As we seek to deploy machine learning models beyond virtual and controlled domains, it is critical to analyze not only the accuracy or the fact that it works most of the time, but if such a model is truly robust and reliable. This paper studies strategies to implement adversary robustly trained algorithms towards guaranteeing safety in machine learning algorithms. We provide a taxonomy to classify adversarial attacks and defenses, formulate the Robust Optimization problem in a min-max setting and divide it into 3 subcategories, namely: Adversarial (re)Training, Regularization Approach, and Certified Defenses. We survey the most recent and important results in adversarial example generation, defense mechanisms with adversarial (re)Training as their main defense against perturbations. We also survey mothods that add regularization terms that change the behavior of the gradient, making it harder for attackers to achieve their objective. Alternatively, we've surveyed methods which formally derive certificates of robustness by exactly solving the optimization problem or by approximations using upper or lower bounds. In addition, we discuss the challenges faced by most of the recent algorithms presenting future research perspectives.

The problem of linking functional connectomics to behavior is extremely challenging due to the complex interactions between the two distinct, but related, data domains. We propose a coupled manifold optimization framework which projects fMRI data onto a low dimensional matrix manifold common to the cohort. The patient specific loadings simultaneously map onto a behavioral measure of interest via a second, non-linear, manifold. By leveraging the kernel trick, we can optimize over a potentially infinite dimensional space without explicitly computing the embeddings. As opposed to conventional manifold learning, which assumes a fixed input representation, our framework directly optimizes for embedding directions that predict behavior. Our optimization algorithm combines proximal gradient descent with the trust region method, which has good convergence guarantees. We validate our framework on resting state fMRI from fifty-eight patients with Autism Spectrum Disorder using three distinct measures of clinical severity. Our method outperforms traditional representation learning techniques in a cross validated setting, thus demonstrating the predictive power of our coupled objective.

In this paper we provide a general framework for designing, analyzing and implementing such algorithms in the episodic reinforcement learning problem. This framework is built upon Lagrangian duality, and demonstrates that every model-optimistic algorithm that constructs an optimistic MDP has an equivalent representation as a value-optimistic dynamic programming algorithm. Typically, it was thought that these two classes of algorithms were distinct, with model-optimistic algorithms benefiting from a cleaner probabilistic analysis while value-optimistic algorithms are easier to implement and thus more practical. With the framework developed in this paper, we show that it is possible to get the best of both worlds by providing a class of algorithms which have a computationally efficient dynamic-programming implementation and also a simple probabilistic analysis. Besides being able to capture many existing algorithms in the tabular setting, our framework can also address largescale problems under realizable function approximation, where it enables a simple model-based analysis of some recently proposed methods.

Wireless connectivity is instrumental in enabling scalable federated learning (FL), yet wireless channels bring challenges for model training, in which channel randomness perturbs each worker's model update while multiple workers' updates incur significant interference under limited bandwidth. To address these challenges, in this work we formulate a novel constrained optimization problem, and propose an FL framework harnessing wireless channel perturbations and interference for improving privacy, bandwidth-efficiency, and scalability. The resultant algorithm is coined analog federated ADMM (A-FADMM) based on analog transmissions and the alternating direct method of multipliers (ADMM). In A-FADMM, all workers upload their model updates to the parameter server (PS) using a single channel via analog transmissions, during which all models are perturbed and aggregated over-the-air. This not only saves communication bandwidth, but also hides each worker's exact model update trajectory from any eavesdropper including the honest-but-curious PS, thereby preserving data privacy against model inversion attacks. We formally prove the convergence and privacy guarantees of A-FADMM for convex functions under time-varying channels, and numerically show the effectiveness of A-FADMM under noisy channels and stochastic non-convex functions, in terms of convergence speed and scalability, as well as communication bandwidth and energy efficiency.

Variance reduction techniques are popular in accelerating gradient descent and stochastic gradient descent for optimization problems defined on both Euclidean space and Riemannian manifold. In this paper, we further improve on existing variance reduction methods for non-convex Riemannian optimization, including R-SVRG and R-SRG/R-SPIDER with batch size adaptation. We show that this strategy can achieve lower total complexities for optimizing both general non-convex and gradient dominated functions under both finite-sum and online settings. As a result, we also provide simpler convergence analysis for R-SVRG and improve complexity bounds for R-SRG under finite-sum setting. Specifically, we prove that R-SRG achieves the same near-optimal complexity as R-SPIDER without requiring a small step size. Empirical experiments on a variety of tasks demonstrate effectiveness of proposed adaptive batch size scheme.

Information-theoretic quantities like entropy and mutual information have found numerous uses in machine learning. It is well known that there is a strong connection between these entropic quantities and submodularity since entropy over a set of random variables is submodular. In this paper, we study combinatorial information measures that generalize independence, (conditional) entropy, (conditional) mutual information, and total correlation defined over sets of (not necessarily random) variables. These measures strictly generalize the corresponding entropic measures since they are all parameterized via submodular functions that themselves strictly generalize entropy. Critically, we show that, unlike entropic mutual information in general, the submodular mutual information is actually submodular in one argument, holding the other fixed, for a large class of submodular functions whose third-order partial derivatives satisfy a non-negativity property. This turns out to include a number of practically useful cases such as the facility location and set-cover functions. We study specific instantiations of the submodular information measures on these, as well as the probabilistic coverage, graph-cut, and saturated coverage functions, and see that they all have mathematically intuitive and practically useful expressions. Regarding applications, we connect the maximization of submodular (conditional) mutual information to problems such as mutual-information-based, query-based, and privacy-preserving summarization -- and we connect optimizing the multi-set submodular mutual information to clustering and robust partitioning.

We develop and analyze a projected particle Langevin optimization method to learn the distribution in the Sch\"{o}nberg integral representation of the radial basis functions from training samples. More specifically, we characterize a distributionally robust optimization method with respect to the Wasserstein distance to optimize the distribution in the Sch\"{o}nberg integral representation. To provide theoretical performance guarantees, we analyze the scaling limits of a projected particle online (stochastic) optimization method in the mean-field regime. In particular, we prove that in the scaling limits, the empirical measure of the Langevin particles converges to the law of a reflected It\^{o} diffusion-drift process. Moreover, the drift is also a function of the law of the underlying process. Using It\^{o} lemma for semi-martingales and Grisanov's change of measure for the Wiener processes, we then derive a Mckean-Vlasov type partial differential equation (PDE) with Robin boundary conditions that describes the evolution of the empirical measure of the projected Langevin particles in the mean-field regime. In addition, we establish the existence and uniqueness of the steady-state solutions of the derived PDE in the weak sense. We apply our learning approach to train radial kernels in the kernel locally sensitive hash (LSH) functions, where the training data-set is generated via a $k$-mean clustering method on a small subset of data-base. We subsequently apply our kernel LSH with a trained kernel for image retrieval task on MNIST data-set, and demonstrate the efficacy of our kernel learning approach. We also apply our kernel learning approach in conjunction with the kernel support vector machines (SVMs) for classification of benchmark data-sets.

In this course you will learn all about the mathematical optimization of linear programming for data science and business analytics. This course is very unique and have its own importance in their respective disciplines. The data science and business study heavily rely on optimization. Optimization is the study of analysis and interpreting mathematical data under the special rules and formula. The length of the course is more than 6 hours and there are total more than 4 sections in this course.

Determination of inspection and maintenance policies for minimizing long-term risks and costs in deteriorating engineering environments constitutes a complex optimization problem. Major computational challenges include the (i) curse of dimensionality, due to exponential scaling of state/action set cardinalities with the number of components; (ii) curse of history, related to exponentially growing decision-trees with the number of decision-steps; (iii) presence of state uncertainties, induced by inherent environment stochasticity and variability of inspection/monitoring measurements; (iv) presence of constraints, pertaining to stochastic long-term limitations, due to resource scarcity and other infeasible/undesirable system responses. In this work, these challenges are addressed within a joint framework of constrained Partially Observable Markov Decision Processes (POMDP) and multi-agent Deep Reinforcement Learning (DRL). POMDPs optimally tackle (ii)-(iii), combining stochastic dynamic programming with Bayesian inference principles. Multi-agent DRL addresses (i), through deep function parametrizations and decentralized control assumptions. Challenge (iv) is herein handled through proper state augmentation and Lagrangian relaxation, with emphasis on life-cycle risk-based constraints and budget limitations. The underlying algorithmic steps are provided, and the proposed framework is found to outperform well-established policy baselines and facilitate adept prescription of inspection and intervention actions, in cases where decisions must be made in the most resource- and risk-aware manner.

White box adversarial perturbations are sought via iterative optimization algorithms most often minimizing an adversarial loss on a $l_p$ neighborhood of the original image, the so-called distortion set. Constraining the adversarial search with different norms results in disparately structured adversarial examples. Here we explore several distortion sets with structure-enhancing algorithms. These new structures for adversarial examples, yet pervasive in optimization, are for instance a challenge for adversarial theoretical certification which again provides only $l_p$ certificates. Because adversarial robustness is still an empirical field, defense mechanisms should also reasonably be evaluated against differently structured attacks. Besides, these structured adversarial perturbations may allow for larger distortions size than their $l_p$ counter-part while remaining imperceptible or perceptible as natural slight distortions of the image. Finally, they allow some control on the generation of the adversarial perturbation, like (localized) bluriness.