Tight Hölderian Error Bounds for p-Cones Without Constraint Qualifications

Core Concepts

The authors prove tight Hölderian error bounds for all p-cones, with exponents that differ from previous conjectures. These results enable the computation of KL exponents for least squares problems with p-norm regularization, and provide a simple proof that most p-cones are neither self-dual nor homogeneous.

Abstract

The paper focuses on the conic feasibility problem (Feas) where K is the p-cone Kn+1
p
, defined as {x = (x0, x̄) ∈ Rn+1 | x0 ≥ ∥x̄∥p}. The authors prove the following key insights:

Explicit Hölderian error bounds hold for all p-cones, with exponents that differ from previous conjectures. The correct exponent depends on the number of zeros in a vector exposing the feasible region.

There is one special case that only happens when p ∈ (1, 2), where the exponent is different from 1/p or 1/2.

The authors also compute Hölderian error bounds for direct products of p-cones.

As an application, the authors compute the KL exponent of the function associated to least squares problems with p-norm regularization, which was previously only known for p ∈ [1, 2] ∪ {∞}.

The authors provide new "easy" proofs of some results about self-duality and homogeneity of p-cones.

The results are obtained using the facial residual function (FRF) framework, which allows computation of error bounds without assuming constraint qualifications. The authors also expand this framework to establish an optimality criterion under which the resulting error bound must be tight.

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Key Insights Distilled From

Optimal error bounds in the absence of constraint qualifications with applications to the $p$-cones and beyond

by Scot... at arxiv.org 03-29-2024

https://arxiv.org/pdf/2109.11729.pdf

Deeper Inquiries

What are some potential applications of the tight Hölderian error bounds for p-cones beyond the examples provided in the paper

The tight Hölderian error bounds for p-cones have various potential applications beyond the examples provided in the paper. One application could be in the field of machine learning, specifically in the optimization of support vector machines (SVMs) with non-linear kernels. By utilizing the error bounds for p-cones, researchers and practitioners could improve the efficiency and accuracy of SVM optimization algorithms, leading to better classification performance in complex datasets. Additionally, these error bounds could be applied in the development of robust optimization techniques for portfolio management in finance, where the goal is to maximize returns while minimizing risk. The precise error bounds provided by the Hölderian framework could help in creating more stable and reliable portfolio optimization models. Furthermore, in the realm of signal processing, the error bounds could be utilized to enhance the performance of sparse signal recovery algorithms, ensuring accurate reconstruction of signals from limited measurements. Overall, the applications of the tight Hölderian error bounds for p-cones extend to various domains where optimization plays a crucial role in decision-making processes.

How do the insights about the special case when p ∈ (1, 2) relate to the underlying geometry and properties of the p-cones in this regime

The insights about the special case when p ∈ (1, 2) shed light on the unique geometry and properties of the p-cones in this regime. Specifically, when p falls within the range of (1, 2), the behavior of the p-cones deviates from the expected patterns observed for other values of p. This deviation highlights the non-linear nature of the p-cones in this specific range, showcasing distinct characteristics that set them apart from the rest of the cone family. The geometric implications of this special case suggest that the structure of the p-cones undergoes a significant transformation, leading to different optimization challenges and opportunities compared to other regions of the parameter space. By understanding the intricacies of the p-cones when p ∈ (1, 2), researchers can gain valuable insights into the underlying geometry of these cones and tailor optimization strategies to effectively navigate this unique landscape.

Are there other classes of cones or optimization problems where the expanded FRF framework and optimality criterion developed in this work could be leveraged to obtain similarly insightful error bound results

The expanded FRF framework and optimality criterion developed in this work can be leveraged in various other classes of cones and optimization problems to obtain similarly insightful error bound results. One potential application could be in the optimization of conic programming problems involving non-symmetric cones, where the framework's optimality criterion could help in deriving tight error bounds for these specialized optimization scenarios. Additionally, the framework could be applied to problems in robust optimization, where uncertainty in the data or constraints necessitates the use of error bounds to ensure the stability and reliability of the optimization solutions. Furthermore, in the context of machine learning, the expanded FRF framework could find applications in training deep neural networks with non-convex activation functions, providing a theoretical basis for understanding the convergence properties and generalization capabilities of such models. Overall, the versatility of the framework makes it a valuable tool in a wide range of optimization problems beyond the scope of p-cones.