Crawling through alien landscapes in the fog, looking for mountain peaks, one step at a time.
Twin to online optimization.
A huge and varied research discipline; this notebook will need to be broken down into subsections as my understanding improves.
But for now, I don’t recommend that you read this! Other people are far more skilled in this area than me; This is not an educational resource so much as keyworddump breadcrumb trail marking my plucking of results I need from someone else’s orchard.
Keywords: Complimentary slackness theorem, High or very high dimensional methods, approximate method, Lagrange multipliers, primal and dual problems, fixed point methods, gradient, subgradient, proximal gradient, optimal control problems, convexity, sparsity, ways to avoid wrecking finding the extrema of perfectly simple little 10000parameter functions before everyone observes that you are a fool in the guise of a mathematician but everyone is not there because you wandered off the optimal path hours ago, and now you are alone and lost in a valley of lowercase Greek letters.
See also automatic differentiation if you don’t want to spend all your time doing mechanical calculus.
TODO: Diagrammatic ontology.
Apropos that, see Zeyuan AllenZhu and Elad Hazan on their teaching strategy:
The following dilemma is encountered by many of my friends when teaching basic optimization: which variant/proof of gradient descent should one start with? Of course, one needs to decide on which depth of convex analysis one should dive into, and decide on issues such as “should I define strongconvexity?”, “discuss smoothness?”, “Nesterov acceleration?”, etc.
This is especially acute for courses that do not deal directly with optimization, which is described as a tool for learning or as a basic building block for other algorithms. Some approaches:
 I teach online gradient descent, in the context of online convex optimization, and then derive the offline counterpart. This is nonstandard, but permits an extremely simple derivation and gets to the online learning component first.
 Sanjeev Arora teaches basic offline GD for the smooth and stronglyconvex case first.
 In OR/optimization courses the smooth (but not stronglyconvex) cas is many times taught first.
All of these variants have different proofs whose connections are perhaps not immediate. If one wishes to go into more depth, usually in convex optimization courses, one covers the full spectrum of different smoothness/ strongconvexity/ acceleration/ stochasticity regimes, each with a separate analysis (a total of 16 possible configurations!)
This year I’ve tried something different in COS511 @ Princeton, which turns out also to have research significance. We’ve covered basic GD for wellconditioned functions, i.e. smooth and stronglyconvex functions, and then extended these result by reduction to all other cases! A (simplified) outline of this teaching strategy is given in chapter 2 of Introduction to Online Convex Optimization.
Classical StrongConvexity and Smoothness Reductions:
Given any optimization algorithm A for the wellconditioned case (i.e., strongly convex and smooth case), we can derive an algorithm for smooth but not strongly functions as follows.
Given a nonstrongly convex but smooth objective \(f\), define a objective by \(f_1(x)=f(x)+e\x\^2\).
It is straightforward to see that \(f_1\) differs from \(f\) by at most ϵ times a distance factor, and in addition it is ϵstrongly convex. Thus, one can apply A to minimize \(f_1\) and get a solution which is not too far from the optimal solution for \(f\) itself. This simplistic reduction yields an almost optimal rate, up to logarithmic factors.
 Also looks interesting: Elad Hazan and Satyan Kale’s tutorial on online convex optimisation.
 casual programmer’s overview
See also geometry of fitness landscapes, expectation maximisation, matrix factorisations, discrete optimisation, natureinspired “metaheuristic” optimisation.
General
Brief intro material
basic but enlightening, John Nash’s graphical explanation of R’s optimization
Martin Jaggi’s Optimization in two hours
Celebrated union of optimisation, computational complexity and commandandcontroleconomics, by that showoff Cosma Shalizi: In Soviet Union, Optimization Problem Solves You
Geoffrey Hinton’s slides are good overview from the artificial neural network perspective, where “good in messy circumstances” is the rule and convexity is not assumed.
Elad Hazan The two cultures of optimization:
The standard curriculum in high school math includes elementary functional analysis, and methods for finding the minima, maxima and saddle points of a single dimensional function. When moving to high dimensions, this becomes beyond the reach of your typical highschool student: mathematical optimization theory spans a multitude of involved techniques in virtually all areas of computer science and mathematics.
Iterative methods, in particular, are the most successful algorithms for largescale optimization and the most widely used in machine learning. Of these, most popular are firstorder gradientbased methods due to their very low periteration complexity.
However, way before these became prominent, physicists needed to solve large scale optimization problems, since the time of the Manhattan project at Los Alamos. The problems that they faced looked very different, essentially simulation of physical experiments, as were the solutions they developed. The Metropolis algorithm is the basis for randomized optimization methods and Markov Chain Monte Carlo algorithms.[…]
In our recent paper (AbHa15), we show that for convex optimization, the heat path and central path for IPM for a particular barrier function (called the entropic barrier, following the terminology of the recent excellent work of Bubeck and Eldan) are identical! Thus, in some precise sense, the two cultures of optimization have been studied the same object in disguise and using different techniques.
Textbooks
Whole free textbooks online. Mostly convex.
 K. Madsen, H.B. Nielsen, O. Tingleff, Methods for Nonlinear Least Squares Problems is super simple for leastsquares type optimisations
 Aharon BenTal and Arkadi Nemirovski’s lectures on modern convex optimization
 Arkadi Nemirovski, Interior point polynomial time methods in convex programming
 Boyd and Vandenberghe’s influential Convex Optimization
 Bubeck, S. (2014). Convex Optimization: Algorithms and Complexity. arXiv:1405.4980 [cs, Math, Stat]. based on Bubeck’s course notes
 Elad Hazan’s Introduction to Online Convex Optimization.
Constrained optimisation
Lagrange multipliers
Constrained optimisation using Lagrange’s one weird trick, and the Karush–Kuhn–Tucker conditions. The search for saddle points and roots.
Alternating Direction Method of Multipliers
Dunno. It’s everywhere, though. Maybe this is a problem of definition, though? (Boyd10)
In this review, we argue that the alternating direction method of multipliers is well suited to distributed convex optimization, and in particular to largescale problems arising in statistics, machine learning, and related areas. The method was developed in the 1970s, with roots in the 1950s, and is equivalent or closely related to many other algorithms, such as dual decomposition, the method of multipliers, Douglas–Rachford splitting, Spingarn’s method of partial inverses, Dykstra’s alternating projections, Bregman iterative algorithms for \(\ell_1\) problems, proximal methods, and others. After briefly surveying the theory and history of the algorithm, we discuss applications to a wide variety of statistical and machine learning problems of recent interest, including the lasso, sparse logistic regression, basis pursuit, covariance selection, support vector machines, and many others. We also discuss general distributed optimization, extensions to the nonconvex setting, and efficient implementation, including some details on distributed MPI and Hadoop Map Reduce implementations.
Reductions
Oh crap, where did I rip this quote from?
The diverse world of machine learning applications has given rise to a plethora of algorithms and optimization methods, finely tuned to the specific regression or classification task at hand. We reduce the complexity of algorithm design for machine learning by reductions: we develop reductions that take a method developed for one setting and apply it to the entire spectrum of smoothness and strongconvexity in applications.
Continuous Relaxations
Solving discrete problems with differentiable continuous approximations.
Convex
…Of composite functions
Hip for sparse regression, compressed sensing. etc. “FISTA” is one option. (Bubeck explains.)
Second order (QuasiNewton)
If you have the second derivative you can be fancy when finding zeros.
Trust region
Pretend the gradient is locally quadratic, then seeing how bad that pretense was.
Least Squares
This particular objective function has some particular shortcuts; e.g. you don’t necessarily need a gradient to do it.
Conjugate gradient method
Finding quadratic minima, as in Least Squares.
And also notquitelinear uses of this.
 Jonathan Richard Shewchuk, An Introduction to the Conjugate Gradient Method Without the Agonizing Pain is fun
…on a manifold
What if your constraints are naturally represented as some kind of smooth manifold? Is that worth thinking about? Apparently sometimes it is. See, e.g. ToKW16, BMAS14.
To file
Miscellaneous optimisation techniques suggested on Linkedin
The whole world of exotic specialized optimisers. See, e.g. Nuit Blanche namedropping Bregmann iteration, alternating method, augmented Lagrangian…
Fixed point methods
Contraction maps are nice when you have them. TBD.
Primal/dual problems
Majorizationminorization
DifferenceofConvexobjectives
When your objective function is not convex but you can represent it in terms of convex functions, use DC optimisation. (GaRC09.) (I don’t think this guarantees you a global optimum, but rather faster convergence to a local one)
Nonconvex optimisation
“How doomed are you?”
Gradientfree optimization
Not all the methods described here use gradient information, but it’s frequently assumed to be something you can access easily. It’s worth considering which objectives you can optimize easily
But not all objectives are easily differentiable, even when parameters are continuous. For example, if you are not getting your measurement from a mathematical model, but from a physical experiment you can’t differentiate it since reality itself is usually not anlytically differentiable. In this latter case, you are getting close to a question of online experiment design, as in ANOVA, and a further constraint that your function evaluations are possibly stupendously expensive. See Bayesian optimisation for one approach to this i the context of experiment design.
In general situations like this we use gradientfree methods, such as simulated annealing or metaheuristic methods or whathaveyou.
“Metaheuristic” methods
Biologicallyinspired or arbitrary. Evolutionary algorithms, particle swarm optimisation, ant colony optimisation, harmony search. A lot of the tricks from these are adopted into mainstream stochastic methods. Some not.
See biometic algorithms for the care and husbandry of such as those.
Annealing and Monte Carlo optimisation methods
Simulated annealing: Constructing a process to yield maximallylikely estimates for the parameters. This has a statistical mechanics justification that makes it attractive to physicists; But it’s generally useful. You don’t necessarily need a gradient here, just the ability to evaluate something interpretable as a “likelihood ratio”. Long story. I don’t yet cover this at Monte carlo methods but I should.
Expectation maximization
My problem: constrained, pathwise sparsifyingpenalty optimisers for nonlinear problems
I’m trialling a bunch of sparse Lassolike regression models. I want them to be fastish to run and fast to develop. I want them to go in python. I would like to be able to vary my regularisation parameter and warmrestart, like the glmnet Lasso. I would like to be able to handle constraints, especially componentwise nonnegativity, and matrix nonnegativedefiniteness.
Notes on that here.
Ideas:
use scipy.optimize
give up the idea of warm restarts, and enforce constraints in the callback.
use cvxpy
Pretty good, but not that fast since it does not in general exploit gradient information. for some problems this is fine, though.
use spams
Wonderful, but only if your problem fits one of their categories. Otherwise you can maybe extract some bits from their code and use them, but that is now a serious project. They have a passable LASSO.
Roll my own algorithm
Potentially yakshaving. But I can work from the example of my colleagues, which are specialpurpose algorithms, usually reasonably fast ones.
 pyspsolve
 ADMM (Alternating Direction Method of Multipliers) for LASSO, which is a Python implementation for an original MATLAB version.
 pytron (python/c++)
 A TrustRegion Newton Method in Python using TRON optimization software (files src/tron.{h,cpp}) distributed from the LIBLINEAR sources (v1.93),
 AdaptiveLasso
 in the incoming scikitlearn looks likes a good reference.
Implementations
Specialised optimisation software.
See also statistical software, and online optimisation
pysporco a Python package for solving optimisation problems with sparsityinducing regularisation. These consist primarily of sparse coding and dictionary learning problems, including convolutional sparse coding and dictionary learning, but there is also support for other problems such as Total Variation regularisation and Robust PCA. In the current version all of the optimisation algorithms are based on the Alternating Direction Method of Multipliers (ADMM).
scipy.optimise.minimize: The python default. Includes many different algorithms than can do whatever you want. Failure modes are opaque, onlineonly and they don’t support warmrestarts, which is a thing for me, but a good starting point unless you have reason to prefer others. (i.e. if all your data does not fit in RAM, don’t bother.)

SPAMS (SPArse Modeling Software) is an optimization toolbox for solving various sparse estimation problems. Dictionary learning and matrix factorization (NMF, sparse PCA, …) Solving sparse decomposition problems with LARS, coordinate descent, OMP, SOMP, proximal methods Solving structured sparse decomposition problems (:math`ell_1/ell_2,` \(\ell_1/\ell_\infty,\) sparse group lasso, treestructured regularization structured sparsity with overlapping groups,…). It is developped by Julien Mairal, with the collaboration of Francis Bach, Jean Ponce, Guillermo Sapiro, Rodolphe Jenatton and Guillaume Obozinski. It is coded in C++ with a Matlab interface. Recently, interfaces for R and Python have been developed by JeanPaul Chieze (INRIA), and archetypal analysis was written by Yuansi Chen (UC Berkeley).

…is a user friendly interface to several conic and integer programming solvers, very much like YALMIP or CVX under MATLAB.
The main motivation for PICOS is to have the possibility to enter an optimization problem as a high level model, and to be able to solve it with several different solvers. Multidimensional and matrix variables are handled in a natural fashion, which makes it painless to formulate a SDP or a SOCP. This is very useful for educational purposes, and to quickly implement some models and test their validity on simple examples.
also maintains a list of other solvers.
Manifold optimisation implementations:

… is a free software package for convex optimization based on the Python programming language. It can be used with the interactive Python interpreter, on the command line by executing Python scripts, or integrated in other software via Python extension modules. Its main purpose is to make the development of software for convex optimization applications straightforward by building on Python’s extensive standard library and on the strengths of Python as a highlevel programming language. […]
 efficient Python classes for dense and sparse matrices (real and complex), with Python indexing and slicing and overloaded operations for matrix arithmetic
 an interface to most of the doubleprecision real and complex BLAS
 an interface to LAPACK routines for solving linear equations and leastsquares problems, matrix factorisations (LU, Cholesky, LDLT and QR), symmetric eigenvalue and singular value decomposition, and Schur factorization
 an interface to the fast Fourier transform routines from FFTW
 interfaces to the sparse LU and Cholesky solvers from UMFPACK and CHOLMOD
 routines for linear, secondorder cone, and semidefinite programming problems
 routines for nonlinear convex optimization
 interfaces to the linear programming solver in GLPK, the semidefinite programming solver in DSDP5, and the linear, quadratic and secondorder cone programming solvers in MOSEK
 a modeling tool for specifying convex piecewiselinear optimization problems.
seems to reinvent half of numpy and scipy. Also seems to be used by the all the other python packages. Including…

…is a Pythonembedded modeling language for convex optimization problems. It allows you to express your problem in a natural way that follows the math, rather than in the restrictive standard form required by solvers.
So it’s a DSL for convex constraint programming. Can be extended heuristically to nonconvex constraints by…

… is a package for modeling and solving problems with convex objectives and decision variables from a nonconvex set. This package provides heuristic such as NCADMM (a variation of alternating direction method of multipliers for nonconvex problems) and relaxroundpolish, which can be viewed as a majorizationminimization algorithm. The solver methods provided and the syntax for constructing problems are discussed in our associated paper.


… is a free/opensource library for nonlinear optimization, providing a common interface for a number of different free optimization routines available online as well as original implementations of various other algorithms. Its features include:
 Callable from C, C++, Fortran, Matlab or GNU Octave, Python, GNU Guile, Julia, GNU R, Lua, and OCaml.
 A common interface for many different algorithms—try a different algorithm just by changing one parameter.
 Support for largescale optimization (some algorithms scalable to millions of parameters and thousands of constraints)…
 Algorithms using function values only (derivativefree) and also algorithms exploiting usersupplied gradients.

…(pronounced teefox) provides a set of Matlab templates, or building blocks, that can be used to construct efficient, customized solvers for a variety of convex models, including in particular those employed in sparse recovery applications. It was conceived and written by Stephen Becker, Emmanuel J. Candès and Michael Grant.
stan is famous for Monte Carlo sampling, but also does deterministic optimisation using automatic differentiation. this is a luxurious “full service” option, although with limited scope for customisation; Curious how it performs in very high dimensions, as LBFGS does not scale forever.
Optimization algorithms:
 Limitedmemory BFGS (Stan’s default optimization algorithm)
 BFGS
 Laplace’s method for classical standard error estimates and approximate Bayesian posteriors
Optim.jl is a generic optimizer for julia
JuMP.jl is a domainspecific modeling language for mathematical optimization embedded in Julia. It currently supports a number of opensource and commercial solvers (Bonmin, Cbc, Clp, Couenne, CPLEX, ECOS, FICO Xpress, GLPK, Gurobi, Ipopt, KNITRO, MOSEK, NLopt, SCS, BARON) for a variety of problem classes, including linear programming, (mixed) integer programming, secondorder conic programming, semidefinite programming, and nonlinear programming.
NLsolve.jl solves systems of nonlinear equations. […]
The package is also able to solve mixed complementarity problems, which are similar to systems of nonlinear equations, except that the equality to zero is allowed to become an inequality if some boundary condition is satisfied. See further below for a formal definition and the related commands.
Since there is some overlap between optimizers and nonlinear solvers, this package borrows some ideas from the Optim package, and depends on it for linesearch algorithms.
Many of these solvers optionally use commercial backends such as Mosek.
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