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Approximated Orthogonal Projection Unit: Stabilizing Regression Network Training Using Natural Gradient
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Proposes AOPU, a regression network that truncates backpropagation at dual parameters so the truncated gradient approximates natural gradient and the output approximates a minimum variance estimator, evaluated on two chemical soft-sensor datasets for stability.
Problem
Neural network regression for industrial soft sensors lacks stable online training; existing NNs show large validation fluctuations, and MVE/natural gradient are impractical for NNs.
Approach
Introduce trackable parameters W and dual parameters D(W)=x x^T W, backpropagate truncated at D, and update W with the dual gradient; on a random-feature augmented input, this is shown to approximate natural gradient descent and a linear minimum variance estimator, with rank ratio (RR) as a stability/interpretability index.
Key results
- On SRU at seq 48, AOPU reaches R^2 0.8209 (final-checkpoint best 0.8069) with very low std (0.0092), while other models drop several-fold in stability under fixed-epoch training.
- On Debutanizer, AOPU's best final-checkpoint R^2 is 0.6054 (from 0.7118 with early stopping), a 14.9% drop, versus drops of 26% to 124% for baselines.
- Ablation shows zero-mean activations (e.g., Tanh) outperform non-zero-mean ones like ReLU and layer normalization hurts AOPU performance.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use AOPU rather than the strongest listed baseline | Regression on Debutanizer, seq 48 and bs 64 · Debutanizer (best epoch) | R²: 0.7182 -> 0.7118 | Mixed |
| Use AOPU rather than the strongest listed baseline | Regression on SRU, seq 48 and bs 64 · SRU (best epoch) | R²: 0.7789 -> 0.8209 | Helped |
| Use AOPU rather than other models with final checkpoints | Regression on Debutanizer and SRU, seq 48 and bs 64 · Debutanizer and SRU (final checkpoint) | R²: 0.6054; 0.8069 | Helped |
| Use Tanh instead of ReLU augmentation (ablation) | AOPU · Debutanizer and SRU | R²: 0.5654 -> 0.7216; 0.8115 -> 0.8026 | Mixed |
| Add layer normalization (ablation) | AOPU with ReLU or Tanh augmentation · Debutanizer and SRU | R²: 0.5654 -> -0.6961; 0.8115 -> -147.98; 0.7216 -> 0.4930; 0.8026 -> 0.4803 | Hurt |
| Replace Tanh with zero-mean activation alternatives (ablation) | AOPU · Debutanizer and SRU | R²: 0.7202, 0.7221, 0.7225, 0.7201; 0.8056, 0.8043, 0.8040, 0.8074 | No clear effect |
| Replace Tanh with Sigmoid, ReLU6, or RReLU (ablation) | AOPU · Debutanizer | R²: 0.5777; 0.5416; 0.5582 | Hurt |
| Replace Tanh with Hard Swish or Mish (ablation) | AOPU · Debutanizer and SRU | R²: 0.7232, 0.7229; 0.8026, 0.7999 | No clear effect |
Surprising findings
- Layer normalization significantly impairs AOPU performance, with R^2 dropping to -0.6961 on Debutanizer.
- Autoformer's performance collapses under fixed-epoch training, with R^2 falling to -0.2396 on Debutanizer.
Limitations
- AOPU is sensitive to batch size and sequence length via rank ratio and is not plug-and-play.
- Matrix inversion with pseudo-inverse via positive singular values loses precision when RR is low, risking divergence.
- Theoretical NG and MVE claims rely on approximations valid only when the augmented input is column-full-rank.
Experimental setup
Datasets: Debutanizer Sulfur Recovery Unit (SRU)
Benchmarks: Debutanizer Sulfur Recovery Unit (SRU)
Metrics: MSE MAPE R^2
Baselines: Autoformer Informer DNN SDAE SVAE LSTM RVFLNN
Builds on: natural gradient descent stochastic mirror descent random vector functional-link network
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