Modern optimisation algorithms are often metaheuristic, and they are very promising in solving NP-hard optimization problems. In this paper, we show how to use the recently developed Firefly Algorithm to solve nonlinear design problems. For the standard pressure vessel design optimisation, the optimal solution found by FA is far better than the best solution obtained previously in literature. In addition, we also propose a few new test functions with either singularity or stochastic components but with known global optimality, and thus they can be used to validate new optimisation algorithms. Possible topics for further research are also discussed.
We analyse the asymptotic properties of a continuous-time, two-timescale stochastic approximation algorithm designed for stochastic bilevel optimisation problems in continuous-time models. We obtain the weak convergence rate of this algorithm in the form of a central limit theorem. We also demonstrate how this algorithm can be applied to several continuous-time bilevel optimisation problems.
Technische Universität Berlin（柏林工业大学）；GT-ARC Gemeinnützige GmbH（GT-ARC公益有限责任公司）
In this paper, we propose and analyse a family of generalised stochastic composite mirror descent algorithms. With adaptive step sizes, the proposed algorithms converge without requiring prior knowledge of the problem. Combined with an entropy-like update-generating function, these algorithms perform gradient descent in the space equipped with the maximum norm, which allows us to exploit the low-dimensional structure of the decision sets for high-dimensional problems. Together with a sampling method based on the Rademacher distribution and variance reduction techniques, the proposed algorithms…
We introduce a simple and efficient algorithm for unconstrained zeroth-order stochastic convex bandits and prove its regret is at most $(1 + r/d)[d^{1.5} \sqrt{n} + d^3] polylog(n, d, r)$ where $n$ is the horizon, $d$ the dimension and $r$ is the radius of a known ball containing the minimiser of the loss.
A Modular Approach to Stochastic Optimisation for Inverse Problems Using the Core Imaging Library
Finden Ltd（Finden有限公司）；Science and Technology Facilities Council, Rutherford Appleton Laboratory（英国科学与技术设施委员会，鲁瑟福阿普尔顿实验室）；Department of Applied Mathematics and Computer Science, Technical University of Denmark（丹麦技术大学应用数学与计算机科学系）；Institute of Nuclear Medicine, University College London（伦敦大学学院核医学研究所）；EDF Lab Paris-Saclay（法国电力集团巴黎萨克雷实验室）；UCL Hawkes Institute, University College London（伦敦大学学院UCL霍克斯研究所）
We study the success probability for a variant of the secretary problem, with noisy observations and multiple offline selection. Our formulation emulates, and is motivated by, problems involving noisy selection arising in the disciplines of stochastic simulation and simulation-based optimisation. In addition, we employ the philosophy of ordinal optimisation - involving an ordinal selection rule, and a percentile notion of goal softening for the success probability. As a result, it is shown that the success probability only depends on the underlying copula of the problem. Other general properti…
Many machine learning models require a training procedure based on running stochastic gradient descent. A key element for the efficiency of those algorithms is the choice of the learning rate schedule. While finding good learning rates schedules using Bayesian optimisation has been tackled by several authors, adapting it dynamically in a data-driven way is an open question. This is of high practical importance to users that need to train a single, expensive model. To tackle this problem, we introduce an original probabilistic model for traces of optimisers, based on latent Gaussian processes a…
We present the Basin Hopping with Skipping (BH-S) algorithm for stochastic optimisation, which replaces the perturbation step of basin hopping (BH) with a so-called skipping proposal from the rare-event sampling literature. Empirical results on benchmark optimisation surfaces demonstrate that BH-S can improve performance relative to BH by encouraging non-local exploration, that is, by hopping between distant basins.
Stochastic Optimisation Framework using the Core Imaging Library and Synergistic Image Reconstruction Framework for PET Reconstruction
Finden Ltd（芬登有限公司）；Rutherford Appleton Laboratory（卢瑟福·阿普尔顿实验室）；Science & Technology Facilities Council（科学与技术设施委员会）；National Physical Laboratory（国家物理实验室）；Électricité de France（法国电力公司）；CSIRO（联邦科学与工业研究组织）；Technical University of Denmark（丹麦技术大学）；University College London（伦敦大学学院）；KU Leuven（鲁汶大学）；Institute of Nuclear Medicine, University College London（伦敦大学学院核医学研究所）
Losing momentum in continuous-time stochastic optimisation
Princeton University（普林斯顿大学）；The University of Manchester（曼彻斯特大学）；Technische Universiteit Delft（代尔夫特理工大学）；Technische Universität München（慕尼黑工业大学）；Munich Center for Machine Learning (MCML)（慕尼黑机器学习中心）
Global optimisation problems in high-dimensional and infinite dimensional spaces arise in various real-world applications such as engineering, economics, geophysics, biology, machine learning, optimal control, etc. Among stochastic approaches to global optimisation, biology-inspired methods are currently popular in the literature, imitating natural ecological and evolutionary processes and reported to be efficient in many practical study cases. However, many of bio-inspired methods have some vital drawbacks. Due to their semi-empirical nature, convergence to the globally optimal solution canno…
Operations research practitioners frequently want to model complicated functions that are are difficult to encode in their underlying optimisation framework. A common approach is to solve an approximate model, and to use a simulation to evaluate the true objective value of one or more solutions. We propose a new approach to integrating simulation into the optimisation model itself. The idea is to run the simulation at each incumbent solution to the master problem. The simulation data is then used to guide the trajectory of the optimisation model itself using logic-based Benders cuts. We test t…
Technische Universität Berlin（柏林工业大学）；GT-ARC Gemeinnützige GmbH（GT-ARC公益有限责任公司）
This paper proposes a new family of algorithms for the online optimisation of composite objectives. The algorithms can be interpreted as the combination of the exponentiated gradient and $p$-norm algorithm. Combined with algorithmic ideas of adaptivity and optimism, the proposed algorithms achieve a sequence-dependent regret upper bound, matching the best-known bounds for sparse target decision variables. Furthermore, the algorithms have efficient implementations for popular composite objectives and constraints and can be converted to stochastic optimisation algorithms with the optimal acceler…
In this paper, we propose and analyze algorithms for zeroth-order optimization of non-convex composite objectives, focusing on reducing the complexity dependence on dimensionality. This is achieved by exploiting the low dimensional structure of the decision set using the stochastic mirror descent method with an entropy alike function, which performs gradient descent in the space equipped with the maximum norm. To improve the gradient estimation, we replace the classic Gaussian smoothing method with a sampling method based on the Rademacher distribution and show that the mini-batch method copes…
We propose an adaptive optimisation approach for tuning stochastic model predictive control (MPC) hyper-parameters while jointly estimating probability distributions of the transition model parameters based on performance rewards. In particular, we develop a Bayesian optimisation (BO) algorithm with a heteroscedastic noise model to deal with varying noise across the MPC hyper-parameter and dynamics model parameter spaces. Typical homoscedastic noise models are unrealistic for tuning MPC since stochastic controllers are inherently noisy, and the level of noise is affected by their hyper-paramet…
We present a classical algorithm to find approximate solutions to instances of quadratic unconstrained binary optimisation. The algorithm can be seen as an analogue of quantum annealing under the restriction of a product state space, where the dynamical evolution in quantum annealing is replaced with a gradient-descent based method. This formulation is able to quickly find high-quality solutions to large-scale problem instances, and can naturally be accelerated by dedicated hardware such as graphics processing units. We benchmark our approach for large scale problem instances with tuneable har…
Stochastic gradient descent (SGD) is a widely adopted iterative method for optimizing differentiable objective functions. In this paper, we propose and discuss a novel approach to scale up SGD in applications involving non-convex functions and large datasets. We address the bottleneck problem arising when using both shared and distributed memory. Typically, the former is bounded by limited computation resources and bandwidth whereas the latter suffers from communication overheads. We propose a unified distributed and parallel implementation of SGD (named DPSGD) that relies on both asynchronous…
Information exchange over networks can be affected by various forms of delay. This causes challenges for using the network by a multi-agent system to solve a distributed optimisation problem. Distributed optimisation schemes, however, typically do not assume network models that are representative for real-world communication networks, since communication links are most of the time abstracted as lossless. Our objective is therefore to formulate a representative network model and provide practically verifiable network conditions that ensure convergence of distributed algorithms in the presence o…
Online Newton Method for Bandit Convex Optimisation
University of Amsterdam（阿姆斯特丹大学）；Leiden University（莱顿大学）；Google DeepMind（谷歌DeepMind）；Korteweg-de Vries Institute for Mathematics（科特韦格-德弗里斯数学研究所）；Mathematical Institute（数学研究所）
We consider a standard distributed optimisation setting where $N$ machines, each holding a $d$-dimensional function $f_i$, aim to jointly minimise the sum of the functions $\sum_{i = 1}^N f_i (x)$. This problem arises naturally in large-scale distributed optimisation, where a standard solution is to apply variants of (stochastic) gradient descent. We focus on the communication complexity of this problem: our main result provides the first fully unconditional bounds on total number of bits which need to be sent and received by the $N$ machines to solve this problem under point-to-point communic…
In this paper, we study a semi-martingale optimal transport problem and its application to the calibration of Local-Stochastic Volatility (LSV) models. Rather than considering the classical constraints on marginal distributions at initial and final time, we optimise our cost function given the prices of a finite number of European options. We formulate the problem as a convex optimisation problem, for which we provide a PDE formulation along with its dual counterpart. Then we solve numerically the dual problem, which involves a fully non-linear Hamilton-Jacobi-Bellman equation. The method is t…
In this paper, we establish the almost sure convergence of two-timescale stochastic gradient descent algorithms in continuous time under general noise and stability conditions, extending well known results in discrete time. We analyse algorithms with additive noise and those with non-additive noise. In the non-additive case, our analysis is carried out under the assumption that the noise is a continuous-time Markov process, controlled by the algorithm states. The algorithms we consider can be applied to a broad class of bilevel optimisation problems. We study one such problem in detail, namely…
Structured Projection-Based Model Reduction with Application to Stochastic Biochemical Networks
Montefiore Institute, University of Liège（列日大学蒙特菲奥尔研究所）；St John’s College, Oxford（牛津大学圣约翰学院）；Department of Engineering Science, University of Oxford（牛津大学工程科学系）
A body of work has been done to automate machine learning algorithm to highlight the importance of model choice. Automating the process of choosing the best forecasting model and its corresponding parameters can result to improve a wide range of real-world applications. Bayesian optimisation (BO) uses a blackbox optimisation methods to propose solutions according to an exploration-exploitation trade-off criterion through acquisition functions. BO framework imposes two key ingredients: a probabilistic surrogate model that consist of prior belief of the unknown objective function(data-dependant)…
