Mien Brabeeba Wang

Chi-Ning Chou, Juspreet Singh Sandhu, Mien Brabeeba Wang et al.

One of the challenges in analyzing learning algorithms is the circular entanglement between the objective value and the stochastic noise. This is also known as the "chicken and egg" phenomenon and traditionally, there is no principled way to tackle this issue. People solve the problem by utilizing the special structure of the dynamic, and hence the analysis would be difficult to generalize. In this work, we present a streamlined three-step recipe to tackle the "chicken and egg" problem and give a general framework for analyzing stochastic dynamics in learning algorithms. Our framework composes standard techniques from probability theory, such as stopping time and martingale concentration. We demonstrate the power and flexibility of our framework by giving a unifying analysis for three very different learning problems with the last iterate and the strong uniform high probability convergence guarantee. The problems are stochastic gradient descent for strongly convex functions, streaming principal component analysis, and linear bandit with stochastic gradient descent updates. We either improve or match the state-of-the-art bounds on all three dynamics.

Chi-Ning Chou, Mien Brabeeba Wang

Oja's rule [Oja, Journal of mathematical biology 1982] is a well-known biologically-plausible algorithm using a Hebbian-type synaptic update rule to solve streaming principal component analysis (PCA). Computational neuroscientists have known that this biological version of Oja's rule converges to the top eigenvector of the covariance matrix of the input in the limit. However, prior to this work, it was open to prove any convergence rate guarantee. In this work, we give the first convergence rate analysis for the biological version of Oja's rule in solving streaming PCA. Moreover, our convergence rate matches the information theoretical lower bound up to logarithmic factors and outperforms the state-of-the-art upper bound for streaming PCA. Furthermore, we develop a novel framework inspired by ordinary differential equations (ODE) to analyze general stochastic dynamics. The framework abandons the traditional step-by-step analysis and instead analyzes a stochastic dynamic in one-shot by giving a closed-form solution to the entire dynamic. The one-shot framework allows us to apply stopping time and martingale techniques to have a flexible and precise control on the dynamic. We believe that this general framework is powerful and should lead to effective yet simple analysis for a large class of problems with stochastic dynamics.

Nancy Lynch, Mien Brabeeba Wang

In this paper, we consider networks of deterministic spiking neurons, firing synchronously at discrete times; such spiking neural networks are inspired by networks of neurons and synapses that occur in brains. We consider the problem of translating temporal information into spatial information in such networks, an important task that is carried out by actual brains. Specifically, we define two problems: "First Consecutive Spikes Counting (FCSC)" and "Total Spikes Counting (TSC)", which model spike and rate coding aspects of translating temporal information into spatial information respectively. Assuming an upper bound of $T$ on the length of the temporal input signal, we design two networks that solve these two problems, each using $O(\log T)$ neurons and terminating in time $1$. We also prove that there is no network with less than $T$ neurons that solves either question in time $0$.