­Business reliance on algorithms is becoming ubiquitous, and companies are increasingly concerned about their algorithms causing major financial or reputational damage. High-profile cases include Google’s AI algorithm for photo classification mistakenly labelling a black couple as gorillas in 2015 (Gebru 2020 In The Oxford handbook of ethics of AI , pp. 251–269), Microsoft’s AI chatbot Tay that spread racist, sexist and antisemitic speech on Twitter (now X) (Wolf et al . 2017 ACM Sigcas Comput. Soc . 47 , 54–64 (doi: 10.1145/3144592.3144598 )), and Amazon’s AI recruiting tool being scrapped after showing bias against women. In response, governments are legislating and imposing bans, regulators fining companies and the judiciary discussing potentially making algorithms artificial ‘persons’ in law. As with financial audits, governments, business and society will require algorithm audits; formal assurance that algorithms are legal, ethical and safe. A new industry is envisaged: Auditing and Assurance of Algorithms (cf. data privacy), with the remit to professionalize and industrialize AI, ML and associated algorithms. The stakeholders range from those working on policy/regulation to industry practitioners and developers. We also anticipate the nature and scope of the auditing levels and framework presented will inform those interested in systems of governance and compliance with regulation/standards. Our goal in this article is to survey the key areas necessary to perform auditing and assurance and instigate the debate in this novel area of research and practice.

We propose and analyse a new Milstein type scheme for simulating stochastic differential equations (SDEs) with highly nonlinear coefficients. Our work is motivated by the need to justify multi-level Monte Carlo simulations for mean-reverting financial models with polynomial growth in the diffusion term. We introduce a double implicit Milstein scheme and show that it possesses desirable properties. It converges strongly and preserves non-negativity for a rich family of financial models and can reproduce linear and nonlinear stability behaviour of the underlying SDE without severe restriction on the time step. Although the scheme is implicit, we point out examples of financial models where an explicit formula for the solution to the scheme can be found.

Optimal control problems are inherently hard to solve as the optimization must be performed simultaneously with updating the underlying system. Starting from an initial guess, Howard's policy impro...

This work uses the entropy-regularized relaxed stochastic control perspective as a principled framework for designing reinforcement learning (RL) algorithms. Herein, an agent interacts with the environment by generating noisy controls distributed according to the optimal relaxed policy. The noisy policies, on the one hand, explore the space and hence facilitate learning, but, on the other hand, they introduce bias by assigning a positive probability to nonoptimal actions. This exploration-exploitation trade-off is determined by the strength of entropy regularization. We study algorithms resulting from two entropy regularization formulations: the exploratory control approach, where entropy is added to the cost objective, and the proximal policy update approach, where entropy penalizes policy divergence between consecutive episodes. We focus on the finite horizon continuous-time linear-quadratic (LQ) RL problem, where a linear dynamics with unknown drift coefficients is controlled subject to quadratic costs. In this setting, both algorithms yield a Gaussian relaxed policy. We quantify the precise difference between the value functions of a Gaussian policy and its noisy evaluation and show that the execution noise must be independent across time. By tuning the frequency of sampling from relaxed policies and the parameter governing the strength of entropy regularization, we prove that the regret, for both learning algorithms, is of the order [Formula: see text] (up to a logarithmic factor) over [Formula: see text] episodes, matching the best known result from the literature.

The classical Method of Successive Approximations (MSA) is an iterative method for solving stochastic control problems and is derived from Pontryagin’s optimality principle. It is known that the MSA may fail to converge. Using careful estimates for the backward stochastic differential equation (BSDE) this paper suggests a modification to the MSA algorithm. This modified MSA is shown to converge for general stochastic control problems with control in both the drift and diffusion coefficients. Under some additional assumptions the rate of convergence is shown. The results are valid without restrictions on the time horizon of the control problem, in contrast to iterative methods based on the theory of forward-backward stochastic differential equations.

This paper studies stochastic control problems with the action space taken to be probability measures, with the objective penalized by the relative entropy. We identify a suitable metric space on which we construct a gradient flow for the measure-valued control process, in the set of admissible controls, along which the cost functional is guaranteed to decrease. It is shown that any invariant measure of this gradient flow satisfies the Pontryagin optimality principle. If the problem we work with is sufficiently convex, the gradient flow converges exponentially fast. Furthermore, the optimal measure-valued control process admits a Bayesian interpretation, which means that one can incorporate prior knowledge when solving such stochastic control problems. This work is motivated by a desire to extend the theoretical underpinning for the convergence of stochastic gradient type algorithms widely employed in the reinforcement learning community to solve control problems.