An upper bound on the error of alignment-based Transfer Learning between two linear, time-invariant, scalar systems

Abstract

Methods from machine learning have successfully been used to improve the performance of control systems in cases when accurate models of the system or the environment are not available. These methods require the use of data generated from physical trials. Transfer Learning (TL) allows for this data to come from a different, similar system. This paper studies a simplified TL scenario with the goal of understanding in which cases a simple, alignment-based transfer of data is possible and beneficial. Two linear, time-invariant (LTI), single-input, single-output systems are tasked to follow the same reference signal. A scalar, LTI transformation is applied to the output from a source system to align with the output from a target system. An upper bound on the 2-norm of the transformation error is derived for a large set of reference signals and is minimized with respect to the transformation scalar. Analysis shows that the minimized error bound is reduced for systems with poles that lie close to each other (that is, for systems with similar response times). This criterion is relaxed for systems with poles that have a larger negative real part (that is, for stable systems with fast response), meaning that poles can be further apart for the same minimized error bound. Additionally, numerical results show that using the reference signal as input to the transformation reduces the minimized bound further.

Multi-robot transfer learning: A dynamical system perspective

“…To make the transfer learning problem tractable, we assume that both the source system D S and the target system D T are input-output stable (this is typically characterized by the BIBO stability notion for LTI systems and by the Input to Output Stability (IOS) notion for nonlinear systems [23]). This is a reasonable assumption, given that input-output stability is necessary for the safe operation of the robot and transfer learning is only efficient for stable systems [9].…”

“…As partially stated in [9], although multi-robot transfer learning has been successfully applied in some robotic examples, there is still an urgent need for a general, theoretical study of when multi-robot transfer learning is beneficial, how the dynamics of the considered robots affect the quality of transfer learning, what form the optimal transfer map takes, and how to efficiently identify the transfer map from a few experiments. To fill this gap, the authors of [9] recently initiated a study along these lines for two first-order, linear time-invariant (LTI), single-input single-output (SISO) systems. In particular, in [9], a simple, constant scalar is applied to align the output of the source system with the output of the target system, and then an upper bound on the Euclidean norm of the transformation error is derived and minimized with respect to (w.r.t.)…”

“…To fill this gap, the authors of [9] recently initiated a study along these lines for two first-order, linear time-invariant (LTI), single-input single-output (SISO) systems. In particular, in [9], a simple, constant scalar is applied to align the output of the source system with the output of the target system, and then an upper bound on the Euclidean norm of the transformation error is derived and minimized with respect to (w.r.t.) the transformation parameter.…”

“…the transformation parameter. The paper [9] also utilizes the derived, minimized upper bound to analyze the effect of the dynamics of the source and target systems on the quality of transfer learning.…”

Data-Efficient Multirobot, Multitask Transfer Learning for Trajectory Tracking

“…In the realm of multi-robot transfer, the work in [4] learns an optimal static gain between the outputs of two linear, timeinvariant (LTI), single-input, single-output (SISO) systems when they try to follow the same reference. The transformation error is minimized when the two systems have stable poles that lie close to each other.…”