Travis E. Gibson

Travis E. Gibson, Amir Bashan, Hong-Tai Cao et al.

Microbiome-based stratification of healthy individuals into compositional categories, referred to as "community types", holds promise for drastically improving personalized medicine. Despite this potential, the existence of community types and the degree of their distinctness have been highly debated. Here we adopted a dynamic systems approach and found that heterogeneity in the interspecific interactions or the presence of strongly interacting species is sufficient to explain community types, independent of the topology of the underlying ecological network. By controlling the presence or absence of these strongly interacting species we can steer the microbial ecosystem to any desired community type. This open-loop control strategy still holds even when the community types are not distinct but appear as dense regions within a continuous gradient. This finding can be used to develop viable therapeutic strategies for shifting the microbial composition to a healthy configuration

Benjamin M. Jenkins, Anuradha M. Annaswamy, Eugene Lavretsky et al.

The convergence properties of adaptive systems in terms of excitation conditions on the regressor vector are well known. With persistent excitation of the regressor vector in model reference adaptive control the state error and the adaptation error are globally exponentially stable, or equivalently, exponentially stable in the large. When the excitation condition however is imposed on the reference input or the reference model state it is often incorrectly concluded that the persistent excitation in those signals also implies exponential stability in the large. The definition of persistent excitation is revisited so as to address some possible confusion in the adaptive control literature. It is then shown that persistent excitation of the reference model only implies local persistent excitation (weak persistent excitation). Weak persistent excitation of the regressor is still sufficient for uniform asymptotic stability in the large, but not exponential stability in the large. We show that there exists an infinite region in the state-space of adaptive systems where the state rate is bounded. This infinite region with finite rate of convergence is shown to exist not only in classic open-loop reference model adaptive systems, but also in a new class of closed-loop reference model adaptive systems.

Eugene Lavretsky, Travis E. Gibson

The projection algorithm is frequently used in adaptive control and this note presents a detailed analysis of its properties.

Travis E. Gibson, Anuradha M. Annaswamy, Eugene Lavretsky

This paper explores the properties of adaptive systems with closed-loop reference models. Using additional design freedom available in closed-loop reference models, we design new adaptive controllers that are (a) stable, and (b) have improved transient properties. Numerical studies that complement theoretical derivations are also reported.

Travis E. Gibson, Anuradha M. Annaswamy, Eugene Lavretsky

Closed-loop reference models have recently been proposed for states accessible adaptive systems. They have been shown to have improved transient response over their open loop counter parts. The results in the states accessible case are extended to single input single output plants of arbitrary relative degree.

Travis E. Gibson

With the rise of network science old topics in ecology and economics are resurfacing. One such topic is structural stability (often referred to as qualitative stability or sign stability). A system is deemed structurally stable if the system remains stable for all possible parameter variations so long as the parameters do not change sign. This type of stability analysis is appealing when studying real systems as the underlying stability result only requires the scientist or engineer to know the sign of the parameters in the model and not the specific values. The necessary and sufficient conditions for qualitative stability however are opaque. In order to shed light on those conditions root locus analysis is employed. This technique allows us to illustrate the necessary conditions for qualitative stability.

Younhun Kim, Georg K. Gerber, Travis E. Gibson

Microbiome functions are encoded within the genes of the community-wide metagenome. A natural question is whether properties of a microbial community can be predicted just from knowing the raw DNA sequences of its members. In this work, we employ set-aggregated genome embeddings (SAGE) to predict community-level abundance profiles, exploiting the few-shot learning capabilities of genomic language models (GLMs). We benchmark this approach to show improved generalization on novel genomes compared to classical bioinformatics approaches. Model ablation shows that community-level latent representations directly result in improved performance. Lastly, we demonstrate the benefits of intermediate transformations between latent representations and demonstrate the differences between GLM embedding choices.

Travis E. Gibson, Sawal Acharya

Online learning and model reference adaptive control have many interesting intersections. One area where they differ however is in how the algorithms are analyzed and what objective or metric is used to discriminate "good" algorithms from "bad" algorithms. In adaptive control there are usually two objectives: 1) prove that all time varying parameters/states of the system are bounded, and 2) that the instantaneous error between the adaptively controlled system and a reference system converges to zero over time (or at least a compact set). For online learning the performance of algorithms is often characterized by the regret the algorithm incurs. Regret is defined as the cumulative loss (cost) over time from the online algorithm minus the cumulative loss (cost) of the single optimal fixed parameter choice in hindsight. Another significant difference between the two areas of research is with regard to the assumptions made in order to obtain said results. Adaptive control makes assumptions about the input-output properties of the control problem and derives solutions for a fixed error model or optimization task. In the online learning literature results are derived for classes of loss functions (i.e. convex) while a priori assuming certain signals are bounded. In this work we discuss these differences in detail through the regret based analysis of gradient descent for convex functions and the control based analysis of a streaming regression problem. We close with a discussion about the newly defined paradigm of online adaptive control.

Travis E. Gibson, Sawal Acharya, Anjali Parashar et al.

Gradient based optimization algorithms deployed in Machine Learning (ML) applications are often analyzed and compared by their convergence rates or regret bounds. While these rates and bounds convey valuable information they don't always directly translate to stability guarantees. Stability and similar concepts, like robustness, will become ever more important as we move towards deploying models in real-time and safety critical systems. In this work we build upon the results in Gaudio et al. 2021 and Moreu & Annaswamy 2022 for gradient descent with second order dynamics when applied to explicitly time varying cost functions and provide more general stability guarantees. These more general results can aid in the design and certification of these optimization schemes so as to help ensure safe and reliable deployment for real-time learning applications. We also hope that the techniques provided here will stimulate and cross-fertilize the analysis that occurs on the same algorithms from the online learning and stochastic optimization communities.

Joseph E. Gaudio, Anuradha M. Annaswamy, José M. Moreu et al.

High order momentum-based parameter update algorithms have seen widespread applications in training machine learning models. Recently, connections with variational approaches have led to the derivation of new learning algorithms with accelerated learning guarantees. Such methods however, have only considered the case of static regressors. There is a significant need for parameter update algorithms which can be proven stable in the presence of adversarial time-varying regressors, as is commonplace in control theory. In this paper, we propose a new discrete time algorithm which 1) provides stability and asymptotic convergence guarantees in the presence of adversarial regressors by leveraging insights from adaptive control theory and 2) provides non-asymptotic accelerated learning guarantees leveraging insights from convex optimization. In particular, our algorithm reaches an $ε$ sub-optimal point in at most $\tilde{\mathcal{O}}(1/\sqrtε)$ iterations when regressors are constant - matching lower bounds due to Nesterov of $Ω(1/\sqrtε)$, up to a $\log(1/ε)$ factor and provides guaranteed bounds for stability when regressors are time-varying. We provide numerical experiments for a variant of Nesterov's provably hard convex optimization problem with time-varying regressors, as well as the problem of recovering an image with a time-varying blur and noise using streaming data.

Joseph E. Gaudio, Travis E. Gibson, Anuradha M. Annaswamy et al.

This paper demonstrates many immediate connections between adaptive control and optimization methods commonly employed in machine learning. Starting from common output error formulations, similarities in update law modifications are examined. Concepts in stability, performance, and learning, common to both fields are then discussed. Building on the similarities in update laws and common concepts, new intersections and opportunities for improved algorithm analysis are provided. In particular, a specific problem related to higher order learning is solved through insights obtained from these intersections.

Joseph E. Gaudio, Travis E. Gibson, Anuradha M. Annaswamy et al.

Features in machine learning problems are often time-varying and may be related to outputs in an algebraic or dynamical manner. The dynamic nature of these machine learning problems renders current higher order accelerated gradient descent methods unstable or weakens their convergence guarantees. Inspired by methods employed in adaptive control, this paper proposes new algorithms for the case when time-varying features are present, and demonstrates provable performance guarantees. In particular, we develop a unified variational perspective within a continuous time algorithm. This variational perspective includes higher order learning concepts and normalization, both of which stem from adaptive control, and allows stability to be established for dynamical machine learning problems where time-varying features are present. These higher order algorithms are also examined for provably correct learning in adaptive control and identification. Simulations are provided to verify the theoretical results.

Travis E. Gibson, Georg K. Gerber

Microbes are everywhere, including in and on our bodies, and have been shown to play key roles in a variety of prevalent human diseases. Consequently, there has been intense interest in the design of bacteriotherapies or "bugs as drugs," which are communities of bacteria administered to patients for specific therapeutic applications. Central to the design of such therapeutics is an understanding of the causal microbial interaction network and the population dynamics of the organisms. In this work we present a Bayesian nonparametric model and associated efficient inference algorithm that addresses the key conceptual and practical challenges of learning microbial dynamics from time series microbe abundance data. These challenges include high-dimensional (300+ strains of bacteria in the gut) but temporally sparse and non-uniformly sampled data; high measurement noise; and, nonlinear and physically non-negative dynamics. Our contributions include a new type of dynamical systems model for microbial dynamics based on what we term interaction modules, or learned clusters of latent variables with redundant interaction structure (reducing the expected number of interaction coefficients from $O(n^2)$ to $O((\log n)^2)$); a fully Bayesian formulation of the stochastic dynamical systems model that propagates measurement and latent state uncertainty throughout the model; and introduction of a temporally varying auxiliary variable technique to enable efficient inference by relaxing the hard non-negativity constraint on states. We apply our method to simulated and real data, and demonstrate the utility of our technique for system identification from limited data and gaining new biological insights into bacteriotherapy design.