Result · SOFT-FIXED-TEMP-STATIONARY
Fixed-temperature optimization reaches only ordinary stationary points
Derived from the literaturepublication: derived from published
Meaning
Standard nonconvex optimization theory supplies stationary guarantees at a fixed temperature, not global hard optimality.
Scientific context
Optimizing a soft, temperature-controlled assignment at a fixed positive temperature has the ordinary guarantees of smooth nonconvex optimization, and no others: a stationary point of the soft objective is not thereby a local or global optimum of the hard objective it approximates. Hardening a soft fit is therefore a step whose result must be checked against the hard objective rather than assumed.
Statement
A positive-temperature softmax affine-max family is smooth on the regular region; line-search gradient/quasi-Newton methods can be monotone in the soft objective and converge only to standard stationary notions.
Where it stops
This gives no hard local/global optimality guarantee.
Relationships
Enables
Raises
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Proof
S4. Fixed-temperature soft optimization has ordinary stationary-point guarantees
A positive-temperature softmax affine-max family is smooth away from empty cells/singular information. Line-search gradient ascent or quasi-Newton optimization can be made monotone in the soft objective; standard nonconvex theory gives stationary-point/gradient-norm guarantees, not hard local/global optimality.