Gradient descent

First order of business

October 4, 2014 — August 24, 2023

functional analysis
Figure 1

Gradient descent, a classic first order optimisation], with many variants, and many things one might wish to understand.

There are only few things I wish to understand for the moment.

Very tidy introduction in Anupam Gupta’s notes for 15-850: CMU Advanced Algorithms, Fall 2020, in particular Lectures 18 and 19.

1 Coordinate descent

Descent each coordinate individually.

Small clever hack for certain domains: log gradient descent.

2 Momentum

Polyak momentum (that’s the heavy ball one, right?), Nesterov momentum.

How and when does it work? and how well? Moritz Hardt, The zen of gradient descent explains it through Chebychev polynomials. Cheng-Soon Ong recommends d’Aspremont, Scieur, and Taylor (2021) as an overview. Gabriel Goh, Why Momentum Really Works (Goh 2017) is an incredible illustrated guide.

Sebastian Bubeck explains it from a different angle, Revisiting Nesterov’s Acceleration to expand upon the rather magical introduction given in his lecture Wibisono et al explain it in terms of variational approximation. See also Accelerated gradient descent 1 and 2.

Trung Vu’s Convergence of Heavy-Ball Method and Nesterov’s Accelerated Gradient on Quadratic Optimization differentiates Nesterov momentum from heavy ball momentum.

3 Continuous approximations of iterations

Recent papers (Wibisono and Wilson 2015; Wibisono, Wilson, and Jordan 2016) argue that the discrete time steps can be viewed as a discrete approximation to a continuous time ODE which approaches the optimum (which in itself is trivial), but moreover that many algorithms fit into the same families of ODEs, that these ODEs explain Nesterov acceleration and generate new, improved optimisation methods. (which is not trivial.)


4 Online versus stochastic

Technically, “online” optimisation in, say, bandit/RL problems might imply that you have to “minimise regret online”, which has a slightly different meaning and, e.g. involves seeing each training only as it arrives along some notional arrow of time, yet wishing to make the “best” decision at the next time, and possibly choosing your next experiment in order to trade-off exploration versus exploitation etc.

In SGD you can see your data as often as you want and in whatever order, but you only look at a bit at a time. Usually the data is given and predictions make no difference to what information is available to you.

Some of the same technology pops up in each of these notions of online optimisation, but I am really thinking about SGD here.

There are many more permutations and variations used in practice.

5 Conditional Gradient

a.k.a. Frank-Wolfe algorithm: Don’t know much about this.

6 Mirror descent

See mirror descent.

7 References

Agarwal, Chapelle, Dudık, et al. 2014. A Reliable Effective Terascale Linear Learning System.” Journal of Machine Learning Research.
Allen-Zhu, and Hazan. 2016. Optimal Black-Box Reductions Between Optimization Objectives.” In Advances in Neural Information Processing Systems 29.
Allen-Zhu, Simchi-Levi, and Wang. 2019. The Lingering of Gradients: How to Reuse Gradients over Time.” arXiv:1901.02871 [Cs, Math, Stat].
Andersson, Gillis, Horn, et al. 2019. CasADi: A Software Framework for Nonlinear Optimization and Optimal Control.” Mathematical Programming Computation.
Bansal, and Gupta. 2019. Potential-Function Proofs for First-Order Methods.”
Beck, and Teboulle. 2003. Mirror Descent and Nonlinear Projected Subgradient Methods for Convex Optimization.” Operations Research Letters.
———. 2009. A Fast Iterative Shrinkage-Thresholding Algorithm for Linear Inverse Problems.” SIAM Journal on Imaging Sciences.
Betancourt, Jordan, and Wilson. 2018. On Symplectic Optimization.” arXiv:1802.03653 [Stat].
Botev, Lever, and Barber. 2016. Nesterov’s Accelerated Gradient and Momentum as Approximations to Regularised Update Descent.” arXiv:1607.01981 [Cs, Stat].
Bubeck. 2015. Convex Optimization: Algorithms and Complexity. Foundations and Trends in Machine Learning.
———. 2019. The Five Miracles of Mirror Descent.
Chen. 2012. Smoothing Methods for Nonsmooth, Nonconvex Minimization.” Mathematical Programming.
Choromanska, Henaff, Mathieu, et al. 2015. The Loss Surfaces of Multilayer Networks.” In Proceedings of the Eighteenth International Conference on Artificial Intelligence and Statistics.
d’Aspremont, Scieur, and Taylor. 2021. Acceleration Methods.” arXiv:2101.09545 [Cs, Math].
Defazio, Bach, and Lacoste-Julien. 2014. SAGA: A Fast Incremental Gradient Method With Support for Non-Strongly Convex Composite Objectives.” In Advances in Neural Information Processing Systems 27.
DeVore. 1998. Nonlinear Approximation.” Acta Numerica.
Domingos. 2020. Every Model Learned by Gradient Descent Is Approximately a Kernel Machine.” arXiv:2012.00152 [Cs, Stat].
Goh. 2017. Why Momentum Really Works.” Distill.
Hinton, Srivastava, and Kevin Swersky. n.d. “Neural Networks for Machine Learning.”
Jacobsen, and Cutkosky. 2022. Parameter-Free Mirror Descent.”
Jakovetic, Freitas Xavier, and Moura. 2014. Convergence Rates of Distributed Nesterov-Like Gradient Methods on Random Networks.” IEEE Transactions on Signal Processing.
Langford, Li, and Zhang. 2009. Sparse Online Learning via Truncated Gradient.” In Advances in Neural Information Processing Systems 21.
Lee, Panageas, Piliouras, et al. 2017. First-Order Methods Almost Always Avoid Saddle Points.” arXiv:1710.07406 [Cs, Math, Stat].
Lee, Simchowitz, Jordan, et al. 2016. Gradient Descent Converges to Minimizers.” arXiv:1602.04915 [Cs, Math, Stat].
Ma, and Belkin. 2017. Diving into the Shallows: A Computational Perspective on Large-Scale Shallow Learning.” arXiv:1703.10622 [Cs, Stat].
Mandt, Hoffman, and Blei. 2017. Stochastic Gradient Descent as Approximate Bayesian Inference.” JMLR.
Nesterov, Y. 2012. Efficiency of Coordinate Descent Methods on Huge-Scale Optimization Problems.” SIAM Journal on Optimization.
Nesterov, Yu. 2012. Gradient Methods for Minimizing Composite Functions.” Mathematical Programming.
Nocedal, and Wright. 2006. Numerical Optimization. Springer Series in Operations Research and Financial Engineering.
Richards, and Rabbat. 2021. Learning with Gradient Descent and Weakly Convex Losses.” arXiv:2101.04968 [Cs, Math, Stat].
Ruder. 2017. An Overview of Gradient Descent Optimization Algorithms.”
Sagun, Guney, Arous, et al. 2014. Explorations on High Dimensional Landscapes.” arXiv:1412.6615 [Cs, Stat].
Wainwright. 2014. Structured Regularizers for High-Dimensional Problems: Statistical and Computational Issues.” Annual Review of Statistics and Its Application.
Wibisono, and Wilson. 2015. On Accelerated Methods in Optimization.” arXiv:1509.03616 [Math].
Wibisono, Wilson, and Jordan. 2016. A Variational Perspective on Accelerated Methods in Optimization.” Proceedings of the National Academy of Sciences.
Wright, and Recht. 2021. Optimization for Data Analysis.
Zinkevich. 2003. Online Convex Programming and Generalized Infinitesimal Gradient Ascent.” In Proceedings of the Twentieth International Conference on International Conference on Machine Learning. ICML’03.