1 Introduction
Neural networks have become the workhorse nonlinear function approximator in a number of machine learning domains, most notably leading to significant advances in reinforcement learning (RL) and supervised learning. However, it is often overlooked that the success of a particular neural network model depends upon the joint tuning of the model structure, the data presented, and the details of how the model is optimised. Each of these components of a learning framework is controlled by a number of parameters, hyperparameters, which influence the learning process and must be properly tuned to fully unlock the network performance. This essential tuning process is computationally expensive, and as neural network systems become more complicated and endowed with even more hyperparameters, the burden of this search process becomes increasingly heavy. Furthermore, there is yet another level of complexity in scenarios such as deep reinforcement learning, where the learning problem itself can be highly nonstationary (e.g. dependent on which parts of an environment an agent is currently able to explore). As a consequence, it might be the case that the ideal hyperparameters for such learning problems are themselves highly nonstationary, and should vary in a way that precludes setting their schedule in advance.
Two common tracks for the tuning of hyperparameters exist: parallel search and sequential optimisation, which tradeoff concurrently used computational resources with the time required to achieve optimal results. Parallel search performs many parallel optimisation processes (by optimisation process we refer to neural network training runs), each with different hyperparameters, with a view to finding a single best output from one of the optimisation processes – examples of this are grid search and random search. Sequential optimisation performs few optimisation processes in parallel, but does so many times sequentially, to gradually perform hyperparameter optimisation using information obtained from earlier training runs to inform later ones – examples of this are hand tuning and Bayesian optimisation. Sequential optimisation will in general provide the best solutions, but requires multiple sequential training runs, which is often unfeasible for lengthy optimisation processes.
In this work, we present a simple method, Population Based Training (PBT) which bridges and extends parallel search methods and sequential optimisation methods. Advantageously, our proposal has a wallclock run time that is no greater than that of a single optimisation process, does not require sequential runs, and is also able to use fewer computational resources than naive search methods such as random or grid search. Our approach leverages information sharing across a population of concurrently running optimisation processes, and allows for online propagation/transfer of parameters and hyperparameters between members of the population based on their performance. Furthermore, unlike most other adaptation schemes, our method is capable of performing online adaptation of hyperparameters – which can be particularly important in problems with highly nonstationary learning dynamics, such as reinforcement learning settings. PBT is decentralised and asynchronous, requiring minimal overhead and infrastructure. While inherently greedy, we show that this metaoptimisation process results in effective and automatic tuning of hyperparameters, allowing them to be adaptive throughout training. In addition, the model selection and propagation process ensures that intermediate good models are given more computational resources, and are used as a basis of further optimisation and hyperparameter search.
We apply PBT to a diverse set of problems and domains to demonstrate its effectiveness and wide applicability. Firstly, we look at the problem of deep reinforcement learning, showing how PBT can be used to optimise UNREAL (Jaderberg et al., 2016) on DeepMind Lab levels (Beattie et al., 2016), Feudal Networks (Vezhnevets et al., 2017) on Atari games (Bellemare et al., 2013), and simple A3C agents for StarCraft II (Vinyals et al., 2017)
. In all three cases we show faster learning and higher performance across a suite of tasks, with PBT allowing discovery of new stateoftheart performance and behaviour, as well as the potential to reduce the computational resources for training. Secondly, we look at its use in supervised learning for machine translation on WMT 2014 EnglishtoGerman with Transformer networks
(Vaswani et al., 2017), showing that PBT can match and even outperform heavily tuned hyperparameter schedules by optimising for BLEU score directly. Finally, we apply PBT to the training of Generative Adversarial Networks (GANs) (Goodfellow et al., 2014) by optimising for the Inception score (Salimans et al., 2016) – the result is stable GAN training with large performance improvements over a strong, welltuned baseline with the same architecture (Radford et al., 2016).The improvements we show empirically are the result of (a) automatic selection of hyperparameters during training, (b) online model selection to maximise the use of computation spent on promising models, and (c) the ability for online adaptation of hyperparameters to enable nonstationary training regimes and the discovery of complex hyperparameter schedules.
In Sect. 2 we review related work on parallel and sequential hyperparameter optimisation techniques, as well as evolutionary optimisation methods which bear a resemblance to PBT. We introduce PBT in Sect. 3, outlining the algorithm in a general manner which is used as a basis for experiments. The specific incarnations of PBT and the results of experiments in different domains are given in Sect. 4, and finally we conclude in Sect. 5.
2 Related Work
First, we review the methods for sequential optimisation and parallel search for hyperparameter optimisation that go beyond grid search, random search, and hand tuning. Next, we note that PBT inherits many ideas from genetic algorithms, and so highlight the literature in this space.
The majority of automatic hyperparameter tuning mechanisms are sequential optimisation methods: the result of each training run with a particular set of hyperparameters is used as knowledge to inform the subsequent hyperparameters searched. A lot of previous work uses a Bayesian optimisation framework to incorporate this information by updating the posterior of a Bayesian model of successful hyperparameters for training – examples include GPUCB (Srinivas et al., 2009), TPE (Bergstra et al., 2011), Spearmint (Snoek et al., 2012), and SMAC (Hutter et al., 2011). As noted in the previous section, their sequential nature makes these methods prohibitively slow for expensive optimisation processes. For this reason, a number of methods look to speed up the updating of the posterior with information from each optimisation process by performing early stopping, using intermediate losses to predict final performance, and even modelling the entire time and data dependent optimisation process in order to reduce the number of optimisation steps required for each optimisation process and to better explore the space of promising hyperparameters (György & Kocsis, 2011; Agarwal et al., 2011; Sabharwal et al., 2016; Swersky et al., 2013, 2014; Domhan et al., 2015; Klein et al., 2016; Snoek et al., 2015; Springenberg et al., 2016).
Many of these Bayesian optimisation approaches can be further sped up by attempting to parallelise them – training independent models in parallel to quicker update the Bayesian model (though potentially introducing bias) (Shah & Ghahramani, 2015; González et al., 2016; Wu & Frazier, 2016; Rasley et al., 2017; Golovin et al., 2017) – but these will still require multiple sequential model optimisations. The same problem is also encountered where genetic algorithms are used in place of Bayesian optimisation for hyperparameter evolution (Young et al., 2015).
Moving away from Bayesian optimisation, even approaches such as Hyperband (Li et al., 2016) which model the problem of hyperparameter selection as a manyarmed bandit, and natively incorporate parallelisation, cannot practically be executed within a single training optimisation process due to the large amount of computational resources they would initially require.
The method we present in this paper only requires a single training optimisation process – it builds upon the unreasonable success of random hyperparameter search which has shown to be very effective (Bergstra & Bengio, 2012). Random search is effective at finding good regions for sensitive hyperparameters – PBT identifies these and ensures that these areas are explored more using partially trained models (e.g. by copying their weights). By bootstrapping the evaluation of new hyperparameters during training on partially trained models we eliminate the need for sequential optimisation processes that plagues the methods previously discussed. PBT also allows for adaptive hyperparameters during training, which has been shown, especially for learning rates, to improve optimisation (Loshchilov & Hutter, 2016; Smith, 2017; Massé & Ollivier, 2015).
PBT is perhaps most analogous to evolutionary strategies that employ selfadaptive hyperparameter tuning to modify how the genetic algorithm itself operates (Bäck, 1998; Spears, 1995; Gloger, 2004; Clune et al., 2008) – in these works, the evolutionary hyperparameters were mutated at a slower rate than the parameters themselves, similar to how hyperparameters in PBT are adjusted at a slower rate than the parameters. In our case however, rather than evolving the parameters of the model, we train them partially with standard optimisation techniques (e.g.
gradient descent). Other similar works to ours are Lamarckian evolutionary algorithms in which parameters are inherited whilst hyperparameters are evolved
(Castillo et al., 2006) or where the hyperparameters, initial weights, and architecture of networks are evolved, but the parameters are learned (Castillo et al., 1999; Husken et al., 2000), or where evolution and learning are both applied to the parameters (Ku & Mak, 1997; Houck et al., 1997; Igel et al., 2005). This mixture of learning and evolutionarylike algorithms has also been explored successfully in other domains (Zhang et al., 2011) such as neural architecture search (Real et al., 2017; Liu et al., 2017)(Xue et al., 2016), and parameter learning (Fernando et al., 2016). Finally, there are parallels to work such as Salustowicz & Schmidhuber (1997) which perform program search with genetic algorithms.3 Population Based Training
The most common formulation in machine learning is to optimise the parameters of a model to maximise a given objective function , (e.g. classification, reconstruction, or prediction). Generally, the trainable parameters
are updated using an optimisation procedure such as stochastic gradient descent. More importantly, the actual performance metric
that we truly care to optimise is often different to , for example could be accuracy on a validation set, or BLEU score as used in machine translation. The main purpose of PBT is to provide a way to optimise both the parameters and the hyperparameters jointly on the actual metric that we care about.To that end, firstly we define a function eval that evaluates the objective function, , using the current state of model . For simplicity we ignore all terms except , and define eval as function of trainable parameters only. The process of finding the optimal set of parameters that maximise is then:
(1) 
Note that in the methods we are proposing, the eval function does not need to be differentiable, nor does it need to be the same as the function used to compute the iterative updates in the optimisation steps which we outline below (although they ought to be correlated).
When our model is a neural network, we generally optimise the weights in an iterative manner, e.g. by using stochastic gradient descent on the objective function . Each step of this iterative optimisation procedure, step, updates the parameters of the model, and is itself conditioned on some parameters (often referred to as hyperparameters).
In more detail, iterations of parameter update steps:
(2) 
are chained to form a sequence of updates that ideally converges to the optimal solution
(3) 
This iterative optimisation process can be computationally expensive, due to the number of steps required to find as well as the computational cost of each individual step, often resulting in the optimisation of taking days, weeks, or even months. In addition, the solution is typically very sensitive to the choice of the sequence of hyperparemeters . Incorrectly chosen hyperparameters can lead to bad solutions or even a failure of the optimisation of to converge. Correctly selecting hyperparameters requires strong prior knowledge on to exist or to be found (most often through multiple optimisation processes with different ). Furthermore, due to the dependence of on iteration step, the number of possible values grows exponentially with time. Consequently, it is common practise to either make all equal to each other (e.g. constant learning rate through entire training, or constant regularisation strength) or to predefine a simple schedule (e.g. learning rate annealing). In both cases one needs to search over multiple possible values of
(4) 
As a way to perform (4) in a fast and computationally efficient manner, we consider training models forming a population which are optimised with different hyperparameters . The objective is to therefore find the optimal model across the entire population. However, rather than taking the approach of parallel search, we propose to use the collection of partial solutions in the population to additionally perform metaoptimisation, where the hyperparameters and weights are additionally adapted according to the performance of the entire population.
In order to achieve this, PBT uses two methods called independently on each member of the population (each worker): exploit, which, given performance of the whole population, can decide whether the worker should abandon the current solution and instead focus on a more promising one; and explore, which given the current solution and hyperparameters proposes new ones to better explore the solution space.
Each member of the population is trained in parallel, with iterative calls to step to update the member’s weights and eval to measure the member’s current performance. However, when a member of the population is deemed ready (for example, by having been optimised for a minimum number of steps or having reached a certain performance threshold), its weights and hyperparameters are updated by exploit and explore. For example, exploit could replace the current weights with the weights that have the highest recorded performance in the rest of the population, and explore could randomly perturb the hyperparameters with noise. After exploit and explore, iterative training continues using step as before. This cycle of local iterative training (with step) and exploitation and exploration using the rest of the population (with exploit and explore) is repeated until convergence of the model. Algorithm 1 describes this approach in more detail, Fig. 1 schematically illustrates this process (and contrasts it with sequential optimisation and parallel search), and Fig. 2 shows a toy example illustrating the efficacy of PBT.
The specific form of exploit and explore depends on the application. In this work we focus on optimising neural networks for reinforcement learning, supervised learning, and generative modelling with PBT (Sect. 4). In these cases, step is a step of gradient descent (with e.g.
SGD or RMSProp
(Tieleman & Hinton, 2012)), eval is the mean episodic return or validation set performance of the metric we aim to optimise, exploit selects another member of the population to copy the weights and hyperparameters from, and explore creates new hyperparameters for the next steps of gradientbased learning by either perturbing the copied hyperparameters or resampling hyperparameters from the originally defined prior distribution. A member of the population is deemed ready to exploit and explore when it has been trained with gradient descent for a number of steps since the last change to the hyperparameters, such that the number of steps is large enough to allow significant gradientbased learning to have occurred.By combining multiple steps of gradient descent followed by weight copying by exploit, and perturbation of hyperparameters by explore, we obtain learning algorithms which benefit from not only local optimisation by gradient descent, but also periodic model selection, and hyperparameter refinement from a process that is more similar to genetic algorithms, creating a twotimescale learning system. An important property of population based training is that it is asynchronous and does not require a centralised process to orchestrate the training of the members of the population. Only the current performance information, weights, and hyperparameters must be globally available for each population member to access – crucially there is no synchronisation of the population required.
4 Experiments
In this section we will apply Population Based Training to different learning problems. We describe the specific form of PBT for deep reinforcement learning in Sect. 4.1 when applied to optimising UNREAL (Jaderberg et al., 2016) on DeepMind Lab 3D environment tasks (Beattie et al., 2016), Feudal Networks (Vezhnevets et al., 2017) on the Atari Learning Environment games (Bellemare et al., 2013), and the StarCraft II environment baseline agents (Vinyals et al., 2017). In Sect. 4.2 we apply PBT to optimising stateoftheart language models, transformer networks (Vaswani et al., 2017), for machine translation task. Next, in Sect. 4.3 we apply PBT to the optimisation of Generative Adversarial Networks (Goodfellow et al., 2014), a notoriously unstable optimisation problem. In all these domains we aim to build upon the strongest baselines, and show improvements in training these stateoftheart models by using PBT. Finally, we analyse the design space of Population Based Training with respect to this rich set of experimental results to draw practical guidelines for further use in Sect. 4.4.
4.1 Deep Reinforcement Learning
In this section we apply Population Based Training to the training of neural network agents with reinforcement learning (RL) where we aim to find a policy to maximise expected episodic return within an environment. We first focus on A3Cstyle methods Mnih et al. (2016) to perform this maximisation, along with some stateoftheart extensions: UNREAL (Jaderberg et al., 2016) and Feudal Networks (FuN) (Vezhnevets et al., 2017).
These algorithms are sensitive to hyperparameters, which include learning rate, entropy cost, and auxiliary loss weights, as well as being sensitive to the reinforcement learning optimisation process which can suffer from local minima or collapse due to the insufficient or unlucky policy exploration. PBT therefore aims to stabilise this process.
4.1.1 PBT for RL
We use population sizes between 10 and 80, where each member of the population is itself a distributed asynchronous actor critic (A3C) style agent (Mnih et al., 2016).
Hyperparameters
We allow PBT to optimise the learning rate, entropy cost, and unroll length for UNREAL on DeepMind Lab, learning rate, entropy cost, and intrinsic reward cost for FuN on Atari, and learning rate only for A3C on StarCraft II.
Step
Each iteration does a step of gradient descent with RMSProp (Tieleman & Hinton, 2012) on the model weights, with the gradient update provided by vanilla A3C, UNREAL or FuN.
Eval
We evaluate the current model with the last 10 episodic rewards during training.
Ready
A member of the population is deemed ready to go through the exploitandexplore process using the rest of the population when between to agent steps have elapsed since the last time that population member was ready.
Exploit
We consider two exploitation strategies. (a) Ttest selection
where we uniformly sample another agent in the population, and compare the last 10 episodic rewards using Welch’s ttest
(Welch, 1947). If the sampled agent has a higher mean episodic reward and satisfies the ttest, the weights and hyperparameters are copied to replace the current agent. (b) Truncation selection where we rank all agents in the population by episodic reward. If the current agent is in the bottom 20% of the population, we sample another agent uniformly from the top 20% of the population, and copy its weights and hyperparameters.Explore
We consider two exploration strategies in hyperparameter space. (a) Perturb, where each hyperparameter independently is randomly perturbed by a factor of 1.2 or 0.8. (b) Resample
, where each hyperparameter is resampled from the original prior distribution defined with some probability.
4.1.2 Results
We evaluate the effect of population size on the performance of PBT when training FuN on Atari. In general we find that a smaller population sizes leads to higher variance and suboptimal results, however using a population of 20 and over achieves consistent improvements, with diminishing returns for larger populations. (b)
Exploiter: We compare the effect of using different exploit functions – binary tournament and truncation selection. When training GANs, we find truncation selection to work better. (c) PBT over: We show the effect of removing parts of PBT when training UNREAL on DeepMind Lab, in particular only performing PBT on hyperparameters (so model weights are not copied between workers during exploit), and only performing PBT on model weights (so hyperparameters are not copied between workers or explored). We find that indeed it is the combination of optimising hyperparameters as well as model selection which leads to best performance. (d) Hyperparameters: Since PBT allows online adaptation of hyperparameters during training, we evaluate how important the adaptation is by comparing full PBT performance compared to using the set of hyperparameters that PBT found by the end of training. When training UNREAL on DeepMind Lab we find that the strength of PBT is in allowing the hyperparameters to be adaptive, not merely in finding a good prior on the space of hyperparameters.The high level results are summarised in Fig. 3. On all three domains – DeepMind Lab, Atari, and StarCraft II – PBT increases the final performance of the agents when trained for the same number of steps, compared to the very strong baseline of performing random search with the same number of workers.
On DeepMind Lab, when UNREAL is trained using a population of 40 workers, using PBT increases the final performance of UNREAL from 93% to 106% human performance, when performance is averaged across all levels. This represents a significant improvement over an already extremely strong baseline, with some levels such as and showing large gains in performance due to PBT, as shown in Fig. 4. PBT seems to help in two main ways: first is that the hyperparameters are clearly being focused on the best part of the sampling range, and adapted over time. For example, on in Fig. 4 one can see the learning rate being annealed by PBT as training progresses, and on in Fig. 4
one can see the unroll length – the number of steps the agent (a recurrent neural network) is unrolled before performing Nstep policy gradient and backpropagation through time – is gradually increased. This allows the agent to learn to utilise the recurrent neural network for memory, allowing it to obtain superhuman performance on
which is a level which requires memorising the location of a goal in a maze, more details can be found in (Jaderberg et al., 2016). Secondly, since PBT is copying the weights of good performing agents during the exploitation phase, agents which are lucky in environment exploration are quickly propagated to more workers, meaning that all members of the population benefit from the exploration luck of the remainder of the population.When we apply PBT to training FuN on 4 Atari levels, we similarly see a boost in final performance, with PBT increasing the average human normalised performance from 147% to 181%. On particular levels, such as PBT increases performance significantly from 6506 to 9001 (Fig. 4), which represents stateoftheart for an agent receiving only pixels as input.
Finally on StarCraft II, we similarly see PBT improving A3C baselines from 36% huamn performance to 39% human performance when averaged over 6 levels, benefiting from automatic online learning rate adaptation and model selection such as shown in Fig. 4.
More detailed experimental results, and details about the specific experimental protocols can be found in Appendix A.2.
4.2 Machine Translation
As an example of applying PBT on a supervised learning problem, we look at training neural machine translation models. In this scenario, the task is to encode a source sequence of words in one language, and output a sequence in a different target language. We focus on English to German translation on the WMT 2014 EnglishtoGerman dataset, and use a stateoftheart model, Transformer networks
(Vaswani et al., 2017), as the baseline model to apply PBT to. Crucially, we also use the highly optimised learning rate schedules and hyperparameter values for our baselines to compare to – these are the result of a huge amount of hand tuning and Bayesian optimisation.4.2.1 PBT for Machine Translation
We use a population size of 32 workers, where each member of the population is a single GPU optimising a Transformer network for steps.
Hyperparameters
We allow PBT to optimise the learning rate, attention dropout, layer dropout, and ReLU dropout rates.
Step
Each step does a step of gradient descent with Adam (Kingma & Ba, 2015).
Eval
We evaluate the current model by computing the BLEU score on the WMT newstest2012 dataset. This allows us to optimise our population for BLEU score, which usually cannot be optimised for directly.
Ready
A member of the population is deemed ready to exploit and explore using the rest of the population every steps.
Exploit
We use the ttest selection exploitation method described in Sect. 4.1.1.
Explore
We explore hyperparameter space by the perturb strategy described in Sect. 4.1.1 with 1.2 and 0.8 perturbation factors.
4.2.2 Results
When we apply PBT to the training of Transformer networks, we find that PBT actually results in models which exceed the performance of this highly tuned baseline model: BLEU score on the validation set (newstest2012) is improved from 23.71 to 24.23, and on the test set of WMT 2014 EnglishtoGerman is improved from 22.30 to 22.65 using PBT (here we report results using the small Transformer network using a reduced batch size, so is not representative of state of the art). PBT is able to automatically discover adaptations of various dropout rates throughout training, and a learning rate schedule that remarkably resembles that of the hand tuned baseline: very initially the learning rate starts small before jumping by three orders of magnitude, followed by something resembling an exponential decay, as shown in the lineage plot of Fig. 7. We can also see from Fig. 4 that the PBT model trains much faster.
4.3 Generative Adversarial Networks
The Generative Adversarial Networks (GAN) (Goodfellow et al., 2014) framework learns generative models via a training paradigm consisting of two competing modules – a generator and a discriminator (alternatively called a critic). The discriminator takes as input samples from the generator and the real data distribution, and is trained to predict whether inputs are real or generated. The generator typically maps samples from a simple noise distribution to a complex data distribution (e.g.
images) with the objective of maximally fooling the discriminator into classifying or scoring its samples as real.
Like RL agent training, GAN training can be remarkably brittle and unstable in the face of suboptimal hyperparameter selection and even unlucky random initialisation, with generators often collapsing to a single mode or diverging entirely. Exacerbating these difficulties, the presence of two modules optimising competing objectives doubles the number of hyperparameters, often forcing researchers to choose the same hyperparameter values for both modules. Given the complex learning dynamics of GANs, this unnecessary coupling of hyperparameters between the two modules is unlikely to be the best choice.
Finally, the various metrics used by the community to evaluate the quality of samples produced by GAN generators are necessarily distinct from those used for the adversarial optimisation itself. We explore whether we can improve the performance of generators under these metrics by directly targeting them as the PBT metaoptimisation evaluation criteria.
4.3.1 PBT for GANs
We train a population of size 45 with each worker being trained for steps.
Hyperparameters
We allow PBT to optimise the discriminator’s learning rate and the generator’s learning rate separately.
Step
Each step consists of gradient descent updates of the discriminator followed by a single update of the generator using the Adam optimiser (Kingma & Ba, 2015). We train using the WGANGP (Gulrajani et al., 2017)
objective where the discriminator estimates the Wasserstein distance between the real and generated data distributions, with the Lipschitz constraint enforced by regularising its input gradient to have a unit norm. See Appendix
A.3 for additional details.Eval
We evaluate the current model by a variant of the Inception score proposed by Salimans et al. (2016) computed from the outputs of a pretrained CIFAR classifier (as used in Rosca et al. (2017)
) rather than an ImageNet classifier, to avoid directly optimising the final performance metric. The CIFAR Inception scorer uses a much smaller network, making evaluation much faster.
Ready
A member of the population is deemed ready to exploit and explore using the rest of the population every steps.
Exploit
We consider two exploitation strategies. (a) Truncation selection as described in Sect. 4.1.1. (b) Binary tournament in which each member of the population randomly selects another member of the population, and copies its parameters if the other member’s score is better. Whenever one member of the population is copied to another, all parameters – the hyperparameters, weights of the generator, and weights of the discriminator – are copied.
Explore
We explore hyperparameter space by the perturb strategy described in Sect. 4.1.1, but with more aggressive perturbation factors of 2.0 or 0.5.
4.3.2 Results
We apply PBT to optimise the learning rates for a CIFARtrained GAN discriminator and generator using architectures similar to those used DCGAN (Radford et al., 2016). For both PBT and the baseline, the population size is , with learning rates initialised to for a total of initial hyperparameter settings, each replicated times with independent random weight initialisations. Additional architectural and hyperparameter details can be found in Appendix A.3. The baseline utilises an exponential learning rate decay schedule, which we found to work the best among a number of handdesigned annealing strategies, detailed in Appendix A.3.
In Fig. 5 (b), we compare the truncation selection exploration strategy with a simpler variant, binary tournament. We find a slight advantage for truncation selection, and our remaining discussion focuses on results obtained by this strategy. Fig. 4 (bottom left) plots the CIFAR Inception scores and shows how the learning rates of the population change throughout the first half of training (both the learning curves and hyperparameters saturate and remain roughly constant in the latter half of training). We observe that PBT automatically learns to decay learning rates, though it does so dynamically, with irregular flattening and steepening, resulting in a learnt schedule that would not be matched by any typical exponential or linear decay protocol. Interestingly, the learned annealing schedule for the discriminator’s learning rate closely tracks that of the best performing baseline (shown in grey), while the generator’s schedule deviates from the baseline schedule substantially.
In Table 4 (Appendix A.3), we report standard ImageNet Inception scores as our main performance measure, but use CIFAR Inception score for model selection, as optimising ImageNet Inception score directly would potentially overfit our models to the test metric. CIFAR Inception score is used to select the best model after training is finished both for the baseline and for PBT. Our bestperforming baseline achieves a CIFAR Inception score of , corresponding to an ImageNet Inception score of , similar to or better than Inception scores for DCGANlike architectures reported in prior work (Rosca et al., 2017; Gulrajani et al., 2017; Yang et al., 2017). Using PBT, we strongly outperform this baseline, achieving a CIFAR Inception score of and corresponding ImageNet Inception score of . For reference, Table 4 (Appendix A.3) also reports the peak ImageNet Inception scores obtained by any population member at any point during training. Fig. 8 shows samples from the best generators with and without PBT.
4.4 Analysis
In this section we analyse and discuss the effects of some of the design choices in setting up PBT, as well as probing some aspects of why the method is so successful.
Population Size
In Fig. 5 (a) we demonstrate the effect of population size on the performance of PBT when training FuN on Atari. In general, we find that if the population size is too small (10 or below) we tend to encounter higher variance and can suffer from poorer results – this is to be expected as PBT is a greedy algorithm and so can get stuck in local optima if there is not sufficient population to maintain diversity and scope for exploration. However, these problems rapidly disappear as population size increases and we see improved results as the population size grows. In our experiments, we observe that a population size of between and is sufficient to see strong and consistent improvements; larger populations tend to fare even better, although we see diminishing returns for the cost of additional population members.
Exploitation Type
In Fig. 5 (b) we compare the effect of using different exploit functions – binary tournament and truncation selection. For GAN training, we find truncation selection to work better. Similar behaviour was seen in our preliminary experiments with FuN on Atari as well. Further work is required to fully understand the impact of different exploit mechanisms, however it appears that truncation selection is a robustly useful choice.
PBT Targets
In Fig. 5 (c) we show the effect of only applying PBT to subsets of parameters when training UNREAL on DM Lab. In particular, we contrast (i) only performing exploitandexplore on hyperparameters (so model weights are not copied between workers during exploit); and (ii) only performing exploitandexplore on model weights (so hyperparameters are not copied between workers or explored). Whilst there is some increase in performance simply from the selection and propagation of good model weights, our results clearly demonstrate that in this setting it is indeed the combination of optimising hyperparameters as well as model selection that lead to our best performance.
Hyperparameter Adaptivity
In Fig. 5 (d) we provide another demonstration that the benefits of PBT go beyond simply finding a single good fixed hyperparameter combination. Since PBT allows online adaptation of hyperparameters during training, we evaluate how important the adaptation is by comparing full PBT performance compared to using the set of hyperparameters that PBT found by the end of training. When training UNREAL on DeepMind Lab we find that the strength of PBT is in allowing the hyperparameters to be adaptive, not merely in finding a good prior on the space of hyperparameters
Lineages
Fig. 7 shows hyperparameter sequences for the final agents in the population for GAN and Machine Translation (MT) experiments. We have chosen these two examples as the community has developed very specific learning rate schedules for these models. In the GAN case, the typical schedule involves linearly annealed learning rates of both generator and discriminator in a synchronised way. As one can see, PBT discovered a similar scheme, however the discriminator annealing is much more severe, while the generator schedule is slightly less aggressive than the typical one.
As mentioned previously, the highly nonstandard learning rate schedule used for machine translation has been to some extent replicated by PBT – but again it seems to be more aggressive than the hand crafted version. Quite interestingly, the dropout regime is also nonmonotonic, and follows an even more complex path – it first increases by almost an order of magnitude, then drops back to the initial value, to slowly crawl back to an order of magnitude bigger value. These kind of schedules seem easy to discover by PBT, while at the same time are beyond the space of hyperparameter schedules considered by human experts.
Phylogenetic trees
Figure 6 shows a phylogenetic forest for the population in the GAN and Atari experiments. A property made noticeable by such plots is that all final agents are descendants of the same, single initial agent. This is on one hand a consequence of the greedy nature of specific instance of PBT used, as exploit only uses a single other agent from the population. In such scenario, the number of original agents can only decrease over time, and so should finally converge to the situation observed. On the other hand one can notice that in both cases there are quite rich subpopulations which survived for a long time, showing that the PBT process considered multiple promising solutions before finally converging on a single winner – an exploratory capability most typical optimisers lack.
One can also note the following interesting qualitative differences between the two plots: in the GAN example, almost all members of the population enjoy an improvement in score relative to their parent; however, in the Atari experiment we note that even quite late in learning there can be exploration steps that result in a sharp drop in cumulative episode reward relative to the parent. This is perhaps indicative of some instability of neural network based RL on complex problems – an instability that PBT is well positioned to correct.
5 Conclusions
We have presented Population Based Training, which represents a practical way to augment the standard training of neural network models. We have shown consistent improvements in accuracy, training time and stability across a wide range of domains by being able to optimise over weights and hyperparameters jointly. It is important to note that contrary to conventional hyperparameter optimisation techniques, PBT discovers an adaptive schedule rather than a fixed set of hyperparameters. We already observed significant improvements on a wide range of challenging problems including state of the art models on deep reinforcement learning and hierarchical reinforcement learning, machine translation, and GANs. While there are many improvements and extensions to be explored going forward, we believe that the ability of PBT to enhance the optimisation process of new, unfamiliar models, to adapt to nonstationary learning problems, and to incorporate the optimisation of indirect performance metrics and auxiliary tasks, results in a powerful platform to propel future research.
Acknowledgments
We would like to thank Yaroslav Ganin, Mihaela Rosca, John Agapiou, Sasha Vehznevets, Vlad Firoiu, and the wider DeepMind team for many insightful discussions, ideas, and support.
References
 Agarwal et al. (2011) Alekh Agarwal, John C Duchi, Peter L Bartlett, and Clement Levrard. Oracle inequalities for computationally budgeted model selection. In Proceedings of the 24th Annual Conference on Learning Theory, pp. 69–86, 2011.
 Bäck (1998) Thomas Bäck. An overview of parameter control methods by selfadaptation in evolutionary algorithms. Fundamenta Informaticae, 35(14):51–66, 1998.
 Beattie et al. (2016) Charles Beattie, Joel Z Leibo, Denis Teplyashin, Tom Ward, Marcus Wainwright, Heinrich Küttler, Andrew Lefrancq, Simon Green, Víctor Valdés, Amir Sadik, et al. Deepmind lab. arXiv preprint arXiv:1612.03801, 2016.
 Bellemare et al. (2013) Marc G Bellemare, Yavar Naddaf, Joel Veness, and Michael Bowling. The arcade learning environment: An evaluation platform for general agents. J. Artif. Intell. Res.(JAIR), 47:253–279, 2013.
 Bergstra & Bengio (2012) James Bergstra and Yoshua Bengio. Random search for hyperparameter optimization. Journal of Machine Learning Research, 13(Feb):281–305, 2012.
 Bergstra et al. (2011) James S Bergstra, Rémi Bardenet, Yoshua Bengio, and Balázs Kégl. Algorithms for hyperparameter optimization. In Advances in Neural Information Processing Systems, pp. 2546–2554, 2011.

Castillo et al. (1999)
PA Castillo, V Rivas, JJ Merelo, Jesús González, Alberto Prieto, and
Gustavo Romero.
Gpropiii: Global optimization of multilayer perceptrons using an evolutionary algorithm.
InProceedings of the 1st Annual Conference on Genetic and Evolutionary ComputationVolume 1
, pp. 942–942. Morgan Kaufmann Publishers Inc., 1999.  Castillo et al. (2006) PA Castillo, MG Arenas, JG Castellano, JJ Merelo, A Prieto, V Rivas, and G Romero. Lamarckian evolution and the Baldwin effect in evolutionary neural networks. arXiv preprint cs/0603004, 2006.
 Clune et al. (2008) Jeff Clune, Dusan Misevic, Charles Ofria, Richard E Lenski, Santiago F Elena, and Rafael Sanjuán. Natural selection fails to optimize mutation rates for longterm adaptation on rugged fitness landscapes. PLoS Computational Biology, 4(9):e1000187, 2008.
 Domhan et al. (2015) Tobias Domhan, Jost Tobias Springenberg, and Frank Hutter. Speeding up automatic hyperparameter optimization of deep neural networks by extrapolation of learning curves. In IJCAI, pp. 3460–3468, 2015.
 Fernando et al. (2016) Chrisantha Fernando, Dylan Banarse, Malcolm Reynolds, Frederic Besse, David Pfau, Max Jaderberg, Marc Lanctot, and Daan Wierstra. Convolution by evolution: Differentiable pattern producing networks. In Proceedings of the 2016 on Genetic and Evolutionary Computation Conference, pp. 109–116. ACM, 2016.
 Gloger (2004) Bartlomiej Gloger. Self adaptive evolutionary algorithms, 2004.
 Golovin et al. (2017) Daniel Golovin, Benjamin Solnik, Subhodeep Moitra, Greg Kochanski, John Karro, and D Sculley. Google vizier: A service for blackbox optimization. In Proceedings of the 23rd ACM SIGKDD International Conference on Knowledge Discovery and Data Mining, pp. 1487–1495. ACM, 2017.
 González et al. (2016) Javier González, Zhenwen Dai, Philipp Hennig, and Neil Lawrence. Batch Bayesian optimization via local penalization. In Artificial Intelligence and Statistics, pp. 648–657, 2016.
 Goodfellow et al. (2014) Ian Goodfellow, Jean PougetAbadie, Mehdi Mirza, Bing Xu, David WardeFarley, Sherjil Ozair, Aaron Courville, and Yoshua Bengio. Generative adversarial nets. In Advances in Neural Information Processing Systems, pp. 2672–2680, 2014.
 Gulrajani et al. (2017) Ishaan Gulrajani, Faruk Ahmed, Martin Arjovsky, Vincent Dumoulin, and Aaron Courville. Improved training of Wasserstein GANs. In Advances in Neural Information Processing Systems, pp. 5763–5773, 2017.
 György & Kocsis (2011) András György and Levente Kocsis. Efficient multistart strategies for local search algorithms. Journal of Artificial Intelligence Research, 41:407–444, 2011.
 Houck et al. (1997) Christopher R Houck, Jeffery A Joines, Michael G Kay, and James R Wilson. Empirical investigation of the benefits of partial Lamarckianism. Evolutionary Computation, 5(1):31–60, 1997.
 Husken et al. (2000) Michael Husken, Jens E Gayko, and Bernhard Sendhoff. Optimization for problem classesneural networks that learn to learn. In Combinations of Evolutionary Computation and Neural Networks, 2000 IEEE Symposium on, pp. 98–109. IEEE, 2000.
 Hutter et al. (2011) Frank Hutter, Holger H Hoos, and Kevin LeytonBrown. Sequential modelbased optimization for general algorithm configuration. LION, 5:507–523, 2011.
 Igel et al. (2005) Christian Igel, Stefan Wiegand, and Frauke Friedrichs. Evolutionary optimization of neural systems: The use of strategy adaptation. Trends and Applications in Constructive Approximation, pp. 103–123, 2005.
 Jaderberg et al. (2016) Max Jaderberg, Volodymyr Mnih, Wojciech Marian Czarnecki, Tom Schaul, Joel Z Leibo, David Silver, and Koray Kavukcuoglu. Reinforcement learning with unsupervised auxiliary tasks. arXiv preprint arXiv:1611.05397, 2016.
 Kingma & Ba (2015) Diederik Kingma and Jimmy Ba. Adam: A method for stochastic optimization. In ICLR, 2015.
 Klein et al. (2016) Aaron Klein, Stefan Falkner, Simon Bartels, Philipp Hennig, and Frank Hutter. Fast Bayesian optimization of machine learning hyperparameters on large datasets. arXiv preprint arXiv:1605.07079, 2016.
 Ku & Mak (1997) Kim WC Ku and ManWai Mak. Exploring the effects of Lamarckian and Baldwinian learning in evolving recurrent neural networks. In Evolutionary Computation, 1997., IEEE International Conference on, pp. 617–621. IEEE, 1997.
 Li et al. (2016) Lisha Li, Kevin Jamieson, Giulia DeSalvo, Afshin Rostamizadeh, and Ameet Talwalkar. Hyperband: A novel banditbased approach to hyperparameter optimization. arXiv preprint arXiv:1603.06560, 2016.
 Liu et al. (2017) Hanxiao Liu, Karen Simonyan, Oriol Vinyals, Chrisantha Fernando, and Koray Kavukcuoglu. Hierarchical representations for efficient architecture search. arXiv preprint arXiv:1711.00436, 2017.
 Loshchilov & Hutter (2016) Ilya Loshchilov and Frank Hutter. SGDR: stochastic gradient descent with warm restarts. arXiv preprint arXiv:1608.03983, 2016.
 Massé & Ollivier (2015) PierreYves Massé and Yann Ollivier. Speed learning on the fly. arXiv preprint arXiv:1511.02540, 2015.
 Mnih et al. (2016) Volodymyr Mnih, Adria Puigdomenech Badia, Mehdi Mirza, Alex Graves, Timothy Lillicrap, Tim Harley, David Silver, and Koray Kavukcuoglu. Asynchronous methods for deep reinforcement learning. In International Conference on Machine Learning, pp. 1928–1937, 2016.
 Radford et al. (2016) Alec Radford, Luke Metz, and Soumith Chintala. Unsupervised representation learning with deep convolutional generative adversarial networks. In ICLR, 2016.
 Rasley et al. (2017) Jeff Rasley, Yuxiong He, Feng Yan, Olatunji Ruwase, and Rodrigo Fonseca. Hyperdrive: Exploring hyperparameters with POP scheduling. In Proceedings of the 18th International Middleware Conference, Middleware, volume 17, 2017.
 Real et al. (2017) Esteban Real, Sherry Moore, Andrew Selle, Saurabh Saxena, Yutaka Leon Suematsu, Quoc Le, and Alex Kurakin. Largescale evolution of image classifiers. arXiv preprint arXiv:1703.01041, 2017.
 Rosca et al. (2017) Mihaela Rosca, Balaji Lakshminarayanan, David WardeFarley, and Shakir Mohamed. Variational approaches for autoencoding generative adversarial networks. arXiv preprint arXiv:1706.04987, 2017.
 Sabharwal et al. (2016) Ashish Sabharwal, Horst Samulowitz, and Gerald Tesauro. Selecting nearoptimal learners via incremental data allocation. In AAAI, pp. 2007–2015, 2016.
 Salimans et al. (2016) Tim Salimans, Ian Goodfellow, Wojciech Zaremba, Vicki Cheung, Alec Radford, and Xi Chen. Improved techniques for training GANs. In Advances in Neural Information Processing Systems, pp. 2234–2242, 2016.
 Salustowicz & Schmidhuber (1997) Rafal Salustowicz and Jürgen Schmidhuber. Probabilistic incremental program evolution: Stochastic search through program space. In ECML, pp. 213–220, 1997.
 Shah & Ghahramani (2015) Amar Shah and Zoubin Ghahramani. Parallel predictive entropy search for batch global optimization of expensive objective functions. In Advances in Neural Information Processing Systems, pp. 3330–3338, 2015.

Smith (2017)
Leslie N Smith.
Cyclical learning rates for training neural networks.
In
Applications of Computer Vision (WACV), 2017 IEEE Winter Conference on
, pp. 464–472. IEEE, 2017.  Snoek et al. (2012) Jasper Snoek, Hugo Larochelle, and Ryan P Adams. Practical Bayesian optimization of machine learning algorithms. In Advances in neural information processing systems, pp. 2951–2959, 2012.
 Snoek et al. (2015) Jasper Snoek, Oren Rippel, Kevin Swersky, Ryan Kiros, Nadathur Satish, Narayanan Sundaram, Mostofa Patwary, Mr Prabhat, and Ryan Adams. Scalable Bayesian optimization using deep neural networks. In International Conference on Machine Learning, pp. 2171–2180, 2015.
 Spears (1995) William M. Spears. Adapting crossover in evolutionary algorithms. In Proceedings of the Fourth Annual Conference on Evolutionary Programming, pp. 367–384. MIT Press, 1995.
 Springenberg et al. (2016) Jost Tobias Springenberg, Aaron Klein, Stefan Falkner, and Frank Hutter. Bayesian optimization with robust Bayesian neural networks. In Advances in Neural Information Processing Systems, pp. 4134–4142, 2016.
 Srinivas et al. (2009) Niranjan Srinivas, Andreas Krause, Sham M Kakade, and Matthias Seeger. Gaussian process optimization in the bandit setting: No regret and experimental design. arXiv preprint arXiv:0912.3995, 2009.
 Swersky et al. (2013) Kevin Swersky, Jasper Snoek, and Ryan P Adams. Multitask Bayesian optimization. In Advances in neural information processing systems, pp. 2004–2012, 2013.
 Swersky et al. (2014) Kevin Swersky, Jasper Snoek, and Ryan Prescott Adams. Freezethaw Bayesian optimization. arXiv preprint arXiv:1406.3896, 2014.
 Tieleman & Hinton (2012) Tijmen Tieleman and Geoffrey Hinton. Lecture 6.5rmsprop: Divide the gradient by a running average of its recent magnitude. COURSERA: Neural networks for machine learning, 4(2):26–31, 2012.
 Vaswani et al. (2017) Ashish Vaswani, Noam Shazeer, Niki Parmar, Jakob Uszkoreit, Llion Jones, Aidan N Gomez, Lukasz Kaiser, and Illia Polosukhin. Attention is all you need. arXiv preprint arXiv:1706.03762, 2017.
 Vezhnevets et al. (2017) Alexander Sasha Vezhnevets, Simon Osindero, Tom Schaul, Nicolas Heess, Max Jaderberg, David Silver, and Koray Kavukcuoglu. Feudal networks for hierarchical reinforcement learning. arXiv preprint arXiv:1703.01161, 2017.
 Vinyals et al. (2017) Oriol Vinyals, Timo Ewalds, Sergey Bartunov, Petko Georgiev, Alexander Sasha Vezhnevets, Michelle Yeo, Alireza Makhzani, Heinrich Küttler, John Agapiou, Julian Schrittwieser, et al. Starcraft II: A new challenge for reinforcement learning. arXiv preprint arXiv:1708.04782, 2017.
 Welch (1947) Bernard L Welch. The generalization of student’s’ problem when several different population variances are involved. Biometrika, 34(1/2):28–35, 1947.
 Wu & Frazier (2016) Jian Wu and Peter Frazier. The parallel knowledge gradient method for batch Bayesian optimization. In Advances in Neural Information Processing Systems, pp. 3126–3134, 2016.
 Xue et al. (2016) Bing Xue, Mengjie Zhang, Will N Browne, and Xin Yao. A survey on evolutionary computation approaches to feature selection. IEEE Transactions on Evolutionary Computation, 20(4):606–626, 2016.
 Yang et al. (2017) Jianwei Yang, Anitha Kanna, Dhruv Batra, and Devi Parikh. LRGAN: Layered recursive generative adversarial networks for image generation. In ICLR, 2017.

Young et al. (2015)
Steven R Young, Derek C Rose, Thomas P Karnowski, SeungHwan Lim, and Robert M
Patton.
Optimizing deep learning hyperparameters through an evolutionary algorithm.
In Proceedings of the Workshop on Machine Learning in HighPerformance Computing Environments, pp. 4. ACM, 2015.  Zhang et al. (2011) Jun Zhang, Zhihui Zhan, Ying Lin, Ni Chen, Yuejiao Gong, Jinghui Zhong, Henry SH Chung, Yun Li, and Yuhui Shi. Evolutionary computation meets machine learning: A survey. IEEE Computational Intelligence Magazine, 6(4):68–75, 2011.
Appendix A Population Based Training of Neural Networks Appendix
a.1 Practical implementations
In this section, we describe at a high level how one might concretely implement PBT on a commodity training platform. Relative to a standard setup in which multiple hyperparameter configurations are searched in parallel (e.g. grid/random search), a practitioner need only add the ability for population members to read and write to a shared datastore (e.g. a keyvalue store, or a simple filesystem). Augmented with such a setup, there are two main types of interaction required:
(1) Members of the population interact with this datastore to update their current performance and can also use it to query the recent performance of other population members.
(2) Population members periodically checkpoint themselves, and when they do so they write their performance to the shared datastore. In the exploitandexplore process, another member of the population may elect to restore its weights from the checkpoint of another population member (i.e. exploiting), and would then modify the restored hyperparameters (i.e. exploring) before continuing to train.
As a final comment, although we present and recommend PBT as an asynchronous parallel algorithm, it could also be executed semiserially or with partial synchrony. Although it would obviously result in longer training times than running fully asynchronously in parallel, one would still gain the performancepertotalcompute benefits of PBT relative to randomsearch. This execution mode may be attractive to researchers who wish to take advantage of commodity compute platforms at cheaper, but less reliable, preemptible/spot tiers. As an example of an approach with partial synchrony, one could imagine the following procedure: each member of the population is advanced in parallel (but without synchrony) for a fixed number of steps; once all (or some large fraction) of the population has advanced, one would call the exploitandexplore procedure; and then repeat. This partial locking would be forgiving of different population members running at widely varying speeds due to preemption events, etc.
a.2 Detailed Results: RL
FuN  Atari
Table 1 shows the breakdown of results per game for FuN on the Atari domain, with and without PBT for different population sizes. The initial hyperparameter ranges for the population in both the FuN and PBTFuN settings were the same as those used for the corresponding games in the original Feudal Networks paper (Vezhnevets et al., 2017), with the exception of intrinsicreward cost (which was sampled in the range , rather than – since we have found that very low intrinsic reward cost weights are rarely effective). We used fixed discounts of in the worker and in the manager since, for the games selected, these were shown in the original FuN paper to give good performance. The subset of games were themselves chosen as ones for which the hierarchical abstractions provided by FuN give particularly strong performance improvements over a flat A3C agent baseline. In these experiments, we used Truncation as the exploitation mechanism, and Perturb as the exploration mechanism. Training was performed for agentenvironmnent interactions, corresponding to total environment steps (due to use of actionrepeat of ). Each experiment condition was repeated times, and the numbers in the table reflect the meanacrossreplicas of the single best performing population member at the end of the training run.
Game  Pop 10  Pop 20  Pop 40  Pop 80  

FuN  PBTFuN  FuN  PBTFuN  FuN  PBTFuN  FuN  PBTFuN  
amidar 
1742 174  1511 400  1645 158  2454 160  1947 80  2752 81  2055 113  3018 269 
gravitar  3970 145  4110 225  3884 362  4202 113  4018 180  4481 540  4279 161  5386 306 
mspacman  5635 656  5509 589  5280 349  5714 347  6177 402  8124 392  6506 354  9001 711 
enduro  1838 37  1958 81  1881 63  2213 6  2079 9  2228 77  2166 34  2292 16 

UNREAL  DM Lab
Table 2 shows the breakdown on results per level for UNREAL and PBTUNREAL on the DM Lab domain. The following three hyperparameters were randomly sampled for both methods and subsequently perturbed in the case of PBTUNREAL: the learning rate was sampled in the loguniform range [0.00001, 0.005], the entropy cost was sampled in the loguniform range [0.0005, 0.01], and the unroll length was sampled uniformly between 5 and 50 steps. Other experimental details were replicated from Jaderberg et al. (2016), except for pixel control weight which was set to 0.1.
Game  Pop 40  Pop 20  
UNREAL  PBTUNREAL  UNREAL  PBTHypersUNREAL  PBTWeightsUNREAL  PBTUNREAL  
emstm_non_match 
66.0  66.0  42.6  14.5  56.0  53.2 
emstm_watermaze  39.0  36.7  27.9  33.2  34.6  36.8 
lt_horse_shoe_color  75.4  90.2  77.4  44.2  56.3  87.5 
nav_maze_random_goal_01  114.2  149.4  82.2  81.5  74.2  96.1 
lt_hallway_slope  90.5  109.7  
nav_maze_all_random_01  59.6  78.1  
nav_maze_all_random_02  62.0  92.4  
nav_maze_all_random_03  92.7  115.3  
nav_maze_random_goal_02  145.4  173.3  
nav_maze_random_goal_03  127.3  153.5  
nav_maze_static_01  131.5  133.5  
nav_maze_static_02  194.5  220.2  
nav_maze_static_03  588.9  594.8  
seekavoid_arena_01  42.5  42.3  
stairway_to_melon_01  189.7  192.4  

A3C  StarCraft II
Table 2 shows the breakdown on results per level for UNREAL on the DM Lab domain, with and without PBT for different population sizes. We take the baseline setup of Vinyals et al. (2017) with feedforward network agents, and sample the learning rate from the loguniform range [0, 0.001].
Game  Pop 30  

A3C  PBTA3C  
CollectMineralShards 
93.7  100.5 
FindAndDefeatZerglings  46.0  49.9 
DefeatRoaches  121.8  131.5 
DefeatZerglingsAndBanelings  94.6  124.7 
CollectMineralsAndGas  3345.8  3345.3 
BuildMarines  0.17  0.37 

a.3 Detailed Results: GAN
This section provides additional details of and results from our GAN experiments introduced in Sect. 4.3. Fig. 8 shows samples generated by the best performing generators with and without PBT. In Table 4, we report GAN Inception scores with and without PBT.
Architecture and hyperparameters
The architecture of the discriminator and the generator are similar to those proposed in DCGAN (Radford et al., 2016), but with convolutions (rather than ), and a smaller generator with half the number of feature maps in each layer. Batch normalisation is used in the generator, but not in the discriminator, as in Gulrajani et al. (2017). Optimisation is performed using the Adam optimiser (Kingma & Ba, 2015), with , , and . The batch size is 128.
GAN  PBTGAN (TS)  PBTGAN (BT)  

CIFAR Inception  6.380.03  6.800.01  6.770.02 
ImageNet Inception (by CIFAR)  6.450.05  6.890.04  6.740.05 
ImageNet Inception (peak)  6.480.05  7.020.07  6.810.04 
Baseline selection
To establish a strong and directly comparable baseline for GAN PBT, we extensively searched over and chose the best by CIFAR Inception score among many learning rate annealing strategies used in the literature. For each strategy, we trained for a total of steps, where learning rates were fixed to their initial settings for the first steps, then for the latter steps either (a) exponentially decayed to times the initial setting, or (b) linearly decayed to , searching over . (Searching over the number of steps is essential due to the instability of GAN training: performance often degrades with additional training.) In each training run to evaluate a given combination of annealing strategy and choice of training steps , we used the same number of independent workers () and initialised the learning rates to the same values used in PBT. The best annealing strategy we found using this procedure was exponential decay with steps. For the final baseline, we then extended this strategy to train for the same number of steps as used in PBT () by continuing training for the remaining steps with constant learning rates fixed to their final values after exponential decay.
a.4 Detailed Results: Machine Translation
We used the opensourced implementation of the Transformer framework^{1}^{1}1https://github.com/tensorflow/tensor2tensor with the provided transformer_base_single_gpu architecture settings. This model has the same number of parameters as the base configuration that runs on 8 GPUs (), but sees, in each training step, of the number of tokens ( vs. ) as it uses a smaller batch size. For both evolution and the randomsearch baseline, we searched over learning rate, and dropout values applied to the attention softmax activations, the outputs of the ReLU units in the feedforward modules, and the output of each layer. We left other hyperparameters unchanged and made no other changes to the implementation. For evaluation, we computed BLEU score based on autoregressive decoding with beamsearch of width 4. We used newstest2012 and newstest2013 respectively as the evaluation set used by PBT, and as the test set for monitoring. At the end of training, we report tokenized BLEU score on newstest2014 as computed by multibleu.pl script^{2}^{2}2https://github.com/mosessmt/mosesdecoder/blob/master/scripts/generic/multibleu.perl. We also evaluated the original hyperparameter configuration trained for the same number of steps and obtained the BLEU score of , which is lower than both our baselines and PBT results.