We consider Bayesian optimization of the output of a network of functions, where each function takes as input the output of its parent nodes, and where the network takes significant time to evaluate. Such problems arise, for example, in reinforcement learning, engineering design, and manufacturing. While the standard Bayesian optimization approach observes only the final output, our approach delivers greater query efficiency by leveraging information that the former ignores: intermediate output within the network. This is achieved by modeling the nodes of the network using Gaussian processes and choosing the points to evaluate using, as our acquisition function, the expected improvement computed with respect to the implied posterior on the objective. Although the non-Gaussian nature of this posterior prevents computing our acquisition function in closed form, we show that it can be efficiently maximized via sample average approximation. In addition, we prove that our method is asymptotically consistent, meaning that it finds a globally optimal solution as the number of evaluations grows to infinity, thus generalizing previously known convergence results for the expected improvement. Notably, this holds even though our method might not evaluate the domain densely, instead leveraging problem structure to leave regions unexplored. Finally, we show that our approach dramatically outperforms standard Bayesian optimization methods in several synthetic and real-world problems.
Bayesian optimization (BO) is a framework for global optimization of expensive-to-evaluate objective functions. Classical BO methods assume that the objective function is a black box. However, internal information about objective function computation is often available. For example, when optimizing a manufacturing line's throughput with simulation, we observe the number of parts waiting at each workstation, in addition to the overall throughput. Recent BO methods leverage such internal information to dramatically improve performance. We call these "grey-box" BO methods because they treat objective computation as partially observable and even modifiable, blending the black-box approach with so-called "white-box" first-principles knowledge of objective function computation. This tutorial describes these methods, focusing on BO of composite objective functions, where one can observe and selectively evaluate individual constituents that feed into the overall objective; and multi-fidelity BO, where one can evaluate cheaper approximations of the objective function by varying parameters of the evaluation oracle.
Bayesian Optimization algorithm has become a promising approach for nonlinear global optimization problems and many machine learning applications. Over the past few years, improvements and enhancements have been brought forward and they have shown some promising results in solving the complex dynamic problems, systems of ordinary differential equations where the objective functions are computationally expensive to evaluate. Besides, the straightforward implementation of the Bayesian Optimization algorithm performs well merely for optimization problems with 10-20 dimensions. The study presented in this paper proposes a new high dimensional Bayesian Optimization algorithm combining Recurrent neural networks, which is expected to predict the optimal solution for the global optimization problems with high dimensional or time series decision models. The proposed RNN-BO algorithm can solve the optimal control problems in the lower dimension space and then learn from the historical data using the recurrent neural network to learn the historical optimal solution data and predict the optimal control strategy for any new initial system value setting. In addition, accurately and quickly providing the optimal control strategy is essential to effectively and efficiently control the epidemic spread while minimizing the associated financial costs. Therefore, to verify the effectiveness of the proposed algorithm, computational experiments are carried out on a deterministic SEIR epidemic model and a stochastic SIS optimal control model. Finally, we also discuss the impacts of different numbers of the RNN layers and training epochs on the trade-off between solution quality and related computational efforts.
In the pursuit of efficient optimization of expensive-to-evaluate systems, this paper investigates a novel approach to Bayesian multi-objective and multi-fidelity (MOMF) optimization. Traditional optimization methods, while effective, often encounter prohibitively high costs in multi-dimensional optimizations of one or more objectives. Multi-fidelity approaches offer potential remedies by utilizing multiple, less costly information sources, such as low-resolution simulations. However, integrating these two strategies presents a significant challenge. We suggest the innovative use of a trust metric to support simultaneous optimization of multiple objectives and data sources. Our method modifies a multi-objective optimization policy to incorporate the trust gain per evaluation cost as one objective in a Pareto optimization problem, enabling simultaneous MOMF at lower costs. We present and compare two MOMF optimization methods: a holistic approach selecting both the input parameters and the trust parameter jointly, and a sequential approach for benchmarking. Through benchmarks on synthetic test functions, our approach is shown to yield significant cost reductions - up to an order of magnitude compared to pure multi-objective optimization. Furthermore, we find that joint optimization of the trust and objective domains outperforms addressing them in sequential manner. We validate our results using the use case of optimizing laser-plasma acceleration simulations, demonstrating our method's potential in Pareto optimization of high-cost black-box functions. Implementing these methods in existing Bayesian frameworks is simple, and they can be readily extended to batch optimization. With their capability to handle various continuous or discrete fidelity dimensions, our techniques offer broad applicability in solving simulation problems in fields such as plasma physics and fluid dynamics.
ByteScheduler partitions and rearranges tensor transmissions to improve the communication efficiency of distributed Deep Neural Network (DNN) training. The configuration of hyper-parameters (i.e., the partition size and the credit size) is critical to the effectiveness of partitioning and rearrangement. Currently, ByteScheduler adopts Bayesian Optimization (BO) to find the optimal configuration for the hyper-parameters beforehand. In practice, however, various runtime factors (e.g., worker node status and network conditions) change over time, making the statically-determined one-shot configuration result suboptimal for real-world DNN training. To address this problem, we present a real-time configuration method (called AutoByte) that automatically and timely searches the optimal hyper-parameters as the training systems dynamically change. AutoByte extends the ByteScheduler framework with a meta-network, which takes the system's runtime statistics as its input and outputs predictions for speedups under specific configurations. Evaluation results on various DNN models show that AutoByte can dynamically tune the hyper-parameters with low resource usage, and deliver up to 33.2\% higher performance than the best static configuration in ByteScheduler.
Reinforcement learning agents perform well when presented with inputs within the distribution of those encountered during training. However, they are unable to respond effectively when faced with novel, out-of-distribution events, until they have undergone additional training. This paper presents an online, data-driven, emergency-response method that aims to provide autonomous agents the ability to react to unexpected situations that are very different from those it has been trained or designed to address. In such situations, learned policies cannot be expected to perform appropriately since the observations obtained in these novel situations would fall outside the distribution of inputs that the agent has been optimized to handle. The proposed approach devises a customized response to the unforeseen situation sequentially, by selecting actions that minimize the rate of increase of the reconstruction error from a variational auto-encoder. This optimization is achieved online in a data-efficient manner (on the order of 30 data-points) using a modified Bayesian optimization procedure. We demonstrate the potential of this approach in a simulated 3D car driving scenario, in which the agent devises a response in under 2 seconds to avoid collisions with objects it has not seen during training.
Current auto-tuning frameworks struggle with tuning computer systems configurations due to their large parameter space, complex interdependencies, and high evaluation cost. Utilizing probabilistic models, Structured Bayesian Optimization (SBO) has recently overcome these difficulties. SBO decomposes the parameter space by utilizing contextual information provided by system experts leading to fast convergence. However, the complexity of building probabilistic models has hindered its wider adoption. We propose BoAnon, a SBO framework that learns the system structure from its logs. BoAnon provides an API enabling experts to encode knowledge of the system as performance models or components dependency. BoAnon takes in the learned structure and transforms it into a probabilistic graph model. Then it applies the expert-provided knowledge to the graph to further contextualize the system behavior. BoAnon probabilistic graph allows the optimizer to find efficient configurations faster than other methods. We evaluate BoAnon via a hardware architecture search problem, achieving an improvement in energy-latency objectives ranging from $5-7$ x-factors improvement over the default architecture. With its novel contextual structure learning pipeline, BoAnon makes using SBO accessible for a wide range of other computer systems such as databases and stream processors.
Black box optimization requires specifying a search space to explore for solutions, e.g. a d-dimensional compact space, and this choice is critical for getting the best results at a reasonable budget. Unfortunately, determining a high quality search space can be challenging in many applications. For example, when tuning hyperparameters for machine learning pipelines on a new problem given a limited budget, one must strike a balance between excluding potentially promising regions and keeping the search space small enough to be tractable. The goal of this work is to motivate -- through example applications in tuning deep neural networks -- the problem of predicting the quality of search spaces conditioned on budgets, as well as to provide a simple scoring method based on a utility function applied to a probabilistic response surface model, similar to Bayesian optimization. We show that the method we present can compute meaningful budget-conditional scores in a variety of situations. We also provide experimental evidence that accurate scores can be useful in constructing and pruning search spaces. Ultimately, we believe scoring search spaces should become standard practice in the experimental workflow for deep learning.
Optimal setting of several hyper-parameters in machine learning algorithms is key to make the most of available data. To this aim, several methods such as evolutionary strategies, random search, Bayesian optimization and heuristic rules of thumb have been proposed. In reinforcement learning (RL), the information content of data gathered by the learning agent while interacting with its environment is heavily dependent on the setting of many hyper-parameters. Therefore, the user of an RL algorithm has to rely on search-based optimization methods, such as grid search or the Nelder-Mead simplex algorithm, that are very inefficient for most RL tasks, slows down significantly the learning curve and leaves to the user the burden of purposefully biasing data gathering. In this work, in order to make an RL algorithm more user-independent, a novel approach for autonomous hyper-parameter setting using Bayesian optimization is proposed. Data from past episodes and different hyper-parameter values are used at a meta-learning level by performing behavioral cloning which helps improving the effectiveness in maximizing a reinforcement learning variant of an acquisition function. Also, by tightly integrating Bayesian optimization in a reinforcement learning agent design, the number of state transitions needed to converge to the optimal policy for a given task is reduced. Computational experiments reveal promising results compared to other manual tweaking and optimization-based approaches which highlights the benefits of changing the algorithm hyper-parameters to increase the information content of generated data.
Bayesian optimization involves "inner optimization" over a new-data acquisition criterion which is non-convex/highly multi-modal, may be non-differentiable, or may otherwise thwart local numerical optimizers. In such cases it is common to replace continuous search with a discrete one over random candidates. Here we propose using candidates based on a Delaunay triangulation of the existing input design. We detail the construction of these "tricands" and demonstrate empirically how they outperform both numerically optimized acquisitions and random candidate-based alternatives, and are well-suited for hybrid schemes, on benchmark synthetic and real simulation experiments.
Automotive companies are increasingly looking for ways to make their products lighter, using novel materials and novel bonding processes to join these materials together. Finding the optimal process parameters for such adhesive bonding process is challenging. In this research, we successfully applied Bayesian optimization using Gaussian Process Regression and Logistic Regression, to efficiently (i.e., requiring few experiments) guide the design of experiments to the Pareto-optimal process parameter settings.
Recent years have witnessed a large amount of decentralized data in multiple (edge) devices of end-users, while the aggregation of the decentralized data remains difficult for machine learning jobs due to laws or regulations. Federated Learning (FL) emerges as an effective approach to handling decentralized data without sharing the sensitive raw data, while collaboratively training global machine learning models. The servers in FL need to select (and schedule) devices during the training process. However, the scheduling of devices for multiple jobs with FL remains a critical and open problem. In this paper, we propose a novel multi-job FL framework to enable the parallel training process of multiple jobs. The framework consists of a system model and two scheduling methods. In the system model, we propose a parallel training process of multiple jobs, and construct a cost model based on the training time and the data fairness of various devices during the training process of diverse jobs. We propose a reinforcement learning-based method and a Bayesian optimization-based method to schedule devices for multiple jobs while minimizing the cost. We conduct extensive experimentation with multiple jobs and datasets. The experimental results show that our proposed approaches significantly outperform baseline approaches in terms of training time (up to 8.67 times faster) and accuracy (up to 44.6% higher).
Accurate uncertainty estimates are important in sequential model-based decision-making tasks such as Bayesian optimization. However, these estimates can be imperfect if the data violates assumptions made by the model (e.g., Gaussianity). This paper studies which uncertainties are needed in model-based decision-making and in Bayesian optimization, and argues that uncertainties can benefit from calibration -- i.e., an 80% predictive interval should contain the true outcome 80% of the time. Maintaining calibration, however, can be challenging when the data is non-stationary and depends on our actions. We propose using simple algorithms based on online learning to provably maintain calibration on non-i.i.d. data, and we show how to integrate these algorithms in Bayesian optimization with minimal overhead. Empirically, we find that calibrated Bayesian optimization converges to better optima in fewer steps, and we demonstrate improved performance on standard benchmark functions and hyperparameter optimization tasks.
The multi-task learning (MTL) paradigm can be traced back to an early paper of Caruana (1997) in which it was argued that data from multiple tasks can be used with the aim to obtain a better performance over learning each task independently. A solution of MTL with conflicting objectives requires modelling the trade-off among them which is generally beyond what a straight linear combination can achieve. A theoretically principled and computationally effective strategy is finding solutions which are not dominated by others as it is addressed in the Pareto analysis. Multi-objective optimization problems arising in the multi-task learning context have specific features and require adhoc methods. The analysis of these features and the proposal of a new computational approach represent the focus of this work. Multi-objective evolutionary algorithms (MOEAs) can easily include the concept of dominance and therefore the Pareto analysis. The major drawback of MOEAs is a low sample efficiency with respect to function evaluations. The key reason for this drawback is that most of the evolutionary approaches do not use models for approximating the objective function. Bayesian Optimization takes a radically different approach based on a surrogate model, such as a Gaussian Process. In this thesis the solutions in the Input Space are represented as probability distributions encapsulating the knowledge contained in the function evaluations. In this space of probability distributions, endowed with the metric given by the Wasserstein distance, a new algorithm MOEA/WST can be designed in which the model is not directly on the objective function but in an intermediate Information Space where the objects from the input space are mapped into histograms. Computational results show that the sample efficiency and the quality of the Pareto set provided by MOEA/WST are significantly better than in the standard MOEA.
Hyper-parameter optimization is a crucial problem in machine learning as it aims to achieve the state-of-the-art performance in any model. Great efforts have been made in this field, such as random search, grid search, Bayesian optimization. In this paper, we model hyper-parameter optimization process as a Markov decision process, and tackle it with reinforcement learning. A novel hyper-parameter optimization method based on soft actor critic and hierarchical mixture regularization has been proposed. Experiments show that the proposed method can obtain better hyper-parameters in a shorter time.
Multi-agent simulation is commonly used across multiple disciplines, specifically in artificial intelligence in recent years, which creates an environment for downstream machine learning or reinforcement learning tasks. In many practical scenarios, however, only the output series that result from the interactions of simulation agents are observable. Therefore, simulators need to be calibrated so that the simulated output series resemble historical -- which amounts to solving a complex simulation optimization problem. In this paper, we propose a simple and efficient framework for calibrating simulator parameters from historical output series observations. First, we consider a novel concept of eligibility set to bypass the potential non-identifiability issue. Second, we generalize the two-sample Kolmogorov-Smirnov (K-S) test with Bonferroni correction to test the similarity between two high-dimensional distributions, which gives a simple yet effective distance metric between the output series sample sets. Third, we suggest using Bayesian optimization (BO) and trust-region BO (TuRBO) to minimize the aforementioned distance metric. Finally, we demonstrate the efficiency of our framework using numerical experiments both on a multi-agent financial market simulator.
Recent advances in computationally efficient non-myopic Bayesian optimization (BO) improve query efficiency over traditional myopic methods like expected improvement while only modestly increasing computational cost. These advances have been largely limited, however, to unconstrained optimization. For constrained optimization, the few existing non-myopic BO methods require heavy computation. For instance, one existing non-myopic constrained BO method [Lam and Willcox, 2017] relies on computationally expensive unreliable brute-force derivative-free optimization of a Monte Carlo rollout acquisition function. Methods that use the reparameterization trick for more efficient derivative-based optimization of non-myopic acquisition functions in the unconstrained setting, like sample average approximation and infinitesimal perturbation analysis, do not extend: constraints introduce discontinuities in the sampled acquisition function surface that hinder its optimization. Moreover, we argue here that being non-myopic is even more important in constrained problems because fear of violating constraints pushes myopic methods away from sampling the boundary between feasible and infeasible regions, slowing the discovery of optimal solutions with tight constraints. In this paper, we propose a computationally efficient two-step lookahead constrained Bayesian optimization acquisition function (2-OPT-C) supporting both sequential and batch settings. To enable fast acquisition function optimization, we develop a novel likelihood-ratio-based unbiased estimator of the gradient of the two-step optimal acquisition function that does not use the reparameterization trick. In numerical experiments, 2-OPT-C typically improves query efficiency by 2x or more over previous methods, and in some cases by 10x or more.
Optimizing expensive to evaluate black-box functions over an input space consisting of all permutations of d objects is an important problem with many real-world applications. For example, placement of functional blocks in hardware design to optimize performance via simulations. The overall goal is to minimize the number of function evaluations to find high-performing permutations. The key challenge in solving this problem using the Bayesian optimization (BO) framework is to trade-off the complexity of statistical model and tractability of acquisition function optimization. In this paper, we propose and evaluate two algorithms for BO over Permutation Spaces (BOPS). First, BOPS-T employs Gaussian process (GP) surrogate model with Kendall kernels and a Tractable acquisition function optimization approach based on Thompson sampling to select the sequence of permutations for evaluation. Second, BOPS-H employs GP surrogate model with Mallow kernels and a Heuristic search approach to optimize expected improvement acquisition function. We theoretically analyze the performance of BOPS-T to show that their regret grows sub-linearly. Our experiments on multiple synthetic and real-world benchmarks show that both BOPS-T and BOPS-H perform better than the state-of-the-art BO algorithm for combinatorial spaces. To drive future research on this important problem, we make new resources and real-world benchmarks available to the community.
An autonomous experimentation platform in manufacturing is supposedly capable of conducting a sequential search for finding suitable manufacturing conditions by itself or even for discovering new materials with minimal human intervention. The core of the intelligent control of such platforms is a policy to decide where to conduct the next experiment based on what has been done thus far. Such policy inevitably trades off between exploitation and exploration. Currently, the prevailing approach is to use various acquisition functions in the Bayesian optimization framework. We discuss whether it is beneficial to trade off exploitation versus exploration by measuring the element and degree of surprise associated with the immediate past observation. We devise a surprise-reacting policy using two existing surprise metrics, known as the Shannon surprise and Bayesian surprise. Our analysis shows that the surprise-reacting policy appears to be better suited for quickly characterizing the overall landscape of a response surface under resource constraints. We do not claim that we have a fully autonomous experimentation system but believe that the surprise-reacting capability benefits the automation of sequential decisions in autonomous experimentation.
Over the last decades, light-emitting diodes (LED) have replaced common light bulbs in almost every application, from flashlights in smartphones to automotive headlights. Illuminating nightly streets requires LEDs to emit a light spectrum that is perceived as pure white by the human eye. The power associated with such a white light spectrum is not only distributed over the contributing wavelengths but also over the angles of vision. For many applications, the usable light rays are required to exit the LED in forward direction, namely under small angles to the perpendicular. In this work, we demonstrate that a specifically designed multi-layer thin film on top of a white LED increases the power of pure white light emitted in forward direction. Therefore, the deduced multi-objective optimization problem is reformulated via a real-valued physics-guided objective function that represents the hierarchical structure of our engineering problem. Variants of Bayesian optimization are employed to maximize this non-deterministic objective function based on ray tracing simulations. Eventually, the investigation of optical properties of suitable multi-layer thin films allowed to identify the mechanism behind the increased directionality of white light: angle and wavelength selective filtering causes the multi-layer thin film to play ping pong with rays of light.