Serge Massar

Antonino Emanuele Scurria, Dimitri Vanden Abeele, Bortolo Matteo Mognetti et al.

Equilibrium Propagation (EP) is a physics-inspired learning algorithm that uses stationary states of a dynamical system both for inference and learning. In its original formulation it is limited to conservative systems, $\textit{i.e.}$ to dynamics which derive from an energy function. Given their importance in applications, it is important to extend EP to nonconservative systems, $\textit{i.e.}$ systems with non-reciprocal interactions. Previous attempts to generalize EP to such systems failed to compute the exact gradient of the cost function. Here we propose a framework that extends EP to arbitrary nonconservative systems, including feedforward networks. We keep the key property of equilibrium propagation, namely the use of stationary states both for inference and learning. However, we modify the dynamics in the learning phase by a term proportional to the non-reciprocal part of the interaction so as to obtain the exact gradient of the cost function. This algorithm can also be derived using a variational formulation that generates the learning dynamics through an energy function defined over an augmented state space. Numerical experiments using the MNIST database show that this algorithm achieves better performance and learns faster than previous proposals.

Rahul Uma Ramachandran, Serge Massar

Physical computing systems provide a promising route toward hardware-native machine learning, but their computational capabilities remain difficult to characterize in a principled, task-independent, and data-efficient way. We extend the Information Processing Capacity (IPC) framework to stationary physical computing systems and establish several fundamental results: individual capacities are bounded between zero and one, their sum over a complete basis is bounded by the number of readouts, and noise strictly reduces this bound. We address the finite-sample estimation of IPC and derive the asymptotic form of the systematic positive bias affecting naive estimators. Building on these results, we introduce data-efficient estimation methods based on Richardson extrapolation and Sobol quasi-random sampling. We validate the framework experimentally using a photonic computing system based on picosecond laser pulses propagating through a nonlinear optical fibre. By varying the laser power and fibre length, we observe systematic shifts of the IPC distribution toward higher-order nonlinear capacities induced by the Kerr effect. Finally, we demonstrate that the total IPC strongly correlates with performance on benchmark machine-learning tasks and provides a reliable estimate of the effective dimensionality of the system. These results establish IPC as a practical bridge between the intrinsic dynamics of physical computing systems and their machine-learning performance.

Serge Massar, Bortolo Matteo Mognetti

Equilibrium propagation is a recently introduced method to use and train artificial neural networks in which the network is at the minimum (more generally extremum) of an energy functional. Equilibrium propagation has shown good performance on a number of benchmark tasks. Here we extend equilibrium propagation in two directions. First we show that there is a natural quantum generalization of equilibrium propagation in which a quantum neural network is taken to be in the ground state (more generally any eigenstate) of the network Hamiltonian, with a similar training mechanism that exploits the fact that the mean energy is extremal on eigenstates. Second we extend the analysis of equilibrium propagation at finite temperature, showing that thermal fluctuations allow one to naturally train the network without having to clamp the output layer during training. We also study the low temperature limit of equilibrium propagation.

Serge Massar

We propose a method for training dynamical systems governed by Lagrangian mechanics using Equilibrium Propagation. Our approach extends Equilibrium Propagation - initially developed for energy-based models - to dynamical trajectories by leveraging the principle of action extremization. Training is achieved by gently nudging trajectories toward desired targets and measuring how the variables conjugate to the parameters to be trained respond. This method is particularly suited to systems with periodic boundary conditions or fixed initial and final states, enabling efficient parameter updates without requiring explicit backpropagation through time. In the case of periodic boundary conditions, this approach yields the semiclassical limit of Quantum Equilibrium Propagation. Applications to systems with dissipation are also discussed.

Enrico Picco, Lina Jaurigue, Kathy Lüdge et al.

We present an experimental validation of a recently proposed optimization technique for reservoir computing, using an optoelectronic setup. Reservoir computing is a robust framework for signal processing applications, and the development of efficient optimization approaches remains a key challenge. The technique we address leverages solely a delayed version of the input signal to identify the optimal operational region of the reservoir, simplifying the traditionally time-consuming task of hyperparameter tuning. We verify the effectiveness of this approach on different benchmark tasks and reservoir operating conditions.

Enrico Picco, Piotr Antonik, Serge Massar

The recognition of human actions in videos is one of the most active research fields in computer vision. The canonical approach consists in a more or less complex preprocessing stages of the raw video data, followed by a relatively simple classification algorithm. Here we address recognition of human actions using the reservoir computing algorithm, which allows us to focus on the classifier stage. We introduce a new training method for the reservoir computer, based on "Timesteps Of Interest", which combines in a simple way short and long time scales. We study the performance of this algorithm using both numerical simulations and a photonic implementation based on a single non-linear node and a delay line on the well known KTH dataset. We solve the task with high accuracy and speed, to the point of allowing for processing multiple video streams in real time. The present work is thus an important step towards developing efficient dedicated hardware for video processing.

Piotr Antonik, Michiel Hermans, Marc Haelterman et al.

Reservoir computing is a bio-inspired computing paradigm for processing time dependent signals. The performance of its analogue implementations matches other digital algorithms on a series of benchmark tasks. Their potential can be further increased by feeding the output signal back into the reservoir, which would allow to apply the algorithm to time series generation. This requires, in principle, implementing a sufficiently fast readout layer for real-time output computation. Here we achieve this with a digital output layer driven by a FPGA chip. We demonstrate the first opto-electronic reservoir computer with output feedback and test it on two examples of time series generation tasks: frequency and random pattern generation. We obtain very good results on the first task, similar to idealised numerical simulations. The performance on the second one, however, suffers from the experimental noise. We illustrate this point with a detailed investigation of the consequences of noise on the performance of a physical reservoir computer with output feedback. Our work thus opens new possible applications for analogue reservoir computing and brings new insights on the impact of noise on the output feedback.

Piotr Antonik, Marc Haelterman, Serge Massar

Introduction. Reservoir Computing is a bio-inspired computing paradigm for processing time-dependent signals. The performance of its hardware implementation is comparable to state-of-the-art digital algorithms on a series of benchmark tasks. The major bottleneck of these implementation is the readout layer, based on slow offline post-processing. Few analogue solutions have been proposed, but all suffered from notice able decrease in performance due to added complexity of the setup. Methods. Here we propose the use of online training to solve these issues. We study the applicability of this method using numerical simulations of an experimentally feasible reservoir computer with an analogue readout layer. We also consider a nonlinear output layer, which would be very difficult to train with traditional methods. Results. We show numerically that online learning allows to circumvent the added complexity of the analogue layer and obtain the same level of performance as with a digital layer. Conclusion. This work paves the way to high-performance fully-analogue reservoir computers through the use of online training of the output layers.

Lorenz Butschek, Akram Akrout, Evangelia Dimitriadou et al.

Reservoir computing is a brain inspired approach for information processing, well suited to analogue implementations. We report a photonic implementation of a reservoir computer that exploits frequency domain multiplexing to encode neuron states. The system processes 25 comb lines simultaneously (i.e. 25 neurons), at a rate of 20 MHz. We illustrate performances on two standard benchmark tasks: channel equalization and time series forecasting. We also demonstrate that frequency multiplexing allows output weights to be implemented in the optical domain, through optical attenuation. We discuss the perspectives for high speed high performance low footprint implementations.

Piotr Antonik, Marvyn Gulina, Jaël Pauwels et al.

Using the machine learning approach known as reservoir computing, it is possible to train one dynamical system to emulate another. We show that such trained reservoir computers reproduce the properties of the attractor of the chaotic system sufficiently well to exhibit chaos synchronisation. That is, the trained reservoir computer, weakly driven by the chaotic system, will synchronise with the chaotic system. Conversely, the chaotic system, weakly driven by a trained reservoir computer, will synchronise with the reservoir computer. We illustrate this behaviour on the Mackey-Glass and Lorenz systems. We then show that trained reservoir computers can be used to crack chaos based cryptography and illustrate this on a chaos cryptosystem based on the Mackey-Glass system. We conclude by discussing why reservoir computers are so good at emulating chaotic systems.

Piotr Antonik, Marc Haelterman, Serge Massar

Reservoir computing is a bio-inspired computing paradigm for processing time-dependent signals. Its hardware implementations have received much attention because of their simplicity and remarkable performance on a series of benchmark tasks. In previous experiments the output was uncoupled from the system and in most cases simply computed offline on a post-processing computer. However, numerical investigations have shown that feeding the output back into the reservoir would open the possibility of long-horizon time series forecasting. Here we present a photonic reservoir computer with output feedback, and demonstrate its capacity to generate periodic time series and to emulate chaotic systems. We study in detail the effect of experimental noise on system performance. In the case of chaotic systems, this leads us to introduce several metrics, based on standard signal processing techniques, to evaluate the quality of the emulation. Our work significantly enlarges the range of tasks that can be solved by hardware reservoir computers, and therefore the range of applications they could potentially tackle. It also raises novel questions in nonlinear dynamics and chaos theory.

Michiel Hermans, Piotr Antonik, Marc Haelterman et al.

Delay-coupled electro-optical systems have received much attention for their dynamical properties and their potential use in signal processing. In particular it has recently been demonstrated, using the artificial intelligence algorithm known as reservoir computing, that photonic implementations of such systems solve complex tasks such as speech recognition. Here we show how the backpropagation algorithm can be physically implemented on the same electro-optical delay-coupled architecture used for computation with only minor changes to the original design. We find that, compared when the backpropagation algorithm is not used, the error rate of the resulting computing device, evaluated on three benchmark tasks, decreases considerably. This demonstrates that electro-optical analog computers can embody a large part of their own training process, allowing them to be applied to new, more difficult tasks.

Piotr Antonik, François Duport, Michiel Hermans et al.

Reservoir Computing is a bio-inspired computing paradigm for processing time dependent signals. The performance of its analogue implementation are comparable to other state of the art algorithms for tasks such as speech recognition or chaotic time series prediction, but these are often constrained by the offline training methods commonly employed. Here we investigated the online learning approach by training an opto-electronic reservoir computer using a simple gradient descent algorithm, programmed on an FPGA chip. Our system was applied to wireless communications, a quickly growing domain with an increasing demand for fast analogue devices to equalise the nonlinear distorted channels. We report error rates up to two orders of magnitude lower than previous implementations on this task. We show that our system is particularly well-suited for realistic channel equalisation by testing it on a drifting and a switching channels and obtaining good performances

Marc Massar, Serge Massar

Dynamical systems driven by strong external signals are ubiquituous in nature and engineering. Here we study "echo state networks", networks of a large number of randomly connected nodes, which represent a simple model of a neural network, and have important applications in machine learning. We develop a mean field theory of echo state networks. The dynamics of the network is captured by the evolution law, similar to a logistic map, for a single collective variable. When the network is driven by many independent external signals, this collective variable reaches a steady state. But when the network is driven by a single external signal, the collective variable is nonstationnary but can be characterised by its time averaged distribution. The predictions of the mean field theory, including the value of the largest Lyaponuov exponent, are compared with the numerical integration of the equations of motion.

Anteo Smerieri, François Duport, Yvan Paquot et al.

Reservoir computing is a new, powerful and flexible machine learning technique that is easily implemented in hardware. Recently, by using a time-multiplexed architecture, hardware reservoir computers have reached performance comparable to digital implementations. Operating speeds allowing for real time information operation have been reached using optoelectronic systems. At present the main performance bottleneck is the readout layer which uses slow, digital postprocessing. We have designed an analog readout suitable for time-multiplexed optoelectronic reservoir computers, capable of working in real time. The readout has been built and tested experimentally on a standard benchmark task. Its performance is better than non-reservoir methods, with ample room for further improvement. The present work thereby overcomes one of the major limitations for the future development of hardware reservoir computers.

Adrian Kent, Serge Massar, Jonathan Silman

An important class of cryptographic applications of relativistic quantum information work as follows. B generates a random qudit and supplies it to A at point P. A is supposed to transmit it at near light speed c to to one of a number of possible pairwise spacelike separated points Q1; : : : ;Qn. A's transmission is supposed to be secure, in the sense that B cannot tell in advance which Qj will be chosen. This poses significant practical challenges, since secure reliable long-range transmission of quantum data at speeds near to c is presently not easy. Here we propose different techniques to overcome these difficulties. We introduce protocols that allow secure long-range implementations even when both parties control only widely separated laboratories of small size. In particular we introduce a protocol in which A needs send the qudit only over a short distance, and securely transmits classical information (for instance using a one time pad) over the remaining distance. We further show that by using parallel implementations of the protocols security can be maintained in the presence of moderate amounts of losses and errors.