Bokun Wang

Xingyu Chen, Bokun Wang, Ming Yang et al.

Finite-sum Coupled Compositional Optimization (FCCO), characterized by its coupled compositional objective structure, emerges as an important optimization paradigm for addressing a wide range of machine learning problems. In this paper, we focus on a challenging class of non-convex non-smooth FCCO, where the outer functions are non-smooth weakly convex or convex and the inner functions are smooth or weakly convex. Existing state-of-the-art result face two key limitations: (1) a high iteration complexity of $O(1/ε^6)$ under the assumption that the stochastic inner functions are Lipschitz continuous in expectation; (2) reliance on vanilla SGD-type updates, which are not suitable for deep learning applications. Our main contributions are two fold: (i) We propose stochastic momentum methods tailored for non-smooth FCCO that come with provable convergence guarantees; (ii) We establish a new state-of-the-art iteration complexity of $O(1/ε^5)$. Moreover, we apply our algorithms to multiple inequality constrained non-convex optimization problems involving smooth or weakly convex functional inequality constraints. By optimizing a smoothed hinge penalty based formulation, we achieve a new state-of-the-art complexity of $O(1/ε^5)$ for finding an (nearly) $ε$-level KKT solution. Experiments on three tasks demonstrate the effectiveness of the proposed algorithms.

Wei Jiang, Bokun Wang, Yibo Wang et al.

In this paper, we investigate the problem of stochastic multi-level compositional optimization, where the objective function is a composition of multiple smooth but possibly non-convex functions. Existing methods for solving this problem either suffer from sub-optimal sample complexities or need a huge batch size. To address these limitations, we propose a Stochastic Multi-level Variance Reduction method (SMVR), which achieves the optimal sample complexity of $\mathcal{O}\left(1 / ε^{3}\right)$ to find an $ε$-stationary point for non-convex objectives. Furthermore, when the objective function satisfies the convexity or Polyak-Łojasiewicz (PL) condition, we propose a stage-wise variant of SMVR and improve the sample complexity to $\mathcal{O}\left(1 / ε^{2}\right)$ for convex functions or $\mathcal{O}\left(1 /\left(με\right)\right)$ for non-convex functions satisfying the $μ$-PL condition. The latter result implies the same complexity for $μ$-strongly convex functions. To make use of adaptive learning rates, we also develop Adaptive SMVR, which achieves the same complexities but converges faster in practice. All our complexities match the lower bounds not only in terms of $ε$ but also in terms of $μ$ (for PL or strongly convex functions), without using a large batch size in each iteration.