Neural Networks (NN) have been proposed in the past as an effective means for both modeling and control of systems with very complex dynamics. However, despite the extensive research, NN-based controllers have not been adopted by the industry for safety critical systems. The primary reason is that systems with learning based controllers are notoriously hard to test and verify. Even harder is the analysis of such systems against system-level specifications. In this paper, we provide a gradient based method for searching the input space of a closed-loop control system in order to find adversarial samples against some system-level requirements. Our experimental results show that combined with randomized search, our method outperforms Simulated Annealing optimization.

We present the first accelerated randomized algorithm for solving linear systems in Euclidean spaces. One essential problem of this type is the matrix inversion problem. In particular, our algorithm can be specialized to invert positive definite matrices in such a way that all iterates (approximate solutions) generated by the algorithm are positive definite matrices themselves. This opens the way for many applications in the field of optimization and machine learning. As an application of our general theory, we develop the first accelerated (deterministic and stochastic) quasi-Newton updates. Our updates lead to provably more aggressive approximations of the inverse Hessian, and lead to speed-ups over classical non-accelerated rules in numerical experiments. Experiments with empirical risk minimization show that our rules can accelerate training of machine learning models.

In order to achieve state-of-the-art performance, modern machine learning techniques require careful data pre-processing and hyperparameter tuning. Moreover, given the ever increasing number of machine learning models being developed, model selection is becoming increasingly important. Automating the selection and tuning of machine learning pipelines, which can include different data pre-processing methods and machine learning models, has long been one of the goals of the machine learning community. In this paper, we propose to solve this meta-learning task by combining ideas from collaborative filtering and Bayesian optimization. Specifically, we use a probabilistic matrix factorization model to transfer knowledge across experiments performed in hundreds of different datasets and use an acquisition function to guide the exploration of the space of possible ML pipelines. In our experiments, we show that our approach quickly identifies high-performing pipelines across a wide range of datasets, significantly outperforming the current state-of-the-art.

We study a generalization of the classic isotonic regression problem where we allow separable nonconvex objective functions, focusing on the case of estimators used in robust regression. A simple dynamic programming approach allows us to solve this problem to within ε-accuracy (of the global minimum) in time linear in 1/ε and the dimension. We can combine techniques from the convex case with branch-and-bound ideas to form a new algorithm for this problem that naturally exploits the shape of the objective function. Our algorithm achieves the best bounds for both the general nonconvex and convex case (linear in log (1/ε)), while performing much faster in practice than a straightforward dynamic programming approach, especially as the desired accuracy increases.

We consider semidefinite programs (SDPs) of size $n$ with equality constraints. In order to overcome scalability issues, Burer and Monteiro proposed a factorized approach based on optimizing over a matrix $Y$ of size $n\times k$ such that $X=YY^*$ is the SDP variable. The advantages of such formulation are twofold: the dimension of the optimization variable is reduced, and positive semidefiniteness is naturally enforced. However, optimization in $Y$ is non-convex. In prior work, it has been shown that, when the constraints on the factorized variable regularly define a smooth manifold, provided $k$ is large enough, for almost all cost matrices, all second-order stationary points (SOSPs) are optimal. Importantly, in practice, one can only compute points which approximately satisfy necessary optimality conditions, leading to the question: are such points also approximately optimal? To this end, and under similar assumptions, we use smoothed analysis to show that approximate SOSPs for a randomly perturbed objective function are approximate global optima, with $k$ scaling like the square root of the number of constraints (up to log factors). We particularize our results to an SDP relaxation of phase retrieval.

We study the decentralized distributed computation of discrete approximations for the regularized Wasserstein barycenter of a finite set of continuous probability measures distributedly stored over a network. We assume there is a network of agents/machines/computers, and each agent holds a private continuous probability measure and seeks to compute the barycenter of all the measures in the network by getting samples from its local measure and exchanging information with its neighbors. Motivated by this problem, we develop, and analyze, a novel accelerated primal-dual stochastic gradient method for general stochastic convex optimization problems with linear equality constraints. Then, we apply this method to the decen- tralized distributed optimization setting to obtain a new algorithm for the distributed semi-discrete regularized Wasserstein barycenter problem. Moreover, we show explicit non-asymptotic complexity for the proposed algorithm. Finally, we show the effectiveness of our method on the distributed computation of the regularized Wasserstein barycenter of univariate Gaussian and von Mises distributions, as well as some applications to image aggregation.

The original simplicial method (OSM), a variant of the classic Kelley’s cutting plane method, has been shown to converge to the minimizer of a composite convex and submodular objective, though no rate of convergence for this method was known. Moreover, OSM is required to solve subproblems in each iteration whose size grows linearly in the number of iterations. We propose a limited memory version of Kelley’s method (L-KM) and of OSM that requires limited memory (at most n+ 1 constraints for an n-dimensional problem) independent of the iteration. We prove convergence for L-KM when the convex part of the objective g is strongly convex and show it converges linearly when g is also smooth. Our analysis relies on duality between minimization of the composite convex and submodular objective and minimization of a convex function over the submodular base polytope. We introduce a limited memory version, L-FCFW, of the Fully-Corrective Frank-Wolfe (FCFW) method with approximate correction, to solve the dual problem. We show that L-FCFW and L-KM are dual algorithms that produce the same sequence of iterates; hence both converge linearly (when g is smooth and strongly convex) and with limited memory. We propose L-KM to minimize composite convex and submodular objectives; however, our results on L-FCFW hold for general polytopes and may be of independent interest.

The \emph{Cheap Gradient Principle}~\citep{Griewank:2008:EDP:1455489} --- the computational cost of computing a $d$-dimensional vector of partial derivatives of a scalar function is nearly the same (often within a factor of $5$) as that of simply computing the scalar function itself --- is of central importance in optimization; it allows us to quickly obtain (high-dimensional) gradients of scalar loss functions which are subsequently used in black box gradient-based optimization procedures. The current state of affairs is markedly different with regards to computing sub-derivatives: widely used ML libraries, including TensorFlow and PyTorch, do \emph{not} correctly compute (generalized) sub-derivatives even on simple differentiable examples. This work considers the question: is there a \emph{Cheap Sub-gradient Principle}? Our main result shows that, under certain restrictions on our library of non-smooth functions (standard in non-linear programming), provably correct generalized sub-derivatives can be computed at a computational cost that is within a (dimension-free) factor of $6$ of the cost of computing the scalar function itself.

We study the decentralized distributed computation of discrete approximations for the regularized Wasserstein barycenter of a finite set of continuous probability measures distributedly stored over a network. We assume there is a network of agents/machines/computers, and each agent holds a private continuous probability measure and seeks to compute the barycenter of all the measures in the network by getting samples from its local measure and exchanging information with its neighbors. Motivated by this problem, we develop, and analyze, a novel accelerated primal-dual stochastic gradient method for general stochastic convex optimization problems with linear equality constraints. Then, we apply this method to the decen- tralized distributed optimization setting to obtain a new algorithm for the distributed semi-discrete regularized Wasserstein barycenter problem. Moreover, we show explicit non-asymptotic complexity for the proposed algorithm. Finally, we show the effectiveness of our method on the distributed computation of the regularized Wasserstein barycenter of univariate Gaussian and von Mises distributions, as well as some applications to image aggregation.

We consider the problem of multi-objective maximization of monotone submodular functions subject to cardinality constraint, often formulated as $\max_{|A|=k}\min_{i\in\{1,\dots,m\}}f_i(A)$. While it is widely known that greedy methods work well for a single objective, the problem becomes much harder with multiple objectives. In fact, Krause et al.\ (2008) showed that when the number of objectives $m$ grows as the cardinality $k$ i.e., $m=\Omega(k)$, the problem is inapproximable (unless $P=NP$). On the other hand, when $m$ is constant Chekuri et al.\ (2010) showed a randomized $(1-1/e)-\epsilon$ approximation with runtime (number of queries to function oracle) $n^{m/\epsilon^3}$. %In fact, the result of Chekuri et al.\ (2010) is for the far more general case of matroid constant. We focus on finding a fast and practical algorithm that has (asymptotic) approximation guarantees even when $m$ is super constant. We first modify the algorithm of Chekuri et al.\ (2010) to achieve a $(1-1/e)$ approximation for $m=o(\frac{k}{\log^3 k})$. This demonstrates a steep transition from constant factor approximability to inapproximability around $m=\Omega(k)$. Then using Multiplicative-Weight-Updates (MWU), we find a much faster $\tilde{O}(n/\delta^3)$ time asymptotic $(1-1/e)^2-\delta$ approximation. While the above results are all randomized, we also give a simple deterministic $(1-1/e)-\epsilon$ approximation with runtime $kn^{m/\epsilon^4}$. Finally, we run synthetic experiments using Kronecker graphs and find that our MWU inspired heuristic outperforms existing heuristics.