Adeyemi D. Adeoye

Adeyemi D. Adeoye, Alberto Bemporad

We introduce a notion of self-concordant smoothing for minimizing the sum of two convex functions, one of which is smooth and the other nonsmooth. The key highlight is a natural property of the resulting problem's structure that yields a variable metric selection method and a step length rule especially suited to proximal quasi-Newton algorithms. Also, we efficiently handle specific structures promoted by the nonsmooth term, such as l1-regularization and group lasso penalties. A convergence analysis for the class of proximal quasi-Newton methods covered by our framework is presented. In particular, we obtain guarantees, under standard assumptions, for two algorithms: Prox-N-SCORE (a proximal Newton method) and Prox-GGN-SCORE (a proximal generalized Gauss-Newton method). The latter uses a low-rank approximation of the Hessian inverse, reducing most of the cost of matrix inversion and making it effective for overparameterized machine learning models. Numerical experiments on synthetic and real data demonstrate the efficiency of both algorithms against state-of-the-art approaches. A Julia implementation is publicly available at https://github.com/adeyemiadeoye/SelfConcordantSmoothOptimization.jl.

Adeyemi D. Adeoye, Philipp Christian Petersen, Alberto Bemporad

The generalized Gauss-Newton (GGN) optimization method incorporates curvature estimates into its solution steps, and provides a good approximation to the Newton method for large-scale optimization problems. GGN has been found particularly interesting for practical training of deep neural networks, not only for its impressive convergence speed, but also for its close relation with neural tangent kernel regression, which is central to recent studies that aim to understand the optimization and generalization properties of neural networks. This work studies a GGN method for optimizing a two-layer neural network with explicit regularization. In particular, we consider a class of generalized self-concordant (GSC) functions that provide smooth approximations to commonly-used penalty terms in the objective function of the optimization problem. This approach provides an adaptive learning rate selection technique that requires little to no tuning for optimal performance. We study the convergence of the two-layer neural network, considered to be overparameterized, in the optimization loop of the resulting GGN method for a given scaling of the network parameters. Our numerical experiments highlight specific aspects of GSC regularization that help to improve generalization of the optimized neural network. The code to reproduce the experimental results is available at https://github.com/adeyemiadeoye/ggn-score-nn.

Mikalai Korbit, Adeyemi D. Adeoye, Alberto Bemporad et al.

We present Exact Gauss-Newton (EGN), a stochastic second-order optimization algorithm that combines the generalized Gauss-Newton (GN) Hessian approximation with low-rank linear algebra to compute the descent direction. Leveraging the Duncan-Guttman matrix identity, the parameter update is obtained by factorizing a matrix which has the size of the mini-batch. This is particularly advantageous for large-scale machine learning problems where the dimension of the neural network parameter vector is several orders of magnitude larger than the batch size. Additionally, we show how improvements such as line search, adaptive regularization, and momentum can be seamlessly added to EGN to further accelerate the algorithm. Moreover, under mild assumptions, we prove that our algorithm converges in expectation to a stationary point of the objective. Finally, our numerical experiments demonstrate that EGN consistently exceeds, or at most matches the generalization performance of well-tuned SGD, Adam, GAF, SQN, and SGN optimizers across various supervised and reinforcement learning tasks.

Adeyemi D. Adeoye, Alberto Bemporad

Optimization problems that include regularization functions in their objectives are regularly solved in many applications. When one seeks second-order methods for such problems, it may be desirable to exploit specific properties of some of these regularization functions when accounting for curvature information in the solution steps to speed up convergence. In this paper, we propose the SCORE (self-concordant regularization) framework for unconstrained minimization problems which incorporates second-order information in the Newton-decrement framework for convex optimization. We propose the generalized Gauss-Newton with Self-Concordant Regularization (GGN-SCORE) algorithm that updates the minimization variables each time it receives a new input batch. The proposed algorithm exploits the structure of the second-order information in the Hessian matrix, thereby reducing computational overhead. GGN-SCORE demonstrates how to speed up convergence while also improving model generalization for problems that involve regularized minimization under the proposed SCORE framework. Numerical experiments show the efficiency of our method and its fast convergence, which compare favorably against baseline first-order and quasi-Newton methods. Additional experiments involving non-convex (overparameterized) neural network training problems show that the proposed method is promising for non-convex optimization.