QSpace Collection:
http://hdl.handle.net/1974/758
Wed, 02 Sep 2015 04:27:45 GMT2015-09-02T04:27:45ZDistributed Online Optimization on time-varying networks
http://hdl.handle.net/1974/13551
Title: Distributed Online Optimization on time-varying networks
Authors: Akbari Varnousfaderani, Mohammad Jr
Abstract: This thesis introduces two classes of discrete-time distributed online optimization algorithms, with a group of agents which communicate over a network. At each time, a private convex objective function is revealed to each agent. In the next time step, each agent updates its state using its own objective function and the information gathered from its immediate in-neighbours at that time. We design algorithms distributed over the network topologies, which guarantee that the individual regret, the diﬀerence between the network cost incurred by the agent’s states estimation and the cost incurred by the best ﬁxed choice, grows only sublinearly. One algorithm is based on gradient-ﬂow, which provably works for a sequence of time-varying uniformly strongly connected graphs. The other one is based on Alternating Direction Method of Multipliers, which works on ﬁxed undirected graphs and gives an explicit
regret bound in terms of the size of the network. We implement the proposed algorithms on a sensor network and the results demonstrate the good performance for both algorithms.
Description: Thesis (Master, Mathematics & Statistics) -- Queen's University, 2015-08-31 09:59:41.261Tue, 01 Sep 2015 04:00:00 GMThttp://hdl.handle.net/1974/135512015-09-01T04:00:00ZCurves of low genus on surfaces and applications to Diophantine problems
http://hdl.handle.net/1974/13545
Title: Curves of low genus on surfaces and applications to Diophantine problems
Authors: Garcia, Natalia
Abstract: We describe in detail a technique due to Vojta for finding the explicit set of curves of low genus on certain algebraic surfaces of general type, and refine some of its aspects. We then provide applications of this method to three Diophantine problems.
We prove under the Bombieri-Lang Conjecture that there are finitely many non-trivial sequences of integers of length 11 whose squares have constant second differences, and we prove unconditionally the analogous result for function fields of characteristic zero.
We prove under the Bombieri-Lang Conjecture that there are finitely many integer sequences of length 8 whose k-th powers have second differences equal to 2, we give an unconditional result for function fields of characteristic zero. Moreover, this gives new examples of surfaces having no curves of genus 0 or 1.
The third application is related to the surface parametrizing perfect cuboids. We give some new properties about their curves of genus 0 or 1 and we give new bounds for the degree of curves in this surface, in terms of their genus.
Description: Thesis (Ph.D, Mathematics & Statistics) -- Queen's University, 2015-08-28 10:03:04.056Mon, 31 Aug 2015 04:00:00 GMThttp://hdl.handle.net/1974/135452015-08-31T04:00:00ZOptimality of Walrand-Varaiya Type Policies and Approximation Results for Zero-Delay Coding of Markov Sources
http://hdl.handle.net/1974/13457
Title: Optimality of Walrand-Varaiya Type Policies and Approximation Results for Zero-Delay Coding of Markov Sources
Authors: Wood, RICHARD
Abstract: Optimal zero-delay coding of a finite state Markov source through quantization is considered. Building on previous literature, the existence and structure of optimal policies are studied using a stochastic control problem formulation. In the literature, the optimality of deterministic Markov coding policies (or Walrand-Varaiya type policies) for infinite horizon problems has been established. This work expands on this result for systems with finite source alphabets, proving the optimality of de- terministic and stationary Markov coding policies for the infinite horizon setup. In addition, the ε-optimality of finite memory quantizers is established and the depen- dence between the memory length and ε is quantified. An algorithm to find the optimal policy for the finite time horizon problem is presented. Numerical results produced using this algorithm are shown.
Description: Thesis (Master, Mathematics & Statistics) -- Queen's University, 2015-07-27 15:57:18.667Tue, 28 Jul 2015 04:00:00 GMThttp://hdl.handle.net/1974/134572015-07-28T04:00:00ZOptimal Quantization and Approximation in Source Coding and Stochastic Control
http://hdl.handle.net/1974/13147
Title: Optimal Quantization and Approximation in Source Coding and Stochastic Control
Authors: Saldi, NACI
Abstract: This thesis deals with non-standard optimal quantization and approximation problems in source coding and stochastic control.
The first part of the thesis considers randomized quantization. Adapted from stochastic control, a general representation of randomized quantizers that is probabilistically equivalent to common models in the literature is proposed via mixtures of joint probability measures induced by deterministic quantizers. Using this general model, we prove the existence of an optimal randomized quantizer for the generalized distribution preserving quantization problem. A Shannon theoretic version of this source coding problem is also considered, in which an optimal (minimum distortion) coding of stationary and memoryless source is studied under the requirement that the quantizer's output distribution also be stationary and memoryless possibly different than source distribution. We provide a characterization of the achievable rate region where the rate region includes both the coding rate and the rate of common randomness shared between the encoder and the decoder.
In the second part of the thesis, we consider the quantization problems in stochastic control from viewpoints of information transmission and computation. The first problem studies the finite-action approximation (via quantization of the action space) of deterministic stationary policies of a discrete time Markov decision process (MDP), while the second problem considers finite-state approximations (via quantization of the state space) of discrete time Markov decision process. Under certain continuity conditions on the components of the MDP, we establish that optimal policies for the finite models can approximate with arbitrary precision optimal deterministic stationary policies for the original MDP. Combining these results leads to a constructive scheme for obtaining near optimal solutions via well known algorithms developed for finite state/action MDPs. For both problems, we also obtain explicit bounds on the approximation error in terms of the number of representation points in the quantizer, under further conditions.
Description: Thesis (Ph.D, Mathematics & Statistics) -- Queen's University, 2015-06-19 12:20:57.086Tue, 23 Jun 2015 04:00:00 GMThttp://hdl.handle.net/1974/131472015-06-23T04:00:00Z