Gradient descent

First order of business

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.

Coordinate descent

Descent each coordinate individually.

Small clever hack for certain domains: log gradient descent.


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.

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.)


Online versus stochastic

Technically, β€œonline” optimisation in, say, bandit/RL problems might imply that you have to β€œminimise 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 β€œbest” 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.

Conditional Gradient

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

Mirror descent

See mirror descent.


Agarwal, Alekh, Olivier Chapelle, Miroslav DudΔ±k, and John Langford. 2014. β€œA Reliable Effective Terascale Linear Learning System.” Journal of Machine Learning Research 15 (1): 1111–33.
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Allen-Zhu, Zeyuan, David Simchi-Levi, and Xinshang Wang. 2019. β€œThe Lingering of Gradients: How to Reuse Gradients over Time.” arXiv:1901.02871 [Cs, Math, Stat], January.
Andersson, Joel A. E., Joris Gillis, Greg Horn, James B. Rawlings, and Moritz Diehl. 2019. β€œCasADi: A Software Framework for Nonlinear Optimization and Optimal Control.” Mathematical Programming Computation 11 (1): 1–36.
Aspremont, Alexandre d’, Damien Scieur, and Adrien Taylor. 2021. β€œAcceleration Methods.” arXiv:2101.09545 [Cs, Math], January.
Bansal, Nikhil, and Anupam Gupta. 2019. β€œPotential-Function Proofs for First-Order Methods.” arXiv.
Beck, Amir, and Marc Teboulle. 2003. β€œMirror Descent and Nonlinear Projected Subgradient Methods for Convex Optimization.” Operations Research Letters 31 (3): 167–75.
β€”β€”β€”. 2009. β€œA Fast Iterative Shrinkage-Thresholding Algorithm for Linear Inverse Problems.” SIAM Journal on Imaging Sciences 2 (1): 183–202.
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Bubeck, SΓ©bastien. 2015. Convex Optimization: Algorithms and Complexity. Vol. 8. Foundations and Trends in Machine Learning. Now Publishers.
β€”β€”β€”. 2019. The Five Miracles of Mirror Descent.
Chen, Xiaojun. 2012. β€œSmoothing Methods for Nonsmooth, Nonconvex Minimization.” Mathematical Programming 134 (1): 71–99.
Choromanska, Anna, MIkael Henaff, Michael Mathieu, Gerard Ben Arous, and Yann LeCun. 2015. β€œThe Loss Surfaces of Multilayer Networks.” In Proceedings of the Eighteenth International Conference on Artificial Intelligence and Statistics, 192–204.
Defazio, Aaron, Francis Bach, and Simon Lacoste-Julien. 2014. β€œSAGA: A Fast Incremental Gradient Method With Support for Non-Strongly Convex Composite Objectives.” In Advances in Neural Information Processing Systems 27.
DeVore, Ronald A. 1998. β€œNonlinear Approximation.” Acta Numerica 7 (January): 51–150.
Domingos, Pedro. 2020. β€œEvery Model Learned by Gradient Descent Is Approximately a Kernel Machine.” arXiv:2012.00152 [Cs, Stat], November.
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Hinton, Geoffrey, Nitish Srivastava, and Kevin Swersky. n.d. β€œNeural Networks for Machine Learning.”
Jacobsen, Andrew, and Ashok Cutkosky. 2022. β€œParameter-Free Mirror Descent.” arXiv.
Jakovetic, D., J.M. Freitas Xavier, and J.M.F. Moura. 2014. β€œConvergence Rates of Distributed Nesterov-Like Gradient Methods on Random Networks.” IEEE Transactions on Signal Processing 62 (4): 868–82.
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Lee, Jason D., Ioannis Panageas, Georgios Piliouras, Max Simchowitz, Michael I. Jordan, and Benjamin Recht. 2017. β€œFirst-Order Methods Almost Always Avoid Saddle Points.” arXiv:1710.07406 [Cs, Math, Stat], October.
Lee, Jason D., Max Simchowitz, Michael I. Jordan, and Benjamin Recht. 2016. β€œGradient Descent Converges to Minimizers.” arXiv:1602.04915 [Cs, Math, Stat], March.
Ma, Siyuan, and Mikhail Belkin. 2017. β€œDiving into the Shallows: A Computational Perspective on Large-Scale Shallow Learning.” arXiv:1703.10622 [Cs, Stat], March.
Mandt, Stephan, Matthew D. Hoffman, and David M. Blei. 2017. β€œStochastic Gradient Descent as Approximate Bayesian Inference.” JMLR, April.
Nesterov, Y. 2012. β€œEfficiency of Coordinate Descent Methods on Huge-Scale Optimization Problems.” SIAM Journal on Optimization 22 (2): 341–62.
Nesterov, Yu. 2012. β€œGradient Methods for Minimizing Composite Functions.” Mathematical Programming 140 (1): 125–61.
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Richards, Dominic, and Mike Rabbat. 2021. β€œLearning with Gradient Descent and Weakly Convex Losses.” arXiv:2101.04968 [Cs, Math, Stat], June.
Ruder, Sebastian. 2016. β€œAn Overview of Gradient Descent Optimization Algorithms.” arXiv:1609.04747 [Cs], September.
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Wibisono, Andre, Ashia C. Wilson, and Michael I. Jordan. 2016. β€œA Variational Perspective on Accelerated Methods in Optimization.” Proceedings of the National Academy of Sciences 113 (47): E7351–58.
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