Georgia Institute of Technology College of Sciences

The Shadowing lemma describes the behaviour of pseudo-orbits near a hyperbolic invariant set. In this talk, I will present an analytic proof of the shadowing lemma for discrete flows. This is a work by K. R. Meyer and George R. Sell.

Abstract In this talk, I will present a popular distributed method, namely, distributed consensus-based gradient (DCG) method, for solving optimal learning problems over a network of agents. Such problems arise in many applications such as, finding optimal parameters over a large dataset distributed among a network of processors or seeking an optimal policy for coverage control problems in robotic networks. The focus is to present our recent results, where we study the performance of DCG when the agents are only allowed to exchange their quantized values due to their finite communication bandwidth. In particular, we develop a novel quantization method, which we refer to as adaptive quantization. The main idea of our approach is to quantize the nodes' estimates based on the progress of the algorithm, which helps to eliminate the quantized errors. Under the adaptive quantization, we then derive the bounds on the convergence rates of the proposed method as a function of the bandwidths and the underlying network topology, for both convex and strongly convex objective functions. Our results suggest that under the adaptive quantization, the rate of convergence of DCG with and without quantization are the same, except for a factor which captures the number of quantization bits. To the best of the authors’ knowledge, the results in this paper are considered better than any existing results for DCG under quantization. This is based on a joint work with Siva Theja Maguluri and Justin Romberg. Bio Thinh T. Doan is a TRIAD postdoctoral fellow at Georgia Institute of Technology, joint between the School of Industrial and Systems Engineering and the School of Electrical and Computer Engineering (ECE). He was born in Vietnam, where he got his Bachelor degree in Automatic Control at Hanoi University of Science and Technology in 2008. He obtained his Master and Ph.D. degrees both in ECE from the University of Oklahoma in 2013 and the University of Illinois at Urbana-Champaign in 2018, respectively. His research interests lie at the intersection of control theory, optimization, distributed algorithms, and applied probability, with the main applications in machine learning, reinforcement learning, power networks, and multi-agent systems.

A graph G is H-free if H is not isomorphic to an induced subgraph of G. Let Pt denote the path on t vertices, and let Kn denote the complete graph on n vertices. For a positive integer r, we use rG to denote the disjoint union of r copies of G. In this talk, we will discuss the result, by Gaspers and Huang, that (2P2, K4)-free graphs are 4-colorable, where the bound is attained by the five-wheel and the complement of seven-cycle. It answers an open question by Wagon in 1980s.

Alesker has introduced the notion of a smooth valuation on a smooth manifold M. This is a special kind of set function, defined on sufficiently regular compact subsets A of M, extending the corresponding idea from classical convexity theory. Formally, a smooth valuation is a kind of curvature integral; informally, it is a sum of Euler characteristics of intersections of A with a collection of objects B. Smooth valuations admit a natural multiplication, again due to Alesker. I will aim to explain the rather abstruse formal definition of this multiplication, and its relation to the ridiculously simple informal counterpart given by intersections of the objects B.