# 304 References
+
+Battaglia, P. W., Hamrick, J. B., Bapst, V., Sanchez-Gonzalez, A., Zambaldi, V., Malinowski, M., Tacchetti, A., Raposo, D., Santoro, A., Faulkner, R., Gulcehre, C., Song, F., Ballard, A., Gilmer, J., Dahl, G., Vaswani, A., Allen, K., Nash, C., Langston, V., Dyer, C., Heess, N., Wierstra, D.,
+
+308 Kohli, P., Botvinick, M., Vinyals, O., Li, Y. & Pascanu, R. (2018), ‘Relational inductive biases,   
+309 deep learning, and graph networks’.   
+310 Bishop, C. (1995), Neural Networks for Pattern Recognition, Oxford University Press, chapter 6,   
+311 pp. 194–225.   
+312 Chollet, F. et al. (2015), ‘Keras’, https://keras.io.   
+313 Dozat, T. (2016), ‘Incorporating nesterov momentum into adam’, International Conference on   
+314 Learning Representations 2016 .   
+315 Duchi, J., Hazan, E. & Singer, Y. (2011), ‘Adaptive subgradient methods for online learning and   
+316 stochastic optimization’, Journal of Machine Learning Research 12(61), 2121–2159.   
+317 Grudniewski, P. A. & Sobey, A. J. (2019), Do general genetic algorithms provide benefits when   
+318 solving real problems?, in ‘2019 IEEE Congress on Evolutionary Computation (CEC)’, IEEE,   
+319 pp. 1822–1829.   
+320 Grudniewski, P. A. & Sobey, A. J. (2021), ‘cMLSGA: a co-evolutionary multi-level selection genetic   
+321 algorithm for multi-objective optimization’.   
+322 Hanson, S. & Burr, D. (1987), Minkowski-r back-propagation: learning in connectionist models with   
+323 non-euclidian error signals., in ‘Neural Information Processing Systems (NIPS 1987)’.   
+Hinton, G., Srivastava, N. & Swersky, K. (2012), ‘Lecture 6a:overview of mini-batch gradient   
+descent’.   
+URL: http://www.cs.toronto.edu/ tijmen/csc321/slides/lectureslideslec6.pdf   
+324 Jeon, M., Noh, Y., Shin, Y., Lim, O., Lee, I. & Cho, D. (2018), ‘Prediction of ship fuel consumption   
+325 by using an artificial neural network’, Journal of Mechanical Science and Technology 32(12), 5785–   
+326 5796.   
+327 Jin, Y., Okabe, T. & Sendhoff, B. (2004), Neural network regularization and ensembling using   
+328 multi-objective evolutionary algorithms, in ‘Proceedings of the 2004 Congress on Evolutionary   
+329 Computation (IEEE Cat. No.04TH8753)’, Vol. 1.   
+330 Kingma, D. P. & Ba, J. (2014), ‘Adam: A method for stochastic optimization’, arXiv preprint   
+331 arXiv:1412.6980 .   
+332 Kumar, P., Batra, S. & Raman, B. (2021), ‘Deep neural network hyper-parameter tuning through   
+333 twofold genetic approach’, Soft Computing pp. 1–25.   
+334 Le, L., Lee, G., Park, K. & Kim, H. (2020), ‘Neural network-based fuel consumption estimation for   
+335 container ships in korea’, Maritime Policy & Management pp. 1–18.   
+336 Leifsson, L., Sævarsdóttir, H., Sigurðsson, S. & Vésteinsson, A. (2008), ‘Grey-box modeling of an   
+337 ocean vessel for operational optimization’, Simulation Modelling Practice and Theory 16(8), 923–   
+338 932.   
+339 Liang, Q., Tvete, H. A. & Brinks, H. W. (2019), Prediction of vessel propulsion power using machine   
+340 learning on ais data, ship performance measurements and weather data, in ‘Journal of Physics:   
+341 Conference Series’, Vol. 1357, p. 012038.   
+342 Luketina, J., Berglund, M., Greff, K. & Raiko, T. (2016), ‘Scalable gradient-based tuning of continu  
+343 ous regularization hyperparameters’.   
+344 Nowlan, S. J. & Hinton, G. E. (1992), ‘Simplifying neural networks by soft weight sharing’, Neural   
+345 Computation .   
+346 Parkes, A. I., Savasta, T. D., Sobey, A. J. & Hudson, D. A. (2019), Efficient vessel power prediction   
+347 in operational conditions using machine learning., in ‘Practical Design of Ships and Other Floating   
+348 Structures(PRADS), September 2019, Yokohama, Japan’.   
+349 Parkes, A. I., Sobey, A. J. & Hudson, D. A. (2018), ‘Physics-based shaft power prediction for large   
+350 merchant ships using neural networks’, Ocean Engineering 166, 92–104.
+
+Parkes, A. I., Sobey, A. J. & Hudson, D. A. (2021), ‘Towards error measures which influence a learners inductive bias to the ground truth’.   
+Pedersen, B. P. & Larsen, J. (2009), Prediction of full-scale propulsion power using artificial neural networks, in ‘Proceedings of the 8th international conference on computer and IT applications in the maritime industries (COMPIT’09), Budapest, Hungary May’, pp. 10–12.   
+Petersen, J. P., Jacobsen, D. J. & Winther, O. (2012), ‘Statistical modelling for ship propulsion efficiency’, Journal of marine science and technology 17(1), 30–39.   
+Smith, C. & Jin, Y. (2014), ‘Evolutionary multi-objective generation of recurrent neural network ensembles for time series prediction’, Neurocomputing 143, 302–311.   
+Srivastava, N., Hinton, G., Krizhevsky, A., Sutskever, I. & Salakhutdinov, R. (2014), ‘Dropout: a simple way to prevent neural networks from overfitting’, The Journal of Machine Learning Research 15(1), 1929–1958.   
+Tani, L., Rand, D., Veelken, C. & Kadastik, M. (2021), ‘Evolutionary algorithms for hyperparameter optimization in machine learning for application in high energy physics’, The European Physical Journal C 81(2), 1–9.   
+Wager, S., Wang, S. & Liang, P. (2013), ‘Dropout training as adaptive regularization’, arXiv preprint arXiv:1307.1493 .   
+Wang, B., Sun, Y., Xue, B. & Zhang, M. (2019), Evolving deep neural networks by multi-objective particle swarm optimization for image classification, in ‘Proceedings of the Genetic and Evolutionary Computation Conference’, GECCO ’19, p. 490–498.   
+Willard, J., Jia, X., Xu, S., Steinbach, M. & Kumar, V. (2020), ‘Integrating physics-based modeling with machine learning: a survey’.   
+Yang, S., Tian, Y., He, C., Zhang, X., Tan, K. C. & Jin, Y. (2021), ‘A gradient-guided evolutionary approach to training deep neural networks’, IEEE Transactions on Neural Networks and Learning Systems pp. 1–15.
+
+376 Zeiler, M. D. (2012), ‘Adadelta: an adaptive learning rate method’.
+
+# Checklist
+
+The checklist follows the references. Please read the checklist guidelines carefully for information on how to answer these questions. For each question, change the default [TODO] to [Yes] , [No] , or [N/A] . You are strongly encouraged to include a justification to your answer, either by referencing the appropriate section of your paper or providing a brief inline description. For example:
+
+• Did you include the license to the code and datasets? [Yes] See Section ??.   
+• Did you include the license to the code and datasets? [No] The code and the data are proprietary.   
+• Did you include the license to the code and datasets? [N/A]
+
+Please do not modify the questions and only use the provided macros for your answers. Note that the Checklist section does not count towards the page limit. In your paper, please delete this instructions block and only keep the Checklist section heading above along with the questions/answers below.
+
+1. For all authors...
+
+(a) Do the main claims made in the abstract and introduction accurately reflect the paper’s contributions and scope? [Yes] The scope of the paper (regression problem automation) is made clear in the abstract. The claims of improved interpretability using the Fit to Median Error measure are reflected in Figure 5, as well as the body of text.   
+(b) Did you describe the limitations of your work? [Yes] The limitations of the Fit to Median are discussed in Section 7, and the limitations relating to compute are discussed in Section 3.   
+(c) Did you discuss any potential negative societal impacts of your work? [Yes] See Section 6   
+(d) Have you read the ethics review guidelines and ensured that your paper conforms to them? [Yes]
+
+2. If you are including theoretical results...
+
+(a) Did you state the full set of assumptions of all theoretical results? [No] Citation of (Bishop 1995) is included instead of explicit statement of all assumptions, although the most pertinent assumptions are discussed in Section 1   
+(b) Did you include complete proofs of all theoretical results? [N/A] No theorems are posited, the paper explores a practical comparison of common methods.
+
+3. If you ran experiments...
+
+(a) Did you include the code, data, and instructions needed to reproduce the main experimental results (either in the supplemental material or as a URL)? [Yes] Instructions needed to replicate the results on a different dataset are provided in the main text; Sections 3 and 2 specify the expermental setup. The code is based on the cMLSGA algorithm, available at https://www.bitbucket. $\sigma \ \mu \mu \mu \mu \mu \mu \mu$ (redacted for anonymity) which is provided in Section 3. The data is proprietary so is not provided.   
+(b) Did you specify all the training details (e.g., data splits, hyperparameters, how they were chosen)? [Yes] See Sections 3 and 2   
+(c) Did you report error bars (e.g., with respect to the random seed after running experiments multiple times)? [Yes] All results are reported to plus/minus the standard deviation, and all figures show multiple runs with respect to either a random seed or an evolutionary method.   
+(d) Did you include the total amount of compute and the type of resources used (e.g., type of GPUs, internal cluster, or cloud provider)? [Yes] See Section 3
+
+4. If you are using existing assets (e.g., code, data, models) or curating/releasing new assets...
+
+(a) If your work uses existing assets, did you cite the creators? [Yes] The creators of the genetic algorithim cMLSGA are cited in Section 3 and the data is acknowledged to belong to Silverstream Technologies Ltd in Section 4   
+(b) Did you mention the license of the assets? [Yes] The code is licensed under the GNU General Public License. The data is not licensed.   
+(c) Did you include any new assets either in the supplemental material or as a URL? [No]   
+(d) Did you discuss whether and how consent was obtained from people whose data you’re using/curating? [No] The work is performed in collaboration with Silverstream Techonologies Ltd.   
+(e) Did you discuss whether the data you are using/curating contains personally identifiable information or offensive content? [No] The shaft power variable is normalised to remove any possibility of identifying the vessel in question.
+
+5. If you used crowdsourcing or conducted research with human subjects...
+
+(a) Did you include the full text of instructions given to participants and screenshots, if applicable? [N/A]   
+(b) Did you describe any potential participant risks, with links to Institutional Review Board (IRB) approvals, if applicable? [N/A]   
+(c) Did you include the estimated hourly wage paid to participants and the total amount spent on participant compensation? [N/A]

parse/train/blRJEZfyem/blRJEZfyem_content_list.json

@@ -0,0 +1,861 @@
+[
+    {
+        "type": "text",
+        "text": "Automation for Interpretable Machine Learning Through a Comparison of Loss Functions to Regularisers ",
+        "text_level": 1,
+        "bbox": [
+            207,
+            122,
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Anonymous Author(s)   \nAffiliation   \nAddress   \nemail ",
+        "bbox": [
+            423,
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Abstract ",
+        "text_level": 1,
+        "bbox": [
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "1 To increase the ubiquity of machine learning it needs to be automated. Automation   \n2 is cost-effective as it allows experts to spend less time tuning the approach, which   \n3 leads to shorter development times. However, while this automation produces   \n4 highly accurate architectures, they can be uninterpretable, acting as ‘black-boxes’   \n5 which produce low conventional errors but fail to model the underlying input-output   \n6 relationships—the ground truth. This paper explores the use of the Fit to Median   \n7 Error measure in machine learning regression automation, using evolutionary   \n8 computation in order to improve the approximation of the ground truth. When used   \n9 alongside conventional error measures it improves interpretability by regularising   \n10 learnt input-output relationships to the conditional median. It is compared to   \n11 traditional regularisers to illustrate that the use of the Fit to Median Error produces   \n12 regression neural networks which model more consistent input-output relationships.   \n13 The problem considered is ship power prediction using a fuel-saving air lubrication   \n14 system, which is highly stochastic in nature. The networks optimised for their   \n15 Fit to Median Error are shown to approximate the ground truth more consistently,   \n16 without sacrificing conventional Minkowski-r error values. ",
+        "bbox": [
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+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "17 1 Development of Interpretable Machine Learning ",
+        "text_level": 1,
+        "bbox": [
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+        "text": "18 Machine learning regression models are increasingly being used in industrial and engineering con  \n19 texts for high-stakes decision making, automation and control. These methods often produce low   \n20 conventional error values, yet are known to produce physically inconsistent results which cannot   \n21 generalise off test set and so cannot be relied upon to model the ground truth of the system. The   \n22 models are designed to be accurate, not interpretable, and so a human cannot understand how changes   \n23 in the inputs change the prediction. In real-world applications, where the output can have a direct   \n24 effect on human life or the environment, model accuracy alone is not sufficient.   \n25 Trust is increased if a trained model approximates the true input-output relationships, performing   \n26 accurately within the bounds of the training data set and beyond it. It has been demonstrated that for   \n27 many applications minimising traditional error measures cannot guarantee an accurate approximation   \n28 of the ground truth (Willard et al. 2020). This is due to a poor inductive bias, the inherent prioritisation   \n29 of one solution over another (Battaglia et al. 2018), produced by conventional error measures which   \n30 are based on Minkowski-r metrics (Hanson & Burr 1987).   \n31 This trust can be increased by manually tuning to remove overfitting or to provide a solution that   \n32 makes more sense to the user. However, the expert knowledge and domain experience required to   \n33 properly tune a machine learning method manually are not always available in industry. Genetic   \n34 algorithms are therefore increasingly used to search a method’s hyperparameter space more efficiently   \n35 (Yang et al. 2021) (Kumar et al. 2021); which minimise conventional error measures on a test set,   \n36 often combined with lowering the complexity of the network. This automation exacerbates the   \n37 lack of interpretability, as models have a large flexibility, and prediction accuracy is prioritised,   \n38 a low conventional error is achieved without certainty that the method has modelled the correct   \n39 internal functions. Regularisation hyperparameters can be optimised alongside other neural network   \n40 parameters (Tani et al. 2021) (Luketina et al. 2016), which increases the search space and creates   \n41 more flexibility for methods to produce ‘accurate’ predictions and avoid overfitting.   \n42 Common regression regularisation methods are l1 and l2 regularisation and dropout. For l1 and l2   \n43 regularisation, large network weights are penalised in the loss function (Nowlan & Hinton 1992). The   \n44 absolute value of the weights is penalised in l1 regularisation and the squared value in l2, meaning   \n45 l1 encourages weights towards zero and l2 encourages weights to be small but non-zero. The l1, l2   \n46 and elastic net (l1+l2) regularisers improve a networks generality, increasing the applications where   \n47 the trained methods can be applied, by penalising complexity. Dropout, where a randomly selected   \n48 subset of weights are optimised at each epoch rather than the full set, improve the generality of the   \n49 trained models by preventing co-adaption of weight values (Srivastava et al. 2014). Dropout has been   \n50 shown to be equivalent to l2 regularisation after scaling by Fisher information (Wager et al. 2013),   \n51 suggesting that the two should not be used in unison. The neural network regularisation methods   \n52 discussed above aim to improve generality, reducing overfitting by simplifying the relationships   \n53 modelled by the networks.   \n54 Regularisers improve the modelling of the ground truth in scenarios adhering to the assumptions in the   \n55 proof in Bishop (1995), under which minimum Minkowski-r error values approximate the conditional   \n56 average of the dataset. This is because the inductive bias from the loss function guides the input  \n57 output relationships towards the conditional average, while the regularisation stops overfitting by   \n58 simplfying the input-output relationships being modelled. However, these assumptions are restrictive   \n59 and it is noted that few regression applications adhere to them. For example, one assumption is that   \n60 the dataset is homoscedastic. In scenarios not adhering to these assumptions, network regularisation   \n61 simplifies the relationships being modelled but this does not necessarily improve the generality, or   \n62 model the ground truth.   \n63 The Fit to Median Error measure (Parkes et al. 2021) produces more interpretable regression, when   \n64 used in conjunction with conventional error measures. This is achieved by regularising the learnt   \n65 input-output relationships to the conditional median of the training dataset: the median output value,   \n66 conditioned on each isolated input variable in turn (Bishop 1995). For many regression applications   \n67 the conditional medians are a good approximation of the ground truth input-output relationships but   \n68 as yet it has not been explored as part of an automated approach.   \n69 A challenging regression problem is ship power prediction for a vessel using air lubrication to reduce   \n70 fuel consumption. It is chosen to be used in this study as it violates the assumptions in Bishop (1995),   \n71 where the noise in the output space is non-Gaussian and heteroscedastic. In this situation, correctly   \n72 modelling the ground truth and accurate prediction is required but there is limited understanding   \n73 of that ground truth (Parkes et al. 2018). The literature shows that shaft powering of a vessel can   \n74 be predicted with average accuracies of between $1 . 5 \\mathrm { - } 5 \\%$ error with the use of a regression neural   \n75 network trained with high frequency data from the vessel (Pedersen & Larsen 2009), (Petersen et al.   \n76 2012), (Le et al. 2020), (Jeon et al. 2018), (Liang et al. 2019). All neural network applications to   \n77 ship power prediction in the literature use a combination of local searches and domain knowledge to   \n78 identify hyperparameter values. The addition of an air lubrication device increases the complexity of   \n79 the regression problem, as the system interacts with a number of interrelated input variables.   \n80 This paper explores the automation of neural network training to a new problem, with a focus   \n81 on producing a network which accurately models the ground truth. It compares the ground truth   \n82 representation of a neural network when a genetic algorithm optimises the network’s hyperparameters   \n83 to reduce the Mean Fit to Median Error measure and compares it to standard regularization using l1,   \n84 l2 and dropout, and to a network optimised to minimise the Maximum Absolute Error. It is illustrated   \n85 that neural network regularisation methods (l1, l2 and dropout) can be replaced by the use of the   \n86 Mean Fit to Median performance measure as an objective in the genetic algorithm, reducing the   \n87 complexity of the search space and producing networks which more consistently model the ground   \n88 truth.   \n90 Previous applications of neural networks to ship power prediction use between 1 and 3 hidden layers   \n91 (Leifsson et al. 2008) (Parkes et al. 2019), and between 5 and 300 neurons in each hidden layer (Jeon   \n92 et al. 2018). To provide a sufficiently large search space to allow verification, or otherwise, of these   \n93 parameters a maximum of 4 hidden layers and 1000 neurons in each layer are used. The majority   \n94 of the literature treats the problem as time-invariant and use feed-forward networks, so no recurrent   \n95 parameters are optimised. As the optimiser or activation functions are rarely documented in the   \n96 literature, the state-of-the-art optimisers and activation functions available in the Keras framework   \n97 (Chollet et al. 2015) are used in the optimisation, Table 1.   \n98 The number of epochs and early stopping procedure are not optimised, as there was a need for   \n99 predictable compute requirements and allowing the optimisation of these parameters leads to unpre  \n100 dictable run times. The number of epochs to train each network increases for increasing generation   \n101 number in the genetic algorithm, from 1 epoch in the first 15 generations to 20 in the final 15. This   \n102 was also implemented to reduce compute and it was validated that when more than 20 epochs were   \n103 allowed, that the early stopping, with a patience of 5, stopped the training within 20 epochs for the   \n104 majority of networks. The loss function is similarly not optimised, the Mean Absolute Error is used,   \n105 as the conditional medians are closer to the ground truth input-output relationships in these datasets   \n106 than the conditional means.   \n107 The performance measures, or the genetic algorithm’s fitness functions, are the Mean Absolute   \n108 Relative Error, the Maximum Absolute Relative Error and the Mean Fit to Median Error. Different   \n109 combinations of these, alongside the use of regularisation parameters in the search space are compared   \n110 to illustrate the effect of different types of regularisation. ",
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+    {
+        "type": "table",
+        "img_path": "images/bec442d053bf5530975461399e564e2e54333194b4dc43126f84c6d451b21283.jpg",
+        "table_caption": [
+            "Table 1: Selected Neural Network Hyperparameters "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=1 colspan=1>Hyperparameter</td><td rowspan=1 colspan=1>Value or set</td></tr><tr><td rowspan=1 colspan=1>Layers</td><td rowspan=1 colspan=1>[1,4]</td></tr><tr><td rowspan=1 colspan=1>Neurons in each layer</td><td rowspan=1 colspan=1>[1,1000]</td></tr><tr><td rowspan=1 colspan=1>Epochs</td><td rowspan=1 colspan=1>Increasing from 1-2O for increasing generations</td></tr><tr><td rowspan=1 colspan=1>Early stopping patience</td><td rowspan=1 colspan=1>5</td></tr><tr><td rowspan=1 colspan=1>Loss function</td><td rowspan=1 colspan=1>Mean Absolute Error</td></tr><tr><td rowspan=1 colspan=1>Performance measures</td><td rowspan=1 colspan=1>Mean Absolute Relative Error,Maximum Absolute Relative Error,Mean Fit to Median Error</td></tr><tr><td rowspan=1 colspan=1>Optimiser</td><td rowspan=1 colspan=1>SGD, Adam (Kingma &amp; Ba 2014), Nadam (Dozat 2016),RMSprop (Hinton et al. 2012), Adagrad (Duchi et al. 2011),Adadelta (Zeiler 2012),Adamax (Kingma &amp; Ba 2014)</td></tr><tr><td rowspan=1 colspan=1>Activation function</td><td rowspan=1 colspan=1>ReLU, sigmoid, softmax,softplus, softsign, tanh, selu, elu</td></tr><tr><td rowspan=1 colspan=1>11 &amp; 12 Rates</td><td rowspan=1 colspan=1>0,0.01,0.001,0.0001,0.00001</td></tr><tr><td rowspan=1 colspan=1>Dropout</td><td rowspan=1 colspan=1>[0,0.9)</td></tr><tr><td rowspan=1 colspan=1>Initialiser</td><td rowspan=1 colspan=1>Random Normal (μ = 0,σ = 0.1)</td></tr></table>",
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+    {
+        "type": "text",
+        "text": "111 3 cMLSGA Parameters ",
+        "text_level": 1,
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+    },
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+        "img_path": "images/70f832039440aba298790e687ce9daee8d5fd02ed4fc885baa41744b73d341b4.jpg",
+        "table_caption": [
+            "Table 2: Selected cMLSGA Hyperparameters "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=1 colspan=1>Hyperparameter</td><td rowspan=1 colspan=1>Value or set</td></tr><tr><td rowspan=1 colspan=1>Algorithm at Individual Level</td><td rowspan=1 colspan=1>HEIA,IBEA</td></tr><tr><td rowspan=1 colspan=1>Crossover Type&amp;Rate</td><td rowspan=1 colspan=1>SBX&amp;DE,1</td></tr><tr><td rowspan=1 colspan=1>Mutation Type&amp; Rate</td><td rowspan=1 colspan=1>Polynomial, 0.08</td></tr><tr><td rowspan=1 colspan=1>Number of eliminated collectives</td><td rowspan=1 colspan=1>1</td></tr><tr><td rowspan=1 colspan=1>Generations between elimination</td><td rowspan=1 colspan=1>10</td></tr><tr><td rowspan=1 colspan=1>Population size</td><td rowspan=1 colspan=1>1000</td></tr><tr><td rowspan=1 colspan=1>Generations</td><td rowspan=1 colspan=1>300</td></tr><tr><td rowspan=1 colspan=1>Proportion elite</td><td rowspan=1 colspan=1>10%</td></tr></table>",
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+        "page_idx": 2
+    },
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+        "text": "112 In this study cMLSGA1 is selected as it shows the top performance on a range of evolutionary   \n113 benchmarking problems (Grudniewski & Sobey 2021) and practical problems (Grudniewski & Sobey   \n114 2019). Genetic algorithms are increasing used to tune neural network hyperparameters including   \n115 regularisation parameters for use on new problems (Jin et al. 2004). Many approaches have multiple   \n116 genetic algorithm objectives, although these all minimise an error measure and a measure of network   \n117 complexity (Wang et al. 2019) and (Smith & Jin 2014). The use of multiple different performance   \n118 measures as objectives is yet to be explored in the literature.   \n119 Four approaches are investigated in this study, summarised in Table 3, for approach (GAi) and   \n120 (GAii) the genetic algorithm cMLSGA optimises all variables in Table 2, including the l1 and l2   \n121 regularisation rate and the dropout rate of the networks. Although it is advised that l2 regularisation   \n122 and dropout are not used in the same network the genetic algorithms are provided with zero options   \n123 for all regularisation parameters, to identify if one is preferable in this scenario.   \n124 Approach (GAi) is a single objective genetic algorithm optimising the Mean Absolute Error which is   \n125 compared to a multi-objective formulation where the (GAii) approach optimises both Mean Absolute   \n126 Error and Maximum Absolute Error. For approaches (GAiii) and (GAiv) no network regularisation   \n127 parameters are optimised: l1, l2 and dropout rates are all set permanently to zero. They avoid   \n128 producing networks that have overfitted by the use of two performance metrics as multi-objectives,   \n129 (GAiii) uses the Mean Fit to Median and Mean Absolute Errors to be minimised and (GAiv) uses the   \n130 Maximum Absolute and Mean Absolute. All approaches use 40 CPUs with $2 . 0 \\mathrm { G H z }$ Intel Skylake   \n131 processors and 192 GB of DDR4 memory, and take less than 3 days, this setup may not be feasible   \n132 for widespread industrial application, although it is suggested it is within reach of some industries. ",
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+            "Table 3: Genetic Algorithm Approaches "
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+        "table_body": "<table><tr><td rowspan=1 colspan=1>Approach</td><td rowspan=1 colspan=1>Objective(s)</td><td rowspan=1 colspan=1>Network Regularisation</td></tr><tr><td rowspan=1 colspan=1>GAi</td><td rowspan=1 colspan=1>Mean Absolute Error</td><td rowspan=1 colspan=1>11,12 and dropout</td></tr><tr><td rowspan=1 colspan=1>GAii</td><td rowspan=1 colspan=1>MeanAbsoluteErrorMaximum Absolute Error</td><td rowspan=1 colspan=1>11,12 and dropout</td></tr><tr><td rowspan=1 colspan=1>GAiii</td><td rowspan=1 colspan=1>MeanFitto MedianErrorMean Absolute Error</td><td rowspan=1 colspan=1>None</td></tr><tr><td rowspan=1 colspan=1>GAiv</td><td rowspan=1 colspan=1>Mean Absolute ErrorMaximum Absolute Error</td><td rowspan=1 colspan=1>None</td></tr></table>",
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+        "type": "text",
+        "text": "33 4 Data ",
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+        "text": "134 The data used in this study are from a large vessel equipped with the Silverstream $\\textsuperscript { \\textregistered }$ Air Lubrication   \n135 System. The air lubrication system works through use of fluid sheering to create an air microbubble   \n136 carpet directly captured within the boundary layer on the ship hull bottom. The bubble carpet reduces   \n137 the frictional resistance thereby increasing the speed and reducing the shaft power. Compressors   \n138 provide a constant supply of air to the hull bottom to maintain a uniform bubble carpet operated at   \n139 the optimal compressor power that maximises the energy balance. The study is performed on both   \n140 system on and system off datasets, however for brevity only results for system off are presented as   \n141 they show similar performance.This prediction is required for a baseline determination of how the   \n142 system is working, but the relationships between the power, weather, ocean and operating conditions   \n143 are complex and difficult to model. ",
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+        "text": "The variables considered in this study are the shaft power, speed through water, relative wind speed and direction, draught and trim, with shaft power the target variable. These are selected based on a detailed study into variable selection for shaft power prediction (Parkes et al. 2019). The speed through water is selected over the speed over ground, for use as an input variable, as it is more hydrodymanically relevant and its accuracy is validated by comparison to the speed over ground. The dataset is cleaned by removing rows with missing or non-physical values and all datapoints below 0.05 normalised shaft power are removed. The dataset is split into two using the air lubrication system status: system on and system off, where system on is defined as air lubrication system power greater than zero. The system on dataset contains 352,690 datapoints and system off contains 237,962. The data is split into training, testing and validation sets of $70 \\%$ , $15 \\%$ and $15 \\%$ respectively. Each ",
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+        "image_caption": [
+            "Figure 1: The distribution of the observed shaft powers for half knot bins of speed through the water for dataset where the system is off. In the box and whisker plots the boxes contain $50 \\%$ of the distribution and the whiskers extend to the datum which is at 1.5 times the interquartile range. "
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+        "text": "154 network in the genetic algorithm trains on a randomly sampled 35,000 datapoints from the training   \n155 set and uses randomly sampled sets of size 7,500 from validation and testing sets for validation during   \n156 training, and testing to produce the fitness of the network for the genetic algorithm. The errors stated   \n157 in the paper are from networks on the Pareto fronts of each approach, which are validated on the full   \n158 testing set.   \n159 The datasets contain large regions of sparse data in all input variable domains, this is exemplified   \n160 by the ship speed domain where each half-knot interval below 16 knots contains less than $0 . 8 \\%$ of   \n161 the data, which accounts for more than half the speed domain, Figure 1. In addition, the boxplot   \n162 ranges and outliers show high heteroscedicity with idiosyncratic noise caused by situations where the   \n163 angle of the propeller blades is varied to achieve the required speed. This highlights the complexity   \n164 in developing models of the powering of this vessel, as the dataset also contains the effects from other   \n165 latent variables, such as piloting behaviour and route taken. ",
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+        "text": "66 5 Optimisation including regularisation parameters: (GAi) and (GAii) ",
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+        "text": "167 Previous studies predicting ship powering using neural networks report that l1, l2 and elastic net   \n168 increase both test set and off-test set errors and that optimal values for both l1 and l2 are zero.   \n169 Therefore the genetic algorithm setup is biased towards low and zero values of regularisation rates by   \n170 using a set of exponentially decreasing values and an explicit zero option.   \n171 The single objective (GAi) fails to identify that zero regularisation rates produce the lowest errors,   \n172 favouring networks with the highest possible rate of l2 (0.01), Figure 2b. (GAi) produces networks   \n173 with the highest Mean Absolute Relative Errors of all the approaches, $( 5 . 1 9 \\pm 0 . 0 0 ) \\%$ from Figure   \n174 4a. In contrast, (GAii) favours lower l1 and l2 rates of 0 or 0.00001, Figures 2c and 2d, which results   \n175 in networks with the lowest Mean Absolute Relative Errors of all four approaches, on average, with a   \n176 value of $( 2 . 8 7 \\pm 0 . 4 5 ) \\%$ , shown in Figure 4a. This is around $0 . 5 \\%$ higher than the lowest documented   \n177 error for ship power prediction.   \n178 It is posited that the high error for the single objective problem is directly related to the use of a   \n179 large l2 regularisation rate, as noted in previous studies for ship power prediction. It is possible   \n180 that the use of a multi-objective search algorithm for a single-objective problem means that the   \n181 optimal hyperparameters can’t be found, resulting in large errors. The implementation also requires   \n182 restrictions in the number of epochs used for training in the initial generations, it is possible this   \n183 biases (GAi) towards certain size networks, where higher l2 rates are preferable. This hypothesis is   \n184 supported by the fact that $7 4 . 4 \\%$ of networks in the first 15 generations of (GAi) have 1 hidden layer,   \n185 and that over $9 9 . 8 \\%$ of the networks in the final 15 generations have 1 hidden layer, with $8 8 0 \\pm 1 7$   \n186 neurons in this layer, Figure 3b. This is significantly more neurons than those in the hidden layer of   \n187 networks in the final 15 generations of (GAii) which range from 3-952 with a median value of 709,   \n188 Figure 3b. The added objective of minimising Maximum Absolute Error in (GAii) may cause these   \n189 slightly smaller networks to be more attractive as they are in a sense regularised by their size, as they   \n190 have reduced modelling flexibility therefore are less likely to overfit and produce high Maximum   \n191 Absolute Errors.   \n192 Another explanation for the difference in l2 rates chosen by (GAi) and (GAii) is the equivalence   \n193 of l2 and dropout. Since l2 and dropout are equivalent up to a Fisher transformation, their use in   \n194 conjunction is not recommended. The evidence for this is that (GAi) favours the highest l2 rate and   \n195 has a median dropout rate in the final 15 generations of 0.116, whereas (GAii) favours the zero l2 rate   \n196 and has a median dropout rate of 0.624, Figure 3a. This illustrates that the genetic algorithms will   \n197 chose either l2 or dropout to minimise the Mean Absolute Relative Error. The l1 rates also support   \n198 this hypothesis, as chosen rates for l1 regularisation in the final 15 generations are more comparable   \n199 for (GAi) and (GAii).   \n201 For approaches (GAiii) and (GAiv) all neural network regularisation parameters are set to zero.   \n202 The regularisation is performed by minimising different network performance measures, the Mean   \n203 Absolute and Mean Fit to Median for (GAiii), and the Mean Absolute and Maximum Absolute for   \n204 (GAiv). The trade-off between the two objectives produces regularised neural networks, without   \n205 explicitly changing the architecture or loss function. The Mean Fit to Median is chosen as it indicates   \n206 how close the relationships modelled by a network are to the conditional averages of the dataset, in   \n207 many regression examples this is akin to the ground truth input-output relationships (Parkes et al.   \n208 2021). The Maximum Absolute is chosen as for many industrial applications of machine learning the   \n209 maximum prediction error is more pertinent than the mean error. The Mean Absolute Error is used   \n210 instead of the Mean Squared Error in both approaches, as the conditional medians are closer to the   \n211 ground truth input-output relationships in these datasets than the conditional means.   \n212 Differently shaped networks are favoured by (GAiii) and (GAiv), compared to (GAi) and (GAii),   \n213 focusing on networks with 3 hidden layers and on average less than 400 neurons in each layer, Figure   \n214 3b. These networks have 51 times the number of connections than the networks chosen in (GAi) and   \n215 (GAii). Apart from (GAi), (GAiii) has the most consistently sized networks in the final 15 generations,   \n216 with an interquartile range of 46 neurons, compared to (GAiv) which have an interquartile range of   \n217 131 neurons. It is suggested that as the Mean Fit to Median Error biases networks towards specific   \n218 input-output relationships, there is a smaller range of potential network architectures which habitually   \n219 model these relationships. Whereas networks which minimise the Maximum Absolute Error are less   \n220 restricted and can model a wider range of input and output relationships.   \n221 The Mean Absolute Relative Errors from networks in the Pareto fronts are $( 2 . 9 7 \\pm 0 . 2 5 ) \\%$ for (GAiii)   \n222 and $( 3 . 1 0 \\pm 0 . 2 8 ) \\%$ for (GAiv). It is expected that (GAiv) would produce higher Mean Absolute   \n223 Relative Errors as discussed above, minimising the Maximum Absolute Error should bias predictions   \n224 towards the midpoint of the conditional output distributions, whereas minimising the Mean Absolute   \n225 Error should bias predictions towards the median of these distributions. As it is established that noise   \n226 in the output distribution is non-Gaussian, Figure 1, these values will not align so some sacrifice   \n227 in Mean Absolute Error is expected from (GAiv). Both (GAiii) and (GAiv) produce comparable   \n228 Maximum Absolute Errors, of $( 4 9 . 5 \\pm 1 . 1 ) \\%$ and $( 5 1 . 9 \\pm 4 . 5 ) \\%$ . It is suggested that this is because,   \n229 although the conditional median output value and conditional midpoint output value do not align   \n230 for the majority of the input domain, they are sufficiently close to produce comparable Maximum   \n231 Absolute Errors.   \n232 Across all four approaches, the genetic algorithm producing networks with the highest Mean Absolute   \n233 Error is the approach which does not provide extra weighting to sparse areas of data. The approaches   \n234 minimising Maximum Absolute Error are implicitly biased away from networks which predict the   \n235 majority of the testing datapoints correctly, but predict one datapoint poorly, favouring networks   \n236 which predict all testing datapoints to a moderate degree of error. Approach (GAiii) more explicitly   \n237 weights prediction in sparse areas of data by favouring networks which model the conditional median   \n238 of the dataset across all input domains, irrespective of the quantity of data across each input domain.   \n239 The regression problem of ship power prediction is chosen in part because of it’s irregular data   \n240 distribution; more than $9 \\%$ of the dataset lies in less than a 0.5 knot interval of ship speed, Figure 1.   \n241 This provides an explanation for the high testing errors from (GAi), where only the Mean Absolute   \n242 Error is minimised, there is little incentive for the genetic algorithm to produce networks which   \n243 generalise across the full range of the input domain well. ",
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+        "image_caption": [
+            "Figure 2: Distribution of regularisation rates for networks in the last 15 generations of (GAi) cMLSGA with multi-objectives of minimising Maximum and Mean Absolute Error for (a) l1 and (b) l2 and (GAii) cMLSGA with the single objective of minimising Mean Absolute Error for (c) l1 and (d) l2 "
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+            "Figure 3: (a) Dropout rate for networks in the last 15 generations of cMLSGA with (GAi) the single objective of minimising Mean Absolute Error and (GAii) multi-objectives of minimising Maximum and Mean Absolute Error and (b) the number of neurons in each layer for networks in the last 15 generations of cMLSGA with (GAi), (GAii),(GAiii) and (GAiv). "
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+            "Figure 4: Mean Relative Absolute Error (a) and Maximum Absolute Error (b) from cMLSGA with (GAi) the single objective of minimising Mean Absolute Error and (GAii) multi-objectives of minimising Maximum and Mean Absolute Error, both optimising the parameters for l1, l2 regularisation and dropout in the networks, and (GAiii) and (GAiv) which do not use network regularisation but minimise Mean Fit to Median and Maximum Absolute Error respectively, alongside Mean Absolute Error "
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+    {
+        "type": "text",
+        "text": "244 7 Comparison of the interpretability ",
+        "text_level": 1,
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+        "text": "245 To asses the interpretability of the networks selected by the four different approaches the learnt   \n246 relationship between an input, the ship speed, and the output, shaft power, for the networks in   \n247 the Pareto front of each approach are visualised, Figure 5. These are extracted with the following   \n248 procedure: set all but one input variable to be constant at the mode; cycle the remaining variable from   \n249 its minimum to its maximum recorded values with 150 points evenly spaced along the domain and   \n250 run the new dataset through the trained network.   \n251 The approach which produces the most consistent speed-power relationships is (GAi), with an average   \n252 variation of $1 . 8 \\% ^ { 2 }$ , Figure 5a. However, the relationship modelled by the 5 networks with the lowest   \n253 Mean Absolute Relative Error in (GAi) all approximate a piece-wise linear relationship which clearly   \n254 underfits the dataset in Figure 1. The expected trend between ship speed through the water and shaft   \n255 power is a cubic polynomial, therefore as well as producing the highest Mean Absolute Relative   \n256 Errors, networks chosen by (GAi) model the ground truth input-output relationships the worst out of   \n257 the four approaches. Both (GAii) and (GAiv) produce 5 fairly consistent speed-power curves, with   \n258 average variations of $5 . 9 \\%$ and $10 \\%$ respectively, Figures 5b and 5d. Both approaches approximate   \n259 smooth polynomial curves, although the degrees of the polynomials might differ, as multiple curves   \n260 intersect at various points along the speed axis. The spread of learnt relationships is greater at the   \n261 highest and lowest speeds for (GAii), with a decrease in spread for speeds of around 15 knots, where   \n262 many of the curves intersect. The curves from (GAiv) show equal spread across the speed domain.   \n263 The approach with both accurate and consistent learnt speed-power curves is (GAiii), with limited   \n264 intersections of curves and an average spread of $3 . 0 \\%$ . It is suggested that the reason using the   \n265 Mean Fit to Median Error as an objective in a multi-objective genetic algorithm produces more   \n266 interpretable results, or more consistent learnt relationships, is because instead of encouraging the   \n267 networks to model more simple relationships. It encourages the networks to model the conditional   \n268 median functions of the dataset, supported by the increase in network connections. Whereas the other   \n269 approaches leave room for networks to fail to model the conditional averages, especially in irregularly   \n270 distributed and non-normally distributed datasets. The Mean Absolute Error values from networks   \n271 selected by (GAiii) are on average $0 . 1 \\%$ higher than those from (GAii), and the Maximum Absolute   \n272 Error values are $1 . 6 \\%$ higher.   \n273 A limitation of the approach is that the Fit to Median Error measure will likely perform best at   \n274 improving interpretability on datasets which violate the assumptions in Bishop (1995); the ship   \n275 powering example is chosen to illustrate this as it provides a clearly heteroscedastic dataset. For   \n276 applications where noise profiles are Gaussian, and there is no effect from latent or interrelated input   \n277 variables, the Fit to Median Error will not improve interpretability, but will perform the same as   \n278 conventional Minkowski-r metrics, either Mean Squared or Mean Absolute Error depending on the   \n279 convexity of input-output relationships.   \n280 Interestingly, the approach producing the lowest Mean Absolute and Maximum Absolute Errors   \n281 does not model the ground truth the most accurately. This creates a potential for negative societal   \n282 impacts, as the standard performance metrics for regression neural networks do not provide a full   \n283 picture of performance or expected behaviour. Interpretability of trained methods is essential for   \n284 safe application of machine learning in the real world, especially when automated methods are   \n285 used to replace experienced professionals. (GAii) demonstrates the same accuracy of approach as   \n286 those with standard network regularisation, but with a better fit to the ground truth. This approach   \n287 bypasses the need to use, and therefore to optimise the parameters of the regularisation methods.   \n288 If evolutionary computation is already being used to optimise network parameters, then compute   \n289 is saved by removing the network regularisation parameters l1, l2 and dropout. (GAiii) completed   \n290 300 generations in 46hours whereas (GAii) required 12 hours more computation to complete 300   \n291 generations. ",
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+            "Figure 5: The learnt speed-power curves from 5 networks on the Pareto fronts of (GAii), (GAiii), (GAiv) and the 5 networks producing lowest Mean Absolute Relative Error from (GAi). "
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+        "type": "text",
+        "text": "92 8 Conclusion ",
+        "text_level": 1,
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+        "type": "text",
+        "text": "293   \n294   \n295   \n296   \n297   \n298   \n299   \n300   \n301   \n302   \n303 ",
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+        "text": "Interpretable and accurate methods are required for widespread application of machine learning to real-world regression problems. To automate the training of neural networks so that the result is interpretable, three different genetic algorithm approaches are compared: one to minimise the Maximum Absolute Error of the networks, which includes standard regularization using l1, l2 and dropout; and two which do not use any network regularisation, one minimising the Mean Fit to Median and one to minimise the Maximum Absolute Error. The results show that all three approaches give similar Mean Absolute Errors from networks on their Pareto fronts, from $2 . 9 \\%$ for the approach with regularisation to $3 . 1 \\%$ for the approach minimising Maximum Absolute Error. However, the Mean Fit to the Median approach shows a considerably better interpretability, or fit to the ground truth, with a spread in predicted input-output curves of $3 \\%$ compared to a spread of $6 \\%$ for the approach using regularisation and $10 \\%$ when minimising the Maximum Absolute Error. ",
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+        "type": "text",
+        "text": "304 References ",
+        "text_level": 1,
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+        "text": "Battaglia, P. W., Hamrick, J. B., Bapst, V., Sanchez-Gonzalez, A., Zambaldi, V., Malinowski, M., Tacchetti, A., Raposo, D., Santoro, A., Faulkner, R., Gulcehre, C., Song, F., Ballard, A., Gilmer, J., Dahl, G., Vaswani, A., Allen, K., Nash, C., Langston, V., Dyer, C., Heess, N., Wierstra, D., ",
+        "bbox": [
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+        "type": "text",
+        "text": "308 Kohli, P., Botvinick, M., Vinyals, O., Li, Y. & Pascanu, R. (2018), ‘Relational inductive biases,   \n309 deep learning, and graph networks’.   \n310 Bishop, C. (1995), Neural Networks for Pattern Recognition, Oxford University Press, chapter 6,   \n311 pp. 194–225.   \n312 Chollet, F. et al. (2015), ‘Keras’, https://keras.io.   \n313 Dozat, T. (2016), ‘Incorporating nesterov momentum into adam’, International Conference on   \n314 Learning Representations 2016 .   \n315 Duchi, J., Hazan, E. & Singer, Y. (2011), ‘Adaptive subgradient methods for online learning and   \n316 stochastic optimization’, Journal of Machine Learning Research 12(61), 2121–2159.   \n317 Grudniewski, P. A. & Sobey, A. J. (2019), Do general genetic algorithms provide benefits when   \n318 solving real problems?, in ‘2019 IEEE Congress on Evolutionary Computation (CEC)’, IEEE,   \n319 pp. 1822–1829.   \n320 Grudniewski, P. A. & Sobey, A. J. (2021), ‘cMLSGA: a co-evolutionary multi-level selection genetic   \n321 algorithm for multi-objective optimization’.   \n322 Hanson, S. & Burr, D. (1987), Minkowski-r back-propagation: learning in connectionist models with   \n323 non-euclidian error signals., in ‘Neural Information Processing Systems (NIPS 1987)’.   \nHinton, G., Srivastava, N. & Swersky, K. (2012), ‘Lecture 6a:overview of mini-batch gradient   \ndescent’.   \nURL: http://www.cs.toronto.edu/ tijmen/csc321/slides/lectureslideslec6.pdf   \n324 Jeon, M., Noh, Y., Shin, Y., Lim, O., Lee, I. & Cho, D. (2018), ‘Prediction of ship fuel consumption   \n325 by using an artificial neural network’, Journal of Mechanical Science and Technology 32(12), 5785–   \n326 5796.   \n327 Jin, Y., Okabe, T. & Sendhoff, B. (2004), Neural network regularization and ensembling using   \n328 multi-objective evolutionary algorithms, in ‘Proceedings of the 2004 Congress on Evolutionary   \n329 Computation (IEEE Cat. No.04TH8753)’, Vol. 1.   \n330 Kingma, D. P. & Ba, J. (2014), ‘Adam: A method for stochastic optimization’, arXiv preprint   \n331 arXiv:1412.6980 .   \n332 Kumar, P., Batra, S. & Raman, B. (2021), ‘Deep neural network hyper-parameter tuning through   \n333 twofold genetic approach’, Soft Computing pp. 1–25.   \n334 Le, L., Lee, G., Park, K. & Kim, H. (2020), ‘Neural network-based fuel consumption estimation for   \n335 container ships in korea’, Maritime Policy & Management pp. 1–18.   \n336 Leifsson, L., Sævarsdóttir, H., Sigurðsson, S. & Vésteinsson, A. (2008), ‘Grey-box modeling of an   \n337 ocean vessel for operational optimization’, Simulation Modelling Practice and Theory 16(8), 923–   \n338 932.   \n339 Liang, Q., Tvete, H. A. & Brinks, H. W. (2019), Prediction of vessel propulsion power using machine   \n340 learning on ais data, ship performance measurements and weather data, in ‘Journal of Physics:   \n341 Conference Series’, Vol. 1357, p. 012038.   \n342 Luketina, J., Berglund, M., Greff, K. & Raiko, T. (2016), ‘Scalable gradient-based tuning of continu  \n343 ous regularization hyperparameters’.   \n344 Nowlan, S. J. & Hinton, G. E. (1992), ‘Simplifying neural networks by soft weight sharing’, Neural   \n345 Computation .   \n346 Parkes, A. I., Savasta, T. D., Sobey, A. J. & Hudson, D. A. (2019), Efficient vessel power prediction   \n347 in operational conditions using machine learning., in ‘Practical Design of Ships and Other Floating   \n348 Structures(PRADS), September 2019, Yokohama, Japan’.   \n349 Parkes, A. I., Sobey, A. J. & Hudson, D. A. (2018), ‘Physics-based shaft power prediction for large   \n350 merchant ships using neural networks’, Ocean Engineering 166, 92–104. ",
+        "bbox": [
+            137,
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+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "Parkes, A. I., Sobey, A. J. & Hudson, D. A. (2021), ‘Towards error measures which influence a learners inductive bias to the ground truth’.   \nPedersen, B. P. & Larsen, J. (2009), Prediction of full-scale propulsion power using artificial neural networks, in ‘Proceedings of the 8th international conference on computer and IT applications in the maritime industries (COMPIT’09), Budapest, Hungary May’, pp. 10–12.   \nPetersen, J. P., Jacobsen, D. J. & Winther, O. (2012), ‘Statistical modelling for ship propulsion efficiency’, Journal of marine science and technology 17(1), 30–39.   \nSmith, C. & Jin, Y. (2014), ‘Evolutionary multi-objective generation of recurrent neural network ensembles for time series prediction’, Neurocomputing 143, 302–311.   \nSrivastava, N., Hinton, G., Krizhevsky, A., Sutskever, I. & Salakhutdinov, R. (2014), ‘Dropout: a simple way to prevent neural networks from overfitting’, The Journal of Machine Learning Research 15(1), 1929–1958.   \nTani, L., Rand, D., Veelken, C. & Kadastik, M. (2021), ‘Evolutionary algorithms for hyperparameter optimization in machine learning for application in high energy physics’, The European Physical Journal C 81(2), 1–9.   \nWager, S., Wang, S. & Liang, P. (2013), ‘Dropout training as adaptive regularization’, arXiv preprint arXiv:1307.1493 .   \nWang, B., Sun, Y., Xue, B. & Zhang, M. (2019), Evolving deep neural networks by multi-objective particle swarm optimization for image classification, in ‘Proceedings of the Genetic and Evolutionary Computation Conference’, GECCO ’19, p. 490–498.   \nWillard, J., Jia, X., Xu, S., Steinbach, M. & Kumar, V. (2020), ‘Integrating physics-based modeling with machine learning: a survey’.   \nYang, S., Tian, Y., He, C., Zhang, X., Tan, K. C. & Jin, Y. (2021), ‘A gradient-guided evolutionary approach to training deep neural networks’, IEEE Transactions on Neural Networks and Learning Systems pp. 1–15. ",
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+        "text": "376 Zeiler, M. D. (2012), ‘Adadelta: an adaptive learning rate method’. ",
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+        "text": "The checklist follows the references. Please read the checklist guidelines carefully for information on how to answer these questions. For each question, change the default [TODO] to [Yes] , [No] , or [N/A] . You are strongly encouraged to include a justification to your answer, either by referencing the appropriate section of your paper or providing a brief inline description. For example: ",
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+        "text": "• Did you include the license to the code and datasets? [Yes] See Section ??.   \n• Did you include the license to the code and datasets? [No] The code and the data are proprietary.   \n• Did you include the license to the code and datasets? [N/A] ",
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+        "text": "Please do not modify the questions and only use the provided macros for your answers. Note that the Checklist section does not count towards the page limit. In your paper, please delete this instructions block and only keep the Checklist section heading above along with the questions/answers below. ",
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+        "text": "(a) Do the main claims made in the abstract and introduction accurately reflect the paper’s contributions and scope? [Yes] The scope of the paper (regression problem automation) is made clear in the abstract. The claims of improved interpretability using the Fit to Median Error measure are reflected in Figure 5, as well as the body of text.   \n(b) Did you describe the limitations of your work? [Yes] The limitations of the Fit to Median are discussed in Section 7, and the limitations relating to compute are discussed in Section 3.   \n(c) Did you discuss any potential negative societal impacts of your work? [Yes] See Section 6   \n(d) Have you read the ethics review guidelines and ensured that your paper conforms to them? [Yes] ",
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+        "text": "(a) Did you state the full set of assumptions of all theoretical results? [No] Citation of (Bishop 1995) is included instead of explicit statement of all assumptions, although the most pertinent assumptions are discussed in Section 1   \n(b) Did you include complete proofs of all theoretical results? [N/A] No theorems are posited, the paper explores a practical comparison of common methods. ",
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+        "text": "(a) Did you include the code, data, and instructions needed to reproduce the main experimental results (either in the supplemental material or as a URL)? [Yes] Instructions needed to replicate the results on a different dataset are provided in the main text; Sections 3 and 2 specify the expermental setup. The code is based on the cMLSGA algorithm, available at https://www.bitbucket. $\\sigma \\ \\mu \\mu \\mu \\mu \\mu \\mu \\mu$ (redacted for anonymity) which is provided in Section 3. The data is proprietary so is not provided.   \n(b) Did you specify all the training details (e.g., data splits, hyperparameters, how they were chosen)? [Yes] See Sections 3 and 2   \n(c) Did you report error bars (e.g., with respect to the random seed after running experiments multiple times)? [Yes] All results are reported to plus/minus the standard deviation, and all figures show multiple runs with respect to either a random seed or an evolutionary method.   \n(d) Did you include the total amount of compute and the type of resources used (e.g., type of GPUs, internal cluster, or cloud provider)? [Yes] See Section 3 ",
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+        "text": "(a) If your work uses existing assets, did you cite the creators? [Yes] The creators of the genetic algorithim cMLSGA are cited in Section 3 and the data is acknowledged to belong to Silverstream Technologies Ltd in Section 4   \n(b) Did you mention the license of the assets? [Yes] The code is licensed under the GNU General Public License. The data is not licensed.   \n(c) Did you include any new assets either in the supplemental material or as a URL? [No]   \n(d) Did you discuss whether and how consent was obtained from people whose data you’re using/curating? [No] The work is performed in collaboration with Silverstream Techonologies Ltd.   \n(e) Did you discuss whether the data you are using/curating contains personally identifiable information or offensive content? [No] The shaft power variable is normalised to remove any possibility of identifying the vessel in question. ",
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