Sharpened Lazy Incremental Quasi-Newton Method
Aakash Sunil Lahoti, Spandan Senapati, Ketan Rajawat, Alec Koppel
Abstract
The problem of minimizing the sum of $n$ functions in $d$ dimensions is ubiquitous in machine learning and statistics. In many applications where the number of observations $n$ is large, it is necessary to use incremental or stochastic methods, as their per-iteration cost is independent of $n$. Of these, Quasi-Newton (QN) methods strike a balance between the per-iteration cost and the convergence rate. Specifically, they exhibit a superlinear rate with $O(d^2)$ cost in contrast to the linear rate of first-order methods with $O(d)$ cost and the quadratic rate of second-order methods with $O(d^3)$ cost. However, existing incremental methods have notable shortcomings: Incremental Quasi-Newton (IQN) only exhibits asymptotic superlinear convergence. In contrast, Incremental Greedy BFGS (IGS) offers explicit superlinear convergence but suffers from poor empirical performance and has a per-iteration cost of $O(d^3)$. To address these issues, we introduce the Sharpened Lazy Incremental Quasi-Newton Method (SLIQN) that achieves the best of both worlds: an explicit superlinear convergence rate, and superior empirical performance at a per-iteration $O(d^2)$ cost. SLIQN features two key changes: first, it incorporates a hybrid strategy of using both classic and greedy BFGS updates, allowing it to empirically outperform both IQN and IGS. Second, it employs a clever constant multiplicative factor along with a lazy propagation strategy, which enables it to have a cost of $O(d^2)$. Additionally, our experiments demonstrate the superiority of SLIQN over other incremental and stochastic Quasi-Newton variants and establish its competitiveness with second-order incremental methods.
BibTeX
@InProceedings{pmlr-v238-sunil-lahoti24a,
title = {Sharpened Lazy Incremental Quasi-{N}ewton Method},
author = {Sunil Lahoti, Aakash and Senapati, Spandan and Rajawat, Ketan and Koppel, Alec},
booktitle = {Proceedings of The 27th International Conference on Artificial Intelligence and Statistics},
pages = {4735--4743},
year = {2024},
editor = {Dasgupta, Sanjoy and Mandt, Stephan and Li, Yingzhen},
volume = {238},
series = {Proceedings of Machine Learning Research},
month = {02--04 May},
publisher = {PMLR},
pdf = {https://proceedings.mlr.press/v238/sunil-lahoti24a/sunil-lahoti24a.pdf},
url = {https://proceedings.mlr.press/v238/sunil-lahoti24a.html},
abstract = {The problem of minimizing the sum of $n$ functions in $d$ dimensions is ubiquitous in machine learning and statistics. In many applications where the number of observations $n$ is large, it is necessary to use incremental or stochastic methods, as their per-iteration cost is independent of $n$. Of these, Quasi-Newton (QN) methods strike a balance between the per-iteration cost and the convergence rate. Specifically, they exhibit a superlinear rate with $O(d^2)$ cost in contrast to the linear rate of first-order methods with $O(d)$ cost and the quadratic rate of second-order methods with $O(d^3)$ cost. However, existing incremental methods have notable shortcomings: Incremental Quasi-Newton (IQN) only exhibits asymptotic superlinear convergence. In contrast, Incremental Greedy BFGS (IGS) offers explicit superlinear convergence but suffers from poor empirical performance and has a per-iteration cost of $O(d^3)$. To address these issues, we introduce the Sharpened Lazy Incremental Quasi-Newton Method (SLIQN) that achieves the best of both worlds: an explicit superlinear convergence rate, and superior empirical performance at a per-iteration $O(d^2)$ cost. SLIQN features two key changes: first, it incorporates a hybrid strategy of using both classic and greedy BFGS updates, allowing it to empirically outperform both IQN and IGS. Second, it employs a clever constant multiplicative factor along with a lazy propagation strategy, which enables it to have a cost of $O(d^2)$. Additionally, our experiments demonstrate the superiority of SLIQN over other incremental and stochastic Quasi-Newton variants and establish its competitiveness with second-order incremental methods.}
}