arXiv subjects

Mohammed Amin Abdullah

Publications and source records attributed to Mohammed Amin Abdullah.

7 recordsLinked to original sources

arXiv2019

Wasserstein Robust Reinforcement Learning

Reinforcement learning algorithms, though successful, tend to over-fit to training environments hampering their application to the real-world. This paper proposes $\text{W}\text{R}^{2}\text{L}$ -- a robust reinforcement learning algorithm with significant robust performance on low and high-dimensional control tasks. Our method formalises robust reinforcement learning as a novel min-max game with a Wasserstein constraint for a correct and convergent solver. Apart from the formulation, we also propose an efficient and scalable solver following a novel zero-order optimisation method that we believe can be useful to numerical optimisation in general. We empirically demonstrate significant gains compared to standard and robust state-of-the-art algorithms on high-dimensional MuJuCo environments.

cs.LG↗

arXiv2019

Global Majority Consensus by Local Majority Polling on Graphs of a Given Degree Sequence

Suppose in a graph $G$ vertices can be either red or blue. Let $k$ be odd. At each time step, each vertex $v$ in $G$ polls $k$ random neighbours and takes the majority colour. If it doesn't have $k$ neighbours, it simply polls all of them, or all less one if the degree of $v$ is even. We study this protocol on graphs of a given degree sequence, in the following setting: initially each vertex of $G$ is red independently with probability $α< \frac{1}{2}$, and is otherwise blue. We show that if $α$ is sufficiently biased, then with high probability consensus is reached on the initial global majority within $O(\log_k \log_k n)$ steps if $5 \leq k \leq d$, and $O(\log_d \log_d n)$ steps if $k > d$. Here, $d\geq 5$ is the effective minimum degree, the smallest integer which occurs $Θ(n)$ times in the degree sequence. We further show that on such graphs, any local protocol in which a vertex does not change colour if all its neighbours have that same colour, takes time at least $Ω(\log_d \log_d n)$, with high probability. Additionally, we demonstrate how the technique for the above sparse graphs can be applied in a straightforward manner to get bounds for the Erdős-Rényi random graphs in the connected regime.

cs.DC↗

arXiv2019

Reinforcement Learning with Wasserstein Distance Regularisation, with Applications to Multipolicy Learning

We describe an application of Wasserstein distance to Reinforcement Learning. The Wasserstein distance in question is between the distribution of mappings of trajectories of a policy into some metric space, and some other fixed distribution (which may, for example, come from another policy). Different policies induce different distributions, so given an underlying metric, the Wasserstein distance quantifies how different policies are. This can be used to learn multiple polices which are different in terms of such Wasserstein distances by using a Wasserstein regulariser. Changing the sign of the regularisation parameter, one can learn a policy for which its trajectory mapping distribution is attracted to a given fixed distribution.

cs.LG↗

arXiv2017

Typical distances in a geometric model for complex networks

We study typical distances in a geometric random graph on the hyperbolic plane. Introduced by Krioukov et al.~\cite{ar:Krioukov} as a model for complex networks, $N$ vertices are drawn randomly within a bounded subset of the hyperbolic plane and any two of them are joined if they are within a threshold hyperbolic distance. With appropriately chosen parameters, the random graph is sparse and exhibits power law degree distribution as well as local clustering. In this paper we show a further property: the distance between two uniformly chosen vertices that belong to the same component is doubly logarithmic in $N$, i.e., the graph is an ~\emph{ultra-small world}. More precisely, we show that the distance rescaled by $\log \log N$ converges in probability to a certain constant that depends on the exponent of the power law. The same constant emerges in an analogous setting with the well-known \emph{Chung-Lu} model for which the degree distribution has a power law tail.

math.CO↗

arXiv2016

Robust On-line Matrix Completion on Graphs

We study online robust matrix completion on graphs. At each iteration a vector with some entries missing is revealed and our goal is to reconstruct it by identifying the underlying low-dimensional subspace from which the vectors are drawn. We assume there is an underlying graph structure to the data, that is, the components of each vector correspond to nodes of a certain (known) graph, and their values are related accordingly. We give algorithms that exploit the graph to reconstruct the incomplete data, even in the presence of outlier noise. The theoretical properties of the algorithms are studied and numerical experiments using both synthetic and real world datasets verify the improved performance of the proposed technique compared to other state of the art algorithms.

cs.IT↗

arXiv2015

Local majority dynamics on preferential attachment graphs

Suppose in a graph $G$ vertices can be either red or blue. Let $k$ be odd. At each time step, each vertex $v$ in $G$ polls $k$ random neighbours and takes the majority colour. If it doesn't have $k$ neighbours, it simply polls all of them, or all less one if the degree of $v$ is even. We study this protocol on the preferential attachment model of Albert and Barabási, which gives rise to a degree distribution that has roughly power-law $P(x) \sim \frac{1}{x^{3}}$, as well as generalisations which give exponents larger than $3$. The setting is as follows: Initially each vertex of $G$ is red independently with probability $α< \frac{1}{2}$, and is otherwise blue. We show that if $α$ is sufficiently biased away from $\frac{1}{2}$, then with high probability, consensus is reached on the initial global majority within $O(\log_d \log_d t)$ steps. Here $t$ is the number of vertices and $d \geq 5$ is the minimum of $k$ and $m$ (or $m-1$ if $m$ is even), $m$ being the number of edges each new vertex adds in the preferential attachment generative process. Additionally, our analysis reduces the required bias of $α$ for graphs of a given degree sequence studied by the first author (which includes, e.g., random regular graphs).

math.PR↗

arXiv2014

A phase transition in the evolution of bootstrap percolation processes on preferential attachment graphs

The theme of this paper is the analysis of bootstrap percolation processes on random graphs generated by preferential attachment. This is a class of infection processes where vertices have two states: they are either infected or susceptible. At each round every susceptible vertex which has at least $r\geq 2$ infected neighbours becomes infected and remains so forever. Assume that initially $a(t)$ vertices are randomly infected, where $t$ is the total number of vertices of the graph. Suppose also that $r < m$, where $2m$ is the average degree. We determine a critical function $a_c(t)$ such that when $a(t) \gg a_c(t)$, complete infection occurs with high probability as $t \rightarrow \infty$, but when $a(t) \ll a_c (t)$, then with high probability the process evolves only for a bounded number of rounds and the final set of infected vertices is asymptotically equal to $a(t)$.

math.PR↗