Thomas Chaffey

Thomas L. Chaffey, Ian R. Manchester

Control Contraction Metrics (CCMs) provide a nonlinear controller design involving an offline search for a Riemannian metric and an online search for a shortest path between the current and desired trajectories. In this paper, we generalize CCMs to Finsler geometry, allowing the use of non-Riemannian metrics. We provide open loop and sampled data controllers. The sampled data control construction presented here does not require real time computation of globally shortest paths, simplifying computation.

Ian R. Manchester, Thomas L. Chaffey

We provide an amendment to the first theorem of "Control Contraction Metrics: Convex and Intrinsic Criteria for Nonlinear Feedback Design" by Manchester & Slotine in the form of an additional technical condition required to show integrability of differential control signals. This technical condition is shown to be satisfied under the original assumptions if the input matrix is constant rank, and also if the strong conditions for a CCM hold. However a simple counterexample shows that if the input matrix drops rank, then the weaker conditions of the original theorem may not imply stabilizability of all trajectories. The remaining claims and illustrative examples of the paper are shown to remain valid with the new condition.

James Li, Philip H. W. Leong, Thomas Chaffey

Monotone operator equilibrium networks are implicit-layer models whose output is the unique equilibrium of a monotone operator, guaranteeing existence, uniqueness, and convergence. When deployed on low-precision hardware, weights are quantized, potentially destroying these guarantees. We analyze weight quantization as a spectral perturbation of the underlying monotone inclusion. Convergence of the quantized solver is guaranteed whenever the spectral-norm weight perturbation is smaller than the monotonicity margin; the displacement between quantized and full-precision equilibria is bounded in terms of the perturbation size and margin; and a condition number characterizing the ratio of the operator norm to the margin links quantization precision to forward error. MNIST experiments confirm a phase transition at the predicted threshold: three- and four-bit post-training quantization diverge, while five-bit and above converge. The backward-pass guarantee enables quantization-aware training, which recovers provable convergence at four bits.

Anne-Men Huijzer, Thomas Chaffey, Bart Besselink et al.

Energy-based learning algorithms are alternatives to backpropagation and are well-suited to distributed implementations in analog electronic devices. However, a rigorous theory of convergence is lacking. We make a first step in this direction by analysing a particular energy-based learning algorithm, Contrastive Learning, applied to a network of linear adjustable resistors. It is shown that, in this setup, Contrastive Learning is equivalent to projected gradient descent on a convex function, for any step size, giving a guarantee of convergence for the algorithm.

Thomas Chaffey

It is shown that the port behavior of a resistor-diode network corresponds to the solution of a ReLU monotone operator equilibrium network (a neural network in the limit of infinite depth), giving a parsimonious construction of a neural network in analog hardware. We furthermore show that the gradient of such a circuit can be computed directly in hardware, using a procedure we call hardware linearization. This allows the network to be trained in hardware, which we demonstrate with a device-level circuit simulation. We extend the results to cascades of resistor-diode networks, which can be used to implement feedforward and other asymmetric networks. We finally show that different nonlinear elements give rise to different activation functions, and introduce the novel diode ReLU which is induced by a non-ideal diode model.