This paper investigates recently proposed approaches for defending against adversarial examples and evaluating adversarial robustness. The existence of adversarial examples in trained neural networks reflects the fact that expected risk alone does not capture the model's performance against worst-case inputs. We motivate the use of adversarial risk as an objective, although it cannot easily be computed exactly. We then frame commonly used attacks and evaluation metrics as defining a tractable surrogate objective to the true adversarial risk. This suggests that models may be obscured to adversaries, by optimizing this surrogate rather than the true adversarial risk. We demonstrate that this is a significant problem in practice by repurposing gradient-free optimization techniques into adversarial attacks, which we use to decrease the accuracy of several recently proposed defenses to near zero. Our hope is that our formulations and results will help researchers to develop more powerful defenses.

Optimal transport as a loss for machine learning optimization problems has recently gained a lot of attention. Building upon recent advances in computational optimal transport, we develop an optimal transport non-negative matrix factorization (NMF) algorithm for supervised speech blind source separation (BSS). Optimal transport allows us to design and leverage a cost between short-time Fourier transform (STFT) spectrogram frequencies, which takes into account how humans perceive sound. We give empirical evidence that using our proposed optimal transport NMF leads to perceptually better results than Euclidean NMF, for both isolated voice reconstruction and BSS tasks. Finally, we demonstrate how to use optimal transport for cross domain sound processing tasks, where frequencies represented in the input spectrograms may be different from one spectrogram to another.

One of the popular approaches for low-rank tensor completion is to use the latent trace norm as a low-rank regularizer. However, most of the existing works learn a sparse combination of tensors. In this work, we fill this gap by proposing a variant of the latent trace norm which helps to learn a non-sparse combination of tensors. We develop a dual framework for solving the problem of latent trace norm regularized low-rank tensor completion. In this framework, we first show a novel characterization of the solution space with a novel factorization, and then, propose two scalable optimization formulations. The problems are shown to lie on a Cartesian product of Riemannian spectrahedron manifolds. We exploit the versatile Riemannian optimization framework for proposing computationally efficient trust-region algorithms. The experiments show the good performance of the proposed algorithms on several real-world data sets in different applications.

We consider variants of trust-region and cubic regularization methods for non-convex optimization, in which the Hessian matrix is approximated. Under mild conditions on the inexact Hessian, and using approximate solution of the corresponding sub-problems, we provide iteration complexity to achieve $ \epsilon $-approximate second-order optimality which have shown to be tight. Our Hessian approximation conditions constitute a major relaxation over the existing ones in the literature. Consequently, we are able to show that such mild conditions allow for the construction of the approximate Hessian through various random sampling methods. In this light, we consider the canonical problem of finite-sum minimization, provide appropriate uniform and non-uniform sub-sampling strategies to construct such Hessian approximations, and obtain optimal iteration complexity for the corresponding sub-sampled trust-region and cubic regularization methods.

Solving inverse problems with iterative algorithms is popular, especially for large data. Due to time constraints, the number of possible iterations is usually limited, potentially affecting the achievable accuracy. Given an error one is willing to tolerate, an important question is whether it is possible to modify the original iterations to obtain faster convergence to a minimizer achieving the allowed error without increasing the computational cost of each iteration considerably. Relying on recent recovery techniques developed for settings in which the desired signal belongs to some low-dimensional set, we show that using a coarse estimate of this set may lead to faster convergence at the cost of an additional reconstruction error related to the accuracy of the set approximation. Our theory ties to recent advances in sparse recovery, compressed sensing, and deep learning. Particularly, it may provide a possible explanation to the successful approximation of the l1-minimization solution by neural networks with layers representing iterations, as practiced in the learned iterative shrinkage-thresholding algorithm (LISTA).

Noisy observations coupled with nonlinear dynamics pose one of the biggest challenges in robot motion planning. By decomposing the nonlinear dynamics into a discrete set of local dynamics models, hybrid dynamics provide a natural way to model nonlinear dynamics, especially in systems with sudden "jumps" in the dynamics, due to factors such as contacts. We propose a hierarchical POMDP planner that develops locally optimal motion plans for hybrid dynamics models. The hierarchical planner first develops a high-level motion plan to sequence the local dynamics models to be visited. The high-level plan is then converted into a detailed cost-optimized continuous state plan. This hierarchical planning approach results in a decomposition of the POMDP planning problem into smaller sub-parts that can be solved with significantly lower computational costs. The ability to sequence the visitation of local dynamics models also provides a powerful way to leverage the hybrid dynamics to reduce state uncertainty. We evaluate the proposed planner for two navigation and localization tasks in simulated domains, as well as an assembly task with a real robotic manipulator.

Feature missing is a serious problem in many applications, which may lead to low quality of training data and further significantly degrade the learning performance. While feature acquisition usually involves special devices or complex process, it is expensive to acquire all feature values for the whole dataset. On the other hand, features may be correlated with each other, and some values may be recovered from the others. It is thus important to decide which features are most informative for recovering the other features as well as improving the learning performance. In this paper, we try to train an effective classification model with least acquisition cost by jointly performing active feature querying and supervised matrix completion. When completing the feature matrix, a novel target function is proposed to simultaneously minimize the reconstruction error on observed entries and the supervised loss on training data. When querying the feature value, the most uncertain entry is actively selected based on the variance of previous iterations. In addition, a bi-objective optimization method is presented for cost-aware active selection when features bear different acquisition costs. The effectiveness of the proposed approach is well validated by both theoretical analysis and experimental study.

Submodular functions have applications throughout machine learning, but in many settings, we do not have direct access to the underlying function $f$. We focus on stochastic functions that are given as an expectation of functions over a distribution $P$. In practice, we often have only a limited set of samples $f_i$ from $P$. The standard approach indirectly optimizes $f$ by maximizing the sum of $f_i$. However, this ignores generalization to the true (unknown) distribution. In this paper, we achieve better performance on the actual underlying function $f$ by directly optimizing a combination of bias and variance. Algorithmically, we accomplish this by showing how to carry out distributionally robust optimization (DRO) for submodular functions, providing efficient algorithms backed by theoretical guarantees which leverage several novel contributions to the general theory of DRO. We also show compelling empirical evidence that DRO improves generalization to the unknown stochastic submodular function.

Inverse classification uses an induced classifier as a queryable oracle to guide test instances towards a preferred posterior class label. The result produced from the process is a set of instance-specific feature perturbations, or recommendations, that optimally improve the probability of the class label. In this work, we adopt a causal approach to inverse classification, eliciting treatment policies (i.e., feature perturbations) for models induced with causal properties. In so doing, we solve a long-standing problem of eliciting multiple, continuously valued treatment policies, using an updated framework and corresponding set of assumptions, which we term the inverse classification potential outcomes framework (ICPOF), along with a new measure, referred to as the individual future estimated effects ($i$FEE). We also develop the approximate propensity score (APS), based on Gaussian processes, to weight treatments, much like the inverse propensity score weighting used in past works. We demonstrate the viability of our methods on student performance.

While optimizing convex objective (loss) functions has been a powerhouse for machine learning for at least two decades, non-convex loss functions have attracted fast growing interests recently, due to many desirable properties such as superior robustness and classification accuracy, compared with their convex counterparts. The main obstacle for non-convex estimators is that it is in general intractable to find the optimal solution. In this paper, we study the computational issues for some non-convex M-estimators. In particular, we show that the stochastic variance reduction methods converge to the global optimal with linear rate, by exploiting the statistical property of the population loss. En route, we improve the convergence analysis for the batch gradient method in \cite{mei2016landscape}.