Felix Berkenkamp

Melrose Roderick, Gaurav Manek, Felix Berkenkamp et al. · cmu

A key problem in off-policy Reinforcement Learning (RL) is the mismatch, or distribution shift, between the dataset and the distribution over states and actions visited by the learned policy. This problem is exacerbated in the fully offline setting. The main approach to correct this shift has been through importance sampling, which leads to high-variance gradients. Other approaches, such as conservatism or behavior-regularization, regularize the policy at the cost of performance. In this paper, we propose a new approach for stable off-policy Q-Learning. Our method, Projected Off-Policy Q-Learning (POP-QL), is a novel actor-critic algorithm that simultaneously reweights off-policy samples and constrains the policy to prevent divergence and reduce value-approximation error. In our experiments, POP-QL not only shows competitive performance on standard benchmarks, but also out-performs competing methods in tasks where the data-collection policy is significantly sub-optimal.

Carlos E. Luis, Alessandro G. Bottero, Julia Vinogradska et al.

Quantifying uncertainty about a policy's long-term performance is important to solve sequential decision-making tasks. We study the problem from a model-based Bayesian reinforcement learning perspective, where the goal is to learn the posterior distribution over value functions induced by parameter (epistemic) uncertainty of the Markov decision process. Previous work restricts the analysis to a few moments of the distribution over values or imposes a particular distribution shape, e.g., Gaussians. Inspired by distributional reinforcement learning, we introduce a Bellman operator whose fixed-point is the value distribution function. Based on our theory, we propose Epistemic Quantile-Regression (EQR), a model-based algorithm that learns a value distribution function. We combine EQR with soft actor-critic (SAC) for policy optimization with an arbitrary differentiable objective function of the learned value distribution. Evaluation across several continuous-control tasks shows performance benefits with respect to both model-based and model-free algorithms. The code is available at https://github.com/boschresearch/dist-mbrl.

Carlos E. Luis, Alessandro G. Bottero, Julia Vinogradska et al.

We consider the problem of quantifying uncertainty over expected cumulative rewards in model-based reinforcement learning. In particular, we focus on characterizing the variance over values induced by a distribution over MDPs. Previous work upper bounds the posterior variance over values by solving a so-called uncertainty Bellman equation, but the over-approximation may result in inefficient exploration. We propose a new uncertainty Bellman equation whose solution converges to the true posterior variance over values and explicitly characterizes the gap in previous work. Moreover, our uncertainty quantification technique is easily integrated into common exploration strategies and scales naturally beyond the tabular setting by using standard deep reinforcement learning architectures. Experiments in difficult exploration tasks, both in tabular and continuous control settings, show that our sharper uncertainty estimates improve sample-efficiency.

Alessandro G. Bottero, Carlos E. Luis, Julia Vinogradska et al.

We consider a sequential decision making task where we are not allowed to evaluate parameters that violate an a priori unknown (safety) constraint. A common approach is to place a Gaussian process prior on the unknown constraint and allow evaluations only in regions that are safe with high probability. Most current methods rely on a discretization of the domain and cannot be directly extended to the continuous case. Moreover, the way in which they exploit regularity assumptions about the constraint introduces an additional critical hyperparameter. In this paper, we propose an information-theoretic safe exploration criterion that directly exploits the GP posterior to identify the most informative safe parameters to evaluate. Our approach is naturally applicable to continuous domains and does not require additional hyperparameters. We theoretically analyze the method and show that we do not violate the safety constraint with high probability and that we explore by learning about the constraint up to arbitrary precision. Empirical evaluations demonstrate improved data-efficiency and scalability.

Jiarong Pan, Stefan Falkner, Felix Berkenkamp et al.

Bayesian optimization (BO) is a popular method to optimize costly black-box functions. While traditional BO optimizes each new target task from scratch, meta-learning has emerged as a way to leverage knowledge from related tasks to optimize new tasks faster. However, existing meta-learning BO methods rely on surrogate models that suffer from scalability issues and are sensitive to observations with different scales and noise types across tasks. Moreover, they often overlook the uncertainty associated with task similarity. This leads to unreliable task adaptation when only limited observations are obtained or when the new tasks differ significantly from the related tasks. To address these limitations, we propose a novel meta-learning BO approach that bypasses the surrogate model and directly learns the utility of queries across tasks. Our method explicitly models task uncertainty and includes an auxiliary model to enable robust adaptation to new tasks. Extensive experiments show that our method demonstrates strong anytime performance and outperforms state-of-the-art meta-learning BO methods in various benchmarks.

Carlos E. Luis, Alessandro G. Bottero, Julia Vinogradska et al.

Optimal decision-making under partial observability requires reasoning about the uncertainty of the environment's hidden state. However, most reinforcement learning architectures handle partial observability with sequence models that have no internal mechanism to incorporate uncertainty in their hidden state representation, such as recurrent neural networks, deterministic state-space models and transformers. Inspired by advances in probabilistic world models for reinforcement learning, we propose a standalone Kalman filter layer that performs closed-form Gaussian inference in linear state-space models and train it end-to-end within a model-free architecture to maximize returns. Similar to efficient linear recurrent layers, the Kalman filter layer processes sequential data using a parallel scan, which scales logarithmically with the sequence length. By design, Kalman filter layers are a drop-in replacement for other recurrent layers in standard model-free architectures, but importantly they include an explicit mechanism for probabilistic filtering of the latent state representation. Experiments in a wide variety of tasks with partial observability show that Kalman filter layers excel in problems where uncertainty reasoning is key for decision-making, outperforming other stateful models.

Aleph Alpha, Adnen Abdessaied, Artur Baranowski et al.

Tokenization is a central component of natural language processing in current large language models (LLMs), enabling models to convert raw text into processable units. Although learned tokenizers are widely adopted, they exhibit notable limitations, including their large, fixed vocabulary sizes and poor adaptability to new domains or languages. We present a family of models with up to 70 billion parameters based on the hierarchical autoregressive transformer (HAT) architecture. In HAT, an encoder transformer aggregates bytes into word embeddings and then feeds them to the backbone, a classical autoregressive transformer. The outputs of the backbone are then cross-attended by the decoder and converted back into bytes. We show that we can reuse available pre-trained models by converting the Llama 3.1 8B and 70B models into the HAT architecture: Llama-3.1-8B-TFree-HAT and Llama-3.1-70B-TFree-HAT are byte-level models whose encoder and decoder are trained from scratch, but where we adapt the pre-trained Llama backbone, i.e., the transformer blocks with the embedding matrix and head removed, to handle word embeddings instead of the original tokens. We also provide a 7B HAT model, Llama-TFree-HAT-Pretrained, trained entirely from scratch on nearly 4 trillion words. The HAT architecture improves text compression by reducing the number of required sequence positions and enhances robustness to intra-word variations, e.g., spelling differences. Through pre-training, as well as subsequent supervised fine-tuning and direct preference optimization in English and German, we show strong proficiency in both languages, improving on the original Llama 3.1 in most benchmarks. We release our models (including 200 pre-training checkpoints) on Hugging Face.

Carlos E. Luis, Alessandro G. Bottero, Julia Vinogradska et al.

We consider the problem of quantifying uncertainty over expected cumulative rewards in model-based reinforcement learning. In particular, we focus on characterizing the variance over values induced by a distribution over Markov decision processes (MDPs). Previous work upper bounds the posterior variance over values by solving a so-called uncertainty Bellman equation (UBE), but the over-approximation may result in inefficient exploration. We propose a new UBE whose solution converges to the true posterior variance over values and leads to lower regret in tabular exploration problems. We identify challenges to apply the UBE theory beyond tabular problems and propose a suitable approximation. Based on this approximation, we introduce a general-purpose policy optimization algorithm, Q-Uncertainty Soft Actor-Critic (QU-SAC), that can be applied for either risk-seeking or risk-averse policy optimization with minimal changes. Experiments in both online and offline RL demonstrate improved performance compared to other uncertainty estimation methods.

Alessandro G. Bottero, Carlos E. Luis, Julia Vinogradska et al.

We consider a sequential decision making task, where the goal is to optimize an unknown function without evaluating parameters that violate an a~priori unknown (safety) constraint. A common approach is to place a Gaussian process prior on the unknown functions and allow evaluations only in regions that are safe with high probability. Most current methods rely on a discretization of the domain and cannot be directly extended to the continuous case. Moreover, the way in which they exploit regularity assumptions about the constraint introduces an additional critical hyperparameter. In this paper, we propose an information-theoretic safe exploration criterion that directly exploits the GP posterior to identify the most informative safe parameters to evaluate. The combination of this exploration criterion with a well known Bayesian optimization acquisition function yields a novel safe Bayesian optimization selection criterion. Our approach is naturally applicable to continuous domains and does not require additional explicit hyperparameters. We theoretically analyze the method and show that we do not violate the safety constraint with high probability and that we learn about the value of the safe optimum up to arbitrary precision. Empirical evaluations demonstrate improved data-efficiency and scalability.

Melrose Roderick, Felix Berkenkamp, Fatemeh Sheikholeslami et al. · cmu

In many real-world problems, there is a limited set of training data, but an abundance of unlabeled data. We propose a new method, Generative Posterior Networks (GPNs), that uses unlabeled data to estimate epistemic uncertainty in high-dimensional problems. A GPN is a generative model that, given a prior distribution over functions, approximates the posterior distribution directly by regularizing the network towards samples from the prior. We prove theoretically that our method indeed approximates the Bayesian posterior and show empirically that it improves epistemic uncertainty estimation and scalability over competing methods.

Petru Tighineanu, Kathrin Skubch, Paul Baireuther et al.

Bayesian optimization is a powerful paradigm to optimize black-box functions based on scarce and noisy data. Its data efficiency can be further improved by transfer learning from related tasks. While recent transfer models meta-learn a prior based on large amount of data, in the low-data regime methods that exploit the closed-form posterior of Gaussian processes (GPs) have an advantage. In this setting, several analytically tractable transfer-model posteriors have been proposed, but the relative advantages of these methods are not well understood. In this paper, we provide a unified view on hierarchical GP models for transfer learning, which allows us to analyze the relationship between methods. As part of the analysis, we develop a novel closed-form boosted GP transfer model that fits between existing approaches in terms of complexity. We evaluate the performance of the different approaches in large-scale experiments and highlight strengths and weaknesses of the different transfer-learning methods.

Lukas P. Fröhlich, Maksym Lefarov, Melanie N. Zeilinger et al.

Model-free reinforcement learning algorithms can compute policy gradients given sampled environment transitions, but require large amounts of data. In contrast, model-based methods can use the learned model to generate new data, but model errors and bias can render learning unstable or suboptimal. In this paper, we present a novel method that combines real-world data and a learned model in order to get the best of both worlds. The core idea is to exploit the real-world data for on-policy predictions and use the learned model only to generalize to different actions. Specifically, we use the data as time-dependent on-policy correction terms on top of a learned model, to retain the ability to generate data without accumulating errors over long prediction horizons. We motivate this method theoretically and show that it counteracts an error term for model-based policy improvement. Experiments on MuJoCo- and PyBullet-benchmarks show that our method can drastically improve existing model-based approaches without introducing additional tuning parameters.

Sebastian Curi, Felix Berkenkamp, Andreas Krause

Model-based reinforcement learning algorithms with probabilistic dynamical models are amongst the most data-efficient learning methods. This is often attributed to their ability to distinguish between epistemic and aleatoric uncertainty. However, while most algorithms distinguish these two uncertainties for learning the model, they ignore it when optimizing the policy, which leads to greedy and insufficient exploration. At the same time, there are no practical solvers for optimistic exploration algorithms. In this paper, we propose a practical optimistic exploration algorithm (H-UCRL). H-UCRL reparameterizes the set of plausible models and hallucinates control directly on the epistemic uncertainty. By augmenting the input space with the hallucinated inputs, H-UCRL can be solved using standard greedy planners. Furthermore, we analyze H-UCRL and construct a general regret bound for well-calibrated models, which is provably sublinear in the case of Gaussian Process models. Based on this theoretical foundation, we show how optimistic exploration can be easily combined with state-of-the-art reinforcement learning algorithms and different probabilistic models. Our experiments demonstrate that optimistic exploration significantly speeds-up learning when there are penalties on actions, a setting that is notoriously difficult for existing model-based reinforcement learning algorithms.

Noah Y. Siegel, Jost Tobias Springenberg, Felix Berkenkamp et al.

Off-policy reinforcement learning algorithms promise to be applicable in settings where only a fixed data-set (batch) of environment interactions is available and no new experience can be acquired. This property makes these algorithms appealing for real world problems such as robot control. In practice, however, standard off-policy algorithms fail in the batch setting for continuous control. In this paper, we propose a simple solution to this problem. It admits the use of data generated by arbitrary behavior policies and uses a learned prior -- the advantage-weighted behavior model (ABM) -- to bias the RL policy towards actions that have previously been executed and are likely to be successful on the new task. Our method can be seen as an extension of recent work on batch-RL that enables stable learning from conflicting data-sources. We find improvements on competitive baselines in a variety of RL tasks -- including standard continuous control benchmarks and multi-task learning for simulated and real-world robots.

Matteo Turchetta, Felix Berkenkamp, Andreas Krause

In Interactive Machine Learning (IML), we iteratively make decisions and obtain noisy observations of an unknown function. While IML methods, e.g., Bayesian optimization and active learning, have been successful in applications, on real-world systems they must provably avoid unsafe decisions. To this end, safe IML algorithms must carefully learn about a priori unknown constraints without making unsafe decisions. Existing algorithms for this problem learn about the safety of all decisions to ensure convergence. This is sample-inefficient, as it explores decisions that are not relevant for the original IML objective. In this paper, we introduce a novel framework that renders any existing unsafe IML algorithm safe. Our method works as an add-on that takes suggested decisions as input and exploits regularity assumptions in terms of a Gaussian process prior in order to efficiently learn about their safety. As a result, we only explore the safe set when necessary for the IML problem. We apply our framework to safe Bayesian optimization and to safe exploration in deterministic Markov Decision Processes (MDP), which have been analyzed separately before. Our method outperforms other algorithms empirically.

Silvan Melchior, Sebastian Curi, Felix Berkenkamp et al.

We propose a new variational inference algorithm for learning in Gaussian Process State-Space Models (GPSSMs). Our algorithm enables learning of unstable and partially observable systems, where previous algorithms fail. Our main algorithmic contribution is a novel approximate posterior that can be calculated efficiently using a single forward and backward pass along the training trajectories. The forward-backward pass is inspired on Kalman smoothing for linear dynamical systems but generalizes to GPSSMs. Our second contribution is a modification of the conditioning step that effectively lowers the Kalman gain. This modification is crucial to attaining good test performance where no measurements are available. Finally, we show experimentally that our learning algorithm performs well in stable and unstable real systems with hidden states.

Torsten Koller, Felix Berkenkamp, Matteo Turchetta et al.

Reinforcement learning has been successfully used to solve difficult tasks in complex unknown environments. However, these methods typically do not provide any safety guarantees during the learning process. This is particularly problematic, since reinforcement learning agent actively explore their environment. This prevents their use in safety-critical, real-world applications. In this paper, we present a learning-based model predictive control scheme that provides high-probability safety guarantees throughout the learning process. Based on a reliable statistical model, we construct provably accurate confidence intervals on predicted trajectories. Unlike previous approaches, we allow for input-dependent uncertainties. Based on these reliable predictions, we guarantee that trajectories satisfy safety constraints. Moreover, we use a terminal set constraint to recursively guarantee the existence of safe control actions at every iteration. We evaluate the resulting algorithm to safely explore the dynamics of an inverted pendulum and to solve a reinforcement learning task on a cart-pole system with safety constraints.

Felix Berkenkamp, Angela P. Schoellig, Andreas Krause

Bayesian optimization (BO) based on Gaussian process models is a powerful paradigm to optimize black-box functions that are expensive to evaluate. While several BO algorithms provably converge to the global optimum of the unknown function, they assume that the hyperparameters of the kernel are known in advance. This is not the case in practice and misspecification often causes these algorithms to converge to poor local optima. In this paper, we present the first BO algorithm that is provably no-regret and converges to the optimum without knowledge of the hyperparameters. During optimization we slowly adapt the hyperparameters of stationary kernels and thereby expand the associated function class over time, so that the BO algorithm considers more complex function candidates. Based on the theoretical insights, we propose several practical algorithms that achieve the empirical sample efficiency of BO with online hyperparameter estimation, but retain theoretical convergence guarantees. We evaluate our method on several benchmark problems.

Nikolay Nikolov, Johannes Kirschner, Felix Berkenkamp et al.

Efficient exploration remains a major challenge for reinforcement learning. One reason is that the variability of the returns often depends on the current state and action, and is therefore heteroscedastic. Classical exploration strategies such as upper confidence bound algorithms and Thompson sampling fail to appropriately account for heteroscedasticity, even in the bandit setting. Motivated by recent findings that address this issue in bandits, we propose to use Information-Directed Sampling (IDS) for exploration in reinforcement learning. As our main contribution, we build on recent advances in distributional reinforcement learning and propose a novel, tractable approximation of IDS for deep Q-learning. The resulting exploration strategy explicitly accounts for both parametric uncertainty and heteroscedastic observation noise. We evaluate our method on Atari games and demonstrate a significant improvement over alternative approaches.

Robin Spiess, Felix Berkenkamp, Jan Poland et al.

The energy output of photovoltaic (PV) power plants depends on the environment and thus fluctuates over time. As a result, PV power can cause instability in the power grid, in particular when increasingly used. Limiting the rate of change of the power output is a common way to mitigate these fluctuations, often with the help of large batteries. A reactive controller that uses these batteries to compensate ramps works in practice, but causes stress on the battery due to a high energy throughput. In this paper, we present a deep learning approach that uses images of the sky to compensate power fluctuations predictively and reduces battery stress. In particular, we show that the optimal control policy can be computed using information that is only available in hindsight. Based on this, we use imitation learning to train a neural network that approximates this hindsight-optimal policy, but uses only currently available sky images and sensor data. We evaluate our method on a large dataset of measurements and images from a real power plant and show that the trained policy reduces stress on the battery.

Spencer M. Richards, Felix Berkenkamp, Andreas Krause

Learning algorithms have shown considerable prowess in simulation by allowing robots to adapt to uncertain environments and improve their performance. However, such algorithms are rarely used in practice on safety-critical systems, since the learned policy typically does not yield any safety guarantees. That is, the required exploration may cause physical harm to the robot or its environment. In this paper, we present a method to learn accurate safety certificates for nonlinear, closed-loop dynamical systems. Specifically, we construct a neural network Lyapunov function and a training algorithm that adapts it to the shape of the largest safe region in the state space. The algorithm relies only on knowledge of inputs and outputs of the dynamics, rather than on any specific model structure. We demonstrate our method by learning the safe region of attraction for a simulated inverted pendulum. Furthermore, we discuss how our method can be used in safe learning algorithms together with statistical models of dynamical systems.

Torsten Koller, Felix Berkenkamp, Matteo Turchetta et al.

Learning-based methods have been successful in solving complex control tasks without significant prior knowledge about the system. However, these methods typically do not provide any safety guarantees, which prevents their use in safety-critical, real-world applications. In this paper, we present a learning-based model predictive control scheme that can provide provable high-probability safety guarantees. To this end, we exploit regularity assumptions on the dynamics in terms of a Gaussian process prior to construct provably accurate confidence intervals on predicted trajectories. Unlike previous approaches, we do not assume that model uncertainties are independent. Based on these predictions, we guarantee that trajectories satisfy safety constraints. Moreover, we use a terminal set constraint to recursively guarantee the existence of safe control actions at every iteration. In our experiments, we show that the resulting algorithm can be used to safely and efficiently explore and learn about dynamic systems.

Shromona Ghosh, Felix Berkenkamp, Gireeja Ranade et al.

Recent successes in reinforcement learning have lead to the development of complex controllers for real-world robots. As these robots are deployed in safety-critical applications and interact with humans, it becomes critical to ensure safety in order to avoid causing harm. A first step in this direction is to test the controllers in simulation. To be able to do this, we need to capture what we mean by safety and then efficiently search the space of all behaviors to see if they are safe. In this paper, we present an active-testing framework based on Bayesian Optimization. We specify safety constraints using logic and exploit structure in the problem in order to test the system for adversarial counter examples that violate the safety specifications. These specifications are defined as complex boolean combinations of smooth functions on the trajectories and, unlike reward functions in reinforcement learning, are expressive and impose hard constraints on the system. In our framework, we exploit regularity assumptions on individual functions in form of a Gaussian Process (GP) prior. We combine these into a coherent optimization framework using problem structure. The resulting algorithm is able to provably verify complex safety specifications or alternatively find counter examples. Experimental results show that the proposed method is able to find adversarial examples quickly.

Felix Berkenkamp, Matteo Turchetta, Angela P. Schoellig et al.

Reinforcement learning is a powerful paradigm for learning optimal policies from experimental data. However, to find optimal policies, most reinforcement learning algorithms explore all possible actions, which may be harmful for real-world systems. As a consequence, learning algorithms are rarely applied on safety-critical systems in the real world. In this paper, we present a learning algorithm that explicitly considers safety, defined in terms of stability guarantees. Specifically, we extend control-theoretic results on Lyapunov stability verification and show how to use statistical models of the dynamics to obtain high-performance control policies with provable stability certificates. Moreover, under additional regularity assumptions in terms of a Gaussian process prior, we prove that one can effectively and safely collect data in order to learn about the dynamics and thus both improve control performance and expand the safe region of the state space. In our experiments, we show how the resulting algorithm can safely optimize a neural network policy on a simulated inverted pendulum, without the pendulum ever falling down.

Alonso Marco, Felix Berkenkamp, Philipp Hennig et al.

In practice, the parameters of control policies are often tuned manually. This is time-consuming and frustrating. Reinforcement learning is a promising alternative that aims to automate this process, yet often requires too many experiments to be practical. In this paper, we propose a solution to this problem by exploiting prior knowledge from simulations, which are readily available for most robotic platforms. Specifically, we extend Entropy Search, a Bayesian optimization algorithm that maximizes information gain from each experiment, to the case of multiple information sources. The result is a principled way to automatically combine cheap, but inaccurate information from simulations with expensive and accurate physical experiments in a cost-effective manner. We apply the resulting method to a cart-pole system, which confirms that the algorithm can find good control policies with fewer experiments than standard Bayesian optimization on the physical system only.

Matteo Turchetta, Felix Berkenkamp, Andreas Krause

In classical reinforcement learning, when exploring an environment, agents accept arbitrary short term loss for long term gain. This is infeasible for safety critical applications, such as robotics, where even a single unsafe action may cause system failure. In this paper, we address the problem of safely exploring finite Markov decision processes (MDP). We define safety in terms of an, a priori unknown, safety constraint that depends on states and actions. We aim to explore the MDP under this constraint, assuming that the unknown function satisfies regularity conditions expressed via a Gaussian process prior. We develop a novel algorithm for this task and prove that it is able to completely explore the safely reachable part of the MDP without violating the safety constraint. To achieve this, it cautiously explores safe states and actions in order to gain statistical confidence about the safety of unvisited state-action pairs from noisy observations collected while navigating the environment. Moreover, the algorithm explicitly considers reachability when exploring the MDP, ensuring that it does not get stuck in any state with no safe way out. We demonstrate our method on digital terrain models for the task of exploring an unknown map with a rover.

Felix Berkenkamp, Andreas Krause, Angela P. Schoellig

Robotic algorithms typically depend on various parameters, the choice of which significantly affects the robot's performance. While an initial guess for the parameters may be obtained from dynamic models of the robot, parameters are usually tuned manually on the real system to achieve the best performance. Optimization algorithms, such as Bayesian optimization, have been used to automate this process. However, these methods may evaluate unsafe parameters during the optimization process that lead to safety-critical system failures. Recently, a safe Bayesian optimization algorithm, called SafeOpt, has been developed, which guarantees that the performance of the system never falls below a critical value; that is, safety is defined based on the performance function. However, coupling performance and safety is often not desirable in robotics. For example, high-gain controllers might achieve low average tracking error (performance), but can overshoot and violate input constraints. In this paper, we present a generalized algorithm that allows for multiple safety constraints separate from the objective. Given an initial set of safe parameters, the algorithm maximizes performance but only evaluates parameters that satisfy safety for all constraints with high probability. To this end, it carefully explores the parameter space by exploiting regularity assumptions in terms of a Gaussian process prior. Moreover, we show how context variables can be used to safely transfer knowledge to new situations and tasks. We provide a theoretical analysis and demonstrate that the proposed algorithm enables fast, automatic, and safe optimization of tuning parameters in experiments on a quadrotor vehicle.

Felix Berkenkamp, Angela P. Schoellig, Andreas Krause

One of the most fundamental problems when designing controllers for dynamic systems is the tuning of the controller parameters. Typically, a model of the system is used to obtain an initial controller, but ultimately the controller parameters must be tuned manually on the real system to achieve the best performance. To avoid this manual tuning step, methods from machine learning, such as Bayesian optimization, have been used. However, as these methods evaluate different controller parameters on the real system, safety-critical system failures may happen. In this paper, we overcome this problem by applying, for the first time, a recently developed safe optimization algorithm, SafeOpt, to the problem of automatic controller parameter tuning. Given an initial, low-performance controller, SafeOpt automatically optimizes the parameters of a control law while guaranteeing safety. It models the underlying performance measure as a Gaussian process and only explores new controller parameters whose performance lies above a safe performance threshold with high probability. Experimental results on a quadrotor vehicle indicate that the proposed method enables fast, automatic, and safe optimization of controller parameters without human intervention.