We present DANTE, a novel method for training neural networks using the alternating minimization principle. DANTE provides an alternate perspective to traditional gradient-based backpropagation techniques commonly used to train deep networks. It utilizes an adaptation of quasi-convexity to cast training a neural network as a bi-quasi-convex optimization problem. We show that for neural network configurations with both differentiable (e.g. sigmoid) and non-differentiable (e.g. ReLU) activation functions, we can perform the alternations very effectively. DANTE can also be extended to networks with multiple hidden layers. In experiments on standard datasets, neural networks trained using the proposed method were found to be very promising and competitive to traditional backpropagation techniques, both in terms of quality of the solution, as well as training speed.
In recent years, Taylor proposed a method to train neural networks using the Alternating Direction Method of Multipliers (ADMM) and Bregman iterations @cite_2 . The focus of this method, however, was on scaling the training of neural networks to a distributed setting on multiple cores across a computing cluster. Jaderberg also proposed the idea of 'synthetic gradients' in @cite_11 . While this approach is interesting, this work is more focused towards a more efficient way to carry out gradient-based parameter updates in a neural network. More recently, Jagatap and Hegde @cite_25 proposed a method to train single hidden layer ReLU networks using an alternating minimization technique. Unlike our method, this method alternates between updating weights, and state variables which indicate which ReLU activations are on, and so is very specific to ReLU activations.
@cite_11 @cite_25 @cite_2
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We propose and analyze a new family of algorithms for training neural networks with ReLU activations. Our algorithms are based on the technique of alternating minimization: estimating the activation patterns of each ReLU for all given samples, interleaved with weight updates via a least-squares step. The main focus of our paper are 1-hidden layer networks with @math hidden neurons and ReLU activation. We show that under standard distributional assumptions on the @math dimensional input data, our algorithm provably recovers the true ground truth' parameters in a linearly convergent fashion. This holds as long as the weights are sufficiently well initialized; furthermore, our method requires only @math samples. We also analyze the special case of 1-hidden layer networks with skipped connections, commonly used in ResNet-type architectures, and propose a novel initialization strategy for the same. For ReLU based ResNet type networks, we provide the first linear convergence guarantee with an end-to-end algorithm. We also extend this framework to deeper networks and empirically demonstrate its convergence to a global minimum. With the growing importance of large network models and enormous training datasets, GPUs have become increasingly necessary to train neural networks. This is largely because conventional optimization algorithms rely on stochastic gradient methods that don't scale well to large numbers of cores in a cluster setting. Furthermore, the convergence of all gradient methods, including batch methods, suffers from common problems like saturation effects, poor conditioning, and saddle points. This paper explores an unconventional training method that uses alternating direction methods and Bregman iteration to train networks without gradient descent steps. The proposed method reduces the network training problem to a sequence of minimization substeps that can each be solved globally in closed form. The proposed method is advantageous because it avoids many of the caveats that make gradient methods slow on highly nonconvex problems. The method exhibits strong scaling in the distributed setting, yielding linear speedups even when split over thousands of cores. One of the mysteries in the success of neural networks is randomly initialized first order methods like gradient descent can achieve zero training loss even though the objective function is non-convex and non-smooth. This paper demystifies this surprising phenomenon for two-layer fully connected ReLU activated neural networks. For an @math hidden node shallow neural network with ReLU activation and @math training data, we show as long as @math is large enough and no two inputs are parallel, randomly initialized gradient descent converges to a globally optimal solution at a linear convergence rate for the quadratic loss function. Our analysis relies on the following observation: over-parameterization and random initialization jointly restrict every weight vector to be close to its initialization for all iterations, which allows us to exploit a strong convexity-like property to show that gradient descent converges at a global linear rate to the global optimum. We believe these insights are also useful in analyzing deep models and other first order methods. As an integral component of blind image deblurring, non-blind deconvolution removes image blur with a given blur kernel, which is essential but difficult due to the ill-posed nature of the inverse problem. The predominant approach is based on optimization subject to regularization functions that are either manually designed, or learned from examples. Existing learning based methods have shown superior restoration quality but are not practical enough due to their restricted model design. They solely focus on learning a prior and require to know the noise level for deconvolution. We address the gap between the optimization-based and learning-based approaches by learning an optimizer. We propose a Recurrent Gradient Descent Network (RGDN) by systematically incorporating deep neural networks into a fully parameterized gradient descent scheme. A parameter-free update unit is used to generate updates from the current estimates, based on a convolutional neural network. By training on diverse examples, the Recurrent Gradient Descent Network learns an implicit image prior and a universal update rule through recursive supervision. Extensive experiments on synthetic benchmarks and challenging real-world images demonstrate that the proposed method is effective and robust to produce favorable results as well as practical for real-world image deblurring applications.