Optimal Asynchronous Stochastic Nonconvex Optimization under Heavy-Tailed Noise

This paper considers the problem of asynchronous stochastic nonconvex optimization with heavy-tailed gradient noise and arbitrarily heterogeneous computation times across workers. We propose an asynchronous normalized stochastic gradient descent algorithm with momentum. The analysis show that our method achieves the optimal time complexity under the assumption of bounded $p$th-order central moment with $p\in(1,2]$. We also provide numerical experiments to show the effectiveness of proposed method.

Stability and Generalization of Nonconvex Optimization with Heavy-Tailed Noise

The empirical evidence indicates that stochastic optimization with heavy-tailed gradient noise is more appropriate to characterize the training of machine learning models than that with standard bounded gradient variance noise. Most existing works on this phenomenon focus on the convergence of optimization errors, while the analysis for generalization bounds under the heavy-tailed gradient noise remains limited. In this paper, we develop a general framework for establishing generalization bounds under heavy-tailed noise. Specifically, we introduce a truncation argument to achieve the generalization error bound based on the algorithmic stability under the assumption of bounded $p$th centered moment with $p\in(1,2]$. Building on this framework, we further provide the stability and generalization analysis for several popular stochastic algorithms under heavy-tailed noise, including clipped and normalized stochastic gradient descent, as well as their mini-batch and momentum variants.

Near-Optimal Decentralized Stochastic Nonconvex Optimization with Heavy-Tailed Noise

This paper studies decentralized stochastic nonconvex optimization problem over row-stochastic networks. We consider the heavy-tailed gradient noise which is empirically observed in many popular real-world applications. Specifically, we propose a decentralized normalized stochastic gradient descent with Pull-Diag gradient tracking, which achieves approximate stationary points with the optimal sample complexity and the near-optimal communication complexity. We further follow our framework to study the setting of undirected networks, also achieving the nearly tight upper complexity bounds. Moreover, we conduct empirical studies to show the practical superiority of the proposed methods.

Accelerated Evolving Set Processes for Local PageRank Computation

This work proposes a novel framework based on nested evolving set processes to accelerate Personalized PageRank (PPR) computation. At each stage of the process, we employ a localized inexact proximal point iteration to solve a simplified linear system. We show that the time complexity of such localized methods is upper bounded by $\min\{\tilde{\mathcal{O}}(R^2/\epsilon^2), \tilde{\mathcal{O}}(m)\}$ to obtain an $\epsilon$-approximation of the PPR vector, where $m$ denotes the number of edges in the graph and $R$ is a constant defined via nested evolving set processes. Furthermore, the algorithms induced by our framework require solving only $\tilde{\mathcal{O}}(1/\sqrt{\alpha})$ such linear systems, where $\alpha$ is the damping factor. When $1/\epsilon^2\ll m$, this implies the existence of an algorithm that computes an $\ epsilon $-approximation of the PPR vector with an overall time complexity of $\tilde{\mathcal{O}}\left(R^2 / (\sqrt{\alpha}\epsilon^2)\right)$, independent of the underlying graph size. Our result resolves an open conjecture from existing literature. Experimental results on real-world graphs validate the efficiency of our methods, demonstrating significant convergence in the early stages.

Stochastic Bilevel Optimization with Heavy-Tailed Noise

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 $\epsilon$-stationary point with the stochastic first-order oracle (SFO) complexity of $\tilde{\mathcal{O}}\big(\kappa^{\frac{7p-3}{p-1}} \sigma^{\frac{p}{p-1}} \epsilon^{-\frac{4 p - 2}{p-1}}\big)$, where $\kappa$ is the condition number, $p\in(1,2]$ is the order of central moment for the noise, and $\sigma$ is the noise level. Furthermore, we specialize our idea to solve the nonconvex-strongly-concave minimax optimization problem, achieving an $\epsilon$-stationary point with the SFO complexity of $\tilde{\mathcal O}\big(\kappa^{\frac{2p-1}{p-1}} \sigma^{\frac{p}{p-1}} \epsilon^{-\frac{3p-2}{p-1}}\big)$. All above upper bounds match the best-known results under the special case of the bounded variance setting, i.e., $p=2$.

Decentralized Stochastic Nonconvex Optimization under the Relaxed Smoothness

This paper studies decentralized optimization problem $f(\mathbf{x})=\frac{1}{m}\sum_{i=1}^m f_i(\mathbf{x})$, where each local function has the form of $f_i(\mathbf{x}) = {\mathbb E}\left[F(\mathbf{x};{\xi}_i)\right]$ which is $(L_0,L_1)$-smooth but possibly nonconvex and the random variable ${\xi}_i$ follows distribution ${\mathcal D}_i$. We propose a novel algorithm called decentralized normalized stochastic gradient descent (DNSGD), which can achieve the $\epsilon$-stationary point on each local agent. We present a new framework for analyzing decentralized first-order methods in the relaxed smooth setting, based on the Lyapunov function related to the product of the gradient norm and the consensus error. The analysis shows upper bounds on sample complexity of ${\mathcal O}(m^{-1}(L_f\sigma^2\Delta_f\epsilon^{-4} + \sigma^2\epsilon^{-2} + L_f^{-2}L_1^3\sigma^2\Delta_f\epsilon^{-1} + L_f^{-2}L_1^2\sigma^2))$ per agent and communication complexity of $\tilde{\mathcal O}((L_f\epsilon^{-2} + L_1\epsilon^{-1})\gamma^{-1/2}\Delta_f)$, where $L_f=L_0 +L_1\zeta$, $\sigma^2$ is the variance of the stochastic gradient, $\Delta_f$ is the initial optimal function value gap, $\gamma$ is the spectral gap of the network, and $\zeta$ is the degree of the gradient dissimilarity. In the special case of $L_1=0$, the above results (nearly) match the lower bounds on decentralized nonconvex optimization in the standard smooth setting. We also conduct numerical experiments to show the empirical superiority of our method.

Solving Convex-Concave Problems with $\tilde{\mathcal{O}}(ε^{-4/7})$ Second-Order Oracle Complexity

Previous algorithms can solve convex-concave minimax problems $\min_{x \in \mathcal{X}} \max_{y \in \mathcal{Y}} f(x,y)$ with $\mathcal{O}(\epsilon^{-2/3})$ second-order oracle calls using Newton-type methods. This result has been speculated to be optimal because the upper bound is achieved by a natural generalization of the optimal first-order method. In this work, we show an improved upper bound of $\tilde{\mathcal{O}}(\epsilon^{-4/7})$ by generalizing the optimal second-order method for convex optimization to solve the convex-concave minimax problem. We further apply a similar technique to lazy Hessian algorithms and show that our proposed algorithm can also be seen as a second-order ``Catalyst'' framework (Lin et al., JMLR 2018) that could accelerate any globally convergent algorithms for solving minimax problems.

Incremental Gauss--Newton Methods with Superlinear Convergence Rates

This paper addresses the challenge of solving large-scale nonlinear equations with H\"older continuous Jacobians. We introduce a novel Incremental Gauss--Newton (IGN) method within explicit superlinear convergence rate, which outperforms existing methods that only achieve linear convergence rate. In particular, we formulate our problem by the nonlinear least squares with finite-sum structure, and our method incrementally iterates with the information of one component in each round. We also provide a mini-batch extension to our IGN method that obtains an even faster superlinear convergence rate. Furthermore, we conduct numerical experiments to show the advantages of the proposed methods.

Near-Optimal Distributed Minimax Optimization under the Second-Order Similarity

This paper considers the distributed convex-concave minimax optimization under the second-order similarity. We propose stochastic variance-reduced optimistic gradient sliding (SVOGS) method, which takes the advantage of the finite-sum structure in the objective by involving the mini-batch client sampling and variance reduction. We prove SVOGS can achieve the $\varepsilon$-duality gap within communication rounds of ${\mathcal O}(\delta D^2/\varepsilon)$, communication complexity of ${\mathcal O}(n+\sqrt{n}\delta D^2/\varepsilon)$, and local gradient calls of $\tilde{\mathcal O}(n+(\sqrt{n}\delta+L)D^2/\varepsilon\log(1/\varepsilon))$, where $n$ is the number of nodes, $\delta$ is the degree of the second-order similarity, $L$ is the smoothness parameter and $D$ is the diameter of the constraint set. We can verify that all of above complexity (nearly) matches the corresponding lower bounds. For the specific $\mu$-strongly-convex-$\mu$-strongly-convex case, our algorithm has the upper bounds on communication rounds, communication complexity, and local gradient calls of $\mathcal O(\delta/\mu\log(1/\varepsilon))$, ${\mathcal O}((n+\sqrt{n}\delta/\mu)\log(1/\varepsilon))$, and $\tilde{\mathcal O}(n+(\sqrt{n}\delta+L)/\mu)\log(1/\varepsilon))$ respectively, which are also nearly tight. Furthermore, we conduct the numerical experiments to show the empirical advantages of proposed method.

On the Complexity of Finite-Sum Smooth Optimization under the Polyak-Łojasiewicz Condition

This paper considers the optimization problem of the form $\min_{{\bf x}\in{\mathbb R}^d} f({\bf x})\triangleq \frac{1}{n}\sum_{i=1}^n f_i({\bf x})$, where $f(\cdot)$ satisfies the Polyak--{\L}ojasiewicz (PL) condition with parameter $\mu$ and $\{f_i(\cdot)\}_{i=1}^n$ is $L$-mean-squared smooth. We show that any gradient method requires at least $\Omega(n+\kappa\sqrt{n}\log(1/\epsilon))$ incremental first-order oracle (IFO) calls to find an $\epsilon$-suboptimal solution, where $\kappa\triangleq L/\mu$ is the condition number of the problem. This result nearly matches upper bounds of IFO complexity for best-known first-order methods. We also study the problem of minimizing the PL function in the distributed setting such that the individuals $f_1(\cdot),\dots,f_n(\cdot)$ are located on a connected network of $n$ agents. We provide lower bounds of $\Omega(\kappa/\sqrt{\gamma}\,\log(1/\epsilon))$, $\Omega((\kappa+\tau\kappa/\sqrt{\gamma}\,)\log(1/\epsilon))$ and $\Omega\big(n+\kappa\sqrt{n}\log(1/\epsilon)\big)$ for communication rounds, time cost and local first-order oracle calls respectively, where $\gamma\in(0,1]$ is the spectral gap of the mixing matrix associated with the network and~$\tau>0$ is the time cost of per communication round. Furthermore, we propose a decentralized first-order method that nearly matches above lower bounds in expectation.