arXiv subjects

Amol Yerudkar

Publications and source records attributed to Amol Yerudkar.

3 recordsLinked to original sources

arXiv2026

On the Nesterov's acceleration: A NAIM perspective

We present a unifying Nearly Asymptotically Invariant Manifold (NAIM) framework for understanding Nesterovs Accelerated Gradient (NAG) method. By lifting the first-order gradient flow into a second-order phase space we construct a NAIM a slow, attracting graph and show that acceleration emerges from a curvature aware perturbation of this graph. The evolving slope of the perturbed manifold is governed by a Differential Riccati Equation (DRE), which enforces strict tangency of the vector field to the manifold surface. In the quadratic case the DRE reduces to an Algebraic Riccati Equation (ARE), and the requirement of spectral resonance equal contraction rates across all curvature modes uniquely determines the damping coefficient, directly yielding the continuous time Nesterov ODE. Fenichels theorem then extends this picture rigorously to general smooth, strongly convex landscapes: normal hyperbolicity guarantees persistence of the accelerated manifold despite varying Hessian curvature. The method is further extended to unified geometric derivation of NAG methods for smooth convex and strongly convex optimization in the discrete case. We exploit the underlying geometric structure and derive both cases from the same principle of preserving the projective structure under discretization process. A Lie Trotter splitting separates the linear dissipative dynamics from the nonlinear gradient flow. The dissipative subsystem is integrated by the Cayley (bilinear) transform, which preserves the underlying projective (Mobius) structure unconditionally and produces the classical Nesterov momentum coefficient as the unique Pade multiplier. For the convex case, projective flatness (vanishing Schwarzian derivative) uniquely selects the time-varying damping recovering the canonical Nesterov ODE for convex functions.

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arXiv2025

On the Dynamics of Acceleration in First order Gradient Methods

Ever since the original algorithm by Nesterov (1983), the true nature of the acceleration phenomenon has remained elusive, with various interpretations of why the method is actually faster. The diagnosis of the algorithm through the lens of Ordinary Differential Equations (ODEs) and the corresponding dynamical system formulation to explain the underlying dynamics has a rich history. In the literature, the ODEs that explain algorithms are typically derived by considering the limiting case of the algorithm maps themselves, that is, an ODE formulation follows the development of an algorithm. This obfuscates the underlying higher order principles and thus provides little evidence of the working of the algorithm. Such has been the case with Nesterov algorithm and the various analogies used to describe the acceleration phenomena, viz, momentum associated with the rolling of a Heavy-Ball down a slope, Hessian damping etc. The main focus of our work is to ideate the genesis of the Nesterov algorithm from the viewpoint of dynamical systems leading to demystifying the mathematical rigour behind the algorithm. Instead of reverse engineering ODEs from discrete algorithms, this work explores tools from the recently developed control paradigm titled Passivity and Immersion approach and the Geometric Singular Perturbation theory which are applied to arrive at the formulation of a dynamical system that explains and models the acceleration phenomena. This perspective helps to gain insights into the various terms present and the sequence of steps used in Nesterovs accelerated algorithm for the smooth strongly convex and the convex case. The framework can also be extended to derive the acceleration achieved using the triple momentum method and provides justifications for the non-convergence to the optimal solution in the Heavy-Ball method.

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arXiv2022

A Model-Based Reinforcement Learning Approach for PID Design

Proportional-integral-derivative (PID) controller is widely used across various industrial process control applications because of its straightforward implementation. However, it can be challenging to fine-tune the PID parameters in practice to achieve robust performance. The paper proposes a model-based reinforcement learning (RL) framework to design PID controllers leveraging the probabilistic inference for learning control (PILCO) method and Kullback-Leibler divergence (KLD). Since PID controllers have a much more interpretable control structure than a network basis function, an optimal policy given by PILCO is transformed into a set of robust PID tuning parameters for underactuated mechanical systems. The presented method is general and can blend with several model-based and model-free algorithms. The performance of the devised PID controllers is demonstrated with simulation studies for a benchmark cart-pole system under disturbances and system parameter uncertainties.

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