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Overfitting Mechanism and Avoidance in Deep Neural Networks
Shaeke Salman, Xiuwen Liu
TL;DR
Deep neural networks can overfit despite strong performance, making their mechanisms important to understand. The paper analyzes training dynamics and proposes consensus-based classification, reporting improved accuracy by rejecting ambiguous samples. Its conclusions assume models capture intrinsic sample structure and treat disagreement as extrinsic variation.
Problem
The paper addresses insufficient understanding of deep neural-network mechanisms and the generalization gap associated with overfitting.
Method
The paper analyzes sample training dynamics and classifies inputs using consensus among multiple trained models, rejecting samples that fail the probability threshold.
Results
The proposed algorithm identifies consistently classified samples and improves classification performance in experiments using limited training data.
Takeaways & Limitations
Consensus-based classification avoids classifying ambiguous inputs and reduces the influence of extrinsic factors across models.
Takeaways & Limitations
The approach assumes all models capture intrinsic sample information and that model-specific differences are extrinsic factors such as noise.
Abstract
from arXiv · showhide
Assisted by the availability of data and high performance computing, deep learning techniques have achieved breakthroughs and surpassed human performance empirically in difficult tasks, including object recognition, speech recognition, and natural language processing. As they are being used in critical applications, understanding underlying mechanisms for their successes and limitations is imperative. In this paper, we show that overfitting, one of the fundamental issues in deep neural networks, is due to continuous gradient updating and scale sensitiveness of cross entropy loss. By separating samples into correctly and incorrectly classified ones, we show that they behave very differently, where the loss decreases in the correct ones and increases in the incorrect ones. Furthermore, by analyzing dynamics during training, we propose a consensus-based classification algorithm that enables us to avoid overfitting and significantly improve the classification accuracy especially when the number of training samples is limited. As each trained neural network depends on extrinsic factors such as initial values as well as training data, requiring consensus among multiple models reduces extrinsic factors substantially; for statistically independent models, the reduction is exponential. Compared to ensemble algorithms, the proposed algorithm avoids overgeneralization by not classifying ambiguous inputs. Systematic experimental results demonstrate the effectiveness of the proposed algorithm. For example, using only 1000 training samples from MNIST dataset, the proposed algorithm achieves 95% accuracy, significantly higher than any of the individual models, with 90% of the test samples classified.
I. INTRODUCTION
Deep learning has achieved strong empirical performance, but its underlying mechanisms remain insufficiently understood. The paper studies solution abundance, overfitting dynamics, and a consensus-based way to avoid overfitting with limited data.
- Deep learning has produced state-of-the-art and human-surpassing performance in object recognition, speech recognition, and natural language processing.
- The paper motivates this analysis by arguing that reliable understanding is important as deep networks are deployed in critical applications.
- The paper highlights a mismatch between classical capacity-based generalization theory and overparameterized neural networks that generalize well.Such networks can have more parameters than training samples while using simple stochastic gradient descent successfully.
- The authors empirically examine abundant good solutions, explain overfitting through continued gradient updating, and propose consensus-based classification.
II. ABUNDANCE OF “GOOD” DEEP NEURAL NETWORK SOLUTIONS
The paper investigates why simple optimization can find good solutions in overparameterized neural networks. Interpolation experiments provide empirical evidence that many good solutions occupy connected regions of parameter space.
- A neural network is represented as a parametrized function f(x; θ), with θ containing its weights and biases.
- The solution found by iterative optimization depends on initialization, optimization dynamics, and random batch ordering.
- Good solutions can be reached with high probability from typical initializations despite neural-network overparameterization.
- For α from -1.0 to 0.4, interpolated parameters between two MNIST solutions yielded many good solutions.The interpolation was w = αw1 + (1 − α)w2.
III. OVERFITTING MECHANISM
The paper explains overfitting as a training-dynamics phenomenon rather than a special failure of the model. Continued parameter updates increase softmax-input magnitudes and raise loss on incorrectly classified samples.
- Overfitting occurs when performance is good on training data but poor on test data, producing a training–test error gap.
- Deep networks are especially prone to overfitting because their flexibility and many parameters allow them to fit training data, including randomized labels.
- Continuous weight and bias updates increase softmax-input magnitudes, while cross-entropy loss is sensitive to input scaling.
- The loss increase comes from incorrectly classified samples, whereas the loss for correctly classified samples decreases.
- Across numerous examples and models, validation accuracy remains almost constant even as validation loss increases.This supports attributing the loss increase to growing softmax-input magnitudes.
IV. CONSENSUS-BASED CLASSIFICATION
The paper addresses the generalization gap by analyzing sample-level training dynamics and using agreement across models to identify reliable classifications. Samples with unstable predictions are treated as ambiguous rather than forced into a class.
- Figure 1 varies α along linear interpolations between two good parameter vectors and reports test loss and test accuracy.
- Figure 2 shows training loss decreasing while validation loss increases after a certain epoch, alongside a training–validation accuracy gap.
- The proposed approach analyzes training dynamics to address the gap between training and validation performance.
- Training Dynamics: Consistent predictions across models indicate greater confidence, whereas changing predictions suggest unstable, random-factor-driven classification.
B. Consensus-based Classification
The consensus-based classifier uses multiple models to identify consistently classified samples and reject ambiguous ones. It reduces the influence of model-specific randomness rather than averaging predictions.
- Consensus principle: Multiple models reduce the chance that classifications caused by random factors persist across models.The paper states that this reduction is exponential for statistically independent models.
- Algorithm: The algorithm retains each model’s class probabilities, computes the class-wise minimum, and classifies only when the maximum minimum probability exceeds threshold p_t.Otherwise, the input is rejected as ambiguous.
- Examples: For the illustrated samples, consistent predictions can produce a class label, whereas unstable predictions prevent classification at the stated thresholds.The examples include threshold values below 0.96 and 0.712 for two samples, while the unstable sample is not classified for p_t ≥ 0.1.
- Comparison with ensembles: Consensus differs from ensemble averaging because it uses inter-model inconsistency to reject samples that cannot be classified consistently.The paper contrasts this with ensemble methods, which classify every sample and therefore do not avoid overgeneralization.
V. EXPERIMENTAL RESULTS
The experiments evaluate the proposed method on representative datasets with varied neural-network architectures and optimization algorithms, including a dataset of normalized handwritten postal digits.
- Experimental setup: The experiments use two representative datasets, different deep neural-network architectures, and different optimization algorithms.One dataset contains 16×16 grayscale images of handwritten zip-code digits, with 7291 training samples and 2007 test or validation samples.
A. Density Distribution of the Samples
The density-distribution analysis compares losses of correctly and incorrectly classified validation samples across saved training epochs to identify overgeneralized samples.
- Density analysis: Twenty saved models produce 40 loss distributions: 20 for correctly classified samples and 20 for incorrectly classified samples.For each model, the validation samples are evaluated for individual loss, accuracy, and classification correctness.
B. Consensus-based Classification Using Different Architectures
Using multiple architectures, the method separates consistently classified samples from samples affected by extrinsic factors and improves performance when training data are limited. Increasing the consensus threshold rejects more uncertain samples while preserving high-confidence classifications.
- Sample classification: Samples are categorized as consistently or extrinsically classified by examining classification dynamics across models.Extrinsic classifications are associated with factors such as random weight initialization and softmax outputs.
- Sample dynamics: Figure 4 contrasts samples classified consistently across three models with a sample whose classification is unstable over the models.The unstable sample is treated as ambiguous because the models assign different classes.
- Threshold effects: When the threshold is below 0.4, few samples are affected and performance is similar; increasing the threshold changes the number of correctly classified samples considered.The threshold is applied to probabilities from three models with different architectures.
- Performance: The method improves performance when the number of training samples is small.The paper relates this result to work arguing that generalization is not necessarily dependent on data volume.
- Multiple-model comparison: The combined model improves accuracy substantially and consistently across threshold values compared with individual models.At p_t = 0.99, model 3 has 2935 CCS samples with 2926 classified correctly, while the combined model has 2851 CCS samples with 2848 classified correctly.
C. Impact of Dropout in Overfitting
Dropout somewhat reduces overfitting but does not necessarily prevent it. Across tested dropout settings, its effects on consistently classified samples are generally small and similar to the no-dropout results.
- Dropout is widely used to reduce overfitting, but the paper argues it does not necessarily prevent overfitting.In overparameterized networks, regularization may select among good solutions rather than eliminate overfitting.
- Three dropout values—0.25, 0.50, and 0.75—were evaluated for their effects on accuracy and the percentage of consistently classified samples.The experiments varied the threshold alongside the dropout parameter.
- At low thresholds, dropout parameter 0.50 performed better than the other two tested values.
- The differences among dropout settings were not significant, and the curves were similar to those without dropout.The comparison concerns both accuracy and consistently classified samples across threshold values.
VI. DISCUSSION
The discussion connects the consensus approach to model randomness, generalization, and trustworthiness. It presents consensus as a way to identify samples affected by extrinsic factors while noting challenges involving adversarial examples and broader generalization theory.
- The method assumes that all models capture intrinsic sample content, while model-specific classifications reflect extrinsic factors such as noise.The paper states that consensus can be generalized to requiring agreement among k of n models.
- Consensus is motivated by the need to understand generalization mechanisms rather than relying only on architectures, optimization algorithms, and hyperparameters.
- Generalization remains difficult to characterize because standard capacity-based theory does not straightforwardly explain strong performance by overparameterized deep networks.
- The paper frames adversarial examples as inputs near decision boundaries whose small changes can alter predicted labels.
- Under the assumption that models are independent, using more models is expected to reduce the probability of adversarial examples occurring across all models exponentially.The paper identifies this as an advantage of consensus-based classification, while stating that real-dataset effects require further study.
- Typical overfitting curves show training accuracy improvement stopping while validation behavior does not follow the expected nearest-neighbor pattern.
VII. CONCLUSION AND FUTURE WORK
The paper concludes that consensus across multiple models can identify samples classified because of random factors and support overfitting avoidance. Future work includes broader model settings, consensus variants, ensemble connections, solution-space analysis, and active learning.
- The proposed consensus-based algorithm identifies samples classified due to random factors by comparing multiple models.
- The authors report that identifying overgeneralized samples through training dynamics enables overfitting avoidance on the remaining samples.
- Future work will extend the method beyond softmax classification and examine consensus among k of n models.
- Planned studies include connections with bagging and boosting, solution spaces in planes, and active learning for improved generalization.