Zhuang‐Hua Liu

Zhuanghua Liu, Luo Luo

This paper considers the smooth bilevel optimization in which the lower-level problem is strongly convex and the upper-level problem is possibly nonconvex. We focus on the stochastic setting that the algorithm can access the unbiased stochastic gradient evaluation with heavy-tailed noise, which is prevalent in many machine learning applications such as training large language models and reinforcement learning. We propose a nested-loop normalized stochastic bilevel approximation (N$^2$SBA) for finding an $ε$-stationary point with the stochastic first-order oracle (SFO) complexity of $\tilde{\mathcal{O}}\big(κ^{\frac{7p-3}{p-1}} σ^{\frac{p}{p-1}} ε^{-\frac{4 p - 2}{p-1}}\big)$, where $κ$ is the condition number, $p\in(1,2]$ is the order of central moment for the noise, and $σ$ is the noise level. Furthermore, we specialize our idea to solve the nonconvex-strongly-concave minimax optimization problem, achieving an $ε$-stationary point with the SFO complexity of $\tilde{\mathcal O}\big(κ^{\frac{2p-1}{p-1}} σ^{\frac{p}{p-1}} ε^{-\frac{3p-2}{p-1}}\big)$. All above upper bounds match the best-known results under the special case of the bounded variance setting, i.e., $p=2$.

Zhuanghua Liu, Bryan Kian Hsiang Low

We consider the optimization problem of minimizing the sum-of-nonconvex function, i.e., a convex function that is the average of nonconvex components. The existing stochastic algorithms for such a problem only focus on a single machine and the centralized scenario. In this paper, we study the sum-of-nonconvex optimization in the decentralized setting. We present a new theoretical analysis of the PMGT-SVRG algorithm for this problem and prove the linear convergence of their approach. However, the convergence rate of the PMGT-SVRG algorithm has a linear dependency on the condition number, which is undesirable for the ill-conditioned problem. To remedy this issue, we propose an accelerated stochastic decentralized first-order algorithm by incorporating the techniques of acceleration, gradient tracking, and multi-consensus mixing into the SVRG algorithm. The convergence rate of the proposed method has a square-root dependency on the condition number. The numerical experiments validate the theoretical guarantee of our proposed algorithms on both synthetic and real-world datasets.

Zhuanghua Liu, Ivor W. Tsang

Despite the huge success of Deep Neural Networks (DNNs) in a wide spectrum of machine learning and data mining tasks, recent research shows that this powerful tool is susceptible to maliciously crafted adversarial examples. Up until now, adversarial training has been the most successful defense against adversarial attacks. To increase adversarial robustness, a DNN can be trained with a combination of benign and adversarial examples generated by first-order methods. However, in state-of-the-art first-order attacks, adversarial examples with sign gradients retain the sign information of each gradient component but discard the relative magnitude between components. In this work, we replace sign gradients with quantized gradients. Gradient quantization not only preserves the sign information, but also keeps the relative magnitude between components. Experiments show white-box first-order attacks with quantized gradients outperform their variants with sign gradients on multiple datasets. Notably, our BLOB\_QG attack achieves an accuracy of $88.32\%$ on the secret MNIST model from the MNIST Challenge and it outperforms all other methods on the leaderboard of white-box attacks.