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Multi-Step Budgeted Bayesian Optimization with Unknown Evaluation Costs

arXiv.org Machine Learning

Bayesian optimization (BO) is a sample-efficient approach to optimizing costly-to-evaluate black-box functions. Most BO methods ignore how evaluation costs may vary over the optimization domain. However, these costs can be highly heterogeneous and are often unknown in advance. This occurs in many practical settings, such as hyperparameter tuning of machine learning algorithms or physics-based simulation optimization. Moreover, those few existing methods that acknowledge cost heterogeneity do not naturally accommodate a budget constraint on the total evaluation cost. This combination of unknown costs and a budget constraint introduces a new dimension to the exploration-exploitation trade-off, where learning about the cost incurs the cost itself. Existing methods do not reason about the various trade-offs of this problem in a principled way, leading often to poor performance. We formalize this claim by proving that the expected improvement and the expected improvement per unit of cost, arguably the two most widely used acquisition functions in practice, can be arbitrarily inferior with respect to the optimal non-myopic policy. To overcome the shortcomings of existing approaches, we propose the budgeted multi-step expected improvement, a non-myopic acquisition function that generalizes classical expected improvement to the setting of heterogeneous and unknown evaluation costs. Finally, we show that our acquisition function outperforms existing methods in a variety of synthetic and real problems.


Numerical influence of ReLU'(0) on backpropagation

arXiv.org Artificial Intelligence

In theory, the choice of ReLU'(0) in [0, 1] for a neural network has a negligible influence both on backpropagation and training. Yet, in the real world, 32 bits default precision combined with the size of deep learning problems makes it a hyperparameter of training methods. We investigate the importance of the value of ReLU'(0) for several precision levels (16, 32, 64 bits), on various networks (fully connected, VGG, ResNet) and datasets (MNIST, CIFAR10, SVHN). We observe considerable variations of backpropagation outputs which occur around half of the time in 32 bits precision. The effect disappears with double precision, while it is systematic at 16 bits. For vanilla SGD training, the choice ReLU'(0) = 0 seems to be the most efficient. We also evidence that reconditioning approaches as batch-norm or ADAM tend to buffer the influence of ReLU'(0)'s value. Overall, the message we want to convey is that algorithmic differentiation of nonsmooth problems potentially hides parameters that could be tuned advantageously.


Adaptive Importance Sampling meets Mirror Descent: a Bias-variance tradeoff

arXiv.org Machine Learning

Adaptive importance sampling is a widely spread Monte Carlo technique that uses a re-weighting strategy to iteratively estimate the so-called target distribution. A major drawback of adaptive importance sampling is the large variance of the weights which is known to badly impact the accuracy of the estimates. This paper investigates a regularization strategy whose basic principle is to raise the importance weights at a certain power. This regularization parameter, that might evolve between zero and one during the algorithm, is shown (i) to balance between the bias and the variance and (ii) to be connected to the mirror descent framework. Using a kernel density estimate to build the sampling policy, the uniform convergence is established under mild conditions. Finally, several practical ways to choose the regularization parameter are discussed and the benefits of the proposed approach are illustrated empirically.


Bridged Adversarial Training

arXiv.org Artificial Intelligence

Adversarial robustness is considered as a required property of deep neural networks. In this study, we discover that adversarially trained models might have significantly different characteristics in terms of margin and smoothness, even they show similar robustness. Inspired by the observation, we investigate the effect of different regularizers and discover the negative effect of the smoothness regularizer on maximizing the margin. Based on the analyses, we propose a new method called bridged adversarial training that mitigates the negative effect by bridging the gap between clean and adversarial examples. We provide theoretical and empirical evidence that the proposed method provides stable and better robustness, especially for large perturbations.


Bounded rationality for relaxing best response and mutual consistency: An information-theoretic model of partial self-reference

arXiv.org Artificial Intelligence

While game theory has been transformative for decision-making, the assumptions made can be overly restrictive in certain instances. In this work, we focus on some of the assumptions underlying rationality such as mutual consistency and best-response, and consider ways to relax these assumptions using concepts from level-$k$ reasoning and quantal response equilibrium (QRE) respectively. Specifically, we provide an information-theoretic two-parameter model that can relax both mutual consistency and best-response, but can recover approximations of level-$k$, QRE, or typical Nash equilibrium behaviour in the limiting cases. The proposed approach is based on a recursive form of the variational free energy principle, representing self-referential games as (pseudo) sequential decisions. Bounds in player processing abilities are captured as information costs, where future chains of reasoning are discounted, implying a hierarchy of players where lower-level players have fewer processing resources.