We address the unsolved algorithm design problem of choosing a justified regularization parameter in unsupervised domain adaptation. This problem is intriguing as no labels are available in the target domain. Our approach starts with the observation that the widely-used method of minimizing the source error, penalized by a distance measure between source and target feature representations, shares characteristics with regularized ill-posed inverse problems. Regularization parameters in inverse problems are optimally chosen by the fundamental principle of balancing approximation and sampling errors. We use this principle to balance learning errors and domain distance in a target error bound. As a result, we obtain a theoretically justified rule for the choice of the regularization parameter. In contrast to the state of the art, our approach allows source and target distributions with disjoint supports. An empirical comparative study on benchmark datasets underpins the performance of our approach.

Generative Adversarial Networks (GANs) excel at creating realistic images with complex models for which maximum likelihood is infeasible. However, the convergence of GAN training has still not been proved. We propose a two time-scale update rule (TTUR) for training GANs with stochastic gradient descent on arbitrary GAN loss functions. TTUR has an individual learning rate for both the discriminator and the generator. Using the theory of stochastic approximation, we prove that the TTUR converges under mild assumptions to a stationary local Nash equilibrium. The convergence carries over to the popular Adam optimization, for which we prove that it follows the dynamics of a heavy ball with friction and thus prefers flat minima in the objective landscape. For the evaluation of the performance of GANs at image generation, we introduce the `Fréchet Inception Distance'' (FID) which captures the similarity of generated images to real ones better than the Inception Score. In experiments, TTUR improves learning for DCGANs and Improved Wasserstein GANs (WGAN-GP) outperforming conventional GAN training on CelebA, CIFAR-10, SVHN, LSUN Bedrooms, and the One Billion Word Benchmark.

Even in the absence of sensory stimulation the brain is spontaneously active. This background "noise" seems to be the dominant cause of the notoriously high trial-to-trial variability of neural recordings. Recent experimental observations have extended our knowledge of trial-to-trial variability and spontaneous activity in several directions: 1. Trial-to-trial variability systematically decreases following the onset of a sensory stimulus or the start of a motor act. 2. Spontaneous activity states in sensory cortex outline the region of evoked sensory responses. 3. Across development, spontaneous activity aligns itself with typical evoked activity patterns. 4. The spontaneous brain activity prior to the presentation of an ambiguous stimulus predicts how the stimulus will be interpreted. At present it is unclear how these observations relate to each other and how they arise in cortical circuits. Here we demonstrate that all of these phenomena can be accounted for by a deterministic self-organizing recurrent neural network model (SORN), which learns a predictive model of its sensory environment. The SORN comprises recurrently coupled populations of excitatory and inhibitory threshold units and learns via a combination of spike-timing dependent plasticity (STDP) and homeostatic plasticity mechanisms. Similar to balanced network architectures, units in the network show irregular activity and variable responses to inputs. Additionally, however, the SORN exhibits sequence learning abilities matching recent findings from visual cortex and the network's spontaneous activity reproduces the experimental findings mentioned above. Intriguingly, the network's behaviour is reminiscent of sampling-based probabilistic inference, suggesting that correlates of sampling-based inference can develop from the interaction of STDP and homeostasis in deterministic networks. We conclude that key observations on spontaneous brain activity and the variability of neural responses can be accounted for by a simple deterministic recurrent neural network which learns a predictive model of its sensory environment via a combination of generic neural plasticity mechanisms.

In order to cross a street without being run over, we need to be able to extract very fast hidden causes of dynamically changing multi-modal sensory stimuli, and to predict their future evolution. We show here that a generic cortical microcircuit motif, pyramidal cells with lateral excitation and inhibition, provides the basis for this difficult but all-important information processing capability. This capability emerges in the presence of noise automatically through effects of STDP on connections between pyramidal cells in Winner-Take-All circuits with lateral excitation. In fact, one can show that these motifs endow cortical microcircuits with functional properties of a hidden Markov model, a generic model for solving such tasks through probabilistic inference. Whereas in engineering applications this model is adapted to specific tasks through offline learning, we show here that a major portion of the functionality of hidden Markov models arises already from online applications of STDP, without any supervision or rewards. We demonstrate the emergent computing capabilities of the model through several computer simulations. The full power of hidden Markov model learning can be attained through reward-gated STDP. This is due to the fact that these mechanisms enable a rejection sampling approximation to theoretically optimal learning. We investigate the possible performance gain that can be achieved with this more accurate learning method for an artificial grammar task.

The principles by which networks of neurons compute, and how spike-timing dependent plasticity (STDP) of synaptic weights generates and maintains their computational function, are unknown. Preceding work has shown that soft winner-take-all (WTA) circuits, where pyramidal neurons inhibit each other via interneurons, are a common motif of cortical microcircuits. We show through theoretical analysis and computer simulations that Bayesian computation is induced in these network motifs through STDP in combination with activity-dependent changes in the excitability of neurons. The fundamental components of this emergent Bayesian computation are priors that result from adaptation of neuronal excitability and implicit generative models for hidden causes that are created in the synaptic weights through STDP. In fact, a surprising result is that STDP is able to approximate a powerful principle for fitting such implicit generative models to high-dimensional spike inputs: Expectation Maximization. Our results suggest that the experimentally observed spontaneous activity and trial-to-trial variability of cortical neurons are essential features of their information processing capability, since their functional role is to represent probability distributions rather than static neural codes. Furthermore it suggests networks of Bayesian computation modules as a new model for distributed information processing in the cortex.

Recent spiking network models of Bayesian inference and unsupervised learning frequently assume either inputs to arrive in a special format or employ complex computations in neuronal activation functions and synaptic plasticity rules. Here we show in a rigorous mathematical treatment how homeostatic processes, which have previously received little attention in this context, can overcome common theoretical limitations and facilitate the neural implementation and performance of existing models. In particular, we show that homeostatic plasticity can be understood as the enforcement of a 'balancing' posterior constraint during probabilistic inference and learning with Expectation Maximization. We link homeostatic dynamics to the theory of variational inference, and show that nontrivial terms, which typically appear during probabilistic inference in a large class of models, drop out. We demonstrate the feasibility of our approach in a spiking Winner-Take-All architecture of Bayesian inference and learning. Finally, we sketch how the mathematical framework can be extended to richer recurrent network architectures. Altogether, our theory provides a novel perspective on the interplay of homeostatic processes and synaptic plasticity in cortical microcircuits, and points to an essential role of homeostasis during inference and learning in spiking networks.

The organization of computations in networks of spiking neurons in the brain is still largely unknown, in particular in view of the inherently stochastic features of their firing activity and the experimentally observed trial-to-trial variability of neural systems in the brain. In principle there exists a powerful computational framework for stochastic computations, probabilistic inference by sampling, which can explain a large number of macroscopic experimental data in neuroscience and cognitive science. But it has turned out to be surprisingly difficult to create a link between these abstract models for stochastic computations and more detailed models of the dynamics of networks of spiking neurons. Here we create such a link and show that under some conditions the stochastic firing activity of networks of spiking neurons can be interpreted as probabilistic inference via Markov chain Monte Carlo (MCMC) sampling. Since common methods for MCMC sampling in distributed systems, such as Gibbs sampling, are inconsistent with the dynamics of spiking neurons, we introduce a different approach based on non-reversible Markov chains that is able to reflect inherent temporal processes of spiking neuronal activity through a suitable choice of random variables. We propose a neural network model and show by a rigorous theoretical analysis that its neural activity implements MCMC sampling of a given distribution, both for the case of discrete and continuous time. This provides a step towards closing the gap between abstract functional models of cortical computation and more detailed models of networks of spiking neurons.

We introduce a framework for decision making in which the learning of decision making is reduced to its simplest and biologically most plausible form: Hebbian learning on a linear neuron. We cast our Bayesian-Hebb learning rule as reinforcement learning in which certain decisions are rewarded and prove that each synaptic weight will on average converge exponentially fast to the log-odd of receiving a reward when its pre- and postsynaptic neurons are active. In our simple architecture, a particular action is selected from the set of candidate actions by a winner-take-all operation. The global reward assigned to this action then modulates the update of each synapse. Apart from this global reward signal, our reward-modulated Bayesian Hebb rule is a pure Hebb update that depends only on the coactivation of the pre- and postsynaptic neurons, not on the weighted sum of all presynaptic inputs to the postsynaptic neuron as in the perceptron learning rule or the Rescorla-Wagner rule. This simple approach to action-selection learning requires that information about sensory inputs be presented to the Bayesian decision stage in a suitably preprocessed form resulting from other adaptive processes (acting on a larger timescale) that detect salient dependencies among input features. Hence our proposed framework for fast learning of decisions also provides interesting new hypotheses regarding neural nodes and computational goals of cortical areas that provide input to the final decision stage.

The principles by which spiking neurons contribute to the astounding computational power of generic cortical microcircuits, and how spike-timing-dependent plasticity (STDP) of synaptic weights could generate and maintain this computational function, are unknown. We show here that STDP, in conjunction with a stochastic soft winner-take-all (WTA) circuit, induces spiking neurons to generate through their synaptic weights implicit internal models for subclasses (or causes") of the high-dimensional spike patterns of hundreds of pre-synaptic neurons. Hence these neurons will fire after learning whenever the current input best matches their internal model. The resulting computational function of soft WTA circuits, a common network motif of cortical microcircuits, could therefore be a drastic dimensionality reduction of information streams, together with the autonomous creation of internal models for the probability distributions of their input patterns. We show that the autonomous generation and maintenance of this computational function can be explained on the basis of rigorous mathematical principles. In particular, we show that STDP is able to approximate a stochastic online Expectation-Maximization (EM) algorithm for modeling the input data. A corresponding result is shown for Hebbian learning in artificial neural networks."

Uncertainty is omnipresent when we perceive or interact with our environment, and the Bayesian framework provides computational methods for dealing with it. Mathematical models for Bayesian decision making typically require datastructures that are hard to implement in neural networks. This article shows that even the simplest and experimentally best supported type of synaptic plasticity, Hebbian learning, in combination with a sparse, redundant neural code, can in principle learn to infer optimal Bayesian decisions. We present a concrete Hebbian learning rule operating on log-probability ratios. Modulated by reward-signals, this Hebbian plasticity rule also provides a new perspective for understanding how Bayesian inference could support fast reinforcement learning in the brain. In particular we show that recent experimental results by Yang and Shadlen [1] on reinforcement learning of probabilistic inference in primates can be modeled in this way.