Softplus Curvature Inflection Location Tool

Description

Return peakCurvatureLocation=0, peakCurvature=beta/4 and atPeak only when supplied input equals zero exactly.

Beta-scaled softplus is log(1+exp(beta*x))/beta for positive beta. Its input slope is a sigmoid, while curvature and parameter sensitivities are different derivatives. Numerical libraries may switch to a linear tail; such a switch is distinct from the smooth mathematical function.

Return peakCurvatureLocation=0, peakCurvature=beta/4 and atPeak only when supplied input equals zero exactly.

When to use this tool

Use Softplus Curvature Inflection Location when checking the stated softplus curvature, parameter sensitivity or branch convention.

How to use the tool and how it works

Enter beta, input. Use the units and conventions below.
Run the tool. Correct any format, range, or cross-field input error before interpreting the output.
Read peakCurvatureLocation, atPeak, peakCurvature. Inspect atPeak before interpreting the associated values.

Require beta>0 for beta-scaled softplus. Return peakCurvatureLocation=0, peakCurvature=beta/4 and atPeak only when supplied input equals zero exactly.^[1]

Input field	Meaning and units	Accepted input
beta	Supplied scalar shape parameter under the explicitly stated function convention.	Required. Finite number; strictly above 0.
input	Input	Required. Finite number.

Output field	Interpretation	Returned shape
peakCurvatureLocation	Location of maximum second derivative, not an inflection of softplus itself.	Required. Finite number.
atPeak	At Peak	Required. Boolean.
peakCurvature	Peak Curvature	Required. Finite number.

json

{
  "input": {
    "beta": 2,
    "input": 0
  },
  "output": {
    "peakCurvatureLocation": 0,
    "atPeak": true,
    "peakCurvature": 0.5
  }
}

Limitations and assumptions

This locates the second-derivative maximum, not an inflection of softplus itself; no approximate equality tolerance is applied.
Inputs must satisfy both the field types and the cross-field conditions above. Invalid input causes an error; a successful false flag or documented placeholder is a different result.
Numeric calculations use binary floating point unless exact decimal-integer arithmetic is explicitly stated. No hidden measurement tolerance or decimal rounding is added; nonfinite or out-of-contract results are rejected. Numeric signed-zero results are exported as zero.

Alternative or Complementary methods

Use the complete explicit function and the intended parameter, branch and numerical conventions for symbolic, tensor or model-level differentiation; assess training behavior independently.

References

Calculation and input contract: activation/softplus-curvature-inflection-location — tool.io

Similar or alternative tools

Activation: Softplus Linear Tail Switch
Return scaledInput=beta*x and useLinearTail when scaledInput is strictly greater than switchThreshold.
Softplus Beta Partial Derivative
Return (z*sigmoid(z)-softplus(z))/beta², the beta partial derivative of log(1+exp(beta*x))/beta at fixed x.