Brownian Bridge™

1. In the Levenberg-Marquardt update $(J^\top J + \lambda D)\,\delta\theta = -J^\top r$ , what is the behaviour of the step $\delta\theta$ as $\lambda \to \infty$ ?

The step becomes the Gauss-Newton directionThe step approaches a scaled steepest-descent direction

-\frac{1}{\lambda} D^{-1} J^\top r

The step size grows without boundThe algorithm terminates because the system becomes singular

2. The Marquardt scaling uses $D = \mathrm{diag}(J^\top J)$ rather than $D = I$ . What is the key advantage?

It guarantees global convergence from any starting pointIt makes the LM step invariant to scaling of the individual parametersIt eliminates the need to compute the JacobianIt forces the damping parameter

\lambda

to decrease monotonically

3. In the LM algorithm, the gain ratio $\rho = (F(\theta) - F(\theta + \delta\theta)) / (\mathcal{L}(\theta) - \mathcal{L}(\theta + \delta\theta))$ is used to accept or reject a step. A step is accepted when $\rho > 0.25$ . What does $\rho < 0$ indicate?

The quadratic model underestimates the actual decrease in

F

The actual objective increased — the proposed step is uphillThe Jacobian is degenerate at the current iterateThe residuals have converged to zero

4. Satisfying the first-order optimality condition $\|J^\top r\|_\infty < \varepsilon$ in LM calibration guarantees that the calibrated parameters are the global minimum of the calibration objective.

truefalse

5. Central-difference finite-difference Jacobian approximation has truncation error $O(h^2)$ , compared to $O(h)$ for forward differences. What is the trade-off that determines the optimal step size $h$ for central differences?

Larger

h

reduces truncation error but increases round-off error from catastrophic cancellationSmaller

h

reduces truncation error but increases the number of function evaluations requiredLarger

h

reduces both truncation and round-off errors simultaneouslyThe optimal

h

is always

10^{-6}

regardless of the function

6. In Heston calibration, the parameters $\kappa$ and $\bar{\nu}$ (long-run variance) are often poorly identified from market implied vol data. What is the formal characterisation of this identifiability problem?

The Jacobian

J

has a column of zeros corresponding to

\kappa

The Hessian approximation

J^\top J

has near-zero eigenvalues in certain directions of parameter space, corresponding to combinations of

\kappa

and

\bar{\nu}

that produce indistinguishable vol surfacesThe objective function

F(\theta)

is non-differentiable along the

\kappa\bar{\nu}

ridgeThe constraint

\kappa > 0

is binding at the optimal solution

Quiz: Levenberg-Marquardt for Model Calibration

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