Hesameddin Mohammadi

Hesameddin Mohammadi, Meisam Razaviyayn, Mihailo R. Jovanović

We study momentum-based first-order optimization algorithms in which the iterations utilize information from the two previous steps and are subject to an additive white noise. This setup uses noise to account for uncertainty in either gradient evaluation or iteration updates, and it includes Polyak's heavy-ball and Nesterov's accelerated methods as special cases. For strongly convex quadratic problems, we use the steady-state variance of the error in the optimization variable to quantify noise amplification and identify fundamental stochastic performance tradeoffs. Our approach utilizes the Jury stability criterion to provide a novel geometric characterization of conditions for linear convergence, and it reveals the relation between the noise amplification and convergence rate as well as their dependence on the condition number and the constant algorithmic parameters. This geometric insight leads to simple alternative proofs of standard convergence results and allows us to establish ``uncertainty principle'' of strongly convex optimization: for the two-step momentum method with linear convergence rate, the lower bound on the product between the settling time and noise amplification scales quadratically with the condition number. Our analysis also identifies a key difference between the gradient and iterate noise models: while the amplification of gradient noise can be made arbitrarily small by sufficiently decelerating the algorithm, the best achievable variance for the iterate noise model increases linearly with the settling time in the decelerating regime. Finally, we introduce two parameterized families of algorithms that strike a balance between noise amplification and settling time while preserving order-wise Pareto optimality for both noise models.

Hesameddin Mohammadi, Mohammad Tinati, Stephen Tu et al.

The symmetric low-rank matrix factorization serves as a building block in many learning tasks, including matrix recovery and training of neural networks. However, despite a flurry of recent research, the dynamics of its training via non-convex factorized gradient-descent-type methods is not fully understood especially in the over-parameterized regime where the fitted rank is higher than the true rank of the target matrix. To overcome this challenge, we characterize equilibrium points of the gradient flow dynamics and examine their local and global stability properties. To facilitate a precise global analysis, we introduce a nonlinear change of variables that brings the dynamics into a cascade connection of three subsystems whose structure is simpler than the structure of the original system. We demonstrate that the Schur complement to a principal eigenspace of the target matrix is governed by an autonomous system that is decoupled from the rest of the dynamics. In the over-parameterized regime, we show that this Schur complement vanishes at an $O(1/t)$ rate, thereby capturing the slow dynamics that arises from excess parameters. We utilize a Lyapunov-based approach to establish exponential convergence of the other two subsystems. By decoupling the fast and slow parts of the dynamics, we offer new insight into the shape of the trajectories associated with local search algorithms and provide a complete characterization of the equilibrium points and their global stability properties. Such an analysis via nonlinear control techniques may prove useful in several related over-parameterized problems.

Hesameddin Mohammadi, Samantha Samuelson, Mihailo R. Jovanović

Optimization algorithms are increasingly being used in applications with limited time budgets. In many real-time and embedded scenarios, only a few iterations can be performed and traditional convergence metrics cannot be used to evaluate performance in these non-asymptotic regimes. In this paper, we examine the transient behavior of accelerated first-order optimization algorithms. For convex quadratic problems, we employ tools from linear systems theory to show that transient growth arises from the presence of non-normal dynamics. We identify the existence of modes that yield an algebraic growth in early iterations and quantify the transient excursion from the optimal solution caused by these modes. For strongly convex smooth optimization problems, we utilize the theory of integral quadratic constraints (IQCs) to establish an upper bound on the magnitude of the transient response of Nesterov's accelerated algorithm. We show that both the Euclidean distance between the optimization variable and the global minimizer and the rise time to the transient peak are proportional to the square root of the condition number of the problem. Finally, for problems with large condition numbers, we demonstrate tightness of the bounds that we derive up to constant factors.

Hesameddin Mohammadi, Armin Zare, Mahdi Soltanolkotabi et al.

Model-free reinforcement learning attempts to find an optimal control action for an unknown dynamical system by directly searching over the parameter space of controllers. The convergence behavior and statistical properties of these approaches are often poorly understood because of the nonconvex nature of the underlying optimization problems and the lack of exact gradient computation. In this paper, we take a step towards demystifying the performance and efficiency of such methods by focusing on the standard infinite-horizon linear quadratic regulator problem for continuous-time systems with unknown state-space parameters. We establish exponential stability for the ordinary differential equation (ODE) that governs the gradient-flow dynamics over the set of stabilizing feedback gains and show that a similar result holds for the gradient descent method that arises from the forward Euler discretization of the corresponding ODE. We also provide theoretical bounds on the convergence rate and sample complexity of the random search method with two-point gradient estimates. We prove that the required simulation time for achieving $ε$-accuracy in the model-free setup and the total number of function evaluations both scale as $\log \, (1/ε)$.

Hesameddin Mohammadi, Meisam Razaviyayn, Mihailo R. Jovanović

We study the robustness of accelerated first-order algorithms to stochastic uncertainties in gradient evaluation. Specifically, for unconstrained, smooth, strongly convex optimization problems, we examine the mean-squared error in the optimization variable when the iterates are perturbed by additive white noise. This type of uncertainty may arise in situations where an approximation of the gradient is sought through measurements of a real system or in a distributed computation over a network. Even though the underlying dynamics of first-order algorithms for this class of problems are nonlinear, we establish upper bounds on the mean-squared deviation from the optimal solution that are tight up to constant factors. Our analysis quantifies fundamental trade-offs between noise amplification and convergence rates obtained via any acceleration scheme similar to Nesterov's or heavy-ball methods. To gain additional analytical insight, for strongly convex quadratic problems, we explicitly evaluate the steady-state variance of the optimization variable in terms of the eigenvalues of the Hessian of the objective function. We demonstrate that the entire spectrum of the Hessian, rather than just the extreme eigenvalues, influence robustness of noisy algorithms. We specialize this result to the problem of distributed averaging over undirected networks and examine the role of network size and topology on the robustness of noisy accelerated algorithms.

Armin Zare, Hesameddin Mohammadi, Neil K. Dhingra et al.

Several problems in modeling and control of stochastically-driven dynamical systems can be cast as regularized semi-definite programs. We examine two such representative problems and show that they can be formulated in a similar manner. The first, in statistical modeling, seeks to reconcile observed statistics by suitably and minimally perturbing prior dynamics. The second seeks to optimally select a subset of available sensors and actuators for control purposes. To address modeling and control of large-scale systems we develop a unified algorithmic framework using proximal methods. Our customized algorithms exploit problem structure and allow handling statistical modeling, as well as sensor and actuator selection, for substantially larger scales than what is amenable to current general-purpose solvers. We establish linear convergence of the proximal gradient algorithm, draw contrast between the proposed proximal algorithms and alternating direction method of multipliers, and provide examples that illustrate the merits and effectiveness of our framework.