Bilevel optimization, the problem of minimizing a value function which involves the arg-minimum of another function, appears in many areas of machine learning. In a large scale empirical risk minimization setting where the number of samples is huge, it is crucial to develop stochastic methods, which only use a few samples at a time to progress. However, computing the gradient of the value function involves solving a linear system, which makes it difficult to derive unbiased stochastic estimates. To overcome this problem we introduce a novel framework, in which the solution of the inner problem, the solution of the linear system, and the main variable evolve at the same time. These directions are written as a sum, making it straightforward to derive unbiased estimates. The simplicity of our approach allows us to develop global variance reduction algorithms, where the dynamics of all variables is subject to variance reduction. We demonstrate that SABA, an adaptation of the celebrated SAGA algorithm in our framework, has $O(\frac1T)$ convergence rate, and that it achieves linear convergence under Polyak-Lojasciewicz assumption. This is the first stochastic algorithm for bilevel optimization that verifies either of these properties. Numerical experiments validate the usefulness of our method.
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A fundamental task in science is to design experiments that yield valuable insights about the system under study. Mathematically, these insights can be represented as a utility or risk function that shapes the value of conducting each experiment. We present PDBAL, a targeted active learning method that adaptively designs experiments to maximize scientific utility. PDBAL takes a user-specified risk function and combines it with a probabilistic model of the experimental outcomes to choose designs that rapidly converge on a high-utility model. We prove theoretical bounds on the label complexity of PDBAL and provide fast closed-form solutions for designing experiments with common exponential family likelihoods. In simulation studies, PDBAL consistently outperforms standard untargeted approaches that focus on maximizing expected information gain over the design space. Finally, we demonstrate the scientific potential of PDBAL through a study on a large cancer drug screen dataset where PDBAL quickly recovers the most efficacious drugs with a small fraction of the total number of experiments.
We present a novel sequential Monte Carlo approach to online smoothing of additive functionals in a very general class of path-space models. Hitherto, the solutions proposed in the literature suffer from either long-term numerical instability due to particle-path degeneracy or, in the case that degeneracy is remedied by particle approximation of the so-called backward kernel, high computational demands. In order to balance optimally computational speed against numerical stability, we propose to furnish a (fast) naive particle smoother, propagating recursively a sample of particles and associated smoothing statistics, with an adaptive backward-sampling-based updating rule which allows the number of (costly) backward samples to be kept at a minimum. This yields a new, function-specific additive smoothing algorithm, AdaSmooth, which is computationally fast, numerically stable and easy to implement. The algorithm is provided with rigorous theoretical results guaranteeing its consistency, asymptotic normality and long-term stability as well as numerical results demonstrating empirically the clear superiority of AdaSmooth to existing algorithms.
The problem of monotone submodular maximization has been studied extensively due to its wide range of applications. However, there are cases where one can only access the objective function in a distorted or noisy form because of the uncertain nature or the errors involved in the evaluation. This paper considers the problem of constrained monotone submodular maximization with noisy oracles introduced by [Hassidim et al., 2017]. For a cardinality constraint, we propose an algorithm achieving a near-optimal $\left(1-\frac{1}{e}-O(\varepsilon)\right)$-approximation guarantee (for arbitrary $\varepsilon > 0$) with only a polynomial number of queries to the noisy value oracle, which improves the exponential query complexity of [Singer et al., 2018]. For general matroid constraints, we show the first constant approximation algorithm in the presence of noise. Our main approaches are to design a novel local search framework that can handle the effect of noise and to construct certain smoothing surrogate functions for noise reduction.
The simulation of human neurons and neurotransmission mechanisms has been realized in deep neural networks based on the theoretical implementations of activation functions. However, recent studies have reported that the threshold potential of neurons exhibits different values according to the locations and types of individual neurons, and that the activation functions have limitations in terms of representing this variability. Therefore, this study proposes a simple yet effective activation function that facilitates different thresholds and adaptive activations according to the positions of units and the contexts of inputs. Furthermore, the proposed activation function mathematically exhibits a more generalized form of Swish activation function, and thus we denoted it as Adaptive SwisH (ASH). ASH highlights informative features that exhibit large values in the top percentiles in an input, whereas it rectifies low values. Most importantly, ASH exhibits trainable, adaptive, and context-aware properties compared to other activation functions. Furthermore, ASH represents general formula of the previously studied activation function and provides a reasonable mathematical background for the superior performance. To validate the effectiveness and robustness of ASH, we implemented ASH into many deep learning models for various tasks, including classification, detection, segmentation, and image generation. Experimental analysis demonstrates that our activation function can provide the benefits of more accurate prediction and earlier convergence in many deep learning applications.
Stochastic epidemic models provide an interpretable probabilistic description of the spread of a disease through a population. Yet, fitting these models to partially observed data is a notoriously difficult task due to intractability of the likelihood for many classical models. To remedy this issue, this article introduces a novel data-augmented MCMC algorithm for exact Bayesian inference under the stochastic SIR model, given only discretely observed counts of infection. In a Metropolis-Hastings step, the latent data are jointly proposed from a surrogate process carefully designed to closely resemble the SIR model, from which we can efficiently generate epidemics consistent with the observed data. This yields a method that explores the high-dimensional latent space efficiently, and scales to outbreaks with hundreds of thousands of individuals. We show that the Markov chain underlying the algorithm is uniformly ergodic, and validate its performance via thorough simulation experiments and a case study on the 2013-2015 outbreak of Ebola Haemorrhagic Fever in Western Africa.
Variational Bayes methods are a scalable estimation approach for many complex state space models. However, existing methods exhibit a trade-off between accurate estimation and computational efficiency. This paper proposes a variational approximation that mitigates this trade-off. This approximation is based on importance densities that have been proposed in the context of efficient importance sampling. By directly conditioning on the observed data, the proposed method produces an accurate approximation to the exact posterior distribution. Because the steps required for its calibration are computationally efficient, the approach is faster than existing variational Bayes methods. The proposed method can be applied to any state space model that has a closed-form measurement density function and a state transition distribution that belongs to the exponential family of distributions. We illustrate the method in numerical experiments with stochastic volatility models and a macroeconomic empirical application using a high-dimensional state space model.
We develop an algorithmic framework for solving convex optimization problems using no-regret game dynamics. By converting the problem of minimizing a convex function into an auxiliary problem of solving a min-max game in a sequential fashion, we can consider a range of strategies for each of the two-players who must select their actions one after the other. A common choice for these strategies are so-called no-regret learning algorithms, and we describe a number of such and prove bounds on their regret. We then show that many classical first-order methods for convex optimization -- including average-iterate gradient descent, the Frank-Wolfe algorithm, the Heavy Ball algorithm, and Nesterov's acceleration methods -- can be interpreted as special cases of our framework as long as each player makes the correct choice of no-regret strategy. Proving convergence rates in this framework becomes very straightforward, as they follow from plugging in the appropriate known regret bounds. Our framework also gives rise to a number of new first-order methods for special cases of convex optimization that were not previously known.
We study differentially private (DP) algorithms for smooth stochastic minimax optimization, with stochastic minimization as a byproduct. The holy grail of these settings is to guarantee the optimal trade-off between the privacy and the excess population loss, using an algorithm with a linear time-complexity in the number of training samples. We provide a general framework for solving differentially private stochastic minimax optimization (DP-SMO) problems, which enables the practitioners to bring their own base optimization algorithm and use it as a black-box to obtain the near-optimal privacy-loss trade-off. Our framework is inspired from the recently proposed Phased-ERM method [22] for nonsmooth differentially private stochastic convex optimization (DP-SCO), which exploits the stability of the empirical risk minimization (ERM) for the privacy guarantee. The flexibility of our approach enables us to sidestep the requirement that the base algorithm needs to have bounded sensitivity, and allows the use of sophisticated variance-reduced accelerated methods to achieve near-linear time-complexity. To the best of our knowledge, these are the first near-linear time algorithms with near-optimal guarantees on the population duality gap for smooth DP-SMO, when the objective is (strongly-)convex--(strongly-)concave. Additionally, based on our flexible framework, we enrich the family of near-linear time algorithms for smooth DP-SCO with the near-optimal privacy-loss trade-off.
Since deep neural networks were developed, they have made huge contributions to everyday lives. Machine learning provides more rational advice than humans are capable of in almost every aspect of daily life. However, despite this achievement, the design and training of neural networks are still challenging and unpredictable procedures. To lower the technical thresholds for common users, automated hyper-parameter optimization (HPO) has become a popular topic in both academic and industrial areas. This paper provides a review of the most essential topics on HPO. The first section introduces the key hyper-parameters related to model training and structure, and discusses their importance and methods to define the value range. Then, the research focuses on major optimization algorithms and their applicability, covering their efficiency and accuracy especially for deep learning networks. This study next reviews major services and toolkits for HPO, comparing their support for state-of-the-art searching algorithms, feasibility with major deep learning frameworks, and extensibility for new modules designed by users. The paper concludes with problems that exist when HPO is applied to deep learning, a comparison between optimization algorithms, and prominent approaches for model evaluation with limited computational resources.
When and why can a neural network be successfully trained? This article provides an overview of optimization algorithms and theory for training neural networks. First, we discuss the issue of gradient explosion/vanishing and the more general issue of undesirable spectrum, and then discuss practical solutions including careful initialization and normalization methods. Second, we review generic optimization methods used in training neural networks, such as SGD, adaptive gradient methods and distributed methods, and theoretical results for these algorithms. Third, we review existing research on the global issues of neural network training, including results on bad local minima, mode connectivity, lottery ticket hypothesis and infinite-width analysis.
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