Joseph Mikael

Naomi Mona Chmielewski, Nina Amini, Joseph Mikael

We propose a way to bound the generalisation errors of several classes of quantum reservoirs using the Rademacher complexity. We give specific, parameter-dependent bounds for two particular quantum reservoir classes. We analyse how the generalisation bounds scale with growing numbers of qubits. Applying our results to classes with polynomial readout functions, we find that the risk bounds converge in the number of training samples. The explicit dependence on the quantum reservoir and readout parameters in our bounds can be used to control the generalisation error to a certain extent. It should be noted that the bounds scale exponentially with the number of qubits $n$. The upper bounds on the Rademacher complexity can be applied to other reservoir classes that fulfill a few hypotheses on the quantum dynamics and the readout function.

Thomas Pochart, Paulin Jacquot, Joseph Mikael

Sampling random variables following a Boltzmann distribution is an NP-hard problem involved in various applications such as training of \textit{Boltzmann machines}, a specific kind of neural network. Several attempts have been made to use a D-Wave quantum computer to sample such a distribution, as this could lead to significant speedup in these applications. Yet, at present, several challenges remain to efficiently perform such sampling. We detail the various obstacles and explain the remaining difficulties in solving the sampling problem on a D-wave machine.

Carl Remlinger, Brière Marie, Alasseur Clémence et al.

Machine learning algorithms dedicated to financial time series forecasting have gained a lot of interest. But choosing between several algorithms can be challenging, as their estimation accuracy may be unstable over time. Online aggregation of experts combine the forecasts of a finite set of models in a single approach without making any assumption about the models. In this paper, a Bernstein Online Aggregation (BOA) procedure is applied to the construction of long-short strategies built from individual stock return forecasts coming from different machine learning models. The online mixture of experts leads to attractive portfolio performances even in environments characterised by non-stationarity. The aggregation outperforms individual algorithms, offering a higher portfolio Sharpe Ratio, lower shortfall, with a similar turnover. Extensions to expert and aggregation specialisations are also proposed to improve the overall mixture on a family of portfolio evaluation metrics.

Carl Remlinger, Joseph Mikael, Romuald Elie

We introduce three new generative models for time series that are based on Euler discretization of Stochastic Differential Equations (SDEs) and Wasserstein metrics. Two of these methods rely on the adaptation of generative adversarial networks (GANs) to time series. The third algorithm, called Conditional Euler Generator (CEGEN), minimizes a dedicated distance between the transition probability distributions over all time steps. In the context of Ito processes, we provide theoretical guarantees that minimizing this criterion implies accurate estimations of the drift and volatility parameters. We demonstrate empirically that CEGEN outperforms state-of-the-art and GAN generators on both marginal and temporal dynamics metrics. Besides, it identifies accurate correlation structures in high dimension. When few data points are available, we verify the effectiveness of CEGEN, when combined with transfer learning methods on Monte Carlo simulations. Finally, we illustrate the robustness of our method on various real-world datasets.

Thomas Deschatre, Joseph Mikael

A new method for stochastic control based on neural networks and using randomisation of discrete random variables is proposed and applied to optimal stopping time problems. The method models directly the policy and does not need the derivation of a dynamic programming principle nor a backward stochastic differential equation. Unlike continuous optimization where automatic differentiation is used directly, we propose a likelihood ratio method for gradient computation. Numerical tests are done on the pricing of American and swing options. The proposed algorithm succeeds in pricing high dimensional American and swing options in a reasonable computation time, which is not possible with classical algorithms.

Quentin Chan-Wai-Nam, Joseph Mikael, Xavier Warin

Recent machine learning algorithms dedicated to solving semi-linear PDEs are improved by using different neural network architectures and different parameterizations. These algorithms are compared to a new one that solves a fixed point problem by using deep learning techniques. This new algorithm appears to be competitive in terms of accuracy with the best existing algorithms.