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Register a new userAn Online Method for Distributionally Deep Robust Optimization
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Qi Qi
Publication date
2020
fields
Informatics Engineering
and research's language is
English
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In this paper, we propose a practical online method for solving a distributionally robust optimization (DRO) for deep learning, which has important applications in machine learning for improving the robustness of neural networks. In the literature, most methods for solving DRO are based on stochastic primal-dual methods. However, primal-dual methods for deep DRO suffer from several drawbacks: (1) manipulating a high-dimensional dual variable corresponding to the size of data is time expensive; (2) they are not friendly to online learning where data is coming sequentially. To address these issues, we transform the min-max formulation into a minimization formulation and propose a practical duality-free online stochastic method for solving deep DRO with KL divergence regularization. The proposed online stochastic method resembles the practical stochastic Nesterovs method in several perspectives that are widely used for learning deep neural networks. Under a Polyak-Lojasiewicz (PL) condition, we prove that the proposed method can enjoy an optimal sample complexity without any requirements on large batch size. Of independent interest, the proposed method can be also used for solving a family of stochastic compositional problems.
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In this article, we study the problem of robust reconfigurable intelligent surface (RIS)-aided downlink communication over heterogeneous RIS types in the supervised learning setting. By modeling downlink communication over heterogeneous RIS designs as different workers that learn how to optimize phase configurations in a distributed manner, we solve this distributed learning problem using a distributionally robust formulation in a communication-efficient manner, while establishing its rate of convergence. By doing so, we ensure that the global model performance of the worst-case worker is close to the performance of other workers. Simulation results show that our proposed algorithm requires fewer communication rounds (about 50% lesser) to achieve the same worst-case distribution test accuracy compared to competitive baselines.
Limiting failures of machine learning systems is vital for safety-critical applications. In order to improve the robustness of machine learning systems, Distributionally Robust Optimization (DRO) has been proposed as a generalization of Empirical Risk Minimization (ERM)aiming at addressing this need. However, its use in deep learning has been severely restricted due to the relative inefficiency of the optimizers available for DRO in comparison to the wide-spread variants of Stochastic Gradient Descent (SGD) optimizers for ERM. We propose SGD with hardness weighted sampling, a principled and efficient optimization method for DRO in machine learning that is particularly suited in the context of deep learning. Similar to a hard example mining strategy in essence and in practice, the proposed algorithm is straightforward to implement and computationally as efficient as SGD-based optimizers used for deep learning, requiring minimal overhead computation. In contrast to typical ad hoc hard mining approaches, and exploiting recent theoretical results in deep learning optimization, we prove the convergence of our DRO algorithm for over-parameterized deep learning networks with ReLU activation and finite number of layers and parameters. Our experiments on brain tumor segmentation in MRI demonstrate the feasibility and the usefulness of our approach. Using our hardness weighted sampling leads to a decrease of 2% of the interquartile range of the Dice scores for the enhanced tumor and the tumor core regions. The code for the proposed hard weighted sampler will be made publicly available.
We propose kernel distributionally robust optimization (Kernel DRO) using insights from the robust optimization theory and functional analysis. Our method uses reproducing kernel Hilbert spaces (RKHS) to construct a wide range of convex ambiguity sets, which can be generalized to sets based on integral probability metrics and finite-order moment bounds. This perspective unifies multiple existing robust and stochastic optimization methods. We prove a theorem that generalizes the classical duality in the mathematical problem of moments. Enabled by this theorem, we reformulate the maximization with respect to measures in DRO into the dual program that searches for RKHS functions. Using universal RKHSs, the theorem applies to a broad class of loss functions, lifting common limitations such as polynomial losses and knowledge of the Lipschitz constant. We then establish a connection between DRO and stochastic optimization with expectation constraints. Finally, we propose practical algorithms based on both batch convex solvers and stochastic functional gradient, which apply to general optimization and machine learning tasks.
We propose and analyze algorithms for distributionally robust optimization of convex losses with conditional value at risk (CVaR) and $chi^2$ divergence uncertainty sets. We prove that our algorithms require a number of gradient evaluations independent of training set size and number of parameters, making them suitable for large-scale applications. For $chi^2$ uncertainty sets these are the first such guarantees in the literature, and for CVaR our guarantees scale linearly in the uncertainty level rather than quadratically as in previous work. We also provide lower bounds proving the worst-case optimality of our algorithms for CVaR and a penalized version of the $chi^2$ problem. Our primary technical contributions are novel bounds on the bias of batch robust risk estimation and the variance of a multilevel Monte Carlo gradient estimator due to [Blanchet & Glynn, 2015]. Experiments on MNIST and ImageNet confirm the theoretical scaling of our algorithms, which are 9--36 times more efficient than full-batch methods.
Distributionally robust supervised learning (DRSL) is emerging as a key paradigm for building reliable machine learning systems for real-world applications -- reflecting the need for classifiers and predictive models that are robust to the distribution shifts that arise from phenomena such as selection bias or nonstationarity. Existing algorithms for solving Wasserstein DRSL -- one of the most popular DRSL frameworks based around robustness to perturbations in the Wasserstein distance -- involve solving complex subproblems or fail to make use of stochastic gradients, limiting their use in large-scale machine learning problems. We revisit Wasserstein DRSL through the lens of min-max optimization and derive scalable and efficiently implementable stochastic extra-gradient algorithms which provably achieve faster convergence rates than existing approaches. We demonstrate their effectiveness on synthetic and real data when compared to existing DRSL approaches. Key to our results is the use of variance reduction and random reshuffling to accelerate stochastic min-max optimization, the analysis of which may be of independent interest.
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