Natural continual learning: success is a journey, not (just) a destination

Biological agents are known to learn many different tasks over the course of their lives, and to be able to revisit previous tasks and behaviors with little to no loss in performance. In contrast, artificial agents are prone to 'catastrophic forgetting' whereby performance on previous tasks deteriorates rapidly as new ones are acquired. This shortcoming has recently been addressed using methods that encourage parameters to stay close to those used for previous tasks. This can be done by (i) using specific parameter regularizers that map out suitable destinations in parameter space, or (ii) guiding the optimization journey by projecting gradients into subspaces that do not interfere with previous tasks. However, parameter regularization has been shown to be relatively ineffective in recurrent neural networks (RNNs), a setting relevant to the study of neural dynamics supporting biological continual learning. Similarly, projection based methods can reach capacity and fail to learn any further as the number of tasks increases. To address these limitations, we propose Natural Continual Learning (NCL), a new method that unifies weight regularization and projected gradient descent. NCL uses Bayesian weight regularization to encourage good performance on all tasks at convergence and combines this with gradient projections designed to prevent catastrophic forgetting during optimization. NCL formalizes gradient projection as a trust region algorithm based on the Fisher information metric, and achieves scalability via a novel Kronecker-factored approximation strategy. Our method outperforms both standard weight regularization techniques and projection based approaches when applied to continual learning problems in RNNs. The trained networks evolve task-specific dynamics that are strongly preserved as new tasks are learned, similar to experimental findings in biological circuits.

Automatic differentiation of Sylvester, Lyapunov, and algebraic Riccati equations

Sylvester, Lyapunov, and algebraic Riccati equations are the bread and butter of control theorists. They are used to compute infinite-horizon Gramians, solve optimal control problems in continuous or discrete time, and design observers. While popular numerical computing frameworks (e.g., scipy) provide efficient solvers for these equations, these solvers are still largely missing from most automatic differentiation libraries. Here, we derive the forward and reverse-mode derivatives of the solutions to all three types of equations, and showcase their application on an inverse control problem.

Manifold GPLVMs for discovering non-Euclidean latent structure in neural data

A common problem in neuroscience is to elucidate the collective neural representations of behaviorally important variables such as head direction, spatial location, upcoming movements, or mental spatial transformations. Often, these latent variables are internal constructs not directly accessible to the experimenter. Here, we propose a new probabilistic latent variable model to simultaneously identify the latent state and the way each neuron contributes to its representation in an unsupervised way. In contrast to previous models which assume Euclidean latent spaces, we embrace the fact that latent states often belong to symmetric manifolds such as spheres, tori, or rotation groups of various dimensions. We therefore propose the manifold Gaussian process latent variable model (mGPLVM), where neural responses arise from (i) a shared latent variable living on a specific manifold, and (ii) a set of non-parametric tuning curves determining how each neuron contributes to the representation. Cross-validated comparisons of models with different topologies can be used to distinguish between candidate manifolds, and variational inference enables quantification of uncertainty. We demonstrate the validity of the approach on several synthetic datasets and on calcium recordings from the ellipsoid body of Drosophila melanogaster. This circuit is known to encode head direction, and mGPLVM correctly recovers the ring topology expected from a neural population representing a single angular variable.