Research in adversarial learning follows a cat and mouse game between attackers and defenders where attacks are proposed, they are mitigated by new defenses, and subsequently new attacks are proposed that break earlier defenses, and so on. However, it has remained unclear as to whether there are conditions under which no better attacks or defenses can be proposed. In this paper, we propose a game-theoretic framework for studying attacks and defenses which exist in equilibrium. Under a locally linear decision boundary model for the underlying binary classifier, we prove that the Fast Gradient Method attack and the Randomized Smoothing defense form a Nash Equilibrium. We then show how this equilibrium defense can be approximated given finitely many samples from a data-generating distribution, and derive a generalization bound for the performance of our approximation.

The objective of the response optimization algorithm is to exploit the mathematical model to look for optimal operating conditions. Indeed, the predictive model allows us to simulate different operating scenarios and adjust the control variables to improve efficiency. For a given set of states, determine the controls that minimize or maximize the performance variables. The next figure illustrates the response optimization process. As we can see, for a given state value, s, the control value, c*, minimizes the performance value.

Nonsmoothness is often a curse for optimization; but it is sometimes a blessing, in particular for applications in machine learning. In this paper, we present the specific structure of nonsmooth optimization problems appearing in machine learning and illustrate how to leverage this structure in practice, for compression, acceleration, or dimension reduction. We pay a special attention to the presentation to make it concise and easily accessible, with both simple examples and general results.

In an effort to improve generalization in deep learning, we propose SALR: a sharpness-aware learning rate update technique designed to recover flat minimizers. Our method dynamically updates the learning rate of gradient-based optimizers based on the local sharpness of the loss function. This allows optimizers to automatically increase learning rates at sharp valleys to increase the chance of escaping them. We demonstrate the effectiveness of SALR when adopted by various algorithms over a broad range of networks. Our experiments indicate that SALR improves generalization, converges faster, and drives solutions to significantly flatter regions.

Methods for supervised principal component analysis (SPCA) aim to incorporate label information into principal component analysis (PCA), so that the extracted features are more useful for a prediction task of interest. Prior work on SPCA has focused primarily on optimizing prediction error, and has neglected the value of maximizing variance explained by the extracted features. We propose a new method for SPCA that addresses both of these objectives jointly, and demonstrate empirically that our approach dominates existing approaches, i.e., outperforms them with respect to both prediction error and variation explained. Our approach accommodates arbitrary supervised learning losses and, through a statistical reformulation, provides a novel low-rank extension of generalized linear models.

One important goal in sequential decision making problems is to construct a policy that maximizes the average reward over a certain amount of the time. Depending on the purpose of applications, the duration of the learned policy for use in the future (i.e., the planning horizon) is often unknown and can be different from what we consider in the stage of policy optimization. In addition, the performance measure used in learning the policy often depends on the choice of the initial state's distribution. It is always of a great interest to learn a policy with strong generalizability and adaptivity. Given a pre-collected data of multiple trajectories consisting of states, actions and rewards, our goal is to learn a robust policy in the sense that it can guarantee the uniform performance over the unknown planning horizon and the distributional change in the initial state.

Learning-based methodologies increasingly find applications in safety-critical domains like autonomous driving and medical robotics. Due to the rare nature of dangerous events, real-world testing is prohibitively expensive and unscalable. In this work, we employ a probabilistic approach to safety evaluation in simulation, where we are concerned with computing the probability of dangerous events. We develop a novel rare-event simulation method that combines exploration, exploitation, and optimization techniques to find failure modes and estimate their rate of occurrence. We provide rigorous guarantees for the performance of our method in terms of both statistical and computational efficiency. Finally, we demonstrate the efficacy of our approach on a variety of scenarios, illustrating its usefulness as a tool for rapid sensitivity analysis and model comparison that are essential to developing and testing safety-critical autonomous systems.

Optimal Transport is a theory that allows to define geometrical notions of distance between probability distributions and to find correspondences, relationships, between sets of points. Many machine learning applications are derived from this theory, at the frontier between mathematics and optimization. This thesis proposes to study the complex scenario in which the different data belong to incomparable spaces. In particular we address the following questions: how to define and apply Optimal Transport between graphs, between structured data? How can it be adapted when the data are varied and not embedded in the same metric space? This thesis proposes a set of Optimal Transport tools for these different cases. An important part is notably devoted to the study of the Gromov-Wasserstein distance whose properties allow to define interesting transport problems on incomparable spaces. More broadly, we analyze the mathematical properties of the various proposed tools, we establish algorithmic solutions to compute them and we study their applicability in numerous machine learning scenarii which cover, in particular, classification, simplification, partitioning of structured data, as well as heterogeneous domain adaptation.

These authors have contributed equally to this work. Abstract Stationary robotic trainers are lower limb rehab robots which often inco rporate an exoskeleton attached to a stationary base. The issue observed in the stationery trainers for simultaneous knee and ankle - foot complex joints is that they restrict the natura l motion of ankle - foot in the rehab trainings due to the insufficient D e grees of F reedom (DOFs) of these trainers. A new stationary knee - ankle - foot rehab robot with all necessary DOFs is developed here . A typical rehab training is first implemented in simulation, and then tested on a healthy subject. R esults show that the prop osed system functions naturally and meets the requirements of the desired rehab training . I ntroduction Damage to the nervou s system, caused by accidents such as stroke and spinal cord injuries, often lead s to movement disorders [1] . This issue is of practical importance, a s it can severely hamper activities of daily living for the survivors, in light of the large number of stroke incidents per year [2] . To address this issue, patients take rehabilitation exercises under the supervision of therapists to regain their a bilities. During the last two decades, rehabilitation exercises have received sig nificant attention from robotic researchers. Many robots have been designed to facilitate rehab trainings for both patients and therapists, and to improve the training result s in terms of repeatability, reliability and accuracy in evaluating the patient ' s progress [3] . Among the developed platforms, lower limb rehab robots are of more impo rtance, since they have a direct effect on gait. The lower limb rehab robots can be divided into two general types: standing/walking and sitting/lying. The standing/walking robots mainly aim to correct the gait pattern in conditions similar to daily life.

Subset selection algorithms are ubiquitous in AI-driven applications, including, online recruiting portals and image search engines, so it is imperative that these tools are not discriminatory on the basis of protected attributes such as gender or race. Currently, fair subset selection algorithms assume that the protected attributes are known as part of the dataset. However, attributes may be noisy due to errors during data collection or if they are imputed (as is often the case in real-world settings). While a wide body of work addresses the effect of noise on the performance of machine learning algorithms, its effect on fairness remains largely unexamined. We find that in the presence of noisy protected attributes, in attempting to increase fairness without considering noise, one can, in fact, decrease the fairness of the result! Towards addressing this, we consider an existing noise model in which there is probabilistic information about the protected attributes (e.g.,[19, 32, 56, 44]), and ask is fair selection is possible under noisy conditions? We formulate a ``denoised'' selection problem which functions for a large class of fairness metrics; given the desired fairness goal, the solution to the denoised problem violates the goal by at most a small multiplicative amount with high probability. Although the denoised problem turns out to be NP-hard, we give a linear-programming based approximation algorithm for it. We empirically evaluate our approach on both synthetic and real-world datasets. Our empirical results show that this approach can produce subsets which significantly improve the fairness metrics despite the presence of noisy protected attributes, and, compared to prior noise-oblivious approaches, has better Pareto-tradeoffs between utility and fairness.