Equilibrium Propagation: Bridging the Gap Between Energy-Based Models and Backpropagation

We introduce Equilibrium Propagation, a learning framework for energy-based models. It involves only one kind of neural computation, performed in both the first phase (when the prediction is made) and the second phase of training (after the target or prediction error is revealed). Although this algorithm computes the gradient of an objective function just like Backpropagation, it does not need a special computation or circuit for the second phase, where errors are implicitly propagated. Equilibrium Propagation shares similarities with Contrastive Hebbian Learning and Contrastive Divergence while solving the theoretical issues of both algorithms: our algorithm computes the gradient of a well defined objective function. Because the objective function is defined in terms of local perturbations, the second phase of Equilibrium Propagation corresponds to only nudging the prediction (fixed point, or stationary distribution) towards a configuration that reduces prediction error. In the case of a recurrent multi-layer supervised network, the output units are slightly nudged towards their target in the second phase, and the perturbation introduced at the output layer propagates backward in the hidden layers. We show that the signal 'back-propagated' during this second phase corresponds to the propagation of error derivatives and encodes the gradient of the objective function, when the synaptic update corresponds to a standard form of spike-timing dependent plasticity. This work makes it more plausible that a mechanism similar to Backpropagation could be implemented by brains, since leaky integrator neural computation performs both inference and error back-propagation in our model. The only local difference between the two phases is whether synaptic changes are allowed or not

Recognition of partially occluded threat objects using the annealed Hopefield network

Recognition of partially occluded objects has been an important issue to airport security because occlusion causes significant problems in identifying and locating objects during baggage inspection. The neural network approach is suitable for the problems in the sense that the inherent parallelism of neural networks pursues many hypotheses in parallel resulting in high computation rates. Moreover, they provide a greater degree of robustness or fault tolerance than conventional computers. The annealed Hopfield network which is derived from the mean field annealing (MFA) has been developed to find global solutions of a nonlinear system. In the study, it has been proven that the system temperature of MFA is equivalent to the gain of the sigmoid function of a Hopfield network. In our early work, we developed the hybrid Hopfield network (HHN) for fast and reliable matching. However, HHN doesn't guarantee global solutions and yields false matching under heavily occluded conditions because HHN is dependent on initial states by its nature. In this paper, we present the annealed Hopfield network (AHN) for occluded object matching problems. In AHN, the mean field theory is applied to the hybird Hopfield network in order to improve computational complexity of the annealed Hopfield network and provide reliable matching under heavily occluded conditions. AHN is slower than HHN. However, AHN provides near global solutions without initial restrictions and provides less false matching than HHN. In conclusion, a new algorithm based upon a neural network approach was developed to demonstrate the feasibility of the automated inspection of threat objects from x-ray images. The robustness of the algorithm is proved by identifying occluded target objects with large tolerance of their features

An analog feedback associative memory

A method for the storage of analog vectors, i.e., vectors whose components are real-valued, is developed for the Hopfield continuous-time network. An important requirement is that each memory vector has to be an asymptotically stable (i.e. attractive) equilibrium of the network. Some of the limitations imposed by the continuous Hopfield model on the set of vectors that can be stored are pointed out. These limitations can be relieved by choosing a network containing visible as well as hidden units. An architecture consisting of several hidden layers and a visible layer, connected in a circular fashion, is considered. It is proved that the two-layer case is guaranteed to store any number of given analog vectors provided their number does not exceed 1 + the number of neurons in the hidden layer. A learning algorithm that correctly adjusts the locations of the equilibria and guarantees their asymptotic stability is developed. Simulation results confirm the effectiveness of the approach

Dreaming neural networks: forgetting spurious memories and reinforcing pure ones

The standard Hopfield model for associative neural networks accounts for biological Hebbian learning and acts as the harmonic oscillator for pattern recognition, however its maximal storage capacity is $\alpha \sim 0.14$, far from the theoretical bound for symmetric networks, i.e. $\alpha =1$. Inspired by sleeping and dreaming mechanisms in mammal brains, we propose an extension of this model displaying the standard on-line (awake) learning mechanism (that allows the storage of external information in terms of patterns) and an off-line (sleep) unlearning$\&$consolidating mechanism (that allows spurious-pattern removal and pure-pattern reinforcement): this obtained daily prescription is able to saturate the theoretical bound $\alpha=1$, remaining also extremely robust against thermal noise. Both neural and synaptic features are analyzed both analytically and numerically. In particular, beyond obtaining a phase diagram for neural dynamics, we focus on synaptic plasticity and we give explicit prescriptions on the temporal evolution of the synaptic matrix. We analytically prove that our algorithm makes the Hebbian kernel converge with high probability to the projection matrix built over the pure stored patterns. Furthermore, we obtain a sharp and explicit estimate for the "sleep rate" in order to ensure such a convergence. Finally, we run extensive numerical simulations (mainly Monte Carlo sampling) to check the approximations underlying the analytical investigations (e.g., we developed the whole theory at the so called replica-symmetric level, as standard in the Amit-Gutfreund-Sompolinsky reference framework) and possible finite-size effects, finding overall full agreement with the theory.Comment: 31 pages, 12 figure