As neural networks have begun performing increasingly critical tasks for society, ranging from driving cars to identifying candidates for drug development, the value of their ability to perform uncertainty quantification (UQ) in their predictions has risen commensurately. Permanent dropout, a popular method for neural network UQ, involves injecting stochasticity into the inference phase of the model and creating many predictions for each of the test data. This shifts the computational and energy burden of deep neural networks from the training phase to the inference phase. Recent work has demonstrated near-lossless conversion of classical deep neural networks to their spiking counterparts. We use these results to demonstrate the feasibility of conducting the inference phase with permanent dropout on spiking neural networks, mitigating the technique's computational and energy burden, which is essential for its use at scale or on edge platforms. We demonstrate the proposed approach via the Nengo spiking neural simulator on a combination drug therapy dataset for cancer treatment, where UQ is critical. Our results indicate that the spiking approximation gives a predictive distribution practically indistinguishable from that given by the classical network.
where @math . Unfortunately, this function is not continuously differentiable, complicating gradient-based optimization methods. To resolve this issue, Hunsberger and Eliasmith @cite_20 suggest replacing @math with a smooth approximation given by
@cite_20
2017938640 2535496140 2963116072 1988795359
We consider incrementally updated gradient methods for minimizing the sum of smooth functions and a convex function. This method can use a (sufficiently small) constant stepsize or, more practically, an adaptive stepsize that is decreased whenever sufficient progress is not made. We show that if the gradients of the smooth functions are Lipschitz continuous on the space of n-dimensional real column vectors or the gradients of the smooth functions are bounded and Lipschitz continuous over a certain level set and the convex function is Lipschitz continuous on its domain, then every cluster point of the iterates generated by the method is a stationary point. If in addition a local Lipschitz error bound assumption holds, then the method is linearly convergent. The usual approach to developing and analyzing first-order methods for smooth convex optimization assumes that the gradient of the objective function is uniformly smooth with some Lipschitz constant @math . However, in many settings the differentiable convex function @math is not uniformly smooth---for example, in @math -optimal design where @math and @math , or even the univariate setting with @math . In this paper we develop a notion of “relative smoothness” and relative strong convexity that is determined relative to a user-specified “reference function” @math (that should be computationally tractable for algorithms), and we show that many differentiable convex functions are relatively smooth with respect to a correspondingly fairly simple reference function @math . We extend two standard algorithms---the primal gradient scheme and the dual averaging scheme---to our new setting, with associated computational guarantees. We apply our new a... In this paper, we study the problem of maximizing continuous submodular functions that naturally arise in many learning applications such as those involving utility functions in active learning and sensing, matrix approximations and network inference. Despite the apparent lack of convexity in such functions, we prove that stochastic projected gradient methods can provide strong approximation guarantees for maximizing continuous submodular functions with convex constraints. More specifically, we prove that for monotone continuous DR-submodular functions, all fixed points of projected gradient ascent provide a factor @math approximation to the global maxima. We also study stochastic gradient methods and show that after @math iterations these methods reach solutions which achieve in expectation objective values exceeding @math . An immediate application of our results is to maximize submodular functions that are defined stochastically, i.e. the submodular function is defined as an expectation over a family of submodular functions with an unknown distribution. We will show how stochastic gradient methods are naturally well-suited for this setting, leading to a factor @math approximation when the function is monotone. In particular, it allows us to approximately maximize discrete, monotone submodular optimization problems via projected gradient ascent on a continuous relaxation, directly connecting the discrete and continuous domains. Finally, experiments on real data demonstrate that our projected gradient methods consistently achieve the best utility compared to other continuous baselines while remaining competitive in terms of computational effort. An incremental aggregated gradient method for minimizing a sum of continuously differentiable functions is presented. The method requires a single gradient evaluation per iteration and uses a constant step size. For the case that the gradient is bounded and Lipschitz continuous, we show that the method visits infinitely often regions in which the gradient is small. Under certain unimodality assumptions, global convergence is established. In the quadratic case, a global linear rate of convergence is shown. The method is applied to distributed optimization problems arising in wireless sensor networks, and numerical experiments compare the new method with other incremental gradient methods.