Asynchronous Distributed Optimization over Lossy Networks via Relaxed ADMM: Stability and Linear Convergence

In this work we focus on the problem of minimizing the sum of convex cost functions in a distributed fashion over a peer-to-peer network. In particular, we are interested in the case in which communications between nodes are prone to failures and the agents are not synchronized among themselves. We address the problem proposing a modified version of the relaxed ADMM, which corresponds to the Peaceman-Rachford splitting method applied to the dual. By exploiting results from operator theory, we are able to prove the almost sure convergence of the proposed algorithm under general assumptions on the distribution of communication loss and node activation events. By further assuming the cost functions to be strongly convex, we prove the linear convergence of the algorithm in mean to a neighborhood of the optimal solution, and provide an upper bound to the convergence rate. Finally, we present numerical results testing the proposed method in different scenarios.Comment: To appear in IEEE Transactions on Automatic Contro

Distributed Convex Optimisation using the Alternating Direction Method of Multipliers (ADMM) in Lossy Scenarios

The Alternating Direction Method of Multipliers (ADMM) is an extensively studied algorithm suitable for solving convex distributed optimisation problems. This Thesis presents a formulation of the ADMM that is guaranteed to converge if the communications among agents are faulty and the agents perform updates asynchronously. With strongly convex costs, the proposed algorithm is shown to converge exponentially fast. The further extension to partition-based problems is presented

Distributed Optimisation with Linear Equality and Inequality Constraints using PDMM

In this paper, we consider the problem of distributed optimisation of a separable convex cost function over a graph, where every edge and node in the graph could carry both linear equality and/or inequality constraints. We show how to modify the primal-dual method of multipliers (PDMM), originally designed for linear equality constraints, such that it can handle inequality constraints as well. In contrast to most existing algorithms for optimisation with inequality constraints, the proposed algorithm does not need any slack variables. Using convex analysis, monotone operator theory and fixed-point theory, we show how to derive the update equations of the modified PDMM algorithm by applying Peaceman-Rachford splitting to the monotonic inclusion related to the extended dual problem. To incorporate the inequality constraints, we impose a non-negativity constraint on the associated dual variables. This additional constraint results in the introduction of a reflection operator to model the data exchange in the network, instead of a permutation operator as derived for equality constraint PDMM. Convergence for both synchronous and stochastic update schemes of PDMM are provided. The latter includes asynchronous update schemes and update schemes with transmission losses.Comment: 9 page

Analysis of distributed ADMM algorithm for consensus optimization in presence of error

ADMM is a popular algorithm for solving convex optimization problems. Applying this algorithm to distributed consensus optimization problem results in a fully distributed iterative solution which relies on processing at the nodes and communication between neighbors. Local computations usually suffer from different types of errors, due to e.g., observation or quantization noise, which can degrade the performance of the algorithm. In this work, we focus on analyzing the convergence behavior of distributed ADMM for consensus optimization in presence of additive node error. We specifically show that (a noisy) ADMM converges linearly under certain conditions and also examine the associated convergence point. Numerical results are provided which demonstrate the effectiveness of the presented analysis

Robust and cheating-resilient power auctioning on Resource Constrained Smart Micro-Grids

The principle of Continuous Double Auctioning (CDA) is known to provide an efficient way of matching supply and demand among distributed selfish participants with limited information. However, the literature indicates that the classic CDA algorithms developed for grid-like applications are centralised and insensitive to the processing resources capacity, which poses a hindrance for their application on resource constrained, smart micro-grids (RCSMG). A RCSMG loosely describes a micro-grid with distributed generators and demand controlled by selfish participants with limited information, power storage capacity and low literacy, communicate over an unreliable infrastructure burdened by limited bandwidth and low computational power of devices. In this thesis, we design and evaluate a CDA algorithm for power allocation in a RCSMG. Specifically, we offer the following contributions towards power auctioning on RCSMGs. First, we extend the original CDA scheme to enable decentralised auctioning. We do this by integrating a token-based, mutual-exclusion (MUTEX) distributive primitive, that ensures the CDA operates at a reasonably efficient time and message complexity of O(N) and O(logN) respectively, per critical section invocation (auction market execution). Our CDA algorithm scales better and avoids the single point of failure problem associated with centralised CDAs (which could be used to adversarially provoke a break-down of the grid marketing mechanism). In addition, the decentralised approach in our algorithm can help eliminate privacy and security concerns associated with centralised CDAs. Second, to handle CDA performance issues due to malfunctioning devices on an unreliable network (such as a lossy network), we extend our proposed CDA scheme to ensure robustness to failure. Using node redundancy, we modify the MUTEX protocol supporting our CDA algorithm to handle fail-stop and some Byzantine type faults of sites. This yields a time complexity of O(N), where N is number of cluster-head nodes; and message complexity of O((logN)+W) time, where W is the number of check-pointing messages. These results indicate that it is possible to add fault tolerance to a decentralised CDA, which guarantees continued participation in the auction while retaining reasonable performance overheads. In addition, we propose a decentralised consumption scheduling scheme that complements the auctioning scheme in guaranteeing successful power allocation within the RCSMG. Third, since grid participants are self-interested we must consider the issue of power theft that is provoked when participants cheat. We propose threat models centred on cheating attacks aimed at foiling the extended CDA scheme. More specifically, we focus on the Victim Strategy Downgrade; Collusion by Dynamic Strategy Change, Profiling with Market Prediction; and Strategy Manipulation cheating attacks, which are carried out by internal adversaries (auction participants). Internal adversaries are participants who want to get more benefits but have no interest in provoking a breakdown of the grid. However, their behaviour is dangerous because it could result in a breakdown of the grid. Fourth, to mitigate these cheating attacks, we propose an exception handling (EH) scheme, where sentinel agents use allocative efficiency and message overheads to detect and mitigate cheating forms. Sentinel agents are tasked to monitor trading agents to detect cheating and reprimand the misbehaving participant. Overall, message complexity expected in light demand is O(nLogN). The detection and resolution algorithm is expected to run in linear time complexity O(M). Overall, the main aim of our study is achieved by designing a resilient and cheating-free CDA algorithm that is scalable and performs well on resource constrained micro-grids. With the growing popularity of the CDA and its resource allocation applications, specifically to low resourced micro-grids, this thesis highlights further avenues for future research. First, we intend to extend the decentralised CDA algorithm to allow for participants’ mobile phones to connect (reconnect) at different shared smart meters. Such mobility should guarantee the desired CDA properties, the reliability and adequate security. Secondly, we seek to develop a simulation of the decentralised CDA based on the formal proofs presented in this thesis. Such a simulation platform can be used for future studies that involve decentralised CDAs. Third, we seek to find an optimal and efficient way in which the decentralised CDA and the scheduling algorithm can be integrated and deployed in a low resourced, smart micro-grid. Such an integration is important for system developers interested in exploiting the benefits of the two schemes while maintaining system efficiency. Forth, we aim to improve on the cheating detection and mitigation mechanism by developing an intrusion tolerance protocol. Such a scheme will allow continued auctioning in the presence of cheating attacks while incurring low performance overheads for applicability in a RCSMG

Distributed Stochastic Subgradient Optimization Algorithms Over Random and Noisy Networks

We study distributed stochastic optimization by networked nodes to cooperatively minimize a sum of convex cost functions. The network is modeled by a sequence of time-varying random digraphs with each node representing a local optimizer and each edge representing a communication link. We consider the distributed subgradient optimization algorithm with noisy measurements of local cost functions' subgradients, additive and multiplicative noises among information exchanging between each pair of nodes. By stochastic Lyapunov method, convex analysis, algebraic graph theory and martingale convergence theory, it is proved that if the local subgradient functions grow linearly and the sequence of digraphs is conditionally balanced and uniformly conditionally jointly connected, then proper algorithm step sizes can be designed so that all nodes' states converge to the global optimal solution almost surely

Improving the Practicality of Model-Based Reinforcement Learning: An Investigation into Scaling up Model-Based Methods in Online Settings

This thesis is a response to the current scarcity of practical model-based control algorithms in the reinforcement learning (RL) framework. As of yet there is no consensus on how best to integrate imperfect transition models into RL whilst mitigating policy improvement instabilities in online settings. Current state-of-the-art policy learning algorithms that surpass human performance often rely on model-free approaches that enjoy unmitigated sampling of transition data. Model-based RL (MBRL) instead attempts to distil experience into transition models that allow agents to plan new policies without needing to return to the environment and sample more data. The initial focus of this investigation is on kernel conditional mean embeddings (CMEs) (Song et al., 2009) deployed in an approximate policy iteration (API) algorithm (Grünewälder et al., 2012a). This existing MBRL algorithm boasts theoretically stable policy updates in continuous state and discrete action spaces. The Bellman operator’s value function and (transition) conditional expectation are modelled and embedded respectively as functions in a reproducing kernel Hilbert space (RKHS). The resulting finite-induced approximate pseudo-MDP (Yao et al., 2014a) can be solved exactly in a dynamic programming algorithm with policy improvement suboptimality guarantees. However model construction and policy planning scale cubically and quadratically respectively with the training set size, rendering the CME impractical for sampleabundant tasks in online settings. Three variants of CME API are investigated to strike a balance between stable policy updates and reduced computational complexity. The first variant models the value function and state-action representation explicitly in a parametric CME (PCME) algorithm with favourable computational complexity. However a soft conservative policy update technique is developed to mitigate policy learning oscillations in the planning process. The second variant returns to the non-parametric embedding and contributes (along with external work) to the compressed CME (CCME); a sparse and computationally more favourable CME. The final variant is a fully end-to-end differentiable embedding trained with stochastic gradient updates. The value function remains modelled in an RKHS such that backprop is driven by a non-parametric RKHS loss function. Actively compressed CME (ACCME) satisfies the pseudo-MDP contraction constraint using a sparse softmax activation function. The size of the pseudo-MDP (i.e. the size of the embedding’s last layer) is controlled by sparsifying the last layer weight matrix by extending the truncated gradient method (Langford et al., 2009) with group lasso updates in a novel ‘use it or lose it’ neuron pruning mechanism. Surprisingly this technique does not require extensive fine-tuning between control tasks