4,552 research outputs found
Model-Independent Pricing of Asian Options via Optimal Martingale Transport
In this article we discuss the problem of calculating optimal
model-independent (robust) bounds for the price of Asian options with discrete
and continuous averaging. We will give geometric characterisations of the
maximising and the minimising pricing model for certain types of Asian options
in discrete and continuous time. In discrete time the problem is reduced to
finding the optimal martingale transport for the cost function . In the
continuous time case we consider the cases with one and two given marginals. We
describe the maximising models in both of these cases as well as the minimising
model in the one-marginal case and relate the two-marginals case to the
discrete time problem with two marginals
Optimal Statistical Rates for Decentralised Non-Parametric Regression with Linear Speed-Up
We analyse the learning performance of Distributed Gradient Descent in the
context of multi-agent decentralised non-parametric regression with the square
loss function when i.i.d. samples are assigned to agents. We show that if
agents hold sufficiently many samples with respect to the network size, then
Distributed Gradient Descent achieves optimal statistical rates with a number
of iterations that scales, up to a threshold, with the inverse of the spectral
gap of the gossip matrix divided by the number of samples owned by each agent
raised to a problem-dependent power. The presence of the threshold comes from
statistics. It encodes the existence of a "big data" regime where the number of
required iterations does not depend on the network topology. In this regime,
Distributed Gradient Descent achieves optimal statistical rates with the same
order of iterations as gradient descent run with all the samples in the
network. Provided the communication delay is sufficiently small, the
distributed protocol yields a linear speed-up in runtime compared to the
single-machine protocol. This is in contrast to decentralised optimisation
algorithms that do not exploit statistics and only yield a linear speed-up in
graphs where the spectral gap is bounded away from zero. Our results exploit
the statistical concentration of quantities held by agents and shed new light
on the interplay between statistics and communication in decentralised methods.
Bounds are given in the standard non-parametric setting with source/capacity
assumptions
Robust Resource Allocations in Temporal Networks
Temporal networks describe workflows of time-consuming tasks whose processing order is constrained by precedence relations. In many cases, the durations of the network tasks can be influenced by the assignment of resources. This leads to the problem of selecting an ‘optimal’ resource allocation, where optimality is measured by network characteristics such as the makespan (i.e., the time required to complete all tasks). In this paper, we study a robust resource allocation problem where the functional relationship between task durations and resource assignments is uncertain, and the goal is to minimise the worst-case makespan. We show that this problem is generically NP-hard. We then develop convergent bounds for the optimal objective value, as well as feasible allocations whose objective values are bracketed by these bounds. Numerical results provide empirical support for the proposed method.Robust Optimisation, Temporal Networks, Resource Allocation Problem
Functional Bandits
We introduce the functional bandit problem, where the objective is to find an
arm that optimises a known functional of the unknown arm-reward distributions.
These problems arise in many settings such as maximum entropy methods in
natural language processing, and risk-averse decision-making, but current
best-arm identification techniques fail in these domains. We propose a new
approach, that combines functional estimation and arm elimination, to tackle
this problem. This method achieves provably efficient performance guarantees.
In addition, we illustrate this method on a number of important functionals in
risk management and information theory, and refine our generic theoretical
results in those cases
Spatial networks with wireless applications
Many networks have nodes located in physical space, with links more common
between closely spaced pairs of nodes. For example, the nodes could be wireless
devices and links communication channels in a wireless mesh network. We
describe recent work involving such networks, considering effects due to the
geometry (convex,non-convex, and fractal), node distribution,
distance-dependent link probability, mobility, directivity and interference.Comment: Review article- an amended version with a new title from the origina
Recommended from our members
Mini-Workshop: Stochastic Analysis for Poisson Point Processes: Malliavin Calculus, Wiener-Ito Chaos Expansions and Stochastic Geometry
Malliavin calculus plays an important role in the stochastic analysis for Poisson point processes. This technique is tightly connected with chaotic expansions, that were introduced in the first half of the last century by Itˆo and Wiener. These techniques found an increasing number of applications, in particular in the field of stochastic geometry. This in turn inspired new research in stochastic analysis. Leading experts and young researchers of both fields met for a week for fruitful discussions and new cooperations
Connections Between Mirror Descent, Thompson Sampling and the Information Ratio
The information-theoretic analysis by Russo and Van Roy (2014) in combination
with minimax duality has proved a powerful tool for the analysis of online
learning algorithms in full and partial information settings. In most
applications there is a tantalising similarity to the classical analysis based
on mirror descent. We make a formal connection, showing that the
information-theoretic bounds in most applications can be derived from existing
techniques for online convex optimisation. Besides this, for -armed
adversarial bandits we provide an efficient algorithm with regret that matches
the best information-theoretic upper bound and improve best known regret
guarantees for online linear optimisation on -balls and bandits with
graph feedback
- …