More Like this

1. Convergence of a Relaxed Variable Splitting Method for Learning Sparse Neural Networks via L1, L0, and transformed-L1 Penalties

   Dinh, Thu; Xin, Jack (September 2020, Intelligent Systems Conference (IntelliSys))

   Sparsification of neural networks is one of the effective complexity reduction methods to improve efficiency and generalizability. We consider the problem of learning a one hidden layer convolutional neural network with ReLU activation function via gradient descent under sparsity promoting penalties. It is known that when the input data is Gaussian distributed, no-overlap networks (without penalties) in regression problems with ground truth can be learned in polynomial time at high probability. We propose a relaxed variable splitting method integrating thresholding and gradient descent to overcome the non-smoothness in the loss function. The sparsity in network weight is realized during the optimization (training) process. We prove that under L1, L0, and transformed-L1 penalties, no-overlap networks can be learned with high probability, and the iterative weights converge to a global limit which is a transformation of the true weight under a novel thresholding operation. Numerical experiments confirm theoretical findings, and compare the accuracy and sparsity trade-off among the penalties. 

   more » « less
2. Convex Geometry and Duality of Over-parameterized Neural Networks

   Ergen, Tolga; Pilanci, Mert (January 2000, International Conference on Artificial Intelligence and Statistics (AISTATS 2020))

   null (Ed.)

   We develop a convex analytic framework for ReLU neural networks which elucidates the inner workings of hidden neurons and their function space characteristics. We show that neural networks with rectified linear units act as convex regularizers, where simple solutions are encouraged via extreme points of a certain convex set. For one dimensional regression and classification, as well as rank-one data matrices, we prove that finite two-layer ReLU networks with norm regularization yield linear spline interpolation. We characterize the classification decision regions in terms of a closed form kernel matrix and minimum L1 norm solutions. This is in contrast to Neural Tangent Kernel which is unable to explain neural network predictions with finitely many neurons. Our convex geometric description also provides intuitive explanations of hidden neurons as auto encoders. In higher dimensions, we show that the training problem for two-layer networks can be cast as a finite dimensional convex optimization problem with infinitely many constraints. We then provide a family of convex relaxations to approximate the solution, and a cutting-plane algorithm to improve the relaxations. We derive conditions for the exactness of the relaxations and provide simple closed form formulas for the optimal neural network weights in certain cases. We also establish a connection to ℓ0-ℓ1 equivalence for neural networks analogous to the minimal cardinality solutions in compressed sensing. Extensive experimental results show that the proposed approach yields interpretable and accurate models. 

   more » « less
3. Convex Geometry and Duality of Over-parameterized Neural Networks

   Ergen, T.; Pilanci, M. (January 2021, Journal of machine learning research)

   We develop a convex analytic approach to analyze finite width two-layer ReLU networks. We first prove that an optimal solution to the regularized training problem can be characterized as extreme points of a convex set, where simple solutions are encouraged via its convex geometrical properties. We then leverage this characterization to show that an optimal set of parameters yield linear spline interpolation for regression problems involving one dimensional or rank-one data. We also characterize the classification decision regions in terms of a kernel matrix and minimum `1-norm solutions. This is in contrast to Neural Tangent Kernel which is unable to explain predictions of finite width networks. Our convex geometric characterization also provides intuitive explanations of hidden neurons as auto-encoders. In higher dimensions, we show that the training problem can be cast as a finite dimensional convex problem with infinitely many constraints. Then, we apply certain convex relaxations and introduce a cutting-plane algorithm to globally optimize the network. We further analyze the exactness of the relaxations to provide conditions for the convergence to a global optimum. Our analysis also shows that optimal network parameters can be also characterized as interpretable closed-form formulas in some practically relevant special cases. 

   more » « less
4. Convergence of a Relaxed Variable Splitting Coarse Gradient Descent Method for Learning Sparse Weight Binarized Activation Neural Network

   https://doi.org/DOI:10.3389/fams.2020.00013  

   Dinh, Thu; Xin, Jack (May 2020, Frontiers in applied mathematics and statistics)

   Sparsification of neural networks is one of the effective complexity reduction methods to improve efficiency and generalizability. Binarized activation offers an additional computational saving for inference. Due to vanishing gradient issue in training networks with binarized activation, coarse gradient (a.k.a. straight through estimator) is adopted in practice. In this paper, we study the problem of coarse gradient descent (CGD) learning of a one hidden layer convolutional neural network (CNN) with binarized activation function and sparse weights. It is known that when the input data is Gaussian distributed, no-overlap one hidden layer CNN with ReLU activation and general weight can be learned by GD in polynomial time at high probability in regression problems with ground truth. We propose a relaxed variable splitting method integrating thresholding and coarse gradient descent. The sparsity in network weight is realized through thresholding during the CGD training process. We prove that under thresholding of L1, L0, and transformed-L1 penalties, no-overlap binary activation CNN can be learned with high probability, and the iterative weights converge to a global limit which is a transformation of the true weight under a novel sparsifying operation. We found explicit error estimates of sparse weights from the true weights. 

   more » « less
5. Provable Complexity Improvement of AdaGrad over SGD: Upper and Lower Bounds in Stochastic Non-Convex Optimization

   Jiang, R; Maladkar, D; Mokhtari, A (June 2025, https://doi.org/10.48550/arXiv.2406.04592)

   Adaptive gradient methods, such as AdaGrad, are among the most successful optimization algorithms for neural network training. While these methods are known to achieve better dimensional dependence than stochastic gradient descent (SGD) for stochastic convex optimization under favorable geometry, the theoretical justification for their success in stochastic non-convex optimization remains elusive. In fact, under standard assumptions of Lipschitz gradients and bounded noise variance, it is known that SGD is worst-case optimal in terms of finding a near-stationary point with respect to the l2-norm, making further improvements impossible. Motivated by this limitation, we introduce refined assumptions on the smoothness structure of the objective and the gradient noise variance, which better suit the coordinate-wise nature of adaptive gradient methods. Moreover, we adopt the l1-norm of the gradient as the stationarity measure, as opposed to the standard l2-norm, to align with the coordinate-wise analysis and obtain tighter convergence guarantees for AdaGrad. Under these new assumptions and the l1-norm stationarity measure, we establish an upper bound on the convergence rate of AdaGrad and a corresponding lower bound for SGD. In particular, we identify non-convex settings in which the iteration complexity of AdaGrad is favorable over SGD and show that, for certain configurations of problem parameters, it outperforms SGD by a factor of d, where d is the problem dimension. To the best of our knowledge, this is the first result to demonstrate a provable gain of adaptive gradient methods over SGD in a non-convex setting. We also present supporting lower bounds, including one specific to AdaGrad and one applicable to general deterministic first-order methods, showing that our upper bound for AdaGrad is tight and unimprovable up to a logarithmic factor under certain conditions. 

   more » « less