Control Meets Learning Seminar
The human brain still largely outperforms robotic algorithms in most tasks, using computational elements 7 orders of magnitude slower than their artificial counterparts. Similarly, current large scale machine learning algorithms require millions of examples and close proximity to power plants, compared to the brain's few examples and 20W consumption. We study how modern nonlinear systems tools, such as contraction analysis and virtual dynamical systems, can yield quantifiable insights about collective computation, adaptation, and learning in large dynamical networks.
In optimization, most elementary results on gradient descent based on convexity of a time-invariant cost can be replaced by much more general results based on contraction. For instance, natural gradient descent converges to a unique equilibrium if it is contracting in some metric, with geodesic convexity of the cost corresponding to the special case of contraction in the natural metric. Semi-contraction of natural gradient in some metric implies convergence to a global minimum, and furthermore that all global minima are path-connected. Similar results apply to primal-dual optimization and game-theoretic contexts.
Stable concurrent learning and control of dynamical systems is the subject of adaptive nonlinear control. When multiple parameter choices are consistent with the data (be it for insufficient richness of the task or aggressive overparametrization), stable Riemannian adaptation laws can be designed to implicitly regularize the learned model. Thus, local geometry imposed during learning may be used to select parameter vectors for desired properties such as sparsity. The results can also be systematically applied to predictors for dynamical systems. Stable implicit sparse regularization can be exploited as well to select relevant dynamic models out of plausible physically-based candidates, as we illustrate in the contexts of Hamiltonian systems and mass-action kinetics, We also derive finite-time regret bounds for adaptive control and prediction with matched uncertainty in the stochastic setting.
Contraction-based adaptive controllers or predictors can also be used in transfer learning or sim2real contexts, where a feedback controller or predictor has been carefully learned for a nominal system, but needs to remain effective in real-time in the presence of significant but structured variations in parameters.
Finally, a key aspect of contraction tools is that they also suggest systematic mechanisms to build progressively more refined networks and novel algorithms through stable accumulation of functional building blocks and motifs.