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1. 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. 

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2. Low-Energy Deep Belief Networks using Intrinsic Sigmoidal Spintronic-based Probabilistic Neurons

   Zand, R.; Camsari, K. Y.; Pyle, S. D.; Ahmed, I.; Kim, C. H. (May 2018, ACM Great Lakes Symposium on VLSI)

   A low-energy hardware implementation of deep belief network (DBN) architecture is developed using near-zero energy barrier probabilistic spin logic devices (p-bits), which are modeled to real- ize an intrinsic sigmoidal activation function. A CMOS/spin based weighted array structure is designed to implement a restricted Boltzmann machine (RBM). Device-level simulations based on precise physics relations are used to validate the sigmoidal relation between the output probability of a p-bit and its input currents. Characteristics of the resistive networks and p-bits are modeled in SPICE to perform a circuit-level simulation investigating the performance, area, and power consumption tradeoffs of the weighted array. In the application-level simulation, a DBN is implemented in MATLAB for digit recognition using the extracted device and circuit behavioral models. The MNIST data set is used to assess the accuracy of the DBN using 5,000 training images for five distinct network topologies. The results indicate that a baseline error rate of 36.8% for a 784x10 DBN trained by 100 samples can be reduced to only 3.7% using a 784x800x800x10 DBN trained by 5,000 input samples. Finally, Power dissipation and accuracy tradeoffs for probabilistic computing mechanisms using resistive devices are identified. 

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3. Understanding straight-through estimator in training activation quantized neural nets

   Yin, P; Lyu, J; Zhang, S; Osher, S; Qi, Y-Y; Xin, J (April 2019, International Conference on Learning Representations)

   Training activation quantized neural networks involves minimizing a piecewise constant function whose gradient vanishes almost everywhere, which is undesirable for the standard back-propagation or chain rule. An empirical way around this issue is to use a straight-through estimator (STE) (Bengio et al., 2013) in the backward pass only, so that the “gradient” through the modified chain rule becomes non-trivial. Since this unusual “gradient” is certainly not the gradient of loss function, the following question arises: why searching in its negative direction minimizes the training loss? In this paper, we provide the theoretical justification of the concept of STE by answering this question. We consider the problem of learning a two-linear-layer network with binarized ReLU activation and Gaussian input data. We shall refer to the unusual “gradient” given by the STE-modifed chain rule as coarse gradient. The choice of STE is not unique. We prove that if the STE is properly chosen, the expected coarse gradient correlates positively with the population gradient (not available for the training), and its negation is a descent direction for minimizing the population loss. We further show the associated coarse gradient descent algorithm converges to a critical point of the population loss minimization problem. Moreover, we show that a poor choice of STE leads to instability of the training algorithm near certain local minima, which is verified with CIFAR-10 experiments. 

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4. A Robust Backpropagation-Free Framework for Images

   Zee, Timothy; Ororbia, Alexander G; Mali, Ankur; Nwogu, Ifeoma (November 2023, Transactions on machine learning research)

   Richards, Blake A (Ed.)

   While current deep learning algorithms have been successful for a wide variety of artificial intelligence (AI) tasks, including those involving structured image data, they present deep neurophysiological conceptual issues due to their reliance on the gradients that are computed by backpropagation of errors (backprop). Gradients are required to obtain synaptic weight adjustments but require knowledge of feed-forward activities in order to conduct backward propagation, a biologically implausible process. This is known as the “weight transport problem”. Therefore, in this work, we present a more biologically plausible approach towards solving the weight transport problem for image data. This approach, which we name the error-kernel driven activation alignment (EKDAA) algorithm, accomplishes through the introduction of locally derived error transmission kernels and error maps. Like standard deep learning networks, EKDAA performs the standard forward process via weights and activation functions; however, its backward error computation involves adaptive error kernels that propagate local error signals through the network. The efficacy of EKDAA is demonstrated by performing visual-recognition tasks on the Fashion MNIST, CIFAR-10 and SVHN benchmarks, along with demonstrating its ability to extract visual features from natural color images. Furthermore, in order to demonstrate its non-reliance on gradient computations, results are presented for an EKDAA-trained CNN that employs a non-differentiable activation function. 

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5. Superpolynomial Lower Bounds for Learning One-Layer Neural Networks using Gradient Descent

   Goel, Surbhi; Gollakota, Aravind; Jin, Zhihan; Karmalkar, Sushrut; Klivans, Adam (July 2020, International Conference on Machine Learning)

   We prove the first superpolynomial lower bounds for learning one-layer neural networks with respect to the Gaussian distribution using gradient descent. We show that any classifier trained using gradient descent with respect to square-loss will fail to achieve small test error in polynomial time given access to samples labeled by a one-layer neural network. For classification, we give a stronger result, namely that any statistical query (SQ) algorithm (including gradient descent) will fail to achieve small test error in polynomial time. Prior work held only for gradient descent run with small batch sizes, required sharp activations, and applied to specific classes of queries. Our lower bounds hold for broad classes of activations including ReLU and sigmoid. The core of our result relies on a novel construction of a simple family of neural networks that are exactly orthogonal with respect to all spherically symmetric distributions. 

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