We consider the problem of Bayesian optimization of a one-dimensional Brownian motion in which the $T$ adaptively chosen observations are corrupted by Gaussian noise. We show that as the smallest possible expected cumulative regret and the smallest possible expected simple regret scale as $\Omega(\sigma\sqrt{T / \log (T)}) \cap \mathcal{O}(\sigma\sqrt{T} \cdot \log T)$ and $\Omega(\sigma / \sqrt{T \log (T)}) \cap \mathcal{O}(\sigma\log T / \sqrt{T})$ respectively, where $\sigma^2$ is the noise variance. Thus, our upper and lower bounds are tight up to a factor of $\mathcal{O}( (\log T)^{1.5} )$. The upper bound uses an algorithm based on confidence bounds and the Markov property of Brownian motion (among other useful properties), and the lower bound is based on a reduction to binary hypothesis testing.
In large domains, Monte-Carlo tree search (MCTS) is required to estimate the values of the states as efficiently and accurately as possible. However, the standard update rule in backpropagation assumes a stationary distribution for the returns, and particularly in min-max trees, convergence to the true value can be slow because of averaging. We present two methods, Softmax MCTS and Monotone MCTS, which generalize previous attempts to improve upon the backpropagation strategy. We demonstrate that both methods reduce to finding optimal monotone functions, which we do so by performing Bayesian optimization with a Gaussian process (GP) prior. We conduct experiments on computer Go, where the returns are given by a deep value neural network, and show that our proposed framework outperforms previous methods.
We generalize the well-studied problem of gait learning in modular robots in two dimensions. Firstly, we address locomotion in a given target direction that goes beyond learning a typical undirected gait. Secondly, rather than studying one fixed robot morphology we consider a test suite of different modular robots. This study is based on our interest in evolutionary robot systems where both morphologies and controllers evolve. In such a system, newborn robots have to learn to control their own body that is a random combination of the bodies of the parents. We apply and compare two learning algorithms, Bayesian optimization and HyperNEAT. The results of the experiments in simulation show that both methods successfully learn good controllers, but Bayesian optimization is more effective and efficient. We validate the best learned controllers by constructing three robots from the test suite in the real world and observe their fitness and actual trajectories. The obtained results indicate a reality gap that depends on the controllers and the shape of the robots, but overall the trajectories are adequate and follow the target directions successfully.
Bayesian Optimization (BO) is an effective method for optimizing expensive-to-evaluate black-box functions with a wide range of applications for example in robotics, system design and parameter optimization. However, scaling BO to problems with large input dimensions (>10) remains an open challenge. In this paper, we propose to leverage results from optimal control to scale BO to higher dimensional control tasks and to reduce the need for manually selecting the optimization domain. The contributions of this paper are twofold: 1) We show how we can make use of a learned dynamics model in combination with a model-based controller to simplify the BO problem by focusing onto the most relevant regions of the optimization domain. 2) Based on (1) we present a method to find an embedding in parameter space that reduces the effective dimensionality of the optimization problem. To evaluate the effectiveness of the proposed approach, we present an experimental evaluation on real hardware, as well as simulated tasks including a 48-dimensional policy for a quadcopter.
The design of machine learning systems often requires trading off different objectives, for example, prediction error and energy consumption for deep neural networks (DNNs). Typically, no single design performs well in all objectives; therefore, finding Pareto-optimal designs is of interest. The search for Pareto-optimal designs involves evaluating designs in an iterative process, and the measurements are used to evaluate an acquisition function that guides the search process. However, measuring different objectives incurs different costs. For example, the cost of measuring the prediction error of DNNs is orders of magnitude higher than that of measuring the energy consumption of a pre-trained DNN, as it requires re-training the DNN. Current state-of-the-art methods do not consider this difference in objective evaluation cost, potentially incurring expensive evaluations of objective functions in the optimization process. In this paper, we develop a novel decoupled and cost-aware multi-objective optimization algorithm, we call Flexible Multi-Objective Bayesian Optimization (FlexiBO) to address this issue. FlexiBO weights the improvement of the hypervolume of the Pareto region by the measurement cost of each objective to balance the expense of collecting new information with the knowledge gained through objective evaluations, preventing us from performing expensive measurements for little to no gain. We evaluate FlexiBO on seven state-of-the-art DNNs for image recognition, natural language processing (NLP), and speech-to-text translation. Our results indicate that, given the same total experimental budget, FlexiBO discovers designs with 4.8$\%$ to 12.4$\%$ lower hypervolume error than the best method in state-of-the-art multi-objective optimization.
Most machine learning methods require careful selection of hyper-parameters in order to train a high performing model with good generalization abilities. Hence, several automatic selection algorithms have been introduced to overcome tedious manual (try and error) tuning of these parameters. Due to its very high sample efficiency, Bayesian Optimization over a Gaussian Processes modeling of the parameter space has become the method of choice. Unfortunately, this approach suffers from a cubic compute complexity due to underlying Cholesky factorization, which makes it very hard to be scaled beyond a small number of sampling steps. In this paper, we present a novel, highly accurate approximation of the underlying Gaussian Process. Reducing its computational complexity from cubic to quadratic allows an efficient strong scaling of Bayesian Optimization while outperforming the previous approach regarding optimization accuracy. The first experiments show speedups of a factor of 162 in single node and further speed up by a factor of 5 in a parallel environment.
Bayesian optimization is normally performed within fixed variable bounds. In cases like hyperparameter tuning for machine learning algorithms, setting the variable bounds is not trivial. It is hard to guarantee that any fixed bounds will include the true global optimum. We propose a Bayesian optimization approach that only needs to specify an initial search space that does not necessarily include the global optimum, and expands the search space when necessary. However, over-exploration may occur during the search space expansion. Our method can adaptively balance exploration and exploitation in an expanding space. Results on a range of synthetic test functions and an MLP hyperparameter optimization task show that the proposed method out-performs or at least as good as the current state-of-the-art methods.
An essential problem in automated machine learning (AutoML) is that of model selection. A unique challenge in the sequential setting is the fact that the optimal model itself may vary over time, depending on the distribution of features and labels available up to each point in time. In this paper, we propose a novel Bayesian optimization (BO) algorithm to tackle the challenge of model selection in this setting. This is accomplished by treating the performance at each time step as its own black-box function. In order to solve the resulting multiple black-box function optimization problem jointly and efficiently, we exploit potential correlations among black-box functions using deep kernel learning (DKL). To the best of our knowledge, we are the first to formulate the problem of stepwise model selection (SMS) for sequence prediction, and to design and demonstrate an efficient joint-learning algorithm for this purpose. Using multiple real-world datasets, we verify that our proposed method outperforms both standard BO and multi-objective BO algorithms on a variety of sequence prediction tasks.
Ultra-cold atomic gases are unique in terms of the degree of controllability, both for internal and external degrees of freedom. This makes it possible to use them for the study of complex quantum many-body phenomena. However in many scenarios, the prerequisite condition of faithfully preparing a desired quantum state despite decoherence and system imperfections is not always adequately met. To path the way to a specific target state, we explore quantum optimal control framework based on Bayesian optimization. The probabilistic modeling and broad exploration aspects of Bayesian optimization is particularly suitable for quantum experiments where data acquisition can be expensive. Using numerical simulations for the superfluid to Mott-insulator transition for bosons in a lattice as well for the formation of Rydberg crystals as explicit examples, we demonstrate that Bayesian optimization is capable of finding better control solutions with regards to finite and noisy data compared to existing methods of optimal control.
Surrogate-based optimization relies on so-called infill criteria (acquisition functions) to decide which point to evaluate next. When Kriging is used as the surrogate model of choice (also called Bayesian optimization), one of the most frequently chosen criteria is expected improvement. We argue that the popularity of expected improvement largely relies on its theoretical properties rather than empirically validated performance. Few results from the literature show evidence, that under certain conditions, expected improvement may perform worse than something as simple as the predicted value of the surrogate model. We benchmark both infill criteria in an extensive empirical study on the `BBOB' function set. This investigation includes a detailed study of the impact of problem dimensionality on algorithm performance. The results support the hypothesis that exploration loses importance with increasing problem dimensionality. A statistical analysis reveals that the purely exploitative search with the predicted value criterion performs better on most problems of five or higher dimensions. Possible reasons for these results are discussed. In addition, we give an in-depth guide for choosing the infill criteria based on prior knowledge about the problem at hand, its dimensionality, and the available budget.
Nuclear fusion is regarded as the energy of the future since it presents the possibility of unlimited clean energy. One obstacle in utilizing fusion as a feasible energy source is the stability of the reaction. Ideally, one would have a controller for the reactor that makes actions in response to the current state of the plasma in order to prolong the reaction as long as possible. In this work, we make preliminary steps to learning such a controller. Since learning on a real world reactor is infeasible, we tackle this problem by attempting to learn optimal controls offline via a simulator, where the state of the plasma can be explicitly set. In particular, we introduce a theoretically grounded Bayesian optimization algorithm that recommends a state and action pair to evaluate at every iteration and show that this results in more efficient use of the simulator.
We describe a method for Bayesian optimization by which one may incorporate data from multiple systems whose quantitative interrelationships are unknown a priori. All general (nonreal-valued) features of the systems are associated with continuous latent variables that enter as inputs into a single metamodel that simultaneously learns the response surfaces of all of the systems. Bayesian inference is used to determine appropriate beliefs regarding the latent variables. We explain how the resulting probabilistic metamodel may be used for Bayesian optimization tasks and demonstrate its implementation on a variety of synthetic and real-world examples, comparing its performance under zero-, one-, and few-shot settings against traditional Bayesian optimization, which usually requires substantially more data from the system of interest.
In this work, we aim to solve data-driven optimization problems, where the goal is to find an input that maximizes an unknown score function given access to a dataset of inputs with corresponding scores. When the inputs are high-dimensional and valid inputs constitute a small subset of this space (e.g., valid protein sequences or valid natural images), such model-based optimization problems become exceptionally difficult, since the optimizer must avoid out-of-distribution and invalid inputs. We propose to address such problem with model inversion networks (MINs), which learn an inverse mapping from scores to inputs. MINs can scale to high-dimensional input spaces and leverage offline logged data for both contextual and non-contextual optimization problems. MINs can also handle both purely offline data sources and active data collection. We evaluate MINs on tasks from the Bayesian optimization literature, high-dimensional model-based optimization problems over images and protein designs, and contextual bandit optimization from logged data.
The neural linear model is a simple adaptive Bayesian linear regression method that has recently been used in a number of problems ranging from Bayesian optimization to reinforcement learning. Despite its apparent successes in these settings, to the best of our knowledge there has been no systematic exploration of its capabilities on simple regression tasks. In this work we characterize these on the UCI datasets, a popular benchmark for Bayesian regression models, as well as on the recently introduced UCI "gap" datasets, which are better tests of out-of-distribution uncertainty. We demonstrate that the neural linear model is a simple method that shows generally good performance on these tasks, but at the cost of requiring good hyperparameter tuning.
This paper studies the problem of Person Re-Identification (ReID)for large-scale applications. Recent research efforts have been devoted to building complicated part models, which introduce considerably high computational cost and memory consumption, inhibiting its practicability in large-scale applications. This paper aims to develop a novel learning strategy to find efficient feature embeddings while maintaining the balance of accuracy and model complexity. More specifically, we find by enhancing the classical triplet loss together with cross-entropy loss, our method can explore the hard examples and build a discriminant feature embedding yet compact enough for large-scale applications. Our method is carried out progressively using Bayesian optimization, and we call it the Progressive Learning Algorithm (PLA). Extensive experiments on three large-scale datasets show that our PLA is comparable or better than the-state-of-the-arts. Especially, on the challenging Market-1501 dataset, we achieve Rank-1=94.7\%/mAP=89.4\% while saving at least 30\% parameters than strong part models.
We extend first-order model agnostic meta-learning algorithms (including FOMAML and Reptile) to image segmentation, present a novel neural network architecture built for fast learning which we call EfficientLab, and leverage a formal definition of the test error of meta-learning algorithms to decrease error on out of distribution tasks. We show state of the art results on the FSS-1000 dataset by meta-training EfficientLab with FOMAML and using Bayesian optimization to infer the optimal test-time adaptation routine hyperparameters. We also construct a small benchmark dataset, FP-k, for the empirical study of how meta-learning systems perform in both few- and many-shot settings. On the FP-k dataset, we show that meta-learned initializations provide value for canonical few-shot image segmentation but their performance is quickly matched by conventional transfer learning with performance being equal beyond 10 labeled examples. Our code, meta-learned model, and the FP-k dataset are available at https://github.com/ml4ai/mliis .
Deep learning models are full of hyperparameters, which are set manually before the learning process can start. To find the best configuration for these hyperparameters in such a high dimensional space, with time-consuming and expensive model training / validation, is not a trivial challenge. Bayesian optimization is a powerful tool for the joint optimization of hyperparameters, efficiently trading off exploration and exploitation of the hyperparameter space. In this paper, we discuss Bayesian hyperparameter optimization, including hyperparameter optimization, Bayesian optimization, and Gaussian processes. We also review BoTorch, GPyTorch and Ax, the new open-source frameworks that we use for Bayesian optimization, Gaussian process inference and adaptive experimentation, respectively. For experimentation, we apply Bayesian hyperparameter optimization, for optimizing group weights, to weighted group pooling, which couples unsupervised tiered graph autoencoders learning and supervised graph prediction learning for molecular graphs. We find that Ax, BoTorch and GPyTorch together provide a simple-to-use but powerful framework for Bayesian hyperparameter optimization, using Ax's high-level API that constructs and runs a full optimization loop and returns the best hyperparameter configuration.
Bayesian optimization with Gaussian process as surrogate model has been successfully applied to analog circuit synthesis. In the traditional Gaussian process regression model, the kernel functions are defined explicitly. The computational complexity of training is O(N 3 ), and the computation complexity of prediction is O(N 2 ), where N is the number of training data. Gaussian process model can also be derived from a weight space view, where the original data are mapped to feature space, and the kernel function is defined as the inner product of nonlinear features. In this paper, we propose a Bayesian optimization approach for analog circuit synthesis using neural network. We use deep neural network to extract good feature representations, and then define Gaussian process using the extracted features. Model averaging method is applied to improve the quality of uncertainty prediction. Compared to Gaussian process model with explicitly defined kernel functions, the neural-network-based Gaussian process model can automatically learn a kernel function from data, which makes it possible to provide more accurate predictions and thus accelerate the follow-up optimization procedure. Also, the neural-network-based model has O(N) training time and constant prediction time. The efficiency of the proposed method has been verified by two real-world analog circuits.
Many real-world functions are defined over both categorical and category-specific continuous variables and thus cannot be optimized by traditional Bayesian optimization (BO) methods. To optimize such functions, we propose a new method that formulates the problem as a multi-armed bandit problem, wherein each category corresponds to an arm with its reward distribution centered around the optimum of the objective function in continuous variables. Our goal is to identify the best arm and the maximizer of the corresponding continuous function simultaneously. Our algorithm uses a Thompson sampling scheme that helps connecting both multi-arm bandit and BO in a unified framework. We extend our method to batch BO to allow parallel optimization when multiple resources are available. We theoretically analyze our method for convergence and prove sub-linear regret bounds. We perform a variety of experiments: optimization of several benchmark functions, hyper-parameter tuning of a neural network, and automatic selection of the best machine learning model along with its optimal hyper-parameters (a.k.a automated machine learning). Comparisons with other methods demonstrate the effectiveness of our proposed method.
Scaling Bayesian optimisation (BO) to high-dimensional search spaces is a active and open research problems particularly when no assumptions are made on function structure. The main reason is that at each iteration, BO requires to find global maximisation of acquisition function, which itself is a non-convex optimization problem in the original search space. With growing dimensions, the computational budget for this maximisation gets increasingly short leading to inaccurate solution of the maximisation. This inaccuracy adversely affects both the convergence and the efficiency of BO. We propose a novel approach where the acquisition function only requires maximisation on a discrete set of low dimensional subspaces embedded in the original high-dimensional search space. Our method is free of any low dimensional structure assumption on the function unlike many recent high-dimensional BO methods. Optimising acquisition function in low dimensional subspaces allows our method to obtain accurate solutions within limited computational budget. We show that in spite of this convenience, our algorithm remains convergent. In particular, cumulative regret of our algorithm only grows sub-linearly with the number of iterations. More importantly, as evident from our regret bounds, our algorithm provides a way to trade the convergence rate with the number of subspaces used in the optimisation. Finally, when the number of subspaces is "sufficiently large", our algorithm's cumulative regret is at most $\mathcal{O}^{*}(\sqrt{T\gamma_T})$ as opposed to $\mathcal{O}^{*}(\sqrt{DT\gamma_T})$ for the GP-UCB of Srinivas et al. (2012), reducing a crucial factor $\sqrt{D}$ where $D$ being the dimensional number of input space.