Research map / Learning theory and algorithms / Sample complexity
MLI Formula: A Nearly Scale-Invariant Solution with Noise Perturbation
Read the paperCheck your idea against 108,967 papers
AI-extracted summary. Written by AI from the paper's full text. Numbers and claims can be wrong or missing context, so check the paper before you cite or build on it.
Explains the monotonic linear interpolation (MLI) phenomenon as a nearly scale-invariant converged network αθF perturbed by noise (1−α)ε, introducing GSI, RSI and NSI scale-invariance variants and showing MLI holds even when θ0 is replaced by random or zero initialization.
Problem
Prior explanations of Monotonic Linear Interpolation tie the phenomenon to the optimization trajectory between paired initial and converged parameters; the paper asks whether MLI is instead governed by properties of the converged model alone.
Approach
Decouple MLI from the initialization by viewing the interpolation as αθF + (1−α)ε: the scaled converged network is nearly scale-invariant (via VSI/GSI/RSI/NSI, with BN giving the strongest form), while the added noise grows as α decreases, so error rises monotonically; MLI is the special case ε=θ0.
Key results
- Test error along the path (1-α)ε + αθF decreases monotonically in α for random, zero, and scaled initializations across FCN4, VGG8, VGG16+BN and ResNet20 settings.
- Scaled converged network αθF attains test error close to θF over a range of small α, especially for networks with BN, supporting the nearly scale-invariant explanation.
- MLI violations in ResNet20 trained with large learning rate or Adam are attributed to a broader bias distribution centered near zero.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| replace the original initialization with an unrelated random initialization (ablation) | MLI interpolation · MNIST, SVHN, CIFAR10, and CIFAR100 | test error monotonicity | No clear effect |
| replace the original initialization with zero initialization (ablation) | MLI interpolation · MNIST, SVHN, CIFAR10, and CIFAR100 | test error monotonicity | No clear effect |
| scale the converged parameters by α (ablation) | converged networks · MNIST, SVHN, CIFAR10, and CIFAR100 | test error | Helped |
| use constant-valued initialization instead of random-distribution initialization (ablation) | MLI interpolation · MNIST, SVHN, CIFAR10, and CIFAR100 | test error monotonicity | No clear effect |
| increase the relative scale of initialization parameters (ablation) | MLI interpolation · MNIST, SVHN, CIFAR10, and CIFAR100 | test error curve plateau length | Hurt |
| reset BN running statistics and warm them up over training data (ablation) | interpolation between training checkpoints and the final network · CIFAR10 and CIFAR100 | test error monotonicity | Helped |
| freeze BN running statistics during interpolation (ablation) | interpolated networks · CIFAR10 and CIFAR100 | test error curve plateau length | Hurt |
| interpolate BN running statistics between initialization and final statistics (ablation) | interpolated networks · CIFAR10 and CIFAR100 | test error curve plateau length | Hurt |
| add scaled initialization noise (1−α)θ0 to the scaled converged network (ablation) | scaled networks αθF · CIFAR10 | test error | Hurt |
| train with a large learning rate (ablation) | ResNet20 · CIFAR100 | train loss monotonicity along interpolation | Hurt |
| use Adam instead of SGD (ablation) | ResNet20 · CIFAR100 | train loss monotonicity along interpolation | Hurt |
Surprising findings
- Replacing the initialization θ0 with zero values still yields a monotonically decreasing test error curve along the interpolation.
- The MLI curve is largely independent of the initialization, suggesting it reflects properties of the converged model rather than the optimization trajectory.
Limitations
- Noise hypothesis does not hold for arbitrary ε (e.g., negative gradient noise), and is only verified empirically.
- Violations of MLI under large learning rates or Adam rely on empirical bias-distribution arguments rather than a proof.
Experimental setup
Datasets: MNIST SVHN CIFAR10 CIFAR100 CUB AG News
Benchmarks: MNIST SVHN CIFAR10 CIFAR100 CUB AG News
Metrics: test error train loss descent interval (alpha_S, alpha_E)
Baselines: Goodfellow & Vinyals 2015 MLI Wang et al. 2023 bias-plateau explanation
Builds on: Monotonic Linear Interpolation (Goodfellow & Vinyals, 2015) Lucas et al. 2021 MLI violations Frankle 2020 plateau and running statistics analysis Li et al. 2018 loss landscape visualization
Closest papers by meaning
- Beyond Benign Overfitting in Nadaraya-Watson Interpolators (NeurIPS 2025 · similarity 0.81)
- Smooth Min-Max Monotonic Networks (ICML 2024 · similarity 0.80)
- Fast Last-Iterate Convergence of SGD in the Smooth Interpolation Regime (NeurIPS 2025 · similarity 0.79)
- Convergence Rates for Gradient Descent on the Edge of Stability for Overparametrised Least Squares (NeurIPS 2025 · similarity 0.79)
- Improving Computational Complexity in Statistical Models with Local Curvature Information (ICML 2024 · similarity 0.79)
- Revisiting Zeroth-Order Optimization: Minimum-Variance Two-Point Estimators and Directionally Aligned Perturbations (ICLR 2025 · similarity 0.79)
Writing a paper in this area? scime draft is a free LaTeX editor with AI included, real-time co-authors and a check of your references and venue rules before you submit. It also imports OpenAI Prism projects.