A data scientist is performing unconstrained optimization on a complex, non-convex loss function for a deep learning model. The optimization algorithm has converged to a point where the gradient of the loss function is the zero vector. What additional analysis is required to confirm that this point represents a local minimum?
Verify that the magnitude of the gradient at the point remains below a small epsilon threshold for several iterations.
Compute the Hessian matrix at this point and confirm it is negative definite.
Compute the Hessian matrix at this point and confirm it is positive definite.
Analyze the Jacobian matrix at this point to ensure all its entries are positive.
The correct answer is to compute the Hessian matrix and check if it is positive definite. In multivariable calculus, the second partial derivative test uses the Hessian matrix to classify critical points. A critical point (where the gradient is zero) is a local minimum if the Hessian matrix evaluated at that point is positive definite (all its eigenvalues are positive). This indicates that the function is curved upwards in all directions around the point. A negative definite Hessian indicates a local maximum, and an indefinite Hessian (with both positive and negative eigenvalues) indicates a saddle point. Checking the gradient's magnitude is a convergence criterion, not a method for classifying the point. The Jacobian matrix is used for vector-valued functions and deals with first-order derivatives, making it unsuitable for analyzing the curvature needed to classify a critical point of a scalar-valued loss function.
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