In the last five years, the financial industry has been impacted by the emergence of digitalization and machine learning. In this article, we explore two methods that have undergone rapid development in recent years: Gaussian processes and Bayesian optimization. Gaussian processes can be seen as a generalization of Gaussian random vectors and are associated with the development of kernel methods. Bayesian optimization is an approach for performing derivative-free global optimization in a small dimension, and uses Gaussian processes to locate the global maximum of a black-box function. The first part of the article reviews these two tools and shows how they are connected. In particular, we focus on the Gaussian process regression, which is the core of Bayesian machine learning, and the issue of hyperparameter selection. The second part is dedicated to two financial applications. We first consider the modeling of the term structure of interest rates. More precisely, we test the fitting method and compare the GP prediction and the random walk model. The second application is the construction of trend-following strategies, in particular the online estimation of trend and covariance windows.
Bayesian optimization is popular for optimizing time-consuming black-box objectives. Nonetheless, for hyperparameter tuning in deep neural networks, the time required to evaluate the validation error for even a few hyperparameter settings remains a bottleneck. Multi-fidelity optimization promises relief using cheaper proxies to such objectives --- for example, validation error for a network trained using a subset of the training points or fewer iterations than required for convergence. We propose a highly flexible and practical approach to multi-fidelity Bayesian optimization, focused on efficiently optimizing hyperparameters for iteratively trained supervised learning models. We introduce a new acquisition function, the trace-aware knowledge-gradient, which efficiently leverages both multiple continuous fidelity controls and trace observations --- values of the objective at a sequence of fidelities, available when varying fidelity using training iterations. We provide a provably convergent method for optimizing our acquisition function and show it outperforms state-of-the-art alternatives for hyperparameter tuning of deep neural networks and large-scale kernel learning.
Data science relies on pipelines that are organized in the form of interdependent computational steps. Each step consists of various candidate algorithms that maybe used for performing a particular function. Each algorithm consists of several hyperparameters. Algorithms and hyperparameters must be optimized as a whole to produce the best performance. Typical machine learning pipelines typically consist of complex algorithms in each of the steps. Not only is the selection process combinatorial, but it is also important to interpret and understand the pipelines. We propose a method to quantify the importance of different layers in the pipeline, by computing an error contribution relative to an agnostic choice of algorithms in that layer. We demonstrate our methodology on image classification pipelines. The agnostic methodology quantifies the error contributions from the computational steps, algorithms and hyperparameters in the image classification pipeline. We show that algorithm selection and hyper-parameter optimization methods can be used to quantify the error contribution and that random search is able to quantify the contribution more accurately than Bayesian optimization. This methodology can be used by domain experts to understand machine learning and data analysis pipelines in terms of their individual components, which can help in prioritizing different components of the pipeline.
We present a new approach for transfer of dynamic robot control policies such as biped locomotion from simulation to real hardware. Key to our approach is to perform system identification of the model parameters {\mu} of the hardware (e.g. friction, center-of-mass) in two distinct stages, before policy learning (pre-sysID) and after policy learning (post-sysID). Pre-sysID begins by collecting trajectories from the physical hardware based on a set of generic motion sequences. Because the trajectories may not be related to the task of interest, presysID does not attempt to accurately identify the true value of {\mu}, but only to approximate the range of {\mu} to guide the policy learning. Next, a Projected Universal Policy (PUP) is created by simultaneously training a network that projects {\mu} to a low-dimensional latent variable {\eta} and a family of policies that are conditioned on {\eta}. The second round of system identification (post-sysID) is then carried out by deploying the PUP on the robot hardware using task-relevant trajectories. We use Bayesian Optimization to determine the values for {\eta} that optimizes the performance of PUP on the real hardware. We have used this approach to create three successful biped locomotion controllers (walk forward, walk backwards, walk sideways) on the Darwin OP2 robot.
Data science relies on pipelines that are organized in the form of interdependent computational steps. Each step consists of various candidate algorithms that maybe used for performing a particular function. Each algorithm consists of several hyperparameters. Algorithms and hyperparameters must be optimized as a whole to produce the best performance. Typical machine learning pipelines consist of complex algorithms in each of the steps. Not only is the selection process combinatorial, but it is also important to interpret and understand the pipelines. We propose a method to quantify the importance of different components in the pipeline, by computing an error contribution relative to an agnostic choice of computational steps, algorithms and hyperparameters. We also propose a methodology to quantify the propagation of error from individual components of the pipeline with the help of a naive set of benchmark algorithms not involved in the pipeline. We demonstrate our methodology on image classification pipelines. The agnostic and naive methodologies quantify the error contribution and propagation respectively from the computational steps, algorithms and hyperparameters in the image classification pipeline. We show that algorithm selection and hyperparameter optimization methods like grid search, random search and Bayesian optimization can be used to quantify the error contribution and propagation, and that random search is able to quantify them more accurately than Bayesian optimization. This methodology can be used by domain experts to understand machine learning and data analysis pipelines in terms of their individual components, which can help in prioritizing different components of the pipeline.
Bayesian optimization (BO) is a powerful approach for seeking the global optimum of expensive black-box functions and has proven successful for fine tuning hyper-parameters of machine learning models. However, BO is practically limited to optimizing 10--20 parameters. To scale BO to high dimensions, we usually make structural assumptions on the decomposition of the objective and\slash or exploit the intrinsic lower dimensionality of the problem, e.g. by using linear projections. We could achieve a higher compression rate with nonlinear projections, but learning these nonlinear embeddings typically requires much data. This contradicts the BO objective of a relatively small evaluation budget. To address this challenge, we propose to learn a low-dimensional feature space jointly with (a) the response surface and (b) a reconstruction mapping. Our approach allows for optimization of BO's acquisition function in the lower-dimensional subspace, which significantly simplifies the optimization problem. We reconstruct the original parameter space from the lower-dimensional subspace for evaluating the black-box function. For meaningful exploration, we solve a constrained optimization problem.
Bayesian optimization has become a popular method for high-throughput computing, like the design of computer experiments or hyperparameter tuning of expensive models, where sample efficiency is mandatory. In these applications, distributed and scalable architectures are a necessity. However, Bayesian optimization is mostly sequential. Even parallel variants require certain computations between samples, limiting the parallelization bandwidth. Thompson sampling has been previously applied for distributed Bayesian optimization. But, when compared with other acquisition functions in the sequential setting, Thompson sampling is known to perform suboptimally. In this paper, we present a new method for fully distributed Bayesian optimization, which can be combined with any acquisition function. Our approach considers Bayesian optimization as a partially observable Markov decision process. In this context, stochastic policies, such as the Boltzmann policy, have some interesting properties which can also be studied for Bayesian optimization. Furthermore, the Boltzmann policy trivially allows a distributed Bayesian optimization implementation with high level of parallelism and scalability. We present results in several benchmarks and applications that shows the performance of our method.
Finding an optimal parameter of a black-box function is important for searching stable material structures and finding optimal neural network structures, and Bayesian optimization algorithms are widely used for the purpose. However, most of existing Bayesian optimization algorithms can only handle vector data and cannot handle complex structured data. In this paper, we propose the topological Bayesian optimization, which can efficiently find an optimal solution from structured data using \emph{topological information}. More specifically, in order to apply Bayesian optimization to structured data, we extract useful topological information from a structure and measure the proper similarity between structures. To this end, we utilize persistent homology, which is a topological data analysis method that was recently applied in machine learning. Moreover, we propose the Bayesian optimization algorithm that can handle multiple types of topological information by using a linear combination of kernels for persistence diagrams. Through experiments, we show that topological information extracted by persistent homology contributes to a more efficient search for optimal structures compared to the random search baseline and the graph Bayesian optimization algorithm.
A novel machine learning optimization process coined Restrictive Federated Model Selection (RFMS) is proposed under the scenario, for example, when data from healthcare units can not leave the site it is situated on and it is forbidden to carry out training algorithms on remote data sites due to either technical or privacy and trust concerns. To carry out a clinical research under this scenario, an analyst could train a machine learning model only on local data site, but it is still possible to execute a statistical query at a certain cost in the form of sending a machine learning model to some of the remote data sites and get the performance measures as feedback, maybe due to prediction being usually much cheaper. Compared to federated learning, which is optimizing the model parameters directly by carrying out training across all data sites, RFMS trains model parameters only on one local data site but optimizes hyper-parameters across other data sites jointly since hyper-parameters play an important role in machine learning performance. The aim is to get a Pareto optimal model with respective to both local and remote unseen prediction losses, which could generalize well across data sites. In this work, we specifically consider high dimensional data with shifted distributions over data sites. As an initial investigation, Bayesian Optimization especially multi-objective Bayesian Optimization is used to guide an adaptive hyper-parameter optimization process to select models under the RFMS scenario. Empirical results show that solely using the local data site to tune hyper-parameters generalizes poorly across data sites, compared to methods that utilize the local and remote performances. Furthermore, in terms of dominated hypervolumes, multi-objective Bayesian Optimization algorithms show increased performance across multiple data sites among other candidates.
The key idea of Bayesian optimization is replacing an expensive target function with a cheap surrogate model. By selection of an acquisition function for Bayesian optimization, we trade off between exploration and exploitation. The acquisition function typically depends on the mean and the variance of the surrogate model at a given point. The most common Gaussian process-based surrogate model assumes that the target with fixed parameters is a realization of a Gaussian process. However, often the target function doesn't satisfy this approximation. Here we consider target functions that come from the binomial distribution with the parameter that depends on inputs. Typically we can vary how many Bernoulli samples we obtain during each evaluation. We propose a general Gaussian process model that takes into account Bernoulli outputs. To make things work we consider a simple acquisition function based on Expected Improvement and a heuristic strategy to choose the number of samples at each point thus taking into account precision of the obtained output.
We consider the problem of learning from sparse and underspecified rewards, where an agent receives a complex input, such as a natural language instruction, and needs to generate a complex response, such as an action sequence, while only receiving binary success-failure feedback. Such success-failure rewards are often underspecified: they do not distinguish between purposeful and accidental success. Generalization from underspecified rewards hinges on discounting spurious trajectories that attain accidental success, while learning from sparse feedback requires effective exploration. We address exploration by using a mode covering direction of KL divergence to collect a diverse set of successful trajectories, followed by a mode seeking KL divergence to train a robust policy. We propose Meta Reward Learning (MeRL) to construct an auxiliary reward function that provides more refined feedback for learning. The parameters of the auxiliary reward function are optimized with respect to the validation performance of a trained policy. The MeRL approach outperforms our alternative reward learning technique based on Bayesian Optimization, and achieves the state-of-the-art on weakly-supervised semantic parsing. It improves previous work by 1.2% and 2.4% on WikiTableQuestions and WikiSQL datasets respectively.
An ongoing aim of research in multiobjective Bayesian optimization is to extend its applicability to a large number of objectives. While coping with a limited budget of evaluations, recovering the set of optimal compromise solutions generally requires numerous observations and is less interpretable since this set tends to grow larger with the number of objectives. We thus propose to focus on a specific solution originating from game theory, the Kalai-Smorodinsky solution, which possesses attractive properties. In particular, it ensures equal marginal gains over all objectives. We further make it insensitive to a monotonic transformation of the objectives by considering the objectives in the copula space. A novel tailored algorithm is proposed to search for the solution, in the form of a Bayesian optimization algorithm: sequential sampling decisions are made based on acquisition functions that derive from an instrumental Gaussian process prior. Our approach is tested on four problems with respectively four, six, eight, and nine objectives. The method is available in the Rpackage GPGame available on CRAN at https://cran.r-project.org/package=GPGame.
High throughput sequencing of RNA (RNA-Seq) can provide us with millions of short fragments of RNA transcripts from a sample. How to better recover the original RNA transcripts from those fragments (RNA-Seq assembly) is still a difficult task. For example, RNA-Seq assembly tools typically require hyper-parameter tuning to achieve good performance for particular datasets. This kind of tuning is usually unintuitive and time-consuming. Consequently, users often resort to default parameters, which do not guarantee consistent good performance for various datasets. Here we propose BOAssembler (https://github.com/olivomao/boassembler), a framework that enables end-to-end automatic tuning of RNA-Seq assemblers, based on Bayesian Optimization principles. Experiments show this data-driven approach is effective to improve the overall assembly performance. The approach would be helpful for downstream (e.g. gene, protein, cell) analysis, and more broadly, for future bioinformatics benchmark studies.
Bayesian optimization (BO) is a powerful paradigm for derivative-free global optimization of a black-box objective function (BOF) that is expensive to evaluate. However, the overhead of BO can still be prohibitive for problems with highly expensive function evaluations. In this paper, we investigate how to reduce the required number of function evaluations for BO without compromise in solution quality. We explore the idea of posterior regularization to harness low fidelity (LF) data within the Gaussian process upper confidence bound (GP-UCB) framework. The LF data can arise from previous evaluations of an LF approximation of the BOF or of a related optimization task. An extra GP model called LF-GP is trained to fit the LF data. We develop an operator termed dynamic weighted product of experts (DW-POE) fusion. The regularization is induced by this operator on the posterior of the BOF. The impact of the LF GP model on the resulting regularized posterior is adaptively adjusted via Bayesian formalism. Extensive experimental results on benchmark BOF optimization tasks demonstrate the superior performance of the proposed algorithm over state-of-the-art.
Bayesian optimization is known to be difficult to scale to high dimensions, because the acquisition step requires solving a non-convex optimization problem in the same search space. In order to scale the method and keep its benefits, we propose an algorithm (LineBO) that restricts the problem to a sequence of iteratively chosen one-dimensional sub-problems that can be solved efficiently. We show that our algorithm converges globally and obtains a fast local rate when the function is strongly convex. Further, if the objective has an invariant subspace, our method automatically adapts to the effective dimension without changing the algorithm. When combined with the SafeOpt algorithm to solve the sub-problems, we obtain the first safe Bayesian optimization algorithm with theoretical guarantees applicable in high-dimensional settings. We evaluate our method on multiple synthetic benchmarks, where we obtain competitive performance. Further, we deploy our algorithm to optimize the beam intensity of the Swiss Free Electron Laser with up to 40 parameters while satisfying safe operation constraints.
In this paper we develop a Bayesian optimization based hyperparameter tuning framework inspired by statistical learning theory for classifiers. We utilize two key facts from PAC learning theory; the generalization bound will be higher for a small subset of data compared to the whole, and the highest accuracy for a small subset of data can be achieved with a simple model. We initially tune the hyperparameters on a small subset of training data using Bayesian optimization. While tuning the hyperparameters on the whole training data, we leverage the insights from the learning theory to seek more complex models. We realize this by using directional derivative signs strategically placed in the hyperparameter search space to seek a more complex model than the one obtained with small data. We demonstrate the performance of our method on the tasks of tuning the hyperparameters of several machine learning algorithms.
Theoretical models of the strong nuclear interaction contain unknown coupling constants (parameters) that must be determined using a pool of calibration data. In cases where the models are complex, leading to time consuming calculations, it is particularly challenging to systematically search the corresponding parameter domain for the best fit to the data. In this paper, we explore the prospect of applying Bayesian optimization to constrain the coupling constants in chiral effective field theory descriptions of the nuclear interaction. We find that Bayesian optimization performs rather well with low-dimensional parameter domains and foresee that it can be particularly useful for optimization of a smaller set of coupling constants. A specific example could be the determination of leading three-nucleon forces using data from finite nuclei or three-nucleon scattering experiments.
We propose a method to incrementally learn an embedding space over the domain of network architectures, to enable the careful selection of architectures for evaluation during compressed architecture search. Given a teacher network, we search for a compressed network architecture by using Bayesian Optimization (BO) with a kernel function defined over our proposed embedding space to select architectures for evaluation. We demonstrate that our search algorithm can significantly outperform various baseline methods, such as random search and reinforcement learning (Ashok et al., 2018). The compressed architectures found by our method are also better than the state-of-the-art manually-designed compact architecture ShuffleNet (Zhang et al., 2018). We also demonstrate that the learned embedding space can be transferred to new settings for architecture search, such as a larger teacher network or a teacher network in a different architecture family, without any training. Code is publicly available here: https://github.com/Friedrich1006/ESNAC .
This paper focuses on Bayesian Optimization (BO) for objectives on combinatorial search spaces, including ordinal and categorical variables. Despite the abundance of potential applications of Combinatorial BO, including chipset configuration search and neural architecture search, only a handful of methods have been proposed. We introduce COMBO, a new Gaussian Process (GP) BO. COMBO quantifies "smoothness" of functions on combinatorial search spaces by utilizing a combinatorial graph. The vertex set of the combinatorial graph consists of all possible joint assignments of the variables, while edges are constructed using the graph Cartesian product of the sub-graphs that represent the individual variables. On this combinatorial graph, we propose an ARD diffusion kernel with which the GP is able to model high-order interactions between variables leading to better performance. Moreover, using the Horseshoe prior for the scale parameter in the ARD diffusion kernel results in an effective variable selection procedure, making COMBO suitable for high dimensional problems. Computationally, in COMBO the graph Cartesian product allows the Graph Fourier Transform calculation to scale linearly instead of exponentially. We validate COMBO in a wide array of realistic benchmarks, including weighted maximum satisfiability problems and neural architecture search. COMBO outperforms consistently the latest state-of-the-art while maintaining computational and statistical efficiency.
Optimizing an expensive-to-query function is a common task in science and engineering, where it is beneficial to keep the number of queries to a minimum. A popular strategy is Bayesian optimization (BO), which leverages probabilistic models for this task. Most BO today uses Gaussian processes (GPs), or a few other surrogate models. However, there is a broad set of Bayesian modeling techniques that could be used to capture complex systems and reduce the number of queries in BO. Probabilistic programming languages (PPLs) are modern tools that allow for flexible model definition, prior specification, model composition, and automatic inference. In this paper, we develop ProBO, a BO procedure that uses only standard operations common to most PPLs. This allows a user to drop in a model built with an arbitrary PPL and use it directly in BO. We describe acquisition functions for ProBO, and strategies for efficiently optimizing these functions given complex models or costly inference procedures. Using existing PPLs, we implement new models to aid in a few challenging optimization settings, and demonstrate these on model hyperparameter and architecture search tasks.