Understanding the fundamental principles behind the success of deep neural networks is one of the most important open questions in the current literature.

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We also use subscripts to denote a certain column or element.

We also use subscripts to denote a certain column or element.

Understanding the fundamental principles behind the success of deep neural networks is one of the most important open questions in the current literature.

Solving non-convex, NP-hard optimization problems is crucial for training machine learning models, including neural networks. However, non-convexity often leads to black-box machine learning models with unclear inner workings. While convex formulations have been used for verifying neural network robustness, their application to training neural networks remains less explored. In response to this challenge, we reformulate the problem of training infinite-width two-layer ReLU networks as a convex completely positive program in a finite-dimensional (lifted) space. Despite the convexity, solving this problem remains NP-hard due to the complete positivity constraint. To overcome this challenge, we introduce a semidefinite relaxation that can be solved in polynomial time. We then experimentally evaluate the tightness of this relaxation, demonstrating its competitive performance in test accuracy across a range of classification tasks.

Deploying large and complex deep neural networks on resource-constrained edge devices poses significant challenges due to their computational demands and the complexities of non-convex optimization. Traditional compression methods such as distillation and pruning often retain non-convexity that complicates fine-tuning in real-time on such devices. Moreover, these methods often necessitate extensive end-to-end network fine-tuning after compression to preserve model performance, which is not only time-consuming but also requires fully annotated datasets, thus potentially negating the benefits of efficient network compression. In this paper, we introduce a novel distillation technique that efficiently compresses the model via convex optimization -- eliminating intermediate non-convex activation functions and using only intermediate activations from the original model. Our approach enables distillation in a label-free data setting and achieves performance comparable to the original model without requiring any post-compression fine-tuning. We demonstrate the effectiveness of our method for image classification models on multiple standard datasets, and further show that in the data limited regime, our method can outperform standard non-convex distillation approaches. Our method promises significant advantages for deploying high-efficiency, low-footprint models on edge devices, making it a practical choice for real-world applications. We show that convex neural networks, when provided with rich feature representations from a large pre-trained non-convex model, can achieve performance comparable to their non-convex counterparts, opening up avenues for future research at the intersection of convex optimization and deep learning.

While there has been a lot of progress in developing deep neural networks (DNNs) to solve practical machine learning problems [1-3], the inner workings of neural networks is not well understood. A foundational theory for understanding how neural networks work is still lacking despite extensive research over several decades. In this paper, we provide a novel analysis of neural networks based on geometric algebra and convex optimization. We show that weights of deep ReLU neural networks learn the wedge product of a subset of training samples when trained by minimizing standard regularized loss functions. Furthermore, the training problem reduces to convex optimization over wedge product features, which encode the geometric structure of the training dataset. This structure is given in terms of signed volumes of triangles and parallelotopes generated by data vectors. Our analysis provides a novel perspective on the inner workings of deep neural networks and sheds light on the role of the hidden layers.

Recently, theoretical analyses of deep neural networks have broadly focused on two directions: 1) Providing insight into neural network training by SGD in the limit of infinite hidden-layer width and infinitesimally small learning rate (also known as gradient flow) via the Neural Tangent Kernel (NTK), and 2) Globally optimizing the regularized training objective via cone-constrained convex reformulations of ReLU networks. The latter research direction also yielded an alternative formulation of the ReLU network, called a gated ReLU network, that is globally optimizable via efficient unconstrained convex programs. In this work, we interpret the convex program for this gated ReLU network as a Multiple Kernel Learning (MKL) model with a weighted data masking feature map and establish a connection to the NTK. Specifically, we show that for a particular choice of mask weights that do not depend on the learning targets, this kernel is equivalent to the NTK of the gated ReLU network on the training data. A consequence of this lack of dependence on the targets is that the NTK cannot perform better than the optimal MKL kernel on the training set. By using iterative reweighting, we improve the weights induced by the NTK to obtain the optimal MKL kernel which is equivalent to the solution of the exact convex reformulation of the gated ReLU network. We also provide several numerical simulations corroborating our theory. Additionally, we provide an analysis of the prediction error of the resulting optimal kernel via consistency results for the group lasso.