Tensor Envelopes in the Nonconvex Setting

Abstract

We introduce a tensor envelope (TEE) for composite objectives $\varphi=f+g$, built from a $q$th-order Taylor model of the smooth term $f$ regularized by a $p$th power with $p>q$. To construct it, we first establish a tensor descent lemma and characterize its behavior in the three regimes $q<p<q+1$, $p=q+1$, and $p>q+1$. Under suitable prox-boundedness and parameter conditions, the associated tensor operator (TOP) has nonempty compact values, and TEE is finite-valued and continuous, preserves global minimizers, and inherits the level-boundedness of $\varphi$. We further show that TEE is locally Lipschitz continuous and directionally differentiable, characterize its Fr\'echet differentiability, and prove that it is $C^1$ whenever TOP is single-valued. As such, TEE preserves the essential optimization landscape properties of $\varphi$ while enjoying a richer analytical structure, which facilitates the design and analysis of tensor-based optimization methods. Finally, TEE serves as a Lyapunov function for the exact tensor iteration: every cluster point is critical, and, for bounded trajectories with $q<p\le q+1$, an $\epsilon$-stationary point is reached within $\mathcal{O}(\epsilon^{-p/(p-1)})$ iterations. To our knowledge, both the tensor descent lemma and this complexity bound are the first of their kind obtained without assuming global Lipschitz or H\"older continuity of the $q$th-order derivative of $f$.