Gradient-Based Adaptive Prediction and Control for Nonlinear Dynamical Systems

This paper investigates gradient-based adaptive prediction and control for nonlinear stochastic dynamical systems under a weak convexity condition on the prediction-based loss. This condition accommodates a broad range of nonlinear models in control and machine learning such as saturation functions, sigmoid, ReLU and tanh activation functions, and standard classification models. Without requiring any persistent excitation of the data, we establish global convergence of the proposed adaptive predictor and derive explicit rates for its asymptotic performance. Furthermore, under a classical nonlinear minimum-phase condition and with a linear growth bound on the nonlinearities, we establish the convergence rate of the resulting closed-loop control error. Finally, we demonstrate the effectiveness of the proposed adaptive prediction algorithm on a real-world judicial sentencing dataset. The adaptive control performance will also be evaluated via a numerical simulation.

💡 Research Summary

This paper addresses the problem of adaptive prediction and control for nonlinear stochastic dynamical systems by introducing a gradient‑based algorithm that operates under a weak convexity condition rather than the traditional strong convexity or persistent excitation (PE) assumptions. The authors first formulate the system as yₖ₊₁ = g(ϕₖ, θ*, eₖ₊₁) and define a prediction‑based loss Jₖ(θ) = L(f(ϕₖ, θ*), f(ϕₖ, θ)). The key theoretical assumption is a weak convexity inequality ∇Jₖ(θ)ᵀ(θ − θ*) ≥ δ Jₖ(θ) with δ∈(0,1] together with a bound on the gradient of the loss. This condition is much milder than strong convexity and is satisfied by a broad class of nonlinearities common in machine learning (ReLU, sigmoid, saturation, etc.) and in control‑oriented models.

The proposed adaptive predictor updates the parameter estimate θₖ via a stochastic gradient step θₖ₊₁ = θₖ − μₖ ∇f(ϕₖ, θₖ) ∇ₓL(yₖ₊₁, f(ϕₖ, θₖ)), where the learning rate μₖ is defined as μ rₖ^{β₁} log(β₂ rₖ) + ‖∇f(ϕₖ, θₖ)‖². Here rₖ = β₃ + ∑_{t=1}^k ‖∇f(ϕ_t, θ_t)‖² accumulates past gradient magnitudes, and the tunable constants β₁∈