SUN-DSBO: A Structured Unified Framework for Nonconvex Decentralized Stochastic Bilevel Optimization

Decentralized stochastic bilevel optimization (DSBO) is a powerful tool for various machine learning tasks, including decentralized meta-learning and hyperparameter tuning. Existing DSBO methods primarily address problems with strongly convex lower-level objective functions. However, nonconvex objective functions are increasingly prevalent in modern deep learning. In this work, we introduce SUN-DSBO, a Structured Unified framework for Nonconvex DSBO, in which both the upper- and lower-level objective functions may be nonconvex. Notably, SUN-DSBO offers the flexibility to incorporate decentralized stochastic gradient descent or various techniques for mitigating data heterogeneity, such as gradient tracking (GT). We demonstrate that SUN-DSBO-GT, an adaptation of the GT technique within our framework, achieves a linear speedup with respect to the number of agents. This is accomplished without relying on restrictive assumptions, such as gradient boundedness or any specific assumptions regarding gradient heterogeneity. Numerical experiments validate the effectiveness of our method.

💡 Research Summary

This paper tackles decentralized stochastic bilevel optimization (DSBO) where both the upper‑level objective F and the lower‑level objective G may be nonconvex, a setting that reflects many modern deep‑learning applications. Existing DSBO algorithms largely rely on the strong convexity of the lower‑level problem, which limits their applicability. To overcome this limitation, the authors propose SUN‑DSBO, a Structured Unified framework for Nonconvex Decentralized Stochastic Bilevel Optimization.

The key technical contribution is a reformulation of the original bilevel problem using a Moreau envelope penalty. Specifically, the lower‑level constraint y ∈ argmin G(x,·) is replaced by a penalty term G(x,y) − Vγ(x,y) ≥ 0, where Vγ is the Moreau envelope of G with parameter γ. This yields a constrained problem (2) that, under the assumption that G(x,·) is L₂‑smooth and γ ∈ (0,1/(2L₂)), is equivalent to the stationarity condition ∇y G(x,y)=0. By further introducing a small penalty coefficient μ, the authors obtain the unconstrained objective Ψμ(x,y)=μF(x,y)+G(x,y)−Vγ(x,y). Crucially, Ψμ can be expressed as a nonconvex‑strongly‑concave min‑max problem (5):
min₍x,y₎ max_θ