Domain randomization (DR) is a successful technique for learning robust policies for robot systems, when the dynamics of the target robot system are unknown. The success of policies trained with domain randomization however, is highly dependent on the correct selection of the randomization distribution. The majority of success stories typically use real world data in order to carefully select the DR distribution, or incorporate real world trajectories to better estimate appropriate randomization distributions. In this paper, we consider the problem of finding good domain randomization parameters for simulation, without prior access to data from the target system. We explore the use of gradient-based search methods to learn a domain randomization with the following properties: 1) The trained policy should be successful in environments sampled from the domain randomization distribution 2) The domain randomization distribution should be wide enough so that the experience similar to the target robot system is observed during training, while addressing the practicality of training finite capacity models. These two properties aim to ensure the trajectories encountered in the target system are close to those observed during training, as existing methods in machine learning are better suited for interpolation than extrapolation. We show how adapting the domain randomization distribution while training context-conditioned policies results in improvements on jump-start and asymptotic performance when transferring a learned policy to the target environment.
@cite_10 propose a related approach for learning robust policies over a distribution of simulator models. The proposed approach, based on the the @math -percentile conditional value at risk (CVaR) @cite_11 objective, improves the policy performance on a small proportion of environments where the policy performs the worst. The authors propose an algorithm that updates the distribution of simulation models to maximize the likelihood of real-world trajectories, via Bayesian inference. The combination of worst-case performance optimization and Bayesian updates ensures that the resulting policy is robust to errors in the estimation of the simulation model parameters. Our method can be combined with the CVaR objective to encourage diversity of the learned DR distribution.
@cite_10 @cite_11
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In many sequential decision-making problems we may want to manage risk by minimizing some measure of variability in costs in addition to minimizing a standard criterion. Conditional value-at-risk (CVaR) is a relatively new risk measure that addresses some of the shortcomings of the well-known variance-related risk measures, and because of its computational efficiencies has gained popularity in finance and operations research. In this paper, we consider the mean-CVaR optimization problem in MDPs. We first derive a formula for computing the gradient of this risk-sensitive objective function. We then devise policy gradient and actor-critic algorithms that each uses a specific method to estimate this gradient and updates the policy parameters in the descent direction. We establish the convergence of our algorithms to locally risk-sensitive optimal policies. Finally, we demonstrate the usefulness of our algorithms in an optimal stopping problem. We propose a method to forecast a vehicle’s ego-motion as a distribution over spatiotemporal paths, conditioned on features (e.g., from LIDAR and images) embedded in an overhead map. The method learns a policy inducing a distribution over simulated trajectories that is both “diverse” (produces most of the likely paths) and “precise” (mostly produces likely paths). This balance is achieved through minimization of a symmetrized cross-entropy between the distribution and demonstration data. By viewing the simulated-outcome distribution as the pushforward of a simple distribution under a simulation operator, we obtain expressions for the cross-entropy metrics that can be efficiently evaluated and differentiated, enabling stochastic-gradient optimization. We propose concrete policy architectures for this model, discuss our evaluation metrics relative to previously-used degenerate metrics, and demonstrate the superiority of our method relative to state-of-the-art methods in both the Kitti dataset and a similar but novel and larger real-world dataset explicitly designed for the vehicle forecasting domain. We consider the problem of learning to act in partially observable, continuous-state-and-action worlds where we have abstract prior knowledge about the structure of the optimal policy in the form of a distribution over policies. Using ideas from planning-as-inference reductions and Bayesian unsupervised learning, we cast Markov Chain Monte Carlo as a stochastic, hill-climbing policy search algorithm. Importantly, this algorithm's search bias is directly tied to the prior and its MCMC proposal kernels, which means we can draw on the full Bayesian toolbox to express the search bias, including nonparametric priors and structured, recursive processes like grammars over action sequences. Furthermore, we can reason about uncertainty in the search bias itself by constructing a hierarchical prior and reasoning about latent variables that determine the abstract structure of the policy. This yields an adaptive search algorithm--our algorithm learns to learn a structured policy efficiently. We show how inference over the latent variables in these policy priors enables intra- and intertask transfer of abstract knowledge. We demonstrate the flexibility of this approach by learning meta search biases, by constructing a nonparametric finite state controller to model memory, by discovering motor primitives using a simple grammar over primitive actions, and by combining all three. Conditional Value at Risk (CVaR) is a prominent risk measure that is being used extensively in various domains. We develop a new formula for the gradient of the CVaR in the form of a conditional expectation. Based on this formula, we propose a novel sampling-based estimator for the CVaR gradient, in the spirit of the likelihood-ratio method. We analyze the bias of the estimator, and prove the convergence of a corresponding stochastic gradient descent algorithm to a local CVaR optimum. Our method allows to consider CVaR optimization in new domains. As an example, we consider a reinforcement learning application, and learn a risk-sensitive controller for the game of Tetris. We propose a new constrained Markov decision process framework with risk-type constraints. The risk metric we use is Conditional Value-at-Risk (CVaR), which is gaining popularity in finance. It is a conditional expectation but the conditioning is defined in terms of the level of the tail probability. We propose an iterative offline algorithm to find the risk-contrained optimal control policy. A stochastic approximation-inspired ‘learning’ variant is also sketched. We investigate the control of constrained stochastic linear systems when faced with limited information regarding the disturbance process, i.e., when only the first two moments of the disturbance distribution are known. We consider two types of distributionally robust constraints. In the first case, we require that the constraints hold with a given probability for all disturbance distributions sharing the known moments. These constraints are commonly referred to as distributionally robust chance constraints. In the second case, we impose conditional value-at-risk (CVaR) constraints to bound the expected constraint violation for all disturbance distributions consistent with the given moment information. Such constraints are referred to as distributionally robust CVaR constraints with second-order moment specifications. We propose a method for designing linear controllers for systems with such constraints that is both computationally tractable and practically meaningful for both finite and infinite horizon problems. We prove in the infinite horizon case that our design procedure produces the globally optimal linear output feedback controller for distributionally robust CVaR and chance constrained problems. The proposed methods are illustrated for a wind blade control design case study for which distributionally robust constraints constitute sensible design objectives. In this work, we take a representation learning perspective on hierarchical reinforcement learning, where the problem of learning lower layers in a hierarchy is transformed into the problem of learning trajectory-level generative models. We show that we can learn continuous latent representations of trajectories, which are effective in solving temporally extended and multi-stage problems. Our proposed model, SeCTAR, draws inspiration from variational autoencoders, and learns latent representations of trajectories. A key component of this method is to learn both a latent-conditioned policy and a latent-conditioned model which are consistent with each other. Given the same latent, the policy generates a trajectory which should match the trajectory predicted by the model. This model provides a built-in prediction mechanism, by predicting the outcome of closed loop policy behavior. We propose a novel algorithm for performing hierarchical RL with this model, combining model-based planning in the learned latent space with an unsupervised exploration objective. We show that our model is effective at reasoning over long horizons with sparse rewards for several simulated tasks, outperforming standard reinforcement learning methods and prior methods for hierarchical reasoning, model-based planning, and exploration. Policy gradient methods have been successfully applied to a variety of reinforcement learning tasks. However, while learning in a simulator, these methods do not utilise the opportunity to improve learning by adjusting certain environment variables: unobservable state features that are randomly determined by the environment in a physical setting, but that are controllable in a simulator. This can lead to slow learning, or convergence to highly suboptimal policies. In this paper, we present contextual policy optimisation (CPO). The central idea is to use Bayesian optimisation to actively select the distribution of the environment variable that maximises the improvement generated by each iteration of the policy gradient method. To make this Bayesian optimisation practical, we contribute two easy-to-compute low-dimensional fingerprints of the current policy. We apply CPO to a number of continuous control tasks of varying difficulty and show that CPO can efficiently learn policies that are robust to significant rare events, which are unlikely to be observable under random sampling but are key to learning good policies. Recently, Bayesian Optimization (BO) has been used to successfully optimize parametric policies in several challenging Reinforcement Learning (RL) applications. BO is attractive for this problem because it exploits Bayesian prior information about the expected return and exploits this knowledge to select new policies to execute. Effectively, the BO framework for policy search addresses the exploration-exploitation tradeoff. In this work, we show how to more effectively apply BO to RL by exploiting the sequential trajectory information generated by RL agents. Our contributions can be broken into two distinct, but mutually beneficial, parts. The first is a new Gaussian process (GP) kernel for measuring the similarity between policies using trajectory data generated from policy executions. This kernel can be used in order to improve posterior estimates of the expected return thereby improving the quality of exploration. The second contribution, is a new GP mean function which uses learned transition and reward functions to approximate the surface of the objective. We show that the model-based approach we develop can recover from model inaccuracies when good transition and reward models cannot be learned. We give empirical results in a standard set of RL benchmarks showing that both our model-based and model-free approaches can speed up learning compared to competing methods. Further, we show that our contributions can be combined to yield synergistic improvement in some domains. In the model-based policy search approach to reinforcement learning (RL), policies are found using a model (or "simulator") of the Markov decision process. However, for high-dimensional continuous-state tasks, it can be extremely difficult to build an accurate model, and thus often the algorithm returns a policy that works in simulation but not in real-life. The other extreme, model-free RL, tends to require infeasibly large numbers of real-life trials. In this paper, we present a hybrid algorithm that requires only an approximate model, and only a small number of real-life trials. The key idea is to successively "ground" the policy evaluations using real-life trials, but to rely on the approximate model to suggest local changes. Our theoretical results show that this algorithm achieves near-optimal performance in the real system, even when the model is only approximate. Empirical results also demonstrate that---when given only a crude model and a small number of real-life trials---our algorithm can obtain near-optimal performance in the real system. In the classic Bayesian restless multi-armed bandit (RMAB) problem, there are N arms, with rewards on all arms evolving at each time as Markov chains with known parameters. A player seeks to activate K ≥ 1 arms at each time in order to maximize the expected total reward obtained over multiple plays. RMAB is a challenging problem that is known to be PSPACE-hard in general. We consider in this work the even harder non-Bayesian RMAB, in which the parameters of the Markov chain are assumed to be unknown a priori. We develop an original approach to this problem that is applicable when the corresponding Bayesian problem has the structure that, depending on the known parameter values, the optimal solution is one of a prescribed finite set of policies. In such settings, we propose to learn the optimal policy for the non-Bayesian RMAB by employing a suitable meta-policy which treats each policy from this finite set as an arm in a different non-Bayesian multi-armed bandit problem for which a single-arm selection policy is optimal. We demonstrate this approach by developing a novel sensing policy for opportunistic spectrum access over unknown dynamic channels. We prove that our policy achieves near-logarithmic regret (the difference in expected reward compared to a model-aware genie), which leads to the same average reward that can be achieved by the optimal policy under a known model. This is the first such result in the literature for a non-Bayesian RMAB. Generative Adversarial Networks (GANs) excel at creating realistic images with complex models for which maximum likelihood is infeasible. However, the convergence of GAN training has still not been proved. We propose a two time-scale update rule (TTUR) for training GANs with stochastic gradient descent on arbitrary GAN loss functions. TTUR has an individual learning rate for both the discriminator and the generator. Using the theory of stochastic approximation, we prove that the TTUR converges under mild assumptions to a stationary local Nash equilibrium. The convergence carries over to the popular Adam optimization, for which we prove that it follows the dynamics of a heavy ball with friction and thus prefers flat minima in the objective landscape. For the evaluation of the performance of GANs at image generation, we introduce the "Frechet Inception Distance" (FID) which captures the similarity of generated images to real ones better than the Inception Score. In experiments, TTUR improves learning for DCGANs and Improved Wasserstein GANs (WGAN-GP) outperforming conventional GAN training on CelebA, CIFAR-10, SVHN, LSUN Bedrooms, and the One Billion Word Benchmark. Abstract This paper proposes a stochastic and dynamic mixed-integer linear program (SD-MILP) for optimal coordinated bidding of a risk-averse profit-maximizing hydropower producer. The day-ahead, intra-day, and real-time markets are considered. To model and predict day-ahead, intra-day, and real-time prices, the Holt–Winter (HW) and the Generalized Autoregressive Conditional Heteroscedastic (GARCH) predictors are combined using a proposed Markov switch. The discrete behavior of intra-day and real-time prices is modeled as different Markov states. The proposed Markov-based HW–GARCH model with a standard scenario generation-reduction technique is used to capture the uncertainty in day-ahead, intra-day, and real-time prices. The time-dependent conditional value at risk (T-CVaR) is proposed to model the risk of trading in different considered markets. The convex combination of the expected profit and T-CVaR is used as the objective of SD-MILP. The Markov-based HW–GARCH is modeled in Matlab and the SD-MILP is coded in GAMS. The Markov-based HW–GARCH predictor and the SD-MILP are used to develop the bidding curve of a three-reservoir hydropower producer using the electricity prices from the Nordic power market. To further examine the developed models a seven-reservoir hydropower producer is also studied. For these two cases, the coordinated bidding curves are derived and discussed. Generative Adversarial Networks (GANs) excel at creating realistic images with complex models for which maximum likelihood is infeasible. However, the convergence of GAN training has still not been proved. We propose a two time-scale update rule (TTUR) for training GANs with stochastic gradient descent on arbitrary GAN loss functions. TTUR has an individual learning rate for both the discriminator and the generator. Using the theory of stochastic approximation, we prove that the TTUR converges under mild assumptions to a stationary local Nash equilibrium. The convergence carries over to the popular Adam optimization, for which we prove that it follows the dynamics of a heavy ball with friction and thus prefers flat minima in the objective landscape. For the evaluation of the performance of GANs at image generation, we introduce the Frechet Inception Distance'' (FID) which captures the similarity of generated images to real ones better than the Inception Score. In experiments, TTUR improves learning for DCGANs and Improved Wasserstein GANs (WGAN-GP) outperforming conventional GAN training on CelebA, CIFAR-10, SVHN, LSUN Bedrooms, and the One Billion Word Benchmark. Many Stochastic Optimal Control (SOC) approaches rely on samples to either obtain an estimate of the value function or a linearisation of the underlying system model. However, these approaches typically neglect the fact that the accuracy of the policy update depends on the closeness of the resulting trajectory distribution to these samples. The greedy operator does not consider such closeness constraint to the samples. Hence, the greedy operator can lead to oscillations or even instabilities in the policy updates. Such undesired behaviour is likely to result in an inferior performance of the estimated policy. We reuse inspiration from the reinforcement learning community and relax the greedy operator used in SOC with an information theoretic bound that limits the ‘distance’ of two subsequent trajectory distributions in a policy update. The introduced bound ensures a smooth and stable policy update. Our method is also well suited for model-based reinforcement learning, where we estimate the system dynamics model from data. As this model is likely to be inaccurate, it might be dangerous to exploit the model greedily. Instead, our bound ensures that we generate new data in the vicinity of the current data, such that we can improve our estimate of the system dynamics model. We show that our approach outperforms several state of the art approaches on challenging simulated robot control tasks. Sample complexity and safety are major challenges when learning policies with reinforcement learning for real-world tasks, especially when the policies are represented using rich function approximators like deep neural networks. Model-based methods where the real-world target domain is approximated using a simulated source domain provide an avenue to tackle the above challenges by augmenting real data with simulated data. However, discrepancies between the simulated source domain and the target domain pose a challenge for simulated training. We introduce the EPOpt algorithm, which uses an ensemble of simulated source domains and a form of adversarial training to learn policies that are robust and generalize to a broad range of possible target domains, including to unmodeled effects. Further, the probability distribution over source domains in the ensemble can be adapted using data from the target domain and approximate Bayesian methods, to progressively make it a better approximation. Thus, learning on a model ensemble, along with source domain adaptation, provides the benefit of both robustness and learning. We present a new approach to the problems of evaluating and learning personalized decision policies from observational data of past contexts, decisions, and outcomes. Only the outcome of the enacted decision is available and the historical policy is unknown. These problems arise in personalized medicine using electronic health records and in internet advertising. Existing approaches use inverse propensity weighting (or, doubly robust versions) to make historical outcome (or, residual) data look like it were generated by a new policy being evaluated or learned. But this relies on a plug-in approach that rejects data points with a decision that disagrees with the new policy, leading to high variance estimates and ineffective learning. We propose a new, balance-based approach that too makes the data look like the new policy but does so directly by finding weights that optimize for balance between the weighted data and the target policy in the given, finite sample, which is equivalent to minimizing worst-case or posterior conditional mean square error. Our policy learner proceeds as a two-level optimization problem over policies and weights. We demonstrate that this approach markedly outperforms existing ones both in evaluation and learning, which is unsurprising given the wider support of balance-based weights. We establish extensive theoretical consistency guarantees and regret bounds that support this empirical success. Sample complexity and safety are major challenges when learning policies with reinforcement learning for real-world tasks, especially when the policies are represented using rich function approximators like deep neural networks. Model-based methods where the real-world target domain is approximated using a simulated source domain provide an avenue to tackle the above challenges by augmenting real data with simulated data. However, discrepancies between the simulated source domain and the target domain pose a challenge for simulated training. We introduce the EPOpt algorithm, which uses an ensemble of simulated source domains and a form of adversarial training to learn policies that are robust and generalize to a broad range of possible target domains, including unmodeled effects. Further, the probability distribution over source domains in the ensemble can be adapted using data from target domain and approximate Bayesian methods, to progressively make it a better approximation. Thus, learning on a model ensemble, along with source domain adaptation, provides the benefit of both robustness and learning adaptation. With the goal to generate more scalable algorithms with higher efficiency and fewer open parameters, reinforcement learning (RL) has recently moved towards combining classical techniques from optimal control and dynamic programming with modern learning techniques from statistical estimation theory. In this vein, this paper suggests to use the framework of stochastic optimal control with path integrals to derive a novel approach to RL with parameterized policies. While solidly grounded in value function estimation and optimal control based on the stochastic Hamilton-Jacobi-Bellman (HJB) equations, policy improvements can be transformed into an approximation problem of a path integral which has no open algorithmic parameters other than the exploration noise. The resulting algorithm can be conceived of as model-based, semi-model-based, or even model free, depending on how the learning problem is structured. The update equations have no danger of numerical instabilities as neither matrix inversions nor gradient learning rates are required. Our new algorithm demonstrates interesting similarities with previous RL research in the framework of probability matching and provides intuition why the slightly heuristically motivated probability matching approach can actually perform well. Empirical evaluations demonstrate significant performance improvements over gradient-based policy learning and scalability to high-dimensional control problems. Finally, a learning experiment on a simulated 12 degree-of-freedom robot dog illustrates the functionality of our algorithm in a complex robot learning scenario. We believe that Policy Improvement with Path Integrals (PI2) offers currently one of the most efficient, numerically robust, and easy to implement algorithms for RL based on trajectory roll-outs. We propose in this paper a new approach to train the Generative Adversarial Nets (GANs) with a mixture of generators to overcome the mode collapsing problem. The main intuition is to employ multiple generators, instead of using a single one as in the original GAN. The idea is simple, yet proven to be extremely effective at covering diverse data modes, easily overcoming the mode collapsing problem and delivering state-of-the-art results. A minimax formulation was able to establish among a classifier, a discriminator, and a set of generators in a similar spirit with GAN. Generators create samples that are intended to come from the same distribution as the training data, whilst the discriminator determines whether samples are true data or generated by generators, and the classifier specifies which generator a sample comes from. The distinguishing feature is that internal samples are created from multiple generators, and then one of them will be randomly selected as final output similar to the mechanism of a probabilistic mixture model. We term our method Mixture Generative Adversarial Nets (MGAN). We develop theoretical analysis to prove that, at the equilibrium, the Jensen-Shannon divergence (JSD) between the mixture of generators’ distributions and the empirical data distribution is minimal, whilst the JSD among generators’ distributions is maximal, hence effectively avoiding the mode collapsing problem. By utilizing parameter sharing, our proposed model adds minimal computational cost to the standard GAN, and thus can also efficiently scale to large-scale datasets. We conduct extensive experiments on synthetic 2D data and natural image databases (CIFAR-10, STL-10 and ImageNet) to demonstrate the superior performance of our MGAN in achieving state-of-the-art Inception scores over latest baselines, generating diverse and appealing recognizable objects at different resolutions, and specializing in capturing different types of objects by the generators.