An Adaptive Graduated Nonconvexity Loss Function for Robust Nonlinear Least Squares Solutions

TL;DR

This work addresses robustness in nonlinear least-squares problems common in robotics by integrating graduated nonconvexity (GNC) with Barron’s generalized adaptive loss, removing the need to hand-pick a loss function. It introduces GNC-ADAPT, which combines GNC with the adaptive loss, and GNC-AMB, which applies mode-shifted residual weighting to Chi-like distributions using a Maxwell–Boltzmann fit. The authors derive a surrogate loss ρ_μ(ε, α^*) with a shape function f(μ, α^*) and provide adaptive weight updates, enabling gradual nonconvexity and convergence from convex to the original loss. Empirical results across point-cloud alignment, mesh registration, and pose-graph optimization demonstrate improved robustness to outliers and initialization compared with fixed-loss GNC methods, with AMB offering reduced variance in errors in challenging settings. The proposed framework is widely applicable to state-estimation problems in robotics and reduces the burden of selecting problem-specific loss functions while delivering enhanced performance.

Abstract

Many problems in robotics, such as estimating the state from noisy sensor data or aligning two point clouds, can be posed and solved as least-squares problems. Unfortunately, vanilla nonminimal solvers for least-squares problems are notoriously sensitive to outliers. As such, various robust loss functions have been proposed to reduce the sensitivity to outliers. Examples of loss functions include pseudo-Huber, Cauchy, and Geman-McClure. Recently, these loss functions have been generalized into a single loss function that enables the best loss function to be found adaptively based on the distribution of the residuals. However, even with the generalized robust loss function, most nonminimal solvers can only be solved locally given a prior state estimate due to the nonconvexity of the problem. The first contribution of this paper is to combine graduated nonconvexity (GNC) with the generalized robust loss function to solve least-squares problems without a prior state estimate and without the need to specify a loss function. Moreover, existing loss functions, including the generalized loss function, are based on Gaussian-like distribution. However, residuals are often defined as the squared norm of a multivariate error and distributed in a Chi-like fashion. The second contribution of this paper is to apply a norm-aware adaptive robust loss function within a GNC framework. The proposed approach enables a GNC formulation of a generalized loss function such that GNC can be readily applied to a wider family of loss functions. Furthermore, simulations and experiments demonstrate that the proposed method is more robust compared to non-GNC counterparts, and yields faster convergence times compared to other GNC formulations.

Figures (11)

• Figure 1: Shape functions with ${\alpha^{\star}=0}$ for Examples \ref{['ex:1']}, \ref{['ex:2']}, and \ref{['ex:3']} shown in blue, orange, and green lines, respectively. Blue line approaches ${f(\cdot)=\alpha^{\star}}$ as ${\mu}$ decreases towards ${1}$, whereas orange and green lines approach ${f(\cdot)=\alpha^{\star}}$ as ${\mu}$ increases towards ${\infty}$. However, the rate at which ${f(\cdot)}$ approaches ${\alpha^{\star}}$, also interpreted as the amount of nonconvexity added to the surrogate loss function ${\rho_{\mu}(\cdot)}$, is higher in the orange line (Example \ref{['ex:2']}).
• Figure 2: Contextual photographs of different environments Pomerleau2012Challenging.
• Figure 3: Three point-cloud scan pairs at different overlap ratios from ETH Hauptgebaude dataset.
• Figure 4: The percentage of ICP convergence and success in different initial perturbation levels (left) and overlap ratios (right) are shown here for each loss function (transparent) and its GNC variant (opaque). ICP is considered successful if both the final rotation and translation errors are less than the initial perturbation, ${\|\delta\hat{\boldsymbol{\phi}}\| < \|\delta\check{\boldsymbol{\phi}}\|}$ and ${\|\delta\hat{\boldsymbol{\rho}}\| < \|\delta\check{\boldsymbol{\rho}}\|}$, and the total number of iterations is less than the maximum specified in Table \ref{['tab:icp_setting']}. The GNC variants show higher convergence and success rates than the traditional loss functions in all initial perturbation levels and overlap ratios.
• Figure 5: Prior and posterior translation error distribution after convergence for easy and medium difficulty levels for all trials with EH dataset.

Theorems & Definitions (4)

• Lemma 3.1: Black-Rangarajan Duality Black1996Unification
• Example 4.1
• Example 4.2
• Example 4.3