Mingrui Zhang

Mingrui Zhang

As a projection-free algorithm, Frank-Wolfe (FW) method, also known as conditional gradient, has recently received considerable attention in the machine learning community. In this dissertation, we study several topics on the FW variants for scalable projection-free optimization. We first propose 1-SFW, the first projection-free method that requires only one sample per iteration to update the optimization variable and yet achieves the best known complexity bounds for convex, non-convex, and monotone DR-submodular settings. Then we move forward to the distributed setting, and develop Quantized Frank-Wolfe (QFW), a general communication-efficient distributed FW framework for both convex and non-convex objective functions. We study the performance of QFW in two widely recognized settings: 1) stochastic optimization and 2) finite-sum optimization. Finally, we propose Black-Box Continuous Greedy, a derivative-free and projection-free algorithm, that maximizes a monotone continuous DR-submodular function over a bounded convex body in Euclidean space.

Lin Chen, Yifei Min, Mingrui Zhang et al.

Despite remarkable success in practice, modern machine learning models have been found to be susceptible to adversarial attacks that make human-imperceptible perturbations to the data, but result in serious and potentially dangerous prediction errors. To address this issue, practitioners often use adversarial training to learn models that are robust against such attacks at the cost of higher generalization error on unperturbed test sets. The conventional wisdom is that more training data should shrink the gap between the generalization error of adversarially-trained models and standard models. However, we study the training of robust classifiers for both Gaussian and Bernoulli models under $\ell_\infty$ attacks, and we prove that more data may actually increase this gap. Furthermore, our theoretical results identify if and when additional data will finally begin to shrink the gap. Lastly, we experimentally demonstrate that our results also hold for linear regression models, which may indicate that this phenomenon occurs more broadly.

Mingrui Zhang, Lin Chen, Hamed Hassani et al.

In this paper, we propose three online algorithms for submodular maximisation. The first one, Mono-Frank-Wolfe, reduces the number of per-function gradient evaluations from $T^{1/2}$ [Chen2018Online] and $T^{3/2}$ [chen2018projection] to 1, and achieves a $(1-1/e)$-regret bound of $O(T^{4/5})$. The second one, Bandit-Frank-Wolfe, is the first bandit algorithm for continuous DR-submodular maximization, which achieves a $(1-1/e)$-regret bound of $O(T^{8/9})$. Finally, we extend Bandit-Frank-Wolfe to a bandit algorithm for discrete submodular maximization, Responsive-Frank-Wolfe, which attains a $(1-1/e)$-regret bound of $O(T^{8/9})$ in the responsive bandit setting.

Mingrui Zhang, Zebang Shen, Aryan Mokhtari et al.

One of the beauties of the projected gradient descent method lies in its rather simple mechanism and yet stable behavior with inexact, stochastic gradients, which has led to its wide-spread use in many machine learning applications. However, once we replace the projection operator with a simpler linear program, as is done in the Frank-Wolfe method, both simplicity and stability take a serious hit. The aim of this paper is to bring them back without sacrificing the efficiency. In this paper, we propose the first one-sample stochastic Frank-Wolfe algorithm, called 1-SFW, that avoids the need to carefully tune the batch size, step size, learning rate, and other complicated hyper parameters. In particular, 1-SFW achieves the optimal convergence rate of $\mathcal{O}(1/ε^2)$ for reaching an $ε$-suboptimal solution in the stochastic convex setting, and a $(1-1/e)-ε$ approximate solution for a stochastic monotone DR-submodular maximization problem. Moreover, in a general non-convex setting, 1-SFW finds an $ε$-first-order stationary point after at most $\mathcal{O}(1/ε^3)$ iterations, achieving the current best known convergence rate. All of this is possible by designing a novel unbiased momentum estimator that governs the stability of the optimization process while using a single sample at each iteration.

Mingrui Zhang, Lin Chen, Aryan Mokhtari et al.

How can we efficiently mitigate the overhead of gradient communications in distributed optimization? This problem is at the heart of training scalable machine learning models and has been mainly studied in the unconstrained setting. In this paper, we propose Quantized-Frank-Wolfe (QFW), the first projection-free and communication-efficient algorithm for solving constrained optimization problems at scale. We consider both convex and non-convex objective functions, expressed as a finite-sum or more generally a stochastic optimization problem, and provide strong theoretical guarantees on the convergence rate of QFW. This is accomplished by proposing novel quantization schemes that efficiently compress gradients while controlling the noise variance introduced during this process. Finally, we empirically validate the efficiency of QFW in terms of communication and the quality of returned solution against natural baselines.

Lin Chen, Mingrui Zhang, Amin Karbasi

In this paper, we propose the first computationally efficient projection-free algorithm for bandit convex optimization (BCO). We show that our algorithm achieves a sublinear regret of $O(nT^{4/5})$ (where $T$ is the horizon and $n$ is the dimension) for any bounded convex functions with uniformly bounded gradients. We also evaluate the performance of our algorithm against baselines on both synthetic and real data sets for quadratic programming, portfolio selection and matrix completion problems.