We propose a second-order (Hessian or Hessian-free) based optimization method for variational inference inspired by Gaussian backpropagation, and argue that quasi-Newton optimization can be developed as well. This is accomplished by generalizing the gradient computation in stochastic backpropagation via a reparametrization trick with lower complexity. As an illustrative example, we apply this approach to the problems of Bayesian logistic regression and variational auto-encoder (VAE). Additionally, we compute bounds on the estimator variance of intractable expectations for the family of Lipschitz continuous function. Our method is practical, scalable and model free.
You must be thinking, another Backprop from scratch blog? Well kinda yes but I thought this through and came up with something that you can use to tinker around along with easy to understand equations that you usually write down to understand the algorithm. This blog will focus on implementing the Backpropagation algorithm step-by-step on mini-batches of the dataset. There are plenty of tutorials and blogs to demonstrate the backpropagation algorithm in detail and all the logic behind calculus and algebra happening. So I'll skip that part and cut to equations in math and implementation using Python (coz why not).
Spiking neural networks (SNNs) are positioned to enable spatio-temporal information processing and ultra-low power event-driven neuromorphic hardware. However, SNNs are yet to reach the same performances of conventional deep artificial neural networks (ANNs), a long-standing challenge due to complex dynamics and non-differentiable spike events encountered in training. The existing SNN error backpropagation (BP) methods are limited in terms of scalability, lack of proper handling of spiking discontinuities, and/or mismatch between the rate-coded loss function and computed gradient. We present a hybrid macro/micro level backpropagation (HM2-BP) algorithm for training multi-layer SNNs. The temporal effects are precisely captured by the proposed spike-train level post-synaptic potential (S-PSP) at the microscopic level. The rate-coded errors are defined at the macroscopic level, computed and back-propagated across both macroscopic and microscopic levels. Different from existing BP methods, HM2-BP directly computes the gradient of the rate-coded loss function w.r.t tunable parameters. We evaluate the proposed HM2-BP algorithm by training deep fully connected and convolutional SNNs based on the static MNIST  and dynamic neuromorphic N-MNIST . HM2-BP achieves an accuracy level of 99.49% and 98.88% for MNIST and N-MNIST, respectively, outperforming the best reported performances obtained from the existing SNN BP algorithms. Furthermore, the HM2-BP produces the highest accuracies based on SNNs for the EMNIST  dataset, and leads to high recognition accuracy for the 16-speaker spoken English letters of TI46 Corpus , a challenging patio-temporal speech recognition benchmark for which no prior success based on SNNs was reported. It also achieves competitive performances surpassing those of conventional deep learning models when dealing with asynchronous spiking streams.
Neural network learning is typically slow since backpropagation needs to compute full gradients and backpropagate them across multiple layers. Despite its success of existing work in accelerating propagation through sparseness, the relevant theoretical characteristics remain unexplored and we empirically find that they suffer from the loss of information contained in unpropagated gradients. To tackle these problems, in this work, we present a unified sparse backpropagation framework and provide a detailed analysis of its theoretical characteristics. Analysis reveals that when applied to a multilayer perceptron, our framework essentially performs gradient descent using an estimated gradient similar enough to the true gradient, resulting in convergence in probability under certain conditions. Furthermore, a simple yet effective algorithm named memorized sparse backpropagation (MSBP) is proposed to remedy the problem of information loss by storing unpropagated gradients in memory for the next learning. The experiments demonstrate that the proposed MSBP is able to effectively alleviate the information loss in traditional sparse backpropagation while achieving comparable acceleration.