Machine Learning / Uncertainty Quantification
Uncertainty propagation through trained multi-layer perceptrons: Exact analytical results
This paper presents analytical results for propagating uncertainty through trained multi-layer perceptrons (MLPs) with a single hidden layer and ReLU activation functions. Specifically, it provides exact expressions for the mean and variance of the MLP output when the input is a multivariate Gaussian distribution, contrasting with previous work that relied on series expansions. The methodology is validated through numerical experiments on a test problem involving the prediction of lithium-ion cell state-of-health using Electrical Impedance Spectroscopy data, showing strong agreement with Monte Carlo sampling.
Key Executive Impact Metrics
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RMSE (Mean Output) with 10⁶ MC trials
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RMSE (Variance Output) with 10⁶ MC trials
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Error Scaling (Mean)
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Error Scaling (Variance)
Deep Analysis & Enterprise Applications
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Core Concepts
Analytical Uncertainty Propagation: Deriving exact mathematical expressions for output uncertainty in ML models.
Multi-layer Perceptrons (MLPs): Feed-forward neural networks with one or more hidden layers. This paper focuses on single hidden layer MLPs with ReLU activations.
ReLU Activation Function: Rectified Linear Unit, defined as `max(x, 0)`, a popular choice in neural networks.
Multivariate Gaussian Distribution: Input data is assumed to follow this distribution, characterized by a mean vector and covariance matrix.
Mean and Variance of Output: The key summary statistics targeted for exact analytical expression.
Methodology
The method involves propagating known uncertainties from multivariate Gaussian input data through a fixed (trained) single-hidden-layer MLP with ReLU activation. The challenge lies in propagating uncertainty through the non-linear ReLU function. Exact closed-form expressions for the mean (EXi), second moment (EXi^2), and cross-moment (EXiXj) of the rectified multivariate Gaussian distribution are derived using standard 1D and 2D Gaussian integrals. These expressions are then used to calculate the mean and covariance of the hidden layer outputs (X), which in turn allows for the calculation of the final output's mean and variance (Y) through standard linear algebra.
Enterprise Applications
Trustworthy AI & Risk Management: Provides a robust method for quantifying uncertainty in ML predictions, essential for compliance with regulations like the EU AI Act requiring transparency and risk management.
Sensitivity Analysis: Propagating uncertainties through a fixed model can be used to perform sensitivity analysis, understanding how input uncertainties affect model output.
Improved Accuracy & Reproducibility: Analytical expressions offer more accurate and precise characterization of output uncertainty compared to sampling-based methods, which approximate true distributions. They are also more reproducible.
Model Transparency & Insight: Provides mathematical insight into the sources of propagated uncertainty, enhancing understanding of model behavior.
Specific Use Case (Lithium-ion Cells): Applied to estimate the State-of-Health (SOH) of lithium-ion cells from Electrical Impedance Spectroscopy (EIS) data, demonstrating practical utility.
| Feature | This Paper's Method | Related Work ([19]) |
|---|---|---|
| Activation Function | ReLU | General activation functions (e.g., ReLU, Heaviside, GELU) |
| Output Expressions | Exact, closed-form functions of 1D/2D Gaussian integrals | Infinite Taylor series expansions |
| Computational Complexity | Simpler, direct computation | Dependent on number of terms for desired precision, painstaking calculation per function |
| Accuracy | Exact results | Arbitrary precision achievable, but still an approximation if series truncated |
Exact Analytical Results
for MLP output mean and variance with Gaussian input, avoiding series expansions.
Validation through Monte Carlo Sampling
The analytical expressions were validated against Monte Carlo sampling on a lithium-ion cell State-of-Health prediction task. For 10⁶ Monte Carlo trials, the RMSE for the mean output was 0.0068 x 10⁻² and for the variance was 0.0100 x 10⁻³, demonstrating strong agreement. The error convergence followed the expected 1/√n rule, with log-log plot gradients of -0.5036 for mean and -0.4966 for variance.
Enterprise Process Flow
Multivariate Gaussian Input (V)
→
Affine Convolution (W=AᵀV+c)
→
ReLU Activation (X=W⁺)
→
Affine Convolution (Y=βᵀX+d)
→
Output Mean & Variance (Y)
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