Wei Jiang

Dingzhi Yu, Wei Jiang, Yuanyu Wan et al.

Smoothness is crucial for attaining fast rates in first-order optimization. However, many optimization problems in modern machine learning involve non-smooth objectives. Recent studies relax the smoothness assumption by allowing the Lipschitz constant of the gradient to grow with respect to the gradient norm, which accommodates a broad range of objectives in practice. Despite this progress, existing generalizations of smoothness are restricted to Euclidean geometry with $\ell_2$-norm and only have theoretical guarantees for optimization in the Euclidean space. In this paper, we address this limitation by introducing a new $\ell*$-smoothness concept that measures the norm of Hessians in terms of a general norm and its dual, and establish convergence for mirror-descent-type algorithms, matching the rates under the classic smoothness. Notably, we propose a generalized self-bounding property that facilitates bounding the gradients via controlling suboptimality gaps, serving as a principal component for convergence analysis. Beyond deterministic optimization, we establish an anytime convergence for stochastic mirror descent based on a new bounded noise condition that encompasses the widely adopted bounded or affine noise assumptions.

Wei Jiang, Dingzhi Yu, Sifan Yang et al.

In this paper, we present enhanced analysis for sign-based optimization algorithms with momentum updates. Traditional sign-based methods, under the separable smoothness assumption, guarantee a convergence rate of $\mathcal{O}(T^{-1/4})$, but they either require large batch sizes or assume unimodal symmetric stochastic noise. To address these limitations, we demonstrate that signSGD with momentum can achieve the same convergence rate using constant batch sizes without additional assumptions. Our analysis, under the standard $l_2$-smoothness condition, improves upon the result of the prior momentum-based signSGD method by a factor of $\mathcal{O}(d^{1/2})$, where $d$ is the problem dimension. Furthermore, we explore sign-based methods with majority vote in distributed settings and show that the proposed momentum-based method yields convergence rates of $\mathcal{O}\left( d^{1/2}T^{-1/2} + dn^{-1/2} \right)$ and $\mathcal{O}\left( \max \{ d^{1/4}T^{-1/4}, d^{1/10}T^{-1/5} \} \right)$, which outperform the previous results of $\mathcal{O}\left( dT^{-1/4} + dn^{-1/2} \right)$ and $\mathcal{O}\left( d^{3/8}T^{-1/8} \right)$, respectively. Numerical experiments further validate the effectiveness of the proposed methods.

Wei Jiang, Lijun Zhang

In this paper, we analyze the convergence properties of the Lion optimizer. First, we establish that the Lion optimizer attains a convergence rate of $\mathcal{O}(d^{1/2}T^{-1/4})$ under standard assumptions, where $d$ denotes the problem dimension and $T$ is the iteration number. To further improve this rate, we introduce the Lion optimizer with variance reduction, resulting in an enhanced convergence rate of $\mathcal{O}(d^{1/2}T^{-1/3})$. We then analyze in distributed settings, where the standard and variance reduced version of the distributed Lion can obtain the convergence rates of $\mathcal{O}(d^{1/2}(nT)^{-1/4})$ and $\mathcal{O}(d^{1/2}(nT)^{-1/3})$, with $n$ denoting the number of nodes. Furthermore, we investigate a communication-efficient variant of the distributed Lion that ensures sign compression in both communication directions. By employing the unbiased sign operations, the proposed Lion variant and its variance reduction counterpart, achieve convergence rates of $\mathcal{O}\left( \max \left\{\frac{d^{1/4}}{T^{1/4}}, \frac{d^{1/10}}{n^{1/5}T^{1/5}} \right\} \right)$ and $\mathcal{O}\left( \frac{d^{1/4}}{T^{1/4}} \right)$, respectively.

Wei Jiang, Sifan Yang, Wenhao Yang et al.

This paper investigates projection-free algorithms for stochastic constrained multi-level optimization. In this context, the objective function is a nested composition of several smooth functions, and the decision set is closed and convex. Existing projection-free algorithms for solving this problem suffer from two limitations: 1) they solely focus on the gradient mapping criterion and fail to match the optimal sample complexities in unconstrained settings; 2) their analysis is exclusively applicable to non-convex functions, without considering convex and strongly convex objectives. To address these issues, we introduce novel projection-free variance reduction algorithms and analyze their complexities under different criteria. For gradient mapping, our complexities improve existing results and match the optimal rates for unconstrained problems. For the widely-used Frank-Wolfe gap criterion, we provide theoretical guarantees that align with those for single-level problems. Additionally, by using a stage-wise adaptation, we further obtain complexities for convex and strongly convex functions. Finally, numerical experiments on different tasks demonstrate the effectiveness of our methods.

Wei Jiang, Sifan Yang, Yibo Wang et al.

This paper explores adaptive variance reduction methods for stochastic optimization based on the STORM technique. Existing adaptive extensions of STORM rely on strong assumptions like bounded gradients and bounded function values, or suffer an additional $\mathcal{O}(\log T)$ term in the convergence rate. To address these limitations, we introduce a novel adaptive STORM method that achieves an optimal convergence rate of $\mathcal{O}(T^{-1/3})$ for non-convex functions with our newly designed learning rate strategy. Compared with existing approaches, our method requires weaker assumptions and attains the optimal convergence rate without the additional $\mathcal{O}(\log T)$ term. We also extend the proposed technique to stochastic compositional optimization, obtaining the same optimal rate of $\mathcal{O}(T^{-1/3})$. Furthermore, we investigate the non-convex finite-sum problem and develop another innovative adaptive variance reduction method that achieves an optimal convergence rate of $\mathcal{O}(n^{1/4} T^{-1/2} )$, where $n$ represents the number of component functions. Numerical experiments across various tasks validate the effectiveness of our method.

Wei Jiang, Sifan Yang, Wenhao Yang et al.

Sign stochastic gradient descent (signSGD) is a communication-efficient method that transmits only the sign of stochastic gradients for parameter updating. Existing literature has demonstrated that signSGD can achieve a convergence rate of $\mathcal{O}(d^{1/2}T^{-1/4})$, where $d$ represents the dimension and $T$ is the iteration number. In this paper, we improve this convergence rate to $\mathcal{O}(d^{1/2}T^{-1/3})$ by introducing the Sign-based Stochastic Variance Reduction (SSVR) method, which employs variance reduction estimators to track gradients and leverages their signs to update. For finite-sum problems, our method can be further enhanced to achieve a convergence rate of $\mathcal{O}(m^{1/4}d^{1/2}T^{-1/2})$, where $m$ denotes the number of component functions. Furthermore, we investigate the heterogeneous majority vote in distributed settings and introduce two novel algorithms that attain improved convergence rates of $\mathcal{O}(d^{1/2}T^{-1/2} + dn^{-1/2})$ and $\mathcal{O}(d^{1/4}T^{-1/4})$ respectively, outperforming the previous results of $\mathcal{O}(dT^{-1/4} + dn^{-1/2})$ and $\mathcal{O}(d^{3/8}T^{-1/8})$, where $n$ represents the number of nodes. Numerical experiments across different tasks validate the effectiveness of our proposed methods.