Here we elaborate on the details of using SNFs as a variational approximation of the posterior distribution of a variational autoencoder (V AE) [21] as presented in our last results section. All experiments were run using PyTorch 1.2 and on GTX1080Ti cards. NSF block consists of two subsequent NSF layers with intermediate swap layers. "Biased data" is defined by running local Metropolis MC in each of the two wells. "Unbiased data" is produced by running Metropolis MC with a large proposal step (standard The other settings are the same as in Table 1.

A biologically plausible method for training an Artificial Neural Network (ANN) involves treating each unit as a stochastic Reinforcement Learning (RL) agent, thereby considering the network as a team of agents. Consequently, all units can learn via REINFORCE, a local learning rule modulated by a global reward signal, which aligns more closely with biologically observed forms of synaptic plasticity. Nevertheless, this learning method is often slow and scales poorly with network size due to inefficient structural credit assignment, since a single reward signal is broadcast to all units without considering individual contributions. Weight Maximization, a proposed solution, replaces a unit's reward signal with the norm of its outgoing weight, thereby allowing each hidden unit to maximize the norm of the outgoing weight instead of the global reward signal. In this research report, we analyze the theoretical properties of Weight Maximization and propose a variant, Unbiased Weight Maximization. This new approach provides an unbiased learning rule that increases learning speed and improves asymptotic performance. Notably, to our knowledge, this is the first learning rule for a network of Bernoulli-logistic units that is unbiased and scales well with the number of network's units in terms of learning speed.

For a network of stochastic units trained on a reinforcement learning task, one biologically plausible way of learning is to treat each unit as a reinforcement learning unit and train each unit by REINFORCE using the same global reward signal. In this case, only a global reward signal has to be broadcast to all units, and the learning rule given is local. Although this learning rule follows the gradient of return in expectation, it suffers from high variance and cannot be used to train a deep network in practice. In this paper, we propose an algorithm called Weight Maximization, which can significantly improve the speed of applying REINFORCE to all units. Essentially, we replace the global reward to each hidden unit with the change in the norm of output weights, such that each hidden unit in the network is trying to maximize the norm of output weights instead of the global reward. We found that the new algorithm can solve simple reinforcement learning tasks significantly faster than the baseline model. We also prove that the resulting learning rule is approximately following gradient ascent on the reward in expectation when applied to a multi-layer network of Bernoulli logistic unit. It illustrates an example of intelligent behavior arising from a population of self-interested hedonistic neurons, which corresponds to Klopf's hedonistic neuron hypothesis.