When trying to find the minimum of a function f using the steepest descent method, which of the following is a plausible termination criteria?

Question

BlackTom AI · Accepted Answer

When examining termination criteria for steepest descent (gradient descent) aimed at minimizing a function f, it's helpful to recall the role of the gradient.
Option 1: The norm of the gradient of f is nearly zero. This is a classic and plausible criterion because, at a local minimum, the gradient vanishes; thus ||∇f(x)|| ≈ 0 indicates that you are at or near a stationary point where further descent is unlikely to yield improvements.
Option 2: The value of f is near zero. While achieving a small objective value can be desirable, there is no guarantee that f can be made arbitrarily close to zero for all problems, nor does a small f imply proximity to a true minimum. Different problems may have minima with f(x) not near zero, so this is not a reliable termination criterion.
Option 3: The gradient is perpendicular to the iso-line at that point. In one-dimensional terms this is less precise, and in higher dimensions the concept of an iso-line (level set) tangency to the gradient relates to gradient being normal to level sets, not to a practical stop condition for iterative minimization. This statement does not translate into a directly usable stopping rule for steepest descent.
Option 4: The norm of the gradient of f approaches infinity. This would indicate divergence or moving far from a stationary point rather than approaching a minimum. It is the opposite of a sensible termination criterion for descent algorithms, which seek small gradient norms, not large ones.
In summary, option 1 directly aligns with standard stopping rules for gradient-based minimization, because near a minimum the gradient vanishes and further steps yield little improvement. The other options either lack generality, misinterpret the geometric meaning, or indicate the opposite behavior of what a descent method aims to achieve.

When trying to find the minimum of a function f using the steepest descent method, which of the following is a plausible termination criteria?单项选择题

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