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Activation: CELU Alpha Partial Derivative Tool

Evaluate the alpha partial of CELU alpha*(exp(x/alpha)-1) on its negative branch and zero on its linear branch, using supplied positive alpha.

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Description

Return zero for x>=0; for x<0 return expm1(z)-z*exp(z), the alpha partial derivative at fixed x.

CELU uses alpha*(exp(x/alpha)-1) for negative input and x otherwise, with positive alpha. Rescaling the exponential argument makes the first input derivative continuous at zero, while second derivatives on its two sides still differ.

Return zero for x>=0; for x<0 return expm1(z)-z*exp(z), the alpha partial derivative at fixed x.

When to use this tool

Use Activation: CELU Alpha Partial Derivative when checking the stated CELU join curvature or alpha sensitivity convention.

How to use the tool and how it works

  1. Enter x, alpha. Use the units and conventions below.
  2. Run the tool. Correct any format, range, or cross-field input error before interpreting the output.
  3. Read result. Expand result to inspect the returned fields.

Require alpha>0 and finite z=x/alpha. Return zero for x>=0; for x<0 return expm1(z)-z*exp(z), the alpha partial derivative at fixed x.1

CELU places alpha in the negative exponential’s argument as well as its amplitude. First-derivative continuity at zero does not imply matching second derivatives at the join.2

Input fieldMeaning and unitsAccepted input
xFinite scalar input coordinate; use the mathematical scale specified above.Required. Finite number.
alphaPositive shape parameter when required by the function convention above.Required. Finite number.
Output fieldInterpretationReturned shape
resultObject containing the operation-specific returned members listed below.Required. Object.
result.alphaPartialAlpha PartialCalculated result member: number.
json
{
  "input": {
    "x": 0,
    "alpha": 2
  },
  "output": {
    "result": {
      "alphaPartial": 0
    }
  }
}

Limitations and assumptions

  • Unlike ELU, alpha also rescales the negative exponential argument; no parameter fitting or tensor gradient is performed.
  • Outputs are nested inside result. 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

  1. Calculation and input contract: activation/celu-alpha-partial-derivative — tool.io

  2. Continuously Differentiable Exponential Linear Units — Jonathan T. Barron

Similar or alternative tools

  • CELU Join Curvature Jump

    At the zero join, return left curvature=1/alpha, right curvature=0 and right-minus-left jump=-1/alpha.

  • ELU Negative Curvature Tool

    Evaluate ELU second derivative on the negative branch and zero on the positive branch; at zero use the positive-side value.

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