Convex-Concave Programming: An Effective Alternative for Optimizing Shallow Neural Networks
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Abstract
In this study, we address the challenges of non-convex optimization in neural networks (NNs) by formulating the training of multilayer perceptron (MLP) NNs as a difference of convex functions (DC) problem. Utilizing the basic convex–concave algorithm to solve our DC problems, we introduce two alternative optimization techniques, <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">DC-GD</i> and <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">DC-OPT</i>, for determining MLP parameters. By leveraging the non-uniqueness property of the convex components in DC functions, we generate strongly convex components for the DC NN cost function. This strong convexity enables our proposed algorithms, <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">DC-GD</i> and <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">DC-OPT</i>, to achieve an <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">iteration complexity</i> of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$O\left(\log \left(\frac{1}{\varepsilon }\right)\right)$</tex-math></inline-formula>, surpassing that of other solvers, such as stochastic gradient descent (<italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">SGD</i>), which has an <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">iteration complexity</i> of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$O\left(\frac{1}{\varepsilon }\right)$</tex-math></inline-formula>. This improvement raises the convergence rate from sublinear (<italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">SGD</i>) to linear (ours) while maintaining comparable <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">total computational costs</i>. Furthermore, conventional NN optimizers like <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">SGD</i>, <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">RMSprop</i>, and <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Adam</i> are highly sensitive to the learning rate, adding computational overhead for practitioners in selecting an appropriate learning rate. In contrast, our <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">DC-OPT</i> algorithm is hyperparameter-free (i.e., it requires no learning rate), and our <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">DC-GD</i> algorithm is less sensitive to the learning rate, offering comparable accuracy to other solvers. Additionally, we extend our approach to a convolutional NN architecture, demonstrating its applicability to modern NNs. We evaluate the performance of our proposed algorithms by comparing them to conventional optimizers such as <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">SGD</i>, <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">RMSprop</i>, and <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Adam</i> across various test cases. The results suggest that our approach is a viable alternative for optimizing shallow MLP NNs.
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Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.000 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.000 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.000 | 0.000 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
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