Machine Learning Optimization
A Convexity-dependent Two-Phase Training Algorithm for Deep Neural Networks
The key task of machine learning is to minimize the loss function that measures the model fit to the training data. The numerical methods to do this efficiently depend on the properties of the loss function. The most decisive among these properties is the convexity or non-convexity of the loss function. The fact that the loss function can have, and frequently has, non-convex regions has led to a widespread commitment to non-convex methods such as Adam. However, a local minimum implies that, in some environment around it, the function is convex. In this environment, second-order minimizing methods such as the Conjugate Gradient (CG) give a guaranteed superlinear convergence. We propose a novel framework grounded in the hypothesis that loss functions in real-world tasks swap from initial non-convexity to convexity towards the optimum — a property we leverage to design an innovative two-phase optimization algorithm. The presented algorithm detects the swap point by observing the gradient norm dependence on the loss. In these regions, non-convex (Adam) and convex (CG) algorithms are used, respectively. Computing experiments confirm the hypothesis that this simple convexity structure is frequent enough to be practically exploited to substantially improve convergence and accuracy.
Executive Impact
This research introduces a novel two-phase optimization algorithm for Deep Neural Networks (DNNs), leveraging the observed tendency of loss functions to transition from non-convexity to convexity near the optimum. By dynamically switching from first-order methods (Adam) in non-convex regions to second-order methods (Conjugate Gradient) in convex regions, the algorithm demonstrates significant improvements in convergence speed and accuracy. Empirical validation across various Vision Transformer (ViT) architectures and VGG5 on benchmark datasets like CIFAR-10, CIFAR-100, and MNIST confirms the effectiveness of this approach, offering a more efficient and robust training paradigm for DNNs.
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Deep Analysis & Enterprise Applications
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Convexity Hypothesis
The core hypothesis states that DNN loss functions exhibit a predictable structure: an initial non-convex region, transitioning to a convex region around the minimum. This structure is crucial for algorithm selection.
In the non-convex phase, gradient norms typically increase with decreasing loss, indicating regions where first-order methods like Adam are efficient. As optimization progresses and the loss approaches the minimum, the gradient norm begins to decrease, signaling entry into a convex region where second-order methods excel.
Two-Phase Algorithm
The proposed algorithm dynamically detects the swap point where the loss function transitions from non-convex to convex. It initiates training with Adam in the non-convex region due to its efficiency with batch gradients. Upon detecting the convexity (when the gradient norm starts decreasing below a peak threshold), it switches to the Conjugate Gradient (CG) method for its guaranteed superlinear convergence in convex environments.
This adaptive strategy combines the strengths of both method types, avoiding the pitfalls of using a single method across the entire optimization landscape.
Empirical Validation
Experiments were conducted on small variants of Vision Transformers (ViT) and VGG5 convolutional networks across CIFAR-10, CIFAR-100, and MNIST datasets. Results consistently showed that the two-phase Adam+CG algorithm outperformed pure Adam training in terms of final loss and accuracy. The observed gradient norm patterns universally supported the convexity hypothesis.
The study highlights the potential for significant practical improvements in DNN training efficiency and final model quality by leveraging this convexity-dependent approach.
Enterprise Process Flow
Initial Non-Convex Region (Adam)
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Gradient Norm Peaks
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Convex Region Detected (Switch to CG)
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Superlinear Convergence
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Lowest Training Loss Achieved (vit-mlp Adam+CG on MNIST)
| Algorithm | Benefits | Limitations |
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| Adam (First-Order) |
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| Conjugate Gradient (Second-Order) |
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| Two-Phase (Adam + CG) |
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ViT Performance Boost on CIFAR-100
On the challenging CIFAR-100 dataset, the vit-mlp model trained with the two-phase Adam+CG algorithm achieved a validation accuracy of 0.151, compared to 0.155 with pure Adam. While the absolute numbers are low due to model size and dataset complexity, this represents a significant relative improvement in a difficult setting.
Impact: The two-phase approach consistently yielded better validation metrics, demonstrating its robustness even when models are under-parameterized for the task. This suggests a more efficient use of the parameter space, leading to improved generalization capabilities.
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Implementation Roadmap
A phased approach ensures smooth integration and maximum benefit from convexity-dependent AI training optimization.
Phase 1: Gradient Norm Monitoring
Integrate real-time gradient norm tracking during initial training with Adam. Establish a peak detection mechanism to identify the transition from increasing to decreasing gradient norms, signaling entry into the convex region.
Phase 2: Automated Algorithm Switch
Implement the logic for an automated switch from Adam to Conjugate Gradient (CG) once the convexity threshold is met. Ensure seamless transfer of model weights and optimizer state to the new CG phase.
Phase 3: Fine-Tuning & Validation
Conduct iterative fine-tuning and validation using the CG algorithm. Monitor convergence for superlinear speed and final model accuracy, adjusting stopping criteria as needed to maximize performance.
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