The approach proposed in this paper enables a more capable DDDAS paradigm by improving the efficiency of the data-model-optimization loop by extracting a generalization bound based on Rademacher complexity that reveals the role of the $k-neighborhoods and related parameters.
Abstract
Data scarcity poses a fundamental challenge in training generative models to produce initial guesses for parametric optimization problems that are otherwise numerically expensive to solve. We therefore study a $k$-neighborhood data collection strategy that augments datasets of converged solutions with intermediate solver iterates, increasing the amount of training data without additional solver runs. To understand the benefits of this approach, we derive a generalization bound based on Rademacher complexity that reveals the role of the $k$-neighborhoods and related parameters. To achieve this result, we focus on one-sided box-constrained quadratic programs solved by projected gradient descent. We illustrate the behavior of this solver on two examples. The approach proposed in this paper enables a more capable DDDAS paradigm by improving the efficiency of the data-model-optimization loop. We finish by discussing two views of learning solver-iterate data and connect our analysis with GLENS, a new data-efficient global search method.
Abstract.
We introduce a machine-learning framework to learn the hyperparameter sequence of first-order methods (e.g. , the step sizes in gradient descent) to quickly solve parametric convex optimization problems. Our computational architecture amounts to running fixed-point iterations where the hyperparameters are the same across all parametric instances and consists of two phases. In the first step-varying phase the hyperparameters vary across iterations, while in the second steady-state phase the hyperparameters are constant across iterations. Our learned optimizer is flexible, in that it can be evaluated on any number of iterations and is guaranteed to converge to an optimal solution. To train, we minimize the mean square error to a ground truth solution. In the case of gradient descent, the one-step optimal step size is the solution to a least squares problem, and in the case of unconstrained quadratic minimization, we can compute the two-step and three-step optimal solutions in closed form. In other cases, we backpropagate through the algorithm steps to minimize the training objective after a given number of steps. We show how to learn hyperparameters for several popular algorithms: gradient descent, proximal gradient descent, and two ADMM-based solvers: OSQP and SCS. We use a sample convergence bound to obtain generalization guarantees for the performance of our learned algorithm for unseen data, providing both lower and upper bounds. We showcase the effectiveness of our method with many examples, including ones from control, signal processing, and machine learning. Remarkably, our approach is highly data-efficient in that we only use 10 problem instances to train the hyperparameters in all of our examples.
Rajiv Sambharya, Bartolomeo Stellato· SIAM Journal on Mathematics...· 0 citations
It is well known that Newton's method converges faster when the initial guess is closer to a root of a system of nonlinear equations. In this paper, a two-stage Newton initial guess strategy is proposed by learning features from a parameter-space sampling and a database of precomputed solutions. The method uses discrete Newton trajectories to construct two complementary reduced spaces: a solution feature space, built from converged states, and a corrective search direction feature space, built from intermediate Newton increments. For an unseen parameter, a regression model is used to predict a surrogate solution approximation. Then, in a second step, a residual-minimizing correction is computed using a dedicated GMRES-based approach. The resulting state is then used as an initial guess for the high-fidelity Newton method, which completes convergence. The corrective step is computationally inexpensive since it only requires residual evaluations and the solution of a small least-squares problem. The methodology is weakly intrusive once the high-fidelity residual fields and a script-based programming interface are available. This strategy reduces the number of Newton iterations and decreases the overall CPU time. Numerical experiments on representative PDE problems show quantifiable speedups compared with standalone surrogate initialization. Significant speedups are observed. This generic approach can be applied to a broad class of large-scale nonlinear problems.
Rémy Vallot, F. Vuyst, T. Dairay et al.· 0 citations
Abstract.
We propose an adaptive randomized truncation estimator for Krylov subspace methods that optimizes the trade-off between the solution variance and the computational cost while remaining unbiased. The estimator solves a constrained optimization problem to compute the truncation probabilities on the fly with minimal computational overhead. The problem has a closed-form solution when the improvement of the deterministic algorithm satisfies a diminishing returns property. We prove that obtaining the optimal adaptive truncation distribution is impossible in the general case. Without the diminishing return condition, our estimator provides a suboptimal but still unbiased solution. We present experimental results in Gaussian process (GP) hyperparameter training and competitive physics-informed neural networks problems to demonstrate the effectiveness of our approach.
Reproducibility of computational results. This paper has been awarded the “SIAM Reproducibility Badge: Code and data available” as a recognition that the authors have followed reproducibility principles valued by SISC and the scientific computing community. Code and data that allow readers to reproduce the results in this paper are available at https://github.com/RockyL7/AdaptivelySubsampledKrylov.jl and in the supplementary materials ( AdaptivelySubsampledKrylov_jl-master.zip [14.5KB]). [Formula: see text]
This work proposes a Hybrid Augmented Lagrangian (HyAL) method that combines the constraint-handling capabilities of the AL framework with the exploratory power of population-based search, and demonstrates that HyAL effectively solves constrained optimization problems, including high-dimensional cases where population-based methods struggle.
Lampros Printzios, Konstantinos I. Chatzilygeroudis· Proceedings of the Genetic a...· 0 citations
Learning-enabled decision systems often use offline data or computation to reduce online compute cost. Despite the empirical success of such approaches, there is limited general understanding of how much offline information is needed to achieve a desired accuracy under a fixed online computation budget. We study this question through the lens of amortized parametric optimization: an offline phase stores a finite memory of solved problem instances, and an online phase produces a solution to a new instance by retrieving a warm start and applying $K$ steps of projected gradient descent. We analyze this setup for smooth convex parametric optimization over a compact domain, using a nonparametric predictor built from the stored offline solutions. For $\mu$-strongly convex objectives, we establish matching upper and lower bounds on the memory required to guarantee $\varepsilon$-accuracy under a fixed online iteration budget $K$. For convex objectives satisfying a $\beta$-growth condition ($\beta>2$), we obtain near-matching bounds and identify a phase transition in $K$ beyond which additional memory provides no benefit. We further provide a general proof framework that (i) explicitly quantifies the memory cost of acceleration---how much offline memory is required to achieve a prescribed speedup over the unaided online optimizer---and (ii) identifies two key quantities driving this cost: the convergence rate of the online optimizer and the Lipschitz sensitivity of the solution map to the problem parameter. Experiments on parameterized ridge regression confirm the predicted memory--computation--accuracy tradeoffs.
Shijie Pan, Agustin Castellano, Zeyu Shen et al.· 0 citations
This work proposes a relative inexact proximal augmented Lagrangian method with a semismooth Newton subproblem solver for solving SRM-based optimization problems and provides explicit generalized Jacobian characterizations and tailor the pool adjacent violators algorithm for their efficient evaluation.
Rufeng Xiao, Rujun Jiang, Xudong Li et al.· 0 citations