Bayesian optimization is a popular method for solving the problem of global optimization of an expensive-to-evaluate black-box function. It relies on a probabilistic surrogate model of the objective function, upon which an acquisition function is bui
lt to determine where next to evaluate the objective function. In general, Bayesian optimization with Gaussian process regression operates on a continuous space. When input variables are categorical or discrete, an extra care is needed. A common approach is to use one-hot encoded or Boolean representation for categorical variables which might yield a {em combinatorial explosion} problem. In this paper we present a method for Bayesian optimization in a combinatorial space, which can operate well in a large combinatorial space. The main idea is to use a random mapping which embeds the combinatorial space into a convex polytope in a continuous space, on which all essential process is performed to determine a solution to the black-box optimization in the combinatorial space. We describe our {em combinatorial Bayesian optimization} algorithm and present its regret analysis. Numerical experiments demonstrate that our method outperforms existing methods.
Unlike in the traditional statistical modeling for which a user typically hand-specify a prior, Neural Processes (NPs) implicitly define a broad class of stochastic processes with neural networks. Given a data stream, NP learns a stochastic process t
hat best describes the data. While this data-driven way of learning stochastic processes has proven to handle various types of data, NPs still rely on an assumption that uncertainty in stochastic processes is modeled by a single latent variable, which potentially limits the flexibility. To this end, we propose the Boostrapping Neural Process (BNP), a novel extension of the NP family using the bootstrap. The bootstrap is a classical data-driven technique for estimating uncertainty, which allows BNP to learn the stochasticity in NPs without assuming a particular form. We demonstrate the efficacy of BNP on various types of data and its robustness in the presence of model-data mismatch.
Sequential assembly with geometric primitives has drawn attention in robotics and 3D vision since it yields a practical blueprint to construct a target shape. However, due to its combinatorial property, a greedy method falls short of generating a seq
uence of volumetric primitives. To alleviate this consequence induced by a huge number of feasible combinations, we propose a combinatorial 3D shape generation framework. The proposed framework reflects an important aspect of human generation processes in real life -- we often create a 3D shape by sequentially assembling unit primitives with geometric constraints. To find the desired combination regarding combination evaluations, we adopt Bayesian optimization, which is able to exploit and explore efficiently the feasible regions constrained by the current primitive placements. An evaluation function conveys global structure guidance for an assembly process and stability in terms of gravity and external forces simultaneously. Experimental results demonstrate that our method successfully generates combinatorial 3D shapes and simulates more realistic generation processes. We also introduce a new dataset for combinatorial 3D shape generation. All the codes are available at url{https://github.com/POSTECH-CVLab/Combinatorial-3D-Shape-Generation}.
We propose a practical Bayesian optimization method over sets, to minimize a black-box function that takes a set as a single input. Because set inputs are permutation-invariant, traditional Gaussian process-based Bayesian optimization strategies whic
h assume vector inputs can fall short. To address this, we develop a Bayesian optimization method with emph{set kernel} that is used to build surrogate functions. This kernel accumulates similarity over set elements to enforce permutation-invariance, but this comes at a greater computational cost. To reduce this burden, we propose two key components: (i) a more efficient approximate set kernel which is still positive-definite and is an unbiased estimator of the true set kernel with upper-bounded variance in terms of the number of subsamples, (ii) a constrained acquisition function optimization over sets, which uses symmetry of the feasible region that defines a set input. Finally, we present several numerical experiments which demonstrate that our method outperforms other methods.
We propose a practical Bayesian optimization method using Gaussian process regression, of which the marginal likelihood is maximized where the number of model selection steps is guided by a pre-defined threshold. Since Bayesian optimization consumes
a large portion of its execution time in finding the optimal free parameters for Gaussian process regression, our simple, but straightforward method is able to mitigate the time complexity and speed up the overall Bayesian optimization procedure. Finally, the experimental results show that our method is effective to reduce the execution time in most of cases, with less loss of optimization quality.
A meta-model is trained on a distribution of similar tasks such that it learns an algorithm that can quickly adapt to a novel task with only a handful of labeled examples. Most of current meta-learning methods assume that the meta-training set consis
ts of relevant tasks sampled from a single distribution. In practice, however, a new task is often out of the task distribution, yielding a performance degradation. One way to tackle this problem is to construct an ensemble of meta-learners such that each meta-learner is trained on different task distribution. In this paper we present a method for constructing a mixture of meta-learners (MxML), where mixing parameters are determined by the weight prediction network (WPN) optimized to improve the few-shot classification performance. Experiments on various datasets demonstrate that MxML significantly outperforms state-of-the-art meta-learners, or their naive ensemble in the case of out-of-distribution as well as in-distribution tasks.
Bayesian optimization is a sample-efficient method for finding a global optimum of an expensive-to-evaluate black-box function. A global solution is found by accumulating a pair of query point and its function value, repeating these two procedures: (
i) modeling a surrogate function; (ii) maximizing an acquisition function to determine where next to query. Convergence guarantees are only valid when the global optimizer of the acquisition function is found at each round and selected as the next query point. In practice, however, local optimizers of an acquisition function are also used, since searching for the global optimizer is often a non-trivial or time-consuming task. In this paper we consider three popular acquisition functions, PI, EI, and GP-UCB induced by Gaussian process regression. Then we present a performance analysis on the behavior of local optimizers of those acquisition functions, in terms of {em instantaneous regrets} over global optimizers. We also introduce an analysis, allowing a local optimization method to start from multiple different initial conditions. Numerical experiments confirm the validity of our theoretical analysis.
Many machine learning tasks such as multiple instance learning, 3D shape recognition, and few-shot image classification are defined on sets of instances. Since solutions to such problems do not depend on the order of elements of the set, models used
to address them should be permutation invariant. We present an attention-based neural network module, the Set Transformer, specifically designed to model interactions among elements in the input set. The model consists of an encoder and a decoder, both of which rely on attention mechanisms. In an effort to reduce computational complexity, we introduce an attention scheme inspired by inducing point methods from sparse Gaussian process literature. It reduces the computation time of self-attention from quadratic to linear in the number of elements in the set. We show that our model is theoretically attractive and we evaluate it on a range of tasks, demonstrating the state-of-the-art performance compared to recent methods for set-structured data.
Hyperparameter optimization aims to find the optimal hyperparameter configuration of a machine learning model, which provides the best performance on a validation dataset. Manual search usually leads to get stuck in a local hyperparameter configurati
on, and heavily depends on human intuition and experience. A simple alternative of manual search is random/grid search on a space of hyperparameters, which still undergoes extensive evaluations of validation errors in order to find its best configuration. Bayesian optimization that is a global optimization method for black-box functions is now popular for hyperparameter optimization, since it greatly reduces the number of validation error evaluations required, compared to random/grid search. Bayesian optimization generally finds the best hyperparameter configuration from random initialization without any prior knowledge. This motivates us to let Bayesian optimization start from the configurations that were successful on similar datasets, which are able to remarkably minimize the number of evaluations. In this paper, we propose deep metric learning to learn meta-features over datasets such that the similarity over them is effectively measured by Euclidean distance between their associated meta-features. To this end, we introduce a Siamese network composed of deep feature and meta-feature extractors, where deep feature extractor provides a semantic representation of each instance in a dataset and meta-feature extractor aggregates a set of deep features to encode a single representation over a dataset. Then, our learned meta-features are used to select a few datasets similar to the new dataset, so that hyperparameters in similar datasets are adopted as initializations to warm-start Bayesian hyperparameter optimization.
Using the angular dependence of the planar Hall effect in GaMnAs ferromagnetic films, we were able to determine the distribution of magnetic domain pinning fields in this material. Interestingly, there is a major difference between the pinning field
distribution in as-grown and in annealed films, the former showing a strikingly narrower distribution than the latter. This conspicuous difference can be attributed to the degree of non-uniformity of magnetic anisotropy in both types of films. This finding provides a better understanding of the magnetic domain landscape in GaMnAs that has been the subject of intense debate.
