A Library of Mirrors: Deep Neural Nets in Low Dimensions are Convex Lasso Models with Reflection Features

We prove that training neural networks on 1-D data is equivalent to solving a convex Lasso problem with a fixed, explicitly defined dictionary matrix of features. The specific dictionary depends on the activation and depth. We consider 2-layer networks with piecewise linear activations, deep narrow ReLU networks with up to 4 layers, and rectangular and tree networks with sign activation and arbitrary depth. Interestingly in ReLU networks, a fourth layer creates features that represent reflections of training data about themselves. The Lasso representation sheds insight to globally optimal networks and the solution landscape.

The Convex Landscape of Neural Networks: Characterizing Global Optima and Stationary Points via Lasso Models

Due to the non-convex nature of training Deep Neural Network (DNN) models, their effectiveness relies on the use of non-convex optimization heuristics. Traditional methods for training DNNs often require costly empirical methods to produce successful models and do not have a clear theoretical foundation. In this study, we examine the use of convex optimization theory and sparse recovery models to refine the training process of neural networks and provide a better interpretation of their optimal weights. We focus on training two-layer neural networks with piecewise linear activations and demonstrate that they can be formulated as a finite-dimensional convex program. These programs include a regularization term that promotes sparsity, which constitutes a variant of group Lasso. We first utilize semi-infinite programming theory to prove strong duality for finite width neural networks and then we express these architectures equivalently as high dimensional convex sparse recovery models. Remarkably, the worst-case complexity to solve the convex program is polynomial in the number of samples and number of neurons when the rank of the data matrix is bounded, which is the case in convolutional networks. To extend our method to training data of arbitrary rank, we develop a novel polynomial-time approximation scheme based on zonotope subsampling that comes with a guaranteed approximation ratio. We also show that all the stationary of the nonconvex training objective can be characterized as the global optimum of a subsampled convex program. Our convex models can be trained using standard convex solvers without resorting to heuristics or extensive hyper-parameter tuning unlike non-convex methods. Through extensive numerical experiments, we show that convex models can outperform traditional non-convex methods and are not sensitive to optimizer hyperparameters.

Fixing the NTK: From Neural Network Linearizations to Exact Convex Programs

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.

Globally Optimal Training of Neural Networks with Threshold Activation Functions

Threshold activation functions are highly preferable in neural networks due to their efficiency in hardware implementations. Moreover, their mode of operation is more interpretable and resembles that of biological neurons. However, traditional gradient based algorithms such as Gradient Descent cannot be used to train the parameters of neural networks with threshold activations since the activation function has zero gradient except at a single non-differentiable point. To this end, we study weight decay regularized training problems of deep neural networks with threshold activations. We first show that regularized deep threshold network training problems can be equivalently formulated as a standard convex optimization problem, which parallels the LASSO method, provided that the last hidden layer width exceeds a certain threshold. We also derive a simplified convex optimization formulation when the dataset can be shattered at a certain layer of the network. We corroborate our theoretical results with various numerical experiments.

Convexifying Transformers: Improving optimization and understanding of transformer networks

Understanding the fundamental mechanism behind the success of transformer networks is still an open problem in the deep learning literature. Although their remarkable performance has been mostly attributed to the self-attention mechanism, the literature still lacks a solid analysis of these networks and interpretation of the functions learned by them. To this end, we study the training problem of attention/transformer networks and introduce a novel convex analytic approach to improve the understanding and optimization of these networks. Particularly, we first introduce a convex alternative to the self-attention mechanism and reformulate the regularized training problem of transformer networks with our alternative convex attention. Then, we cast the reformulation as a convex optimization problem that is interpretable and easier to optimize. Moreover, as a byproduct of our convex analysis, we reveal an implicit regularization mechanism, which promotes sparsity across tokens. Therefore, we not only improve the optimization of attention/transformer networks but also provide a solid theoretical understanding of the functions learned by them. We also demonstrate the effectiveness of our theory through several numerical experiments.

GLEAM: Greedy Learning for Large-Scale Accelerated MRI Reconstruction

Unrolled neural networks have recently achieved state-of-the-art accelerated MRI reconstruction. These networks unroll iterative optimization algorithms by alternating between physics-based consistency and neural-network based regularization. However, they require several iterations of a large neural network to handle high-dimensional imaging tasks such as 3D MRI. This limits traditional training algorithms based on backpropagation due to prohibitively large memory and compute requirements for calculating gradients and storing intermediate activations. To address this challenge, we propose Greedy LEarning for Accelerated MRI (GLEAM) reconstruction, an efficient training strategy for high-dimensional imaging settings. GLEAM splits the end-to-end network into decoupled network modules. Each module is optimized in a greedy manner with decoupled gradient updates, reducing the memory footprint during training. We show that the decoupled gradient updates can be performed in parallel on multiple graphical processing units (GPUs) to further reduce training time. We present experiments with 2D and 3D datasets including multi-coil knee, brain, and dynamic cardiac cine MRI. We observe that: i) GLEAM generalizes as well as state-of-the-art memory-efficient baselines such as gradient checkpointing and invertible networks with the same memory footprint, but with 1.3x faster training; ii) for the same memory footprint, GLEAM yields 1.1dB PSNR gain in 2D and 1.8 dB in 3D over end-to-end baselines.

Unraveling Attention via Convex Duality: Analysis and Interpretations of Vision Transformers

Vision transformers using self-attention or its proposed alternatives have demonstrated promising results in many image related tasks. However, the underpinning inductive bias of attention is not well understood. To address this issue, this paper analyzes attention through the lens of convex duality. For the non-linear dot-product self-attention, and alternative mechanisms such as MLP-mixer and Fourier Neural Operator (FNO), we derive equivalent finite-dimensional convex problems that are interpretable and solvable to global optimality. The convex programs lead to {\it block nuclear-norm regularization} that promotes low rank in the latent feature and token dimensions. In particular, we show how self-attention networks implicitly clusters the tokens, based on their latent similarity. We conduct experiments for transferring a pre-trained transformer backbone for CIFAR-100 classification by fine-tuning a variety of convex attention heads. The results indicate the merits of the bias induced by attention compared with the existing MLP or linear heads.

Path Regularization: A Convexity and Sparsity Inducing Regularization for Parallel ReLU Networks

Despite several attempts, the fundamental mechanisms behind the success of deep neural networks still remain elusive. To this end, we introduce a novel analytic framework to unveil hidden convexity in training deep neural networks. We consider a parallel architecture with multiple ReLU sub-networks, which includes many standard deep architectures and ResNets as its special cases. We then show that the training problem with path regularization can be cast as a single convex optimization problem in a high-dimensional space. We further prove that the equivalent convex program is regularized via a group sparsity inducing norm. Thus, a path regularized parallel architecture with ReLU sub-networks can be viewed as a parsimonious feature selection method in high-dimensions. More importantly, we show that the computational complexity required to globally optimize the equivalent convex problem is polynomial-time with respect to the number of data samples and feature dimension. Therefore, we prove exact polynomial-time trainability for path regularized deep ReLU networks with global optimality guarantees. We also provide several numerical experiments corroborating our theory.