Fast Sampling-Based Inference in Balanced Neuronal Networks
Multiple lines of evidence support the notion that the brain performs probabilistic inference in multiple cognitive domains, including perception and decision making. There is also evidence that probabilistic inference may be implemented in the brain through the (quasi-)stochastic activity of neural circuits, producing samples from the appropriate posterior distributions, effectively implementing a Markov chain Monte Carlo algorithm. However, time becomes a fundamental bottleneck in such sampling-based probabilistic representations: the quality of inferences depends on how fast the neural circuit generates new, uncorrelated samples from its stationary distribution (the posterior). We explore this bottleneck in a simple, linear-Gaussian latent variable model, in which posterior sampling can be achieved by stochastic neural networks with linear dynamics. The well-known Langevin sampling (LS) recipe, so far the only sampling algorithm for continuous variables of which a neural implementation has been suggested, naturally fits into this dynamical framework. However, we first show analytically and through simulations that the symmetry of the synaptic weight matrix implied by LS yields critically slow mixing when the posterior is high-dimensional. Next, using methods from control theory, we construct and inspect networks that are optimally fast, and hence orders of magnitude faster than LS, while being far more biologically plausible. In these networks, strong -but transient -selective amplification of external noise generates the spatially correlated activity fluctuations prescribed by the posterior. Intriguingly, although a detailed balance of excitation and inhibition is dynamically maintained, detailed balance of Markov chain steps in the resulting sampler is violated, consistent with recent findings on how statistical irreversibility can overcome the speed limitation of random walks in other domains.
Summary: This paper deals with sampling methods based on linear rate-based neural-networks. First, it shows that symmetric weights (a common constraint in many models) significantly hurt the mixing rate. Then it shows that a (more physiological) non-normal network can have a much faster mixing rate, if the connectivity is optimized for this purpose. This works even if more biological constraints (Dale's law) are imposed.Clarity: Very clear and well organized paper.Originality: To the best of my knowledge, these results are novel.Significance: This is not completely clear too me. The importance of this paper mainly depends on how does this generalizes to non-linear networks - which are more physiological, and can be used to generate more sophisticated samplers. Perhaps this could be shown in a simulation. If this does not generalize, then the significance critically depends on how the issues below are resolved.Quality: The derivations appear to be correct and elegant. However, I'm quite confused as to what is the advantage of the suggested optimized networks. Given a linear network as defined in the paper, it is straightforward to construct a network that mixes as fast as possible (with timescale \tau_m) - if we are allowed to use more neurons in the network than components in the sampled vector (r). Simply have two layers (r & h) of neurons with no connections within each layer, and a feedforward connectivity \sqrt(\Sigma-I) from the 'hidden' layer h to the visible layer r. Then it straightforward to show that we can sample from the r layer with the timescale of a single neuron. Clearly, this is similar to the pre-conditioned solution, except the noise sources are not external. Note that you can make it more physiological by adding recurrent connections to the hidden layer (and modifying accordingly the feedforward connectivity to \sqrt(\Sigma-I)*A, where A is the appropriate 'whitening' matrix) and outgoing connections from the r layer (to some other, external neurons). Also we can maintain Dale's Law by summing together several hidden layers. I'm not sure what is the problem with such a simple solution. My final assessment of this paper will mainly depend on the answer to this question, as well the following (closely related) questions:1) What is the asymptotic scaling of slowing factor (\tau_max / \tau_m) for the optimized networks? Can you always maintain a single neuron timescale for the mixing, as N increases? Or do you have to decrease the simulation timestep in order to maintain numerical stability? If so, by how much?2) Why do you need the L2-norm penalty on Eq. 13? Is it to maintain numerical stability? Does the network has to be close the edge of stability in order to mix fast?*** After Author's feedback ***The author feed back have mostly satisfied my objections. I'm still not convinced that my suggestion is not biophysically plausible. If the hidden neurons have recurrent connections, which are then "compensated for" by the feedforward connectivity to the visible neurons, it is not necessarily wasteful, since the hidden neurons can simultaneously do something else (e.g., not sampling). In other words, using feedforward connectivity, you can use any other part of the brain as a stochastic "source" which generates fast sampling. It is not clear to me how this would be non-consistent with literature. Anyway, if you can show that your method works also for non-linear networks, this makes this issue less important.