Iterative Hessian sketch (IHS) is an effective sketching method for modeling large-scale data. It was originally proposed by Pilanci and Wainwright (2016; JMLR) based on randomized sketching matrices. However, it is computationally intensive due to the iterative sketch process. In this paper, we analyze the IHS algorithm under the unconstrained least squares problem setting, then propose a deterministic approach for improving IHS via A-optimal subsampling. Our contributions are three-fold: (1) a good initial estimator based on the $A$-optimal design is suggested; (2) a novel ridged preconditioner is developed for repeated sketching; and (3) an exact line search method is proposed for determining the optimal step length adaptively. Extensive experimental results demonstrate that our proposed A-optimal IHS algorithm outperforms the existing accelerated IHS methods.

Pre-conditioning is a well-known concept that can significantly improve the convergence of optimization algorithms. For noise-free problems, where good pre-conditioners are not known a priori, iterative linear algebra methods offer one way to efficiently construct them. For the stochastic optimization problems that dominate contemporary machine learning, however, this approach is not readily available. We propose an iterative algorithm inspired by classic iterative linear solvers that uses a probabilistic model to actively infer a pre-conditioner in situations where Hessian-projections can only be constructed with strong Gaussian noise. The algorithm is empirically demonstrated to efficiently construct effective pre-conditioners for stochastic gradient descent and its variants. Experiments on problems of comparably low dimensionality show improved convergence. In very high-dimensional problems, such as those encountered in deep learning, the pre-conditioner effectively becomes an automatic learning-rate adaptation scheme, which we also empirically show to work well.

Fair classification has been a topic of intense study in machine learning, and several algorithms have been proposed towards this important task. However, in a recent study, Friedler et al. observed that fair classification algorithms may not be stable with respect to variations in the training dataset -- a crucial consideration in several real-world applications. Motivated by their work, we study the problem of designing classification algorithms that are both fair and stable. We propose an extended framework based on fair classification algorithms that are formulated as optimization problems, by introducing a stability-focused regularization term. Theoretically, we prove a stability guarantee, that was lacking in fair classification algorithms, and also provide an accuracy guarantee for our extended framework. Our accuracy guarantee can be used to inform the selection of the regularization parameter in our framework. To the best of our knowledge, this is the first work that combines stability and fairness in automated decision-making tasks. We assess the benefits of our approach empirically by extending several fair classification algorithms that are shown to achieve the best balance between fairness and accuracy over the Adult dataset. Our empirical results show that our framework indeed improves the stability at only a slight sacrifice in accuracy.

This paper studies noisy low-rank matrix completion: given partial and corrupted entries of a large low-rank matrix, the goal is to estimate the underlying matrix faithfully and efficiently. Arguably one of the most popular paradigms to tackle this problem is convex relaxation, which achieves remarkable efficacy in practice. However, the theoretical support of this approach is still far from optimal in the noisy setting, falling short of explaining the empirical success. We make progress towards demystifying the practical efficacy of convex relaxation vis-\`a-vis random noise. When the rank of the unknown matrix is a constant, we demonstrate that the convex programming approach achieves near-optimal estimation errors --- in terms of the Euclidean loss, the entrywise loss, and the spectral norm loss --- for a wide range of noise levels. All of this is enabled by bridging convex relaxation with the nonconvex Burer-Monteiro approach, a seemingly distinct algorithmic paradigm that is provably robust against noise. More specifically, we show that an approximate critical point of the nonconvex formulation serves as an extremely tight approximation of the convex solution, allowing us to transfer the desired statistical guarantees of the nonconvex approach to its convex counterpart.

Thompson sampling (TS) is a class of algorithms for sequential decision-making, which requires maintaining a posterior distribution over a model. However, calculating exact posterior distributions is intractable for all but the simplest models. Consequently, efficient computation of an approximate posterior distribution is a crucial problem for scalable TS with complex models, such as neural networks. In this paper, we use distribution optimization techniques to approximate the posterior distribution, solved via Wasserstein gradient flows. Based on the framework, a principled particle-optimization algorithm is developed for TS to approximate the posterior efficiently. Our approach is scalable and does not make explicit distribution assumptions on posterior approximations. Extensive experiments on both synthetic data and real large-scale data demonstrate the superior performance of the proposed methods.

We provide a framework to approximate the 2-Wasserstein distance and the optimal transport map, amenable to efficient training as well as statistical and geometric analysis. With the quadratic cost and considering the Kantorovich dual form of the optimal transportation problem, the Brenier theorem states that the optimal potential function is convex and the optimal transport map is the gradient of the optimal potential function. Using this geometric structure, we restrict the optimization problem to different parametrized classes of convex functions and pay special attention to the class of input-convex neural networks. We analyze the statistical generalization and the discriminative power of the resulting approximate metric, and we prove a restricted moment-matching property for the approximate optimal map. Finally, we discuss a numerical algorithm to solve the restricted optimization problem and provide numerical experiments to illustrate and compare the proposed approach with the established regularization-based approaches. We further discuss practical implications of our proposal in a modular and interpretable design for GANs which connects the generator training with discriminator computations to allow for learning an overall composite generator.

Machine learning models are vulnerable to adversarial examples. In this paper, we are concerned with black-box adversarial attacks, where only loss-oracle access to a model is available. At the heart of black-box adversarial attack is the gradient estimation problem with query complexity O(n), where n is the number of data features. Recent work has developed query-efficient gradient estimation schemes by exploiting data- and/or time-dependent priors. Practically, sign-based optimization has shown to be effective in both training deep nets as well as attacking them in a white-box setting. Therefore, instead of a gradient estimation view of black-box adversarial attacks, we view the black-box adversarial attack problem as estimating the gradient's sign bits. This shifts the view from continuous to binary black-box optimization and theoretically guarantees a lower query complexity of $\Omega(n/ \log_2(n+1))$ when given access to a Hamming loss oracle. We present three algorithms to estimate the gradient sign bits given a limited number of queries to the loss oracle. Using one of our proposed algorithms to craft black-box adversarial examples, we demonstrate evasion rate experiments on standard models trained on the MNIST, CIFAR10, and IMAGENET datasets that set new state-of-the-art results for query-efficient black-box attacks. Averaged over all the datasets and metrics, our attack fails $3.8\times$ less often and spends in total $2.5\times$ fewer queries than the current state-of-the-art attacks combined given a budget of 10,000 queries per attack attempt. On a public MNIST black-box attack challenge, our attack achieves the highest evasion rate surpassing all of the submitted attacks. Notably, our attack is hyperparameter-free (no hyperparameter tuning) and does not employ any data-/time-dependent prior, the latter fact suggesting that the number of queries can further be reduced.

For complex real-world systems, designing controllers are a difficult task. With the advent of neural networks as a proxy for complex function approximators, it has become popular to learn the controller directly. However, these controllers are specific to a given task and need to be relearned for a new task. Alternatively, one can learn just the model of the dynamical system and compose it with external controllers. Such a model is task (and controller) agnostic and must generalize well across the state space. This paper proposes learning a "sufficiently accurate" model of the dynamics that explicitly enforces small residual error on pre-defined parts of the state-space. We formulate task agnostic controller design for this learned model as an optimization problem with state and control constraints that is solved in an online fashion. We validate this approach in simulation using a challenging contact-based Ball-Paddle system.

An ongoing aim of research in multiobjective Bayesian optimization is to extend its applicability to a large number of objectives. While coping with a limited budget of evaluations, recovering the set of optimal compromise solutions generally requires numerous observations and is less interpretable since this set tends to grow larger with the number of objectives. We thus propose to focus on a specific solution originating from game theory, the Kalai-Smorodinsky solution, which possesses attractive properties. In particular, it ensures equal marginal gains over all objectives. We further make it insensitive to a monotonic transformation of the objectives by considering the objectives in the copula space. A novel tailored algorithm is proposed to search for the solution, in the form of a Bayesian optimization algorithm: sequential sampling decisions are made based on acquisition functions that derive from an instrumental Gaussian process prior. Our approach is tested on three problems with respectively four, six, and ten objectives. The method is available in the package GPGame available on CRAN at https://cran.r-project.org/package=GPGame.

In complex simulation environments, certain parameter space regions may result in non-convergent or unphysical outcomes. All parameters can therefore be labeled with a binary class describing whether or not they lead to valid results. In general, it can be very difficult to determine feasible parameter regions, especially without previous knowledge. We propose a novel algorithm to explore such an unknown parameter space and improve its feasibility classification in an iterative way. Moreover, we include an additional optimization target in the algorithm to guide the exploration towards regions of interest and to improve the classification therein. In our method we make use of well-established concepts from the field of machine learning like kernel support vector machines and kernel ridge regression. From a comparison with a Kriging-based exploration approach based on recently published results we can show the advantages of our algorithm in a binary feasibility classification scenario with a discrete feasibility constraint violation. In this context, we also propose an improvement of the Kriging-based exploration approach. We apply our novel method to a fully realistic, industrially relevant chemical process simulation to demonstrate its practical usability and find a comparably good approximation of the data space topology from relatively few data points.