We study online optimization of smoothed piecewise constant functions over the domain [0, 1). This is motivated by the problem of adaptively picking parameters of learning algorithms as in the recently introduced framework by Gupta and Roughgarden (2016). Majority of the machine learning literature has focused on Lipschitz-continuous functions or functions with bounded gradients. This is with good reason—any learning algorithm suffers linear regret even against piecewise constant functions that are chosen adversarially, arguably the simplest of non-Lipschitz continuous functions. The smoothed setting we consider is inspired by the seminal work of Spielman and Teng (2004) and the recent work of Gupta and Roughgarden (2016)—in this setting, the sequence of functions may be chosen by an adversary, however, with some uncertainty in the location of discontinuities. We give algorithms that achieve sublinear regret in the full information and bandit settings.
There is a substantial body of work that seeks to use learning mechanisms to choose the parameters or of algorithms. @cite_2 @cite_1 suggest using Bayesian optimization techniques to choose hyperparameters effectively. Yet other papers (see ) suggest various techniques to choose parameters for algorithms (not necessarily in the context of learning). However, except for the work of @cite_4 , most work is not theoretical in nature.
@cite_1 @cite_4 @cite_2
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Bayesian optimization has recently been proposed as a framework for automatically tuning the hyperparameters of machine learning models and has been shown to yield state-of-the-art performance with impressive ease and efficiency. In this paper, we explore whether it is possible to transfer the knowledge gained from previous optimizations to new tasks in order to find optimal hyperparameter settings more efficiently. Our approach is based on extending multi-task Gaussian processes to the framework of Bayesian optimization. We show that this method significantly speeds up the optimization process when compared to the standard single-task approach. We further propose a straightforward extension of our algorithm in order to jointly minimize the average error across multiple tasks and demonstrate how this can be used to greatly speed up k-fold cross-validation. Lastly, we propose an adaptation of a recently developed acquisition function, entropy search, to the cost-sensitive, multi-task setting. We demonstrate the utility of this new acquisition function by leveraging a small dataset to explore hyper-parameter settings for a large dataset. Our algorithm dynamically chooses which dataset to query in order to yield the most information per unit cost. Bayesian optimization has become a successful tool for hyperparameter optimization of machine learning algorithms, such as support vector machines or deep neural networks. Despite its success, for large datasets, training and validating a single configuration often takes hours, days, or even weeks, which limits the achievable performance. To accelerate hyperparameter optimization, we propose a generative model for the validation error as a function of training set size, which is learned during the optimization process and allows exploration of preliminary configurations on small subsets, by extrapolating to the full dataset. We construct a Bayesian optimization procedure, dubbed Fabolas, which models loss and training time as a function of dataset size and automatically trades off high information gain about the global optimum against computational cost. Experiments optimizing support vector machines and deep neural networks show that Fabolas often finds high-quality solutions 10 to 100 times faster than other state-of-the-art Bayesian optimization methods or the recently proposed bandit strategy Hyperband. Performance of machine learning algorithms depends critically on identifying a good set of hyperparameters. While current methods offer efficiencies by adaptively choosing new configurations to train, an alternative strategy is to adaptively allocate resources across the selected configurations. We formulate hyperparameter optimization as a pure-exploration non-stochastic infinitely many armed bandit problem where allocation of additional resources to an arm corresponds to training a configuration on larger subsets of the data. We introduce Hyperband for this framework and analyze its theoretical properties, providing several desirable guarantees. We compare Hyperband with state-of-the-art Bayesian optimization methods and a random search baseline on a comprehensive benchmark including 117 datasets. Our results on this benchmark demonstrate that while Bayesian optimization methods do not outperform random search trained for twice as long, Hyperband in favorable settings offers valuable speedups. Tuning hyperparameters of machine learning models is important for their performance. Bayesian optimization has recently emerged as a de-facto method for this task. The hyperparameter tuning is usually performed by looking at model performance on a validation set. Bayesian optimization is used to find the hyperparameter set corresponding to the best model performance. However, in many cases, the function representing the model performance on the validation set contains several spurious sharp peaks due to limited datapoints. The Bayesian optimization, in such cases, has a tendency to converge to sharp peaks instead of other more stable peaks. When a model trained using these hyperparameters is deployed in the real world, its performance suffers dramatically. We address this problem through a novel stable Bayesian optimization framework. We construct two new acquisition functions that help Bayesian optimization to avoid the convergence to the sharp peaks. We conduct a theoretical analysis and guarantee that Bayesian optimization using the proposed acquisition functions prefers stable peaks over unstable ones. Experiments with synthetic function optimization and hyperparameter tuning for support vector machines show the effectiveness of our proposed framework.