Gradient Descent vs. Gauss-Newton for Robotics Least-Squares

Many robotics estimation problems (e.g., bundle adjustment, pose-graph SLAM, inverse kinematics by optimization) are posed as nonlinear least-squares: minimize $f(x) = \tfrac{1}{2}\sum_i \| r_i(x) \|^2$, where each $r_i$ is a residual.

Compare gradient descent, Gauss-Newton, and Levenberg-Marquardt for solving this problem. Address: (1) what curvature information each method uses and how it relates to the true Hessian; (2) why Gauss-Newton can fail or diverge, and how Levenberg-Marquardt fixes it; and (3) what property of the residuals determines whether the Gauss-Newton approximation is accurate. Conclude with which method you would default to for a typical SLAM back-end and why.