arXiv subjects

Jiawen Bi

Publications and source records attributed to Jiawen Bi.

5 recordsLinked to original sources

arXiv2026

On Nonsmooth and Relatively Weakly Convex Minimization

Composite optimization plays a central role in modern machine learning and signal processing, as it offers a natural balance between data fidelity and structural properties. In this paper, we study composite optimization in the setting where both components are nonsmooth and nonconvex. We start with a deterministic Bregman proximal subgradient method that converges under subgradient upper-bound conditions. This approach relaxes the standard requirement on the convexity of the regularization term, thus accommodating a broader range of applications. To extend this to the stochastic regime, we develop a model-based minimization method under a relative Lipschitz condition and establish a convergence rate of $\mathcal{O}(\varepsilon^{-4})$. We also extend the framework with convergence guarantees to the setting where the distance generating function and its gradient are accessible only through a stochastic oracle.

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arXiv2026

On Second-Order Methods for Bilevel Optimization

Bilevel optimization is an indispensable modeling tool for modern machine learning and engineering design. However, the theory and practice for finding second order stationary points in the context of bilevel optimization still remain largely unsettled. Even for bilevel optimization with strongly convex lower-level problem, the hyperfunction it induces is in general nonconvex. Although the Cubic Regularized Newton methods (CRN) famously achieve the optimal $\mathcal{O}(\varepsilon^{-1.5})$ SOSP (second-order stationary point) rate in single-level optimization, it is unclear how to control the accuracy of the hypergradient and hyper-Hessian computations in the context of applying the second-order methods to bilevel problems in order for the overall process to be efficient. In this paper, we set out to answer this question. In particular, we first formulate a double loop CRN baseline that achieves the optimal outer rate but requires repeated lower level solves. Next, we propose a single loop cubic regularized Newton algorithm that combines one lower-level gradient step with one Newton step for the hypergradient, and prove an overall deterministic $\mathcal{O}(\varepsilon^{-1.5})$ total oracle complexity, which is optimal. In addition, we illustrate that some intuitively simple modifications of our method may fail to hold up the convergence result. To the best of our knowledge, this is the first deterministic single loop method for unconstrained NCSC (non-convex upper-level and strongly convex lower-level) bilevel optimization setting that achieves the $\mathcal{O}(\varepsilon^{-1.5})$ optimal convergence rate for finding an $\varepsilon$-SOSP of the hyperfunction.

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arXiv2025

On Tackling High-Dimensional Nonconvex Stochastic Optimization via Stochastic First-Order Methods with Non-smooth Proximal Terms and Variance Reduction

When the nonconvex problem is complicated by stochasticity, the sample complexity of stochastic first-order methods may depend linearly on the problem dimension, which is undesirable for large-scale problems. To alleviate this linear dependence, we adopt non-Euclidean settings and propose two choices of non-smooth proximal terms when taking the stochastic gradient steps. This approach leads to stronger convergence metric, incremental computational overhead, and potentially dimension-insensitive sample complexity. We also consider further acceleration through variance reduction which achieves near optimal sample complexity and, to our best knowledge, is the first such result in the $\ell_1/\ell_\infty$ setting. Since the use of non-smooth proximal terms is unconventional, the convergence analysis deviates much from algorithms in Euclidean settings or employing Bregman divergence, providing tools for analyzing other non-Euclidean choices of distance functions. Efficient resolution of the subproblems in various scenarios is also discussed and simulated. We illustrate the dimension-insensitive property of the proposed methods via preliminary numerical experiments.

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arXiv2025

A Correspondence-Driven Approach for Bilevel Decision-making with Nonconvex Lower-Level Problems

We consider bilevel optimization problems with general nonconvex lower-level objectives and show that the classical hyperfunction-based formulation is unsettled, since the global minimizer of the lower-level problem is generally unattainable. To address this issue, we propose a correspondence-driven hyperfunction $\phi^{\text{cd}}$. In this formulation, the follower is modeled not as a rational agent always attaining a global minimizer, but as an algorithm-based bounded rational agent whose decisions are produced by a fixed algorithm with initialization and step size. Since $\phi^{\text{cd}}$ is generally discontinuous, we apply Gaussian smoothing to obtain a smooth approximation $\phi^{\text{cd}}_\xi$, then show that its value and gradient converge to those of $\phi^{\text{cd}}$. In the nonconvex setting, we identify that bifurcation phenomena, which arise when $g(x,\cdot)$ has a degenerate stationary point, pose a key challenge for hyperfunction-based methods. This is especially the case when $\phi^{\text{cd}}_\xi$ is solved using gradient methods. To overcome this challenge, we analyze the geometric structure of the bifurcation set under some weak assumptions. Building on these results, we design a biased projected SGD-based algorithm SCiNBiO to solve $\phi^{\text{cd}}_\xi$ with a cubic-regularized Newton lower-level solver. We also provide convergence guarantees and oracle complexity bounds for the upper level. Finally, we connect bifurcation theory from dynamical systems to the bilevel setting and define the notion of fold bifurcation points in this setting. Under the assumption that all degenerate stationary points are fold bifurcation points, we establish the oracle complexity of SCiNBiO for the lower-level problem.

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arXiv2024

Stochastic First-Order Methods with Non-smooth and Non-Euclidean Proximal Terms for Nonconvex High-Dimensional Stochastic Optimization

When the nonconvex problem is complicated by stochasticity, the sample complexity of stochastic first-order methods may depend linearly on the problem dimension, which is undesirable for large-scale problems. In this work, we propose dimension-insensitive stochastic first-order methods (DISFOMs) to address nonconvex optimization with expected-valued objective function. Our algorithms allow for non-Euclidean and non-smooth distance functions as the proximal terms. Under mild assumptions, we show that DISFOM using minibatches to estimate the gradient enjoys sample complexity of $ \mathcal{O} ( (\log d) / \epsilon^4 ) $ to obtain an $\epsilon$-stationary point. Furthermore, we prove that DISFOM employing variance reduction can sharpen this bound to $\mathcal{O} ( (\log d)^{2/3}/\epsilon^{10/3} )$, which perhaps leads to the best-known sample complexity result in terms of $d$. We provide two choices of the non-smooth distance functions, both of which allow for closed-form solutions to the proximal step. Numerical experiments are conducted to illustrate the dimension insensitive property of the proposed frameworks.

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