From 7582697fbe93c8f8852a1a2c1dff4c0d8f2bd5bc Mon Sep 17 00:00:00 2001 From: "Documenter.jl" Date: Thu, 10 Aug 2023 09:59:34 +0000 Subject: [PATCH] build based on bff0a3f --- previews/PR109/api/index.html | 7 + previews/PR109/assets/documenter.js | 331 ++++++++++++++++++ previews/PR109/assets/search.js | 267 ++++++++++++++ .../PR109/assets/themes/documenter-dark.css | 7 + .../PR109/assets/themes/documenter-light.css | 9 + previews/PR109/assets/themeswap.js | 66 ++++ previews/PR109/assets/warner.js | 49 +++ previews/PR109/examples/.gitkeep | 0 previews/PR109/examples/0_intro/index.html | 30 ++ previews/PR109/examples/1_basic/index.html | 74 ++++ previews/PR109/examples/2_advanced/index.html | 41 +++ previews/PR109/examples/3_tricks/index.html | 57 +++ previews/PR109/faq/index.html | 4 + previews/PR109/index.html | 2 + previews/PR109/search/index.html | 2 + previews/PR109/search_index.js | 3 + previews/PR109/siteinfo.js | 1 + 17 files changed, 950 insertions(+) create mode 100644 previews/PR109/api/index.html create mode 100644 previews/PR109/assets/documenter.js create mode 100644 previews/PR109/assets/search.js create mode 100644 previews/PR109/assets/themes/documenter-dark.css create mode 100644 previews/PR109/assets/themes/documenter-light.css create mode 100644 previews/PR109/assets/themeswap.js create mode 100644 previews/PR109/assets/warner.js create mode 100644 previews/PR109/examples/.gitkeep create mode 100644 previews/PR109/examples/0_intro/index.html create mode 100644 previews/PR109/examples/1_basic/index.html create mode 100644 previews/PR109/examples/2_advanced/index.html create mode 100644 previews/PR109/examples/3_tricks/index.html create mode 100644 previews/PR109/faq/index.html create mode 100644 previews/PR109/index.html create mode 100644 previews/PR109/search/index.html create mode 100644 previews/PR109/search_index.js create mode 100644 previews/PR109/siteinfo.js diff --git a/previews/PR109/api/index.html b/previews/PR109/api/index.html new file mode 100644 index 0000000..b84fd79 --- /dev/null +++ b/previews/PR109/api/index.html @@ -0,0 +1,7 @@ + +API reference · ImplicitDifferentiation.jl
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API reference

Index

Docstrings

ImplicitDifferentiation.AbstractLinearSolver — Type
AbstractLinearSolver

All linear solvers used within an ImplicitFunction must satisfy this interface.

It can be useful to roll out your own solver if you need more fine-grained control on convergence / speed / behavior in case of singularity. Check out the source code of IterativeLinearSolver and DirectLinearSolver for implementation examples.

Required methods

  • presolve(linear_solver, A, y): Returns a matrix-like object A for which it is cheaper to solve several linear systems with different vectors b (a typical example would be to perform LU factorization).
  • solve(linear_solver, A, b): Returns a vector x satisfying Ax = b. If the linear system has not been solved to satisfaction, every element of x should be a NaN of the appropriate floating point type.
source
ImplicitDifferentiation.DirectLinearSolver — Type
DirectLinearSolver

An implementation of AbstractLinearSolver using the built-in backslash operator.

Fields

  • verbose::Bool: Whether to throw a warning when the solver fails (defaults to true)
source
ImplicitDifferentiation.ImplicitFunction — Type
ImplicitFunction{F,C,L,B}

Wrapper for an implicit function defined by a forward mapping y and a set of conditions c.

An ImplicitFunction object behaves like a function, and every call is differentiable with respect to the first argument x. When a derivative is queried, the Jacobian of y is computed using the implicit function theorem:

∂/∂y c(x, y(x)) * ∂/∂x y(x) = -∂/∂x c(x, y(x))

This requires solving a linear system A * J = -B where A = ∂c/∂y, B = ∂c/∂x and J = ∂y/∂x.

Fields

  • forward::F: a callable, does not need to be compatible with automatic differentiation
  • conditions::C: a callable, must be compatible with automatic differentiation
  • linear_solver::L: a subtype of AbstractLinearSolver, defines how the linear system will be solved
  • conditions_backend::B: either nothing or a subtype of AbstractDifferentiation.AbstractBackend, defines how the conditions will be differentiated within the implicit function theorem

There are two possible signatures for forward and conditions, which must be consistent with one another:

  1. Standard: forward(x, args...; kwargs...) = y and conditions(x, y, args...; kwargs...) = c
  2. Byproduct: forward(x, args...; kwargs...) = (y, z) and conditions(x, y, z, args...; kwargs...) = c.

In both cases, x, y and c must be arrays, with size(y) = size(c). In the second case, the byproduct z can be an arbitrary object generated by forward. The positional arguments args... and keyword arguments kwargs... must be the same for both forward and conditions.

Warning

The byproduct z and the other positional arguments args... beyond x are considered constant for differentiation purposes.

source
ImplicitDifferentiation.ImplicitFunction — Method
(implicit::ImplicitFunction)(x::AbstractArray, args...; kwargs...)

Return implicit.forward(x, args...; kwargs...), which can be either an array y or a tuple (y, z).

This call is differentiable.

source
ChainRulesCore.rrule — Function
rrule(rc, implicit, x, args...; kwargs...)

Custom reverse rule for an ImplicitFunction, to ensure compatibility with reverse mode autodiff.

This is only available if ChainRulesCore.jl is loaded (extension), except on Julia < 1.9 where it is always available.

We compute the vector-Jacobian product Jᵀv by solving Aᵀu = v and setting Jᵀv = -Bᵀu. Positional and keyword arguments are passed to both implicit.forward and implicit.conditions.

source
diff --git a/previews/PR109/assets/documenter.js b/previews/PR109/assets/documenter.js new file mode 100644 index 0000000..6adfbbb --- /dev/null +++ b/previews/PR109/assets/documenter.js @@ -0,0 +1,331 @@ +// Generated by Documenter.jl +requirejs.config({ + paths: { + 'highlight-julia': 'https://cdnjs.cloudflare.com/ajax/libs/highlight.js/11.5.1/languages/julia.min', + 'headroom': 'https://cdnjs.cloudflare.com/ajax/libs/headroom/0.12.0/headroom.min', + 'jqueryui': 'https://cdnjs.cloudflare.com/ajax/libs/jqueryui/1.12.1/jquery-ui.min', + 'katex-auto-render': 'https://cdnjs.cloudflare.com/ajax/libs/KaTeX/0.13.24/contrib/auto-render.min', + 'jquery': 'https://cdnjs.cloudflare.com/ajax/libs/jquery/3.6.0/jquery.min', + 'headroom-jquery': 'https://cdnjs.cloudflare.com/ajax/libs/headroom/0.12.0/jQuery.headroom.min', + 'katex': 'https://cdnjs.cloudflare.com/ajax/libs/KaTeX/0.13.24/katex.min', + 'highlight': 'https://cdnjs.cloudflare.com/ajax/libs/highlight.js/11.5.1/highlight.min', + 'highlight-julia-repl': 'https://cdnjs.cloudflare.com/ajax/libs/highlight.js/11.5.1/languages/julia-repl.min', + }, + shim: { + "highlight-julia": { + "deps": [ + "highlight" + ] + }, + "katex-auto-render": { + "deps": [ + "katex" + ] + }, + "headroom-jquery": { + "deps": [ + "jquery", + "headroom" + ] + }, + "highlight-julia-repl": { + "deps": [ + "highlight" + ] + } +} +}); +//////////////////////////////////////////////////////////////////////////////// +require(['jquery', 'katex', 'katex-auto-render'], function($, katex, renderMathInElement) { +$(document).ready(function() { + renderMathInElement( + document.body, + { + "delimiters": [ + { + "left": "$", + "right": "$", + "display": false + }, + { + "left": "$$", + "right": "$$", + "display": true + }, + { + "left": "\\[", + "right": "\\]", + "display": true + } + ] +} + + ); +}) + +}) +//////////////////////////////////////////////////////////////////////////////// +require(['jquery', 'highlight', 'highlight-julia', 'highlight-julia-repl'], function($) { +$(document).ready(function() { + hljs.highlightAll(); +}) + +}) +//////////////////////////////////////////////////////////////////////////////// +require([], function() { +function addCopyButtonCallbacks() { + for (const el of document.getElementsByTagName("pre")) { + const button = document.createElement("button"); + button.classList.add("copy-button", "fas", "fa-copy"); + el.appendChild(button); + + const success = function () { + button.classList.add("success", "fa-check"); + button.classList.remove("fa-copy"); + }; + + const failure = function () { + button.classList.add("error", "fa-times"); + button.classList.remove("fa-copy"); + }; + + button.addEventListener("click", function () { + copyToClipboard(el.innerText).then(success, failure); + + setTimeout(function () { + button.classList.add("fa-copy"); + button.classList.remove("success", "fa-check", "fa-times"); + }, 5000); + }); + } +} + +function copyToClipboard(text) { + // clipboard API is only available in secure contexts + if (window.navigator && window.navigator.clipboard) { + return window.navigator.clipboard.writeText(text); + } else { + return new Promise(function (resolve, reject) { + try { + const el = document.createElement("textarea"); + el.textContent = text; + el.style.position = "fixed"; + el.style.opacity = 0; + document.body.appendChild(el); + el.select(); + document.execCommand("copy"); + + resolve(); + } catch (err) { + reject(err); + } finally { + document.body.removeChild(el); + } + }); + } +} + +if (document.readyState === "loading") { + document.addEventListener("DOMContentLoaded", addCopyButtonCallbacks); +} else { + addCopyButtonCallbacks(); +} + +}) +//////////////////////////////////////////////////////////////////////////////// +require(['jquery', 'headroom', 'headroom-jquery'], function($, Headroom) { + +// Manages the top navigation bar (hides it when the user starts scrolling down on the +// mobile). +window.Headroom = Headroom; // work around buggy module loading? +$(document).ready(function() { + $('#documenter .docs-navbar').headroom({ + "tolerance": {"up": 10, "down": 10}, + }); +}) + +}) +//////////////////////////////////////////////////////////////////////////////// +require(['jquery'], function($) { + +// Modal settings dialog +$(document).ready(function() { + var settings = $('#documenter-settings'); + $('#documenter-settings-button').click(function(){ + settings.toggleClass('is-active'); + }); + // Close the dialog if X is clicked + $('#documenter-settings button.delete').click(function(){ + settings.removeClass('is-active'); + }); + // Close dialog if ESC is pressed + $(document).keyup(function(e) { + if (e.keyCode == 27) settings.removeClass('is-active'); + }); +}); + +}) +//////////////////////////////////////////////////////////////////////////////// +require(['jquery'], function($) { + +// Manages the showing and hiding of the sidebar. +$(document).ready(function() { + var sidebar = $("#documenter > .docs-sidebar"); + var sidebar_button = $("#documenter-sidebar-button") + sidebar_button.click(function(ev) { + ev.preventDefault(); + sidebar.toggleClass('visible'); + if (sidebar.hasClass('visible')) { + // Makes sure that the current menu item is visible in the sidebar. + $("#documenter .docs-menu a.is-active").focus(); + } + }); + $("#documenter > .docs-main").bind('click', function(ev) { + if ($(ev.target).is(sidebar_button)) { + return; + } + if (sidebar.hasClass('visible')) { + sidebar.removeClass('visible'); + } + }); +}) + +// Resizes the package name / sitename in the sidebar if it is too wide. +// Inspired by: https://github.com/davatron5000/FitText.js +$(document).ready(function() { + e = $("#documenter .docs-autofit"); + function resize() { + var L = parseInt(e.css('max-width'), 10); + var L0 = e.width(); + if(L0 > L) { + var h0 = parseInt(e.css('font-size'), 10); + e.css('font-size', L * h0 / L0); + // TODO: make sure it survives resizes? + } + } + // call once and then register events + resize(); + $(window).resize(resize); + $(window).on('orientationchange', resize); +}); + +// Scroll the navigation bar to the currently selected menu item +$(document).ready(function() { + var sidebar = $("#documenter .docs-menu").get(0); + var active = $("#documenter .docs-menu .is-active").get(0); + if(typeof active !== 'undefined') { + sidebar.scrollTop = active.offsetTop - sidebar.offsetTop - 15; + } +}) + +}) +//////////////////////////////////////////////////////////////////////////////// +require(['jquery'], function($) { + +function set_theme(theme) { + var active = null; + var disabled = []; + for (var i = 0; i < document.styleSheets.length; i++) { + var ss = document.styleSheets[i]; + var themename = ss.ownerNode.getAttribute("data-theme-name"); + if(themename === null) continue; // ignore non-theme stylesheets + // Find the active theme + if(themename === theme) active = ss; + else disabled.push(ss); + } + if(active !== null) { + active.disabled = false; + if(active.ownerNode.getAttribute("data-theme-primary") === null) { + document.getElementsByTagName('html')[0].className = "theme--" + theme; + } else { + document.getElementsByTagName('html')[0].className = ""; + } + disabled.forEach(function(ss){ + ss.disabled = true; + }); + } + + // Store the theme in localStorage + if(typeof(window.localStorage) !== "undefined") { + window.localStorage.setItem("documenter-theme", theme); + } else { + console.error("Browser does not support window.localStorage"); + } +} + +// Theme picker setup +$(document).ready(function() { + // onchange callback + $('#documenter-themepicker').change(function themepick_callback(ev){ + var themename = $('#documenter-themepicker option:selected').attr('value'); + set_theme(themename); + }); + + // Make sure that the themepicker displays the correct theme when the theme is retrieved + // from localStorage + if(typeof(window.localStorage) !== "undefined") { + var theme = window.localStorage.getItem("documenter-theme"); + if(theme !== null) { + $('#documenter-themepicker option').each(function(i,e) { + e.selected = (e.value === theme); + }) + } else { + $('#documenter-themepicker option').each(function(i,e) { + e.selected = $("html").hasClass(`theme--${e.value}`); + }) + } + } +}) + +}) +//////////////////////////////////////////////////////////////////////////////// +require(['jquery'], function($) { + +// update the version selector with info from the siteinfo.js and ../versions.js files +$(document).ready(function() { + // If the version selector is disabled with DOCUMENTER_VERSION_SELECTOR_DISABLED in the + // siteinfo.js file, we just return immediately and not display the version selector. + if (typeof DOCUMENTER_VERSION_SELECTOR_DISABLED === 'boolean' && DOCUMENTER_VERSION_SELECTOR_DISABLED) { + return; + } + + var version_selector = $("#documenter .docs-version-selector"); + var version_selector_select = $("#documenter .docs-version-selector select"); + + version_selector_select.change(function(x) { + target_href = version_selector_select.children("option:selected").get(0).value; + window.location.href = target_href; + }); + + // add the current version to the selector based on siteinfo.js, but only if the selector is empty + if (typeof DOCUMENTER_CURRENT_VERSION !== 'undefined' && $('#version-selector > option').length == 0) { + var option = $(""); + version_selector_select.append(option); + } + + if (typeof DOC_VERSIONS !== 'undefined') { + var existing_versions = version_selector_select.children("option"); + var existing_versions_texts = existing_versions.map(function(i,x){return x.text}); + DOC_VERSIONS.forEach(function(each) { + var version_url = documenterBaseURL + "/../" + each; + var existing_id = $.inArray(each, existing_versions_texts); + // if not already in the version selector, add it as a new option, + // otherwise update the old option with the URL and enable it + if (existing_id == -1) { + var option = $(""); + version_selector_select.append(option); + } else { + var option = existing_versions[existing_id]; + option.value = version_url; + option.disabled = false; + } + }); + } + + // only show the version selector if the selector has been populated + if (version_selector_select.children("option").length > 0) { + version_selector.toggleClass("visible"); + } +}) + +}) diff --git a/previews/PR109/assets/search.js b/previews/PR109/assets/search.js new file mode 100644 index 0000000..c133f74 --- /dev/null +++ b/previews/PR109/assets/search.js @@ -0,0 +1,267 @@ +// Generated by Documenter.jl +requirejs.config({ + paths: { + 'lunr': 'https://cdnjs.cloudflare.com/ajax/libs/lunr.js/2.3.9/lunr.min', + 'lodash': 'https://cdnjs.cloudflare.com/ajax/libs/lodash.js/4.17.21/lodash.min', + 'jquery': 'https://cdnjs.cloudflare.com/ajax/libs/jquery/3.6.0/jquery.min', + } +}); +//////////////////////////////////////////////////////////////////////////////// +require(['jquery', 'lunr', 'lodash'], function($, lunr, _) { + +$(document).ready(function() { + // parseUri 1.2.2 + // (c) Steven Levithan + // MIT License + function parseUri (str) { + var o = parseUri.options, + m = o.parser[o.strictMode ? "strict" : "loose"].exec(str), + uri = {}, + i = 14; + + while (i--) uri[o.key[i]] = m[i] || ""; + + uri[o.q.name] = {}; + uri[o.key[12]].replace(o.q.parser, function ($0, $1, $2) { + if ($1) uri[o.q.name][$1] = $2; + }); + + return uri; + }; + parseUri.options = { + strictMode: false, + key: ["source","protocol","authority","userInfo","user","password","host","port","relative","path","directory","file","query","anchor"], + q: { + name: "queryKey", + parser: /(?:^|&)([^&=]*)=?([^&]*)/g + }, + parser: { + strict: /^(?:([^:\/?#]+):)?(?:\/\/((?:(([^:@]*)(?::([^:@]*))?)?@)?([^:\/?#]*)(?::(\d*))?))?((((?:[^?#\/]*\/)*)([^?#]*))(?:\?([^#]*))?(?:#(.*))?)/, + loose: /^(?:(?![^:@]+:[^:@\/]*@)([^:\/?#.]+):)?(?:\/\/)?((?:(([^:@]*)(?::([^:@]*))?)?@)?([^:\/?#]*)(?::(\d*))?)(((\/(?:[^?#](?![^?#\/]*\.[^?#\/.]+(?:[?#]|$)))*\/?)?([^?#\/]*))(?:\?([^#]*))?(?:#(.*))?)/ + } + }; + + $("#search-form").submit(function(e) { + e.preventDefault() + }) + + // list below is the lunr 2.1.3 list minus the intersect with names(Base) + // (all, any, get, in, is, only, which) and (do, else, for, let, where, while, with) + // ideally we'd just filter the original list but it's not available as a variable + lunr.stopWordFilter = lunr.generateStopWordFilter([ + 'a', + 'able', + 'about', + 'across', + 'after', + 'almost', + 'also', + 'am', + 'among', + 'an', + 'and', + 'are', + 'as', + 'at', + 'be', + 'because', + 'been', + 'but', + 'by', + 'can', + 'cannot', + 'could', + 'dear', + 'did', + 'does', + 'either', + 'ever', + 'every', + 'from', + 'got', + 'had', + 'has', + 'have', + 'he', + 'her', + 'hers', + 'him', + 'his', + 'how', + 'however', + 'i', + 'if', + 'into', + 'it', + 'its', + 'just', + 'least', + 'like', + 'likely', + 'may', + 'me', + 'might', + 'most', + 'must', + 'my', + 'neither', + 'no', + 'nor', + 'not', + 'of', + 'off', + 'often', + 'on', + 'or', + 'other', + 'our', + 'own', + 'rather', + 'said', + 'say', + 'says', + 'she', + 'should', + 'since', + 'so', + 'some', + 'than', + 'that', + 'the', + 'their', + 'them', + 'then', + 'there', + 'these', + 'they', + 'this', + 'tis', + 'to', + 'too', + 'twas', + 'us', + 'wants', + 'was', + 'we', + 'were', + 'what', + 'when', + 'who', + 'whom', + 'why', + 'will', + 'would', + 'yet', + 'you', + 'your' + ]) + + // add . as a separator, because otherwise "title": "Documenter.Anchors.add!" + // would not find anything if searching for "add!", only for the entire qualification + lunr.tokenizer.separator = /[\s\-\.]+/ + + // custom trimmer that doesn't strip @ and !, which are used in julia macro and function names + lunr.trimmer = function (token) { + return token.update(function (s) { + return s.replace(/^[^a-zA-Z0-9@!]+/, '').replace(/[^a-zA-Z0-9@!]+$/, '') + }) + } + + lunr.Pipeline.registerFunction(lunr.stopWordFilter, 'juliaStopWordFilter') + lunr.Pipeline.registerFunction(lunr.trimmer, 'juliaTrimmer') + + var index = lunr(function () { + this.ref('location') + this.field('title',{boost: 100}) + this.field('text') + documenterSearchIndex['docs'].forEach(function(e) { + this.add(e) + }, this) + }) + var store = {} + + documenterSearchIndex['docs'].forEach(function(e) { + store[e.location] = {title: e.title, category: e.category, page: e.page} + }) + + $(function(){ + searchresults = $('#documenter-search-results'); + searchinfo = $('#documenter-search-info'); + searchbox = $('#documenter-search-query'); + searchform = $('.docs-search'); + sidebar = $('.docs-sidebar'); + function update_search(querystring) { + tokens = lunr.tokenizer(querystring) + results = index.query(function (q) { + tokens.forEach(function (t) { + q.term(t.toString(), { + fields: ["title"], + boost: 100, + usePipeline: true, + editDistance: 0, + wildcard: lunr.Query.wildcard.NONE + }) + q.term(t.toString(), { + fields: ["title"], + boost: 10, + usePipeline: true, + editDistance: 2, + wildcard: lunr.Query.wildcard.NONE + }) + q.term(t.toString(), { + fields: ["text"], + boost: 1, + usePipeline: true, + editDistance: 0, + wildcard: lunr.Query.wildcard.NONE + }) + }) + }) + searchinfo.text("Number of results: " + results.length) + searchresults.empty() + results.forEach(function(result) { + data = store[result.ref] + link = $(''+data.title+'') + link.attr('href', documenterBaseURL+'/'+result.ref) + if (data.category != "page"){ + cat = $('('+data.category+', '+data.page+')') + } else { + cat = $('('+data.category+')') + } + li = $('
  • ').append(link).append(" ").append(cat) + searchresults.append(li) + }) + } + + function update_search_box() { + querystring = searchbox.val() + update_search(querystring) + } + + searchbox.keyup(_.debounce(update_search_box, 250)) + searchbox.change(update_search_box) + + // Disable enter-key form submission for the searchbox on the search page + // and just re-run search rather than refresh the whole page. + searchform.keypress( + function(event){ + if (event.which == '13') { + if (sidebar.hasClass('visible')) { + sidebar.removeClass('visible'); + } + update_search_box(); + event.preventDefault(); + } + } + ); + + search_query_uri = parseUri(window.location).queryKey["q"] + if(search_query_uri !== undefined) { + search_query = decodeURIComponent(search_query_uri.replace(/\+/g, '%20')) + searchbox.val(search_query) + } + update_search_box(); + }) +}) + +}) diff --git a/previews/PR109/assets/themes/documenter-dark.css b/previews/PR109/assets/themes/documenter-dark.css new file mode 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kbd.button.is-outlined.is-loading.is-hovered::after,html.theme--documenter-dark .button.is-dark.is-outlined.is-loading:focus::after,html.theme--documenter-dark .content kbd.button.is-outlined.is-loading:focus::after,html.theme--documenter-dark .button.is-dark.is-outlined.is-loading.is-focused::after,html.theme--documenter-dark .content kbd.button.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #ecf0f1 #ecf0f1 !important}html.theme--documenter-dark .button.is-dark.is-outlined[disabled],html.theme--documenter-dark .content kbd.button.is-outlined[disabled],fieldset[disabled] html.theme--documenter-dark .button.is-dark.is-outlined,fieldset[disabled] html.theme--documenter-dark .content kbd.button.is-outlined{background-color:transparent;border-color:#282f2f;box-shadow:none;color:#282f2f}html.theme--documenter-dark .button.is-dark.is-inverted.is-outlined,html.theme--documenter-dark .content 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form.docs-search>input.button{border-radius:290486px;padding-left:1em;padding-right:1em}html.theme--documenter-dark .buttons{align-items:center;display:flex;flex-wrap:wrap;justify-content:flex-start}html.theme--documenter-dark .buttons .button{margin-bottom:0.5rem}html.theme--documenter-dark .buttons .button:not(:last-child):not(.is-fullwidth){margin-right:0.5rem}html.theme--documenter-dark .buttons:last-child{margin-bottom:-0.5rem}html.theme--documenter-dark .buttons:not(:last-child){margin-bottom:1rem}html.theme--documenter-dark .buttons.are-small .button:not(.is-normal):not(.is-medium):not(.is-large){border-radius:3px;font-size:.85em}html.theme--documenter-dark .buttons.are-medium .button:not(.is-small):not(.is-normal):not(.is-large){font-size:1.25rem}html.theme--documenter-dark .buttons.are-large .button:not(.is-small):not(.is-normal):not(.is-medium){font-size:1.5rem}html.theme--documenter-dark .buttons.has-addons 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.pagination.is-rounded .pagination-previous,html.theme--documenter-dark #documenter .docs-sidebar form.docs-search>input.pagination .pagination-previous,html.theme--documenter-dark .pagination.is-rounded .pagination-next,html.theme--documenter-dark #documenter .docs-sidebar form.docs-search>input.pagination .pagination-next{padding-left:1em;padding-right:1em;border-radius:290486px}html.theme--documenter-dark .pagination.is-rounded .pagination-link,html.theme--documenter-dark #documenter .docs-sidebar form.docs-search>input.pagination .pagination-link{border-radius:290486px}html.theme--documenter-dark .pagination,html.theme--documenter-dark .pagination-list{align-items:center;display:flex;justify-content:center;text-align:center}html.theme--documenter-dark .pagination-previous,html.theme--documenter-dark .pagination-next,html.theme--documenter-dark .pagination-link,html.theme--documenter-dark .pagination-ellipsis{font-size:1em;justify-content:center;margin:.25rem;padding-left:.5em;padding-right:.5em;text-align:center}html.theme--documenter-dark .pagination-previous,html.theme--documenter-dark .pagination-next,html.theme--documenter-dark .pagination-link{border-color:#5e6d6f;color:#1abc9c;min-width:2.25em}html.theme--documenter-dark .pagination-previous:hover,html.theme--documenter-dark .pagination-next:hover,html.theme--documenter-dark .pagination-link:hover{border-color:#8c9b9d;color:#1dd2af}html.theme--documenter-dark .pagination-previous:focus,html.theme--documenter-dark .pagination-next:focus,html.theme--documenter-dark .pagination-link:focus{border-color:#8c9b9d}html.theme--documenter-dark .pagination-previous:active,html.theme--documenter-dark .pagination-next:active,html.theme--documenter-dark .pagination-link:active{box-shadow:inset 0 1px 2px rgba(10,10,10,0.2)}html.theme--documenter-dark .pagination-previous[disabled],html.theme--documenter-dark 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ul.is-center{flex:none;justify-content:center;padding-left:0.75em;padding-right:0.75em}html.theme--documenter-dark .tabs ul.is-right{justify-content:flex-end;padding-left:0.75em}html.theme--documenter-dark .tabs .icon:first-child{margin-right:0.5em}html.theme--documenter-dark .tabs .icon:last-child{margin-left:0.5em}html.theme--documenter-dark .tabs.is-centered ul{justify-content:center}html.theme--documenter-dark .tabs.is-right ul{justify-content:flex-end}html.theme--documenter-dark .tabs.is-boxed a{border:1px solid transparent;border-radius:.4em .4em 0 0}html.theme--documenter-dark .tabs.is-boxed a:hover{background-color:#282f2f;border-bottom-color:#5e6d6f}html.theme--documenter-dark .tabs.is-boxed li.is-active a{background-color:#fff;border-color:#5e6d6f;border-bottom-color:rgba(0,0,0,0) !important}html.theme--documenter-dark .tabs.is-fullwidth li{flex-grow:1;flex-shrink:0}html.theme--documenter-dark .tabs.is-toggle 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If `active` stays `null` + // we either could not find the theme or it is the default (primary) theme anyway. + // Either way, we then need to stick to the primary theme. + // + // - disabled: style sheets that should be disabled (i.e. all the theme style sheets + // that are not the currently active theme) + var active = null; var disabled = []; var darkTheme = null; + for (var i = 0; i < document.styleSheets.length; i++) { + var ss = document.styleSheets[i]; + // The tag of each style sheet is expected to have a data-theme-name attribute + // which must contain the name of the theme. 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    Click here to go to the documentation for the latest stable release.'; + div.appendChild(closer); + document.body.appendChild(div); +}; + +if (document.readyState === 'loading') { + document.addEventListener('DOMContentLoaded', maybeAddWarning); +} else { + maybeAddWarning(); +}; diff --git a/previews/PR109/examples/.gitkeep b/previews/PR109/examples/.gitkeep new file mode 100644 index 0000000..e69de29 diff --git a/previews/PR109/examples/0_intro/index.html b/previews/PR109/examples/0_intro/index.html new file mode 100644 index 0000000..67ceb65 --- /dev/null +++ b/previews/PR109/examples/0_intro/index.html @@ -0,0 +1,30 @@ + +Introduction · ImplicitDifferentiation.jl
    View on GitHub

    Introduction

    We explain the basics of our package on a simple function that is not amenable to naive automatic differentiation.

    using ForwardDiff
+using ImplicitDifferentiation
+using LinearAlgebra
+using Random
+using Zygote
+
+Random.seed!(63);

    Why do we bother?

    ForwardDiff.jl and Zygote.jl are two prominent packages for automatic differentiation in Julia. While they are very generic, there are simple language constructs that they cannot differentiate through.

    function badsqrt(x::AbstractArray)
+    a = [0.0]
+    a[1] = first(x)
+    return sqrt.(x)
+end
    badsqrt (generic function with 1 method)

    This is essentially the componentwise square root function but with an additional twist: a::Vector{Float64} is created internally, and its only element is replaced with the first element of x. We can check that it does what it's supposed to do.

    x = rand(2)
+badsqrt(x) ≈ sqrt.(x)
    true

    Of course the Jacobian has an explicit formula.

    J = Diagonal(0.5 ./ sqrt.(x))
    2×2 LinearAlgebra.Diagonal{Float64, Vector{Float64}}:
+ 1.05545   ⋅ 
+  ⋅       0.874736

    However, things start to go wrong when we compute it with autodiff, due to the limitations of ForwardDiff.jl and those of Zygote.jl.

    try
+    ForwardDiff.jacobian(badsqrt, x)
+catch e
+    e
+end
    MethodError(Float64, (Dual{ForwardDiff.Tag{typeof(Main.badsqrt), Float64}}(0.22442135286865494,1.0,0.0),), 0x0000000000008304)

    ForwardDiff.jl throws an error because it tries to call badsqrt with an array of dual numbers, and cannot use one of these numbers to fill a (which has element type Float64).

    try
+    Zygote.jacobian(badsqrt, x)
+catch e
+    e
+end
    ErrorException("Mutating arrays is not supported -- called setindex!(Vector{Float64}, ...)\nThis error occurs when you ask Zygote to differentiate operations that change\nthe elements of arrays in place (e.g. setting values with x .= ...)\n\nPossible fixes:\n- avoid mutating operations (preferred)\n- or read the documentation and solutions for this error\n  https://fluxml.ai/Zygote.jl/latest/limitations\n")

    Zygote.jl also throws an error because it cannot handle mutation.

    Implicit function

    The first possible use of ImplicitDifferentiation.jl is to overcome the limitations of automatic differentiation packages by defining functions (and computing their derivatives) implicitly. An implicit function is a mapping

    \[x \in \mathbb{R}^n \longmapsto y(x) \in \mathbb{R}^m\]

    whose output is defined by conditions

    \[F(x,y(x)) = 0 \in \mathbb{R}^m\]

    We represent it using a type called ImplicitFunction, which you will see in action shortly.

    First we define a forward mapping correponding to the function we consider. It returns the actual output $y(x)$ of the function, and can be thought of as a black box solver. Importantly, this Julia callable doesn't need to be differentiable by automatic differentiation packages but the underlying function still needs to be mathematically differentiable.

    forward(x) = badsqrt(x)
    forward (generic function with 1 method)

    Then we define conditions $c(x, y) = 0$ that the output $y(x)$ is supposed to satisfy. These conditions must be array-valued, with the same size as $y$. Unlike the forward mapping, the conditions need to be differentiable by automatic differentiation packages with respect to both $x$ and $y$. Here the conditions are very obvious: the square of the square root should be equal to the original value.

    function conditions(x, y)
+    c = y .^ 2 .- x
+    return c
+end
    conditions (generic function with 1 method)

    Finally, we construct a wrapper implicit around the previous objects. By default, forward is assumed to return a single output and conditions is assumed to accept 2 arguments.

    implicit = ImplicitFunction(forward, conditions)
    ImplicitFunction(forward, conditions, IterativeLinearSolver(true), nothing)

    What does this wrapper do? When we call it as a function, it just falls back on first ∘ implicit.forward, so unsurprisingly we get the first output $y(x)$.

    implicit(x) ≈ sqrt.(x)
    true

    And when we try to compute its Jacobian, the implicit function theorem is applied in the background to circumvent the lack of differentiablility of the forward mapping.

    Forward and reverse mode autodiff

    Now ForwardDiff.jl works seamlessly.

    ForwardDiff.jacobian(implicit, x) ≈ J
    true

    And so does Zygote.jl. Hurray!

    Zygote.jacobian(implicit, x)[1] ≈ J
    true

    Second derivative

    We can even go higher-order by mixing the two packages (forward-over-reverse mode). The only technical requirement is to switch the linear solver to something that can handle dual numbers:

    implicit_higher_order = ImplicitFunction(
+    forward, conditions; linear_solver=DirectLinearSolver()
+)
    ImplicitFunction(forward, conditions, DirectLinearSolver(true), nothing)

    Then the Jacobian itself is differentiable.

    h = rand(2)
+J_Z(t) = Zygote.jacobian(implicit_higher_order, x .+ t .* h)[1]
+ForwardDiff.derivative(J_Z, 0) ≈ Diagonal((-0.25 .* h) ./ (x .^ 1.5))
    true

    This page was generated using Literate.jl.

    diff --git a/previews/PR109/examples/1_basic/index.html b/previews/PR109/examples/1_basic/index.html new file mode 100644 index 0000000..a5b57db --- /dev/null +++ b/previews/PR109/examples/1_basic/index.html @@ -0,0 +1,74 @@ + +Basic use cases · ImplicitDifferentiation.jl
    View on GitHub

    Basic use cases

    We show how to differentiate through very common routines:

    • an unconstrained optimization problem
    • a nonlinear system of equations
    • a fixed point iteration
    using ForwardDiff
+using ImplicitDifferentiation
+using LinearAlgebra
+using NLsolve
+using Optim
+using Random
+using Zygote
+
+Random.seed!(63);

    In all three cases, we will use the square root as our forward mapping, but expressed in three different ways. Here's our heroic test vector:

    x = rand(2);

    Since we already know the mathematical expression of the Jacobian, we will be able to compare it with our numerical results.

    J = Diagonal(0.5 ./ sqrt.(x))
    2×2 LinearAlgebra.Diagonal{Float64, Vector{Float64}}:
+ 1.05545   ⋅ 
+  ⋅       0.874736

    Unconstrained optimization

    First, we show how to differentiate through the solution of an unconstrained optimization problem:

    \[y(x) = \underset{y \in \mathbb{R}^m}{\mathrm{argmin}} ~ f(x, y)\]

    The optimality conditions are given by gradient stationarity:

    \[\nabla_2 f(x, y) = 0\]

    To make verification easy, we minimize the following objective:

    \[f(x, y) = \lVert y \odot y - x \rVert^2\]

    In this case, the optimization problem boils down to the componentwise square root function, but we implement it using a black box solver from Optim.jl. Note the presence of a keyword argument.

    function forward_optim(x; method)
+    f(y) = sum(abs2, y .^ 2 .- x)
+    y0 = ones(eltype(x), size(x))
+    result = optimize(f, y0, method)
+    return Optim.minimizer(result)
+end
    forward_optim (generic function with 1 method)

    Even though they are defined as a gradient, it is better to provide optimality conditions explicitly: that way we avoid nesting autodiff calls. By default, the conditions should accept two arguments as input. The forward mapping and the conditions should accept the same set of keyword arguments.

    function conditions_optim(x, y; method)
+    ∇₂f = @. 4 * (y^2 - x) * y
+    return ∇₂f
+end
    conditions_optim (generic function with 1 method)

    We now have all the ingredients to construct our implicit function.

    implicit_optim = ImplicitFunction(forward_optim, conditions_optim)
    ImplicitFunction(forward_optim, conditions_optim, IterativeLinearSolver(true), nothing)

    And indeed, it behaves as it should when we call it:

    implicit_optim(x; method=LBFGS()) .^ 2
    2-element Vector{Float64}:
+ 0.22442135282677544
+ 0.3267275093803709

    Forward mode autodiff

    ForwardDiff.jacobian(_x -> implicit_optim(_x; method=LBFGS()), x)
    2×2 Matrix{Float64}:
+ 1.05545  0.0
+ 0.0      0.874736

    In this instance, we could use ForwardDiff.jl directly on the solver, but it returns the wrong result (not sure why).

    ForwardDiff.jacobian(_x -> forward_optim(x; method=LBFGS()), x)
    2×2 Matrix{Float64}:
+ 0.0  0.0
+ 0.0  0.0

    Reverse mode autodiff

    Zygote.jacobian(_x -> implicit_optim(_x; method=LBFGS()), x)[1]
    2×2 Matrix{Float64}:
+  1.05545  -0.0
+ -0.0       0.874736

    In this instance, we cannot use Zygote.jl directly on the solver (due to unsupported try/catch statements).

    try
+    Zygote.jacobian(_x -> forward_optim(x; method=LBFGS()), x)[1]
+catch e
+    e
+end
    Zygote.CompileError(Tuple{typeof(Optim.perform_linesearch!), Optim.LBFGSState{Vector{Float64}, Vector{Vector{Float64}}, Vector{Vector{Float64}}, Float64, Vector{Float64}}, Optim.LBFGS{Nothing, LineSearches.InitialStatic{Float64}, LineSearches.HagerZhang{Float64, Base.RefValue{Bool}}, Optim.var"#19#21"}, Optim.ManifoldObjective{NLSolversBase.OnceDifferentiable{Float64, Vector{Float64}, Vector{Float64}}}}, ErrorException("try/catch is not supported.\nRefer to the Zygote documentation for fixes.\nhttps://fluxml.ai/Zygote.jl/latest/limitations\n"))

    Nonlinear system

    Next, we show how to differentiate through the solution of a nonlinear system of equations:

    \[\text{find} \quad y(x) \quad \text{such that} \quad F(x, y(x)) = 0\]

    The optimality conditions are pretty obvious:

    \[F(x, y) = 0\]

    To make verification easy, we solve the following system:

    \[F(x, y) = y \odot y - x = 0\]

    In this case, the optimization problem boils down to the componentwise square root function, but we implement it using a black box solver from NLsolve.jl.

    function forward_nlsolve(x; method)
+    F!(storage, y) = (storage .= y .^ 2 .- x)
+    initial_y = similar(x)
+    initial_y .= 1
+    result = nlsolve(F!, initial_y; method)
+    return result.zero
+end
    forward_nlsolve (generic function with 1 method)
    function conditions_nlsolve(x, y; method)
+    c = y .^ 2 .- x
+    return c
+end
    conditions_nlsolve (generic function with 1 method)
    implicit_nlsolve = ImplicitFunction(forward_nlsolve, conditions_nlsolve)
    ImplicitFunction(forward_nlsolve, conditions_nlsolve, IterativeLinearSolver(true), nothing)
    implicit_nlsolve(x; method=:newton) .^ 2
    2-element Vector{Float64}:
+ 0.2244213528686593
+ 0.3267275094228514

    Forward mode autodiff

    ForwardDiff.jacobian(_x -> implicit_nlsolve(_x; method=:newton), x)
    2×2 Matrix{Float64}:
+ 1.05545  0.0
+ 0.0      0.874736
    ForwardDiff.jacobian(_x -> forward_nlsolve(_x; method=:newton), x)
    2×2 Matrix{Float64}:
+ 1.05545  0.0
+ 0.0      0.874736

    Reverse mode autodiff

    Zygote.jacobian(_x -> implicit_nlsolve(_x; method=:newton), x)[1]
    2×2 Matrix{Float64}:
+  1.05545  -0.0
+ -0.0       0.874736
    try
+    Zygote.jacobian(_x -> forward_nlsolve(_x; method=:newton), x)[1]
+catch e
+    e
+end
    Zygote.CompileError(Tuple{typeof(NLsolve.newton_), NLSolversBase.OnceDifferentiable{Vector{Float64}, Matrix{Float64}, Vector{Float64}}, Vector{Float64}, Float64, Float64, Int64, Bool, Bool, Bool, LineSearches.Static, NLsolve.var"#27#29", NLsolve.NewtonCache{Vector{Float64}}}, ErrorException("try/catch is not supported.\nRefer to the Zygote documentation for fixes.\nhttps://fluxml.ai/Zygote.jl/latest/limitations\n"))

    Fixed point

    Finally, we show how to differentiate through the limit of a fixed point iteration:

    \[y \longmapsto T(x, y)\]

    The optimality conditions are pretty obvious:

    \[y = T(x, y)\]

    To make verification easy, we consider Heron's method:

    \[T(x, y) = \frac{1}{2} \left(y + \frac{x}{y}\right)\]

    In this case, the fixed point algorithm boils down to the componentwise square root function, but we implement it manually.

    function forward_fixedpoint(x; iterations)
+    y = ones(eltype(x), size(x))
+    for _ in 1:iterations
+        y .= 0.5 .* (y .+ x ./ y)
+    end
+    return y
+end
    forward_fixedpoint (generic function with 1 method)
    function conditions_fixedpoint(x, y; iterations)
+    T = 0.5 .* (y .+ x ./ y)
+    return T .- y
+end
    conditions_fixedpoint (generic function with 1 method)
    implicit_fixedpoint = ImplicitFunction(forward_fixedpoint, conditions_fixedpoint)
    ImplicitFunction(forward_fixedpoint, conditions_fixedpoint, IterativeLinearSolver(true), nothing)
    implicit_fixedpoint(x; iterations=10) .^ 2
    2-element Vector{Float64}:
+ 0.22442135286865494
+ 0.3267275094228514

    Forward mode autodiff

    ForwardDiff.jacobian(_x -> implicit_fixedpoint(_x; iterations=10), x)
    2×2 Matrix{Float64}:
+ 1.05545  0.0
+ 0.0      0.874736
    ForwardDiff.jacobian(_x -> forward_fixedpoint(_x; iterations=10), x)
    2×2 Matrix{Float64}:
+ 1.05545  0.0
+ 0.0      0.874736

    Reverse mode autodiff

    Zygote.jacobian(_x -> implicit_fixedpoint(_x; iterations=10), x)[1]
    2×2 Matrix{Float64}:
+ 1.05545  0.0
+ 0.0      0.874736
    try
+    Zygote.jacobian(_x -> forward_fixedpoint(_x; iterations=10), x)[1]
+catch e
+    e
+end
    ErrorException("Mutating arrays is not supported -- called copyto!(Vector{Float64}, ...)\nThis error occurs when you ask Zygote to differentiate operations that change\nthe elements of arrays in place (e.g. setting values with x .= ...)\n\nPossible fixes:\n- avoid mutating operations (preferred)\n- or read the documentation and solutions for this error\n  https://fluxml.ai/Zygote.jl/latest/limitations\n")

    This page was generated using Literate.jl.

    diff --git a/previews/PR109/examples/2_advanced/index.html b/previews/PR109/examples/2_advanced/index.html new file mode 100644 index 0000000..9bc8ce1 --- /dev/null +++ b/previews/PR109/examples/2_advanced/index.html @@ -0,0 +1,41 @@ + +Advanced use cases · ImplicitDifferentiation.jl
    View on GitHub

    Advanced use cases

    We dive into more advanced applications of implicit differentiation:

    • constrained optimization problems
    using ForwardDiff
+using ImplicitDifferentiation
+using LinearAlgebra
+using Optim
+using Random
+using Zygote
+
+Random.seed!(63);

    Constrained optimization

    First, we show how to differentiate through the solution of a constrained optimization problem:

    \[y(x) = \underset{y \in \mathbb{R}^m}{\mathrm{argmin}} ~ f(x, y) \quad \text{subject to} \quad g(x, y) \leq 0\]

    The optimality conditions are a bit trickier than in the previous cases. We can projection on the feasible set $\mathcal{C}(x) = \{y: g(x, y) \leq 0 \}$ and exploit the convergence of projected gradient descent with step size $\eta$:

    \[y = \mathrm{proj}_{\mathcal{C}(x)} (y - \eta \nabla_2 f(x, y))\]

    To make verification easy, we minimize the following objective:

    \[f(x, y) = \lVert y \odot y - x \rVert^2\]

    on the hypercube $\mathcal{C}(x) = [0, 1]^n$. In this case, the optimization problem boils down to a thresholded componentwise square root function, but we implement it using a black box solver from Optim.jl.

    function forward_cstr_optim(x)
+    f(y) = sum(abs2, y .^ 2 - x)
+    lower = zeros(size(x))
+    upper = ones(size(x))
+    y0 = ones(eltype(x), size(x)) ./ 2
+    res = optimize(f, lower, upper, y0, Fminbox(GradientDescent()))
+    y = Optim.minimizer(res)
+    return y
+end
    forward_cstr_optim (generic function with 1 method)
    function proj_hypercube(p)
+    return max.(0, min.(1, p))
+end
+
+function conditions_cstr_optim(x, y)
+    ∇₂f = @. 4 * (y^2 - x) * y
+    η = 0.1
+    return y .- proj_hypercube(y .- η .* ∇₂f)
+end
    conditions_cstr_optim (generic function with 1 method)

    We now have all the ingredients to construct our implicit function.

    implicit_cstr_optim = ImplicitFunction(forward_cstr_optim, conditions_cstr_optim)
    ImplicitFunction(forward_cstr_optim, conditions_cstr_optim, IterativeLinearSolver(true), nothing)

    And indeed, it behaves as it should when we call it:

    x = rand(2) .+ [0, 1]
    2-element Vector{Float64}:
+ 0.22442135286865494
+ 1.3267275094228514

    The second component of $x$ is $> 1$, so its square root will be thresholded to one, and the corresponding derivative will be $0$.

    implicit_cstr_optim(x) .^ 2
    2-element Vector{Float64}:
+ 0.22442135289492518
+ 0.999999999578276
    J_thres = Diagonal([0.5 / sqrt(x[1]), 0])
    2×2 LinearAlgebra.Diagonal{Float64, Vector{Float64}}:
+ 1.05545   ⋅ 
+  ⋅       0.0

    Forward mode autodiff

    ForwardDiff.jacobian(implicit_cstr_optim, x)
    2×2 Matrix{Float64}:
+ 1.05545  0.0
+ 0.0      0.0
    ForwardDiff.jacobian(forward_cstr_optim, x)
    2×2 Matrix{Float64}:
+ 1.07645     1.04174
+ 4.07705e-7  2.02198e-5

    Reverse mode autodiff

    Zygote.jacobian(implicit_cstr_optim, x)[1]
    2×2 Matrix{Float64}:
+  1.05545  -0.0
+ -0.0       0.0
    try
+    Zygote.jacobian(forward_cstr_optim, x)[1]
+catch e
+    e
+end
    Zygote.CompileError(Tuple{typeof(Optim.optimize), NLSolversBase.OnceDifferentiable{Float64, Vector{Float64}, Vector{Float64}}, Vector{Float64}, Vector{Float64}, Vector{Float64}, Optim.Fminbox{Optim.GradientDescent{LineSearches.InitialPrevious{Float64}, LineSearches.HagerZhang{Float64, Base.RefValue{Bool}}, Nothing, Optim.var"#13#15"}, Float64, Optim.var"#49#51"}, Optim.Options{Float64, Nothing}}, ErrorException("try/catch is not supported.\nRefer to the Zygote documentation for fixes.\nhttps://fluxml.ai/Zygote.jl/latest/limitations\n"))

    This page was generated using Literate.jl.

    diff --git a/previews/PR109/examples/3_tricks/index.html b/previews/PR109/examples/3_tricks/index.html new file mode 100644 index 0000000..698ec98 --- /dev/null +++ b/previews/PR109/examples/3_tricks/index.html @@ -0,0 +1,57 @@ + +Tricks · ImplicitDifferentiation.jl
    View on GitHub

    Tricks

    We demonstrate several features that may come in handy for some users.

    using ComponentArrays
+using ForwardDiff
+using ImplicitDifferentiation
+using Krylov
+using LinearAlgebra
+using Random
+using Zygote
+
+Random.seed!(63);

    ComponentArrays

    For when you need derivatives with respect to multiple inputs or outputs.

    function forward_components_aux(a::AbstractVector, b::AbstractVector, m::Number)
+    d = m * sqrt.(a)
+    e = sqrt.(b)
+    return d, e
+end
+
+function conditions_components_aux(a, b, m, d, e)
+    c_d = (d ./ m) .^ 2 .- a
+    c_e = (e .^ 2) .- b
+    return c_d, c_e
+end;

    You can use ComponentVector as an intermediate storage.

    function forward_components(x::ComponentVector)
+    d, e = forward_components_aux(x.a, x.b, x.m)
+    y = ComponentVector(; d=d, e=e)
+    return y
+end
+
+function conditions_components(x::ComponentVector, y::ComponentVector)
+    c_d, c_e = conditions_components_aux(x.a, x.b, x.m, y.d, y.e)
+    c = ComponentVector(; c_d=c_d, c_e=c_e)
+    return c
+end;

    And build your implicit function like so.

    implicit_components = ImplicitFunction(forward_components, conditions_components);

    Since ComponentVectors are not yet compatible with iterative solvers from Krylov.jl, we (temporarily) need a bit of type piracy to make it work

    Krylov.ktypeof(::ComponentVector{T,V}) where {T,V} = V

    Now we're good to go.

    a, b, m = rand(2), rand(3), 7
+x = ComponentVector(; a=a, b=b, m=m)
+implicit_components(x)
    ComponentVector{Float64}(d = [3.3161191610923892, 4.001205813466701], e = [0.7883634009110729, 0.8810257447318314, 0.5913404265334673])

    And it works with ForwardDiff.jl but not Zygote.jl (see documentation).

    ForwardDiff.jacobian(implicit_components, x)
    5×6 Matrix{Float64}:
+ 7.38815  0.0      0.0       0.0      0.0       0.473731
+ 0.0      6.12315  0.0       0.0      0.0       0.571601
+ 0.0      0.0      0.634225  0.0      0.0       0.0
+ 0.0      0.0      0.0       0.56752  0.0       0.0
+ 0.0      0.0      0.0       0.0      0.845537  0.0
    function full_pipeline(a, b, m)
+    x = ComponentVector(; a=a, b=b, m=m)
+    y = implicit_components(x)
+    return y.d, y.e
+end;

    Byproducts

    For when you need an additional output but don't care about its derivative.

    function forward_byproduct(x)
+    z = rand((2, 2))  # "randomized" choice
+    y = x .^ (1 / z)
+    return y, z
+end
+
+function conditions_byproduct(x, y, z)
+    c = y .^ z .- x
+    return c
+end;

    The syntax for building the implicit function is the same.

    implicit_byproduct = ImplicitFunction(forward_byproduct, conditions_byproduct);

    But this time the return value is a tuple (y, z)

    x = rand(3)
+implicit_byproduct(x)
    ([0.4261353686793698, 0.6583207204780436, 0.7160144800048425], 2)

    And it works with both ForwardDiff.jl and Zygote.jl

    ForwardDiff.jacobian(first ∘ implicit_byproduct, x)
    3×3 Matrix{Float64}:
+ 1.17334  0.0       0.0
+ 0.0      0.759508  0.0
+ 0.0      0.0       0.69831
    Zygote.jacobian(first ∘ implicit_byproduct, x)[1]
    3×3 Matrix{Float64}:
+  1.17334  -0.0       -0.0
+ -0.0       0.759508  -0.0
+ -0.0      -0.0        0.69831

    This page was generated using Literate.jl.

    diff --git a/previews/PR109/faq/index.html b/previews/PR109/faq/index.html new file mode 100644 index 0000000..4acfee9 --- /dev/null +++ b/previews/PR109/faq/index.html @@ -0,0 +1,4 @@ + +FAQ · ImplicitDifferentiation.jl
    View on GitHub

    Frequently Asked Questions

    Supported autodiff backends

    To differentiate an ImplicitFunction, the following backends are supported.

    BackendForward modeReverse mode
    ForwardDiff.jlyes-
    ChainRules.jl-compatibleyessoon
    Enzyme.jlsomedaysomeday

    By default, the conditions are differentiated with the same backend as the ImplicitFunction that contains them. However, this can be switched to any backend compatible with AbstractDifferentiation.jl (i.e. a subtype of AD.AbstractBackend). You can specify it with the conditions_backend keyword argument when constructing an ImplicitFunction.

    Warning

    At the moment, conditions_backend can only be nothing or AD.ForwardDiffBackend(). We are investigating why the other backends fail.

    Input and output types

    Arrays

    Functions that eat or spit out arbitrary arrays are supported, as long as the forward mapping and conditions return arrays of the same size.

    If the output is a small array (say, less than 100 elements), consider using StaticArrays.jl for increased performance.

    Scalars

    Functions that eat or spit out a single number are not supported. The forward mapping and conditions need arrays: instead of returning val you should return [val] (a 1-element Vector). Or better yet, wrap it in a static vector: SVector(val).

    Sparse arrays

    Danger

    Sparse arrays are not supported and might give incorrect values or NaNs!

    Number of inputs and outputs

    Most of the documentation is written for the simple case where the forward mapping is x -> y, i.e. one input and one output. What can you do to handle multiple inputs or outputs? Well, it depends whether you want their derivatives or not.

    Derivatives neededDerivatives not needed
    Multiple inputsMake x a ComponentVectorSupply args and kwargs to forward
    Multiple outputsMake y and c two ComponentVectorsLet forward return a byproduct

    We now detail each of these options.

    Multiple inputs or outputs | Derivatives needed

    Say your forward mapping takes multiple input arrays and returns multiple output arrays, such that you want derivatives for all of them.

    The trick is to leverage ComponentArrays.jl to wrap all the inputs inside a single a ComponentVector, and do the same for all the outputs. See the examples for a demonstration.

    Warning

    You may run into issues trying to differentiate through the ComponentVector constructor. For instance, Zygote.jl will throw ERROR: Mutating arrays is not supported. Check out this issue for a dirty workaround involving custom chain rules for the constructor.

    Multiple inputs | Derivatives not needed

    If your forward mapping (or conditions) takes multiple inputs but you don't care about derivatives, then you can add further positional and keyword arguments beyond x. It is important to make sure that the forward mapping and conditions accept the same set of arguments, even if each of these functions only uses a subset of them.

    forward(x, arg1, arg2; kwarg1, kwarg2) = y
+conditions(x, arg1, arg2; kwarg1, kwarg2) = c

    All of the positional and keyword arguments apart from x will get zero tangents during differentiation of the implicit function.

    Multiple outputs | Derivatives not needed

    The last and most tricky situation is when your forward mapping returns multiple outputs, but you only care about some of their derivatives. Then, you need to group the objects you don't want to differentiate into a "byproduct" z, returned alongside the actual output y. This way, derivatives of z will not be computed: the byproduct is considered constant during differentiation.

    The signatures of your functions will need to be be slightly different from the previous cases:

    forward(x, arg1, arg2; kwarg1, kwarg2) = (y, z)
+conditions(x, y, z, arg1, arg2; kwarg1, kwarg2) =  c

    See the examples for a demonstration.

    This is mainly useful when the solution procedure creates objects such as Jacobians, which we want to reuse when computing or differentiating the conditions. In that case, you may want to write the conditions differentiation rules yourself. A more advanced application is given by DifferentiableFrankWolfe.jl.

    Modeling tips

    Writing conditions

    We recommend that the conditions themselves do not involve calls to autodiff, even when they describe a gradient. Otherwise, you will need to make sure that nested autodiff works well in your case. For instance, if you're differentiating your implicit function (and your conditions) in reverse mode with Zygote.jl, you may want to use ForwardDiff.jl mode to compute gradients inside the conditions.

    Dealing with constraints

    To express constrained optimization problems as implicit functions, you might need differentiable projections or proximal operators to write the optimality conditions. See Efficient and modular implicit differentiation for precise formulations.

    In case these operators are too complicated to code them yourself, here are a few places you can look:

    An alternative is differentiating through the KKT conditions, which is exactly what DiffOpt.jl does for JuMP models.

    diff --git a/previews/PR109/index.html b/previews/PR109/index.html new file mode 100644 index 0000000..5d33fe3 --- /dev/null +++ b/previews/PR109/index.html @@ -0,0 +1,2 @@ + +Home · ImplicitDifferentiation.jl
    View on GitHub

    ImplicitDifferentiation.jl

    Stable Dev Build Status Coverage Code Style: Blue Aqua QA

    ImplicitDifferentiation.jl is a package for automatic differentiation of functions defined implicitly, i.e., forward mappings

    \[x \in \mathbb{R}^n \longmapsto y(x) \in \mathbb{R}^m\]

    whose output is defined by conditions

    \[c(x,y(x)) = 0 \in \mathbb{R}^m\]

    Background

    Implicit differentiation is useful to differentiate through two types of functions:

    • Those for which automatic differentiation fails. Reasons can vary depending on your backend, but the most common include calls to external solvers, mutating operations or type restrictions.
    • Those for which automatic differentiation is very slow. A common example is iterative procedures like fixed point equations or optimization algorithms.

    If you just need a quick overview, check out our JuliaCon 2022 talk. If you want a deeper dive into the theory, you can refer to the paper Efficient and modular implicit differentiation by Blondel et al. (2022).

    Getting started

    To install the stable version, open a Julia REPL and run:

    julia> using Pkg; Pkg.add("ImplicitDifferentiation")

    For the latest version, run this instead:

    julia> using Pkg; Pkg.add(url="https://github.com/gdalle/ImplicitDifferentiation.jl")

    Please read the documentation, especially the examples and FAQ.

    In Julia:

    In Python:

    • google/jaxopt: hardware accelerated, batchable and differentiable optimizers in JAX
    diff --git a/previews/PR109/search/index.html b/previews/PR109/search/index.html new file mode 100644 index 0000000..9606a71 --- /dev/null +++ b/previews/PR109/search/index.html @@ -0,0 +1,2 @@ + +Search · ImplicitDifferentiation.jl

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      diff --git a/previews/PR109/search_index.js b/previews/PR109/search_index.js new file mode 100644 index 0000000..3bbc2d9 --- /dev/null +++ b/previews/PR109/search_index.js @@ -0,0 +1,3 @@ +var documenterSearchIndex = {"docs": +[{"location":"examples/0_intro/","page":"Introduction","title":"Introduction","text":"EditURL = \"../../../examples/0_intro.jl\"","category":"page"},{"location":"examples/0_intro/#Introduction","page":"Introduction","title":"Introduction","text":"","category":"section"},{"location":"examples/0_intro/","page":"Introduction","title":"Introduction","text":"We explain the basics of our package on a simple function that is not amenable to naive automatic differentiation.","category":"page"},{"location":"examples/0_intro/","page":"Introduction","title":"Introduction","text":"using ForwardDiff\nusing ImplicitDifferentiation\nusing LinearAlgebra\nusing Random\nusing Zygote\n\nRandom.seed!(63);\nnothing #hide","category":"page"},{"location":"examples/0_intro/#Why-do-we-bother?","page":"Introduction","title":"Why do we bother?","text":"","category":"section"},{"location":"examples/0_intro/","page":"Introduction","title":"Introduction","text":"ForwardDiff.jl and Zygote.jl are two prominent packages for automatic differentiation in Julia. While they are very generic, there are simple language constructs that they cannot differentiate through.","category":"page"},{"location":"examples/0_intro/","page":"Introduction","title":"Introduction","text":"function badsqrt(x::AbstractArray)\n a = [0.0]\n a[1] = first(x)\n return sqrt.(x)\nend","category":"page"},{"location":"examples/0_intro/","page":"Introduction","title":"Introduction","text":"This is essentially the componentwise square root function but with an additional twist: a::Vector{Float64} is created internally, and its only element is replaced with the first element of x. We can check that it does what it's supposed to do.","category":"page"},{"location":"examples/0_intro/","page":"Introduction","title":"Introduction","text":"x = rand(2)\nbadsqrt(x) ≈ sqrt.(x)","category":"page"},{"location":"examples/0_intro/","page":"Introduction","title":"Introduction","text":"Of course the Jacobian has an explicit formula.","category":"page"},{"location":"examples/0_intro/","page":"Introduction","title":"Introduction","text":"J = Diagonal(0.5 ./ sqrt.(x))","category":"page"},{"location":"examples/0_intro/","page":"Introduction","title":"Introduction","text":"However, things start to go wrong when we compute it with autodiff, due to the limitations of ForwardDiff.jl and those of Zygote.jl.","category":"page"},{"location":"examples/0_intro/","page":"Introduction","title":"Introduction","text":"try\n ForwardDiff.jacobian(badsqrt, x)\ncatch e\n e\nend","category":"page"},{"location":"examples/0_intro/","page":"Introduction","title":"Introduction","text":"ForwardDiff.jl throws an error because it tries to call badsqrt with an array of dual numbers, and cannot use one of these numbers to fill a (which has element type Float64).","category":"page"},{"location":"examples/0_intro/","page":"Introduction","title":"Introduction","text":"try\n Zygote.jacobian(badsqrt, x)\ncatch e\n e\nend","category":"page"},{"location":"examples/0_intro/","page":"Introduction","title":"Introduction","text":"Zygote.jl also throws an error because it cannot handle mutation.","category":"page"},{"location":"examples/0_intro/#Implicit-function","page":"Introduction","title":"Implicit function","text":"","category":"section"},{"location":"examples/0_intro/","page":"Introduction","title":"Introduction","text":"The first possible use of ImplicitDifferentiation.jl is to overcome the limitations of automatic differentiation packages by defining functions (and computing their derivatives) implicitly. An implicit function is a mapping","category":"page"},{"location":"examples/0_intro/","page":"Introduction","title":"Introduction","text":"x in mathbbR^n longmapsto y(x) in mathbbR^m","category":"page"},{"location":"examples/0_intro/","page":"Introduction","title":"Introduction","text":"whose output is defined by conditions","category":"page"},{"location":"examples/0_intro/","page":"Introduction","title":"Introduction","text":"F(xy(x)) = 0 in mathbbR^m","category":"page"},{"location":"examples/0_intro/","page":"Introduction","title":"Introduction","text":"We represent it using a type called ImplicitFunction, which you will see in action shortly.","category":"page"},{"location":"examples/0_intro/","page":"Introduction","title":"Introduction","text":"First we define a forward mapping correponding to the function we consider. It returns the actual output y(x) of the function, and can be thought of as a black box solver. Importantly, this Julia callable doesn't need to be differentiable by automatic differentiation packages but the underlying function still needs to be mathematically differentiable.","category":"page"},{"location":"examples/0_intro/","page":"Introduction","title":"Introduction","text":"forward(x) = badsqrt(x)","category":"page"},{"location":"examples/0_intro/","page":"Introduction","title":"Introduction","text":"Then we define conditions c(x y) = 0 that the output y(x) is supposed to satisfy. These conditions must be array-valued, with the same size as y. Unlike the forward mapping, the conditions need to be differentiable by automatic differentiation packages with respect to both x and y. Here the conditions are very obvious: the square of the square root should be equal to the original value.","category":"page"},{"location":"examples/0_intro/","page":"Introduction","title":"Introduction","text":"function conditions(x, y)\n c = y .^ 2 .- x\n return c\nend","category":"page"},{"location":"examples/0_intro/","page":"Introduction","title":"Introduction","text":"Finally, we construct a wrapper implicit around the previous objects. By default, forward is assumed to return a single output and conditions is assumed to accept 2 arguments.","category":"page"},{"location":"examples/0_intro/","page":"Introduction","title":"Introduction","text":"implicit = ImplicitFunction(forward, conditions)","category":"page"},{"location":"examples/0_intro/","page":"Introduction","title":"Introduction","text":"What does this wrapper do? When we call it as a function, it just falls back on first ∘ implicit.forward, so unsurprisingly we get the first output y(x).","category":"page"},{"location":"examples/0_intro/","page":"Introduction","title":"Introduction","text":"implicit(x) ≈ sqrt.(x)","category":"page"},{"location":"examples/0_intro/","page":"Introduction","title":"Introduction","text":"And when we try to compute its Jacobian, the implicit function theorem is applied in the background to circumvent the lack of differentiablility of the forward mapping.","category":"page"},{"location":"examples/0_intro/#Forward-and-reverse-mode-autodiff","page":"Introduction","title":"Forward and reverse mode autodiff","text":"","category":"section"},{"location":"examples/0_intro/","page":"Introduction","title":"Introduction","text":"Now ForwardDiff.jl works seamlessly.","category":"page"},{"location":"examples/0_intro/","page":"Introduction","title":"Introduction","text":"ForwardDiff.jacobian(implicit, x) ≈ J","category":"page"},{"location":"examples/0_intro/","page":"Introduction","title":"Introduction","text":"And so does Zygote.jl. Hurray!","category":"page"},{"location":"examples/0_intro/","page":"Introduction","title":"Introduction","text":"Zygote.jacobian(implicit, x)[1] ≈ J","category":"page"},{"location":"examples/0_intro/#Second-derivative","page":"Introduction","title":"Second derivative","text":"","category":"section"},{"location":"examples/0_intro/","page":"Introduction","title":"Introduction","text":"We can even go higher-order by mixing the two packages (forward-over-reverse mode). The only technical requirement is to switch the linear solver to something that can handle dual numbers:","category":"page"},{"location":"examples/0_intro/","page":"Introduction","title":"Introduction","text":"implicit_higher_order = ImplicitFunction(\n forward, conditions; linear_solver=DirectLinearSolver()\n)","category":"page"},{"location":"examples/0_intro/","page":"Introduction","title":"Introduction","text":"Then the Jacobian itself is differentiable.","category":"page"},{"location":"examples/0_intro/","page":"Introduction","title":"Introduction","text":"h = rand(2)\nJ_Z(t) = Zygote.jacobian(implicit_higher_order, x .+ t .* h)[1]\nForwardDiff.derivative(J_Z, 0) ≈ Diagonal((-0.25 .* h) ./ (x .^ 1.5))","category":"page"},{"location":"examples/0_intro/","page":"Introduction","title":"Introduction","text":"","category":"page"},{"location":"examples/0_intro/","page":"Introduction","title":"Introduction","text":"This page was generated using Literate.jl.","category":"page"},{"location":"examples/3_tricks/","page":"Tricks","title":"Tricks","text":"EditURL = \"../../../examples/3_tricks.jl\"","category":"page"},{"location":"examples/3_tricks/#Tricks","page":"Tricks","title":"Tricks","text":"","category":"section"},{"location":"examples/3_tricks/","page":"Tricks","title":"Tricks","text":"We demonstrate several features that may come in handy for some users.","category":"page"},{"location":"examples/3_tricks/","page":"Tricks","title":"Tricks","text":"using ComponentArrays\nusing ForwardDiff\nusing ImplicitDifferentiation\nusing Krylov\nusing LinearAlgebra\nusing Random\nusing Zygote\n\nRandom.seed!(63);\nnothing #hide","category":"page"},{"location":"examples/3_tricks/#ComponentArrays","page":"Tricks","title":"ComponentArrays","text":"","category":"section"},{"location":"examples/3_tricks/","page":"Tricks","title":"Tricks","text":"For when you need derivatives with respect to multiple inputs or outputs.","category":"page"},{"location":"examples/3_tricks/","page":"Tricks","title":"Tricks","text":"function forward_components_aux(a::AbstractVector, b::AbstractVector, m::Number)\n d = m * sqrt.(a)\n e = sqrt.(b)\n return d, e\nend\n\nfunction conditions_components_aux(a, b, m, d, e)\n c_d = (d ./ m) .^ 2 .- a\n c_e = (e .^ 2) .- b\n return c_d, c_e\nend;\nnothing #hide","category":"page"},{"location":"examples/3_tricks/","page":"Tricks","title":"Tricks","text":"You can use ComponentVector as an intermediate storage.","category":"page"},{"location":"examples/3_tricks/","page":"Tricks","title":"Tricks","text":"function forward_components(x::ComponentVector)\n d, e = forward_components_aux(x.a, x.b, x.m)\n y = ComponentVector(; d=d, e=e)\n return y\nend\n\nfunction conditions_components(x::ComponentVector, y::ComponentVector)\n c_d, c_e = conditions_components_aux(x.a, x.b, x.m, y.d, y.e)\n c = ComponentVector(; c_d=c_d, c_e=c_e)\n return c\nend;\nnothing #hide","category":"page"},{"location":"examples/3_tricks/","page":"Tricks","title":"Tricks","text":"And build your implicit function like so.","category":"page"},{"location":"examples/3_tricks/","page":"Tricks","title":"Tricks","text":"implicit_components = ImplicitFunction(forward_components, conditions_components);\nnothing #hide","category":"page"},{"location":"examples/3_tricks/","page":"Tricks","title":"Tricks","text":"Since ComponentVectors are not yet compatible with iterative solvers from Krylov.jl, we (temporarily) need a bit of type piracy to make it work","category":"page"},{"location":"examples/3_tricks/","page":"Tricks","title":"Tricks","text":"Krylov.ktypeof(::ComponentVector{T,V}) where {T,V} = V","category":"page"},{"location":"examples/3_tricks/","page":"Tricks","title":"Tricks","text":"Now we're good to go.","category":"page"},{"location":"examples/3_tricks/","page":"Tricks","title":"Tricks","text":"a, b, m = rand(2), rand(3), 7\nx = ComponentVector(; a=a, b=b, m=m)\nimplicit_components(x)","category":"page"},{"location":"examples/3_tricks/","page":"Tricks","title":"Tricks","text":"And it works with ForwardDiff.jl but not Zygote.jl (see documentation).","category":"page"},{"location":"examples/3_tricks/","page":"Tricks","title":"Tricks","text":"ForwardDiff.jacobian(implicit_components, x)","category":"page"},{"location":"examples/3_tricks/","page":"Tricks","title":"Tricks","text":"function full_pipeline(a, b, m)\n x = ComponentVector(; a=a, b=b, m=m)\n y = implicit_components(x)\n return y.d, y.e\nend;\nnothing #hide","category":"page"},{"location":"examples/3_tricks/#Byproducts","page":"Tricks","title":"Byproducts","text":"","category":"section"},{"location":"examples/3_tricks/","page":"Tricks","title":"Tricks","text":"For when you need an additional output but don't care about its derivative.","category":"page"},{"location":"examples/3_tricks/","page":"Tricks","title":"Tricks","text":"function forward_byproduct(x)\n z = rand((2, 2)) # \"randomized\" choice\n y = x .^ (1 / z)\n return y, z\nend\n\nfunction conditions_byproduct(x, y, z)\n c = y .^ z .- x\n return c\nend;\nnothing #hide","category":"page"},{"location":"examples/3_tricks/","page":"Tricks","title":"Tricks","text":"The syntax for building the implicit function is the same.","category":"page"},{"location":"examples/3_tricks/","page":"Tricks","title":"Tricks","text":"implicit_byproduct = ImplicitFunction(forward_byproduct, conditions_byproduct);\nnothing #hide","category":"page"},{"location":"examples/3_tricks/","page":"Tricks","title":"Tricks","text":"But this time the return value is a tuple (y, z)","category":"page"},{"location":"examples/3_tricks/","page":"Tricks","title":"Tricks","text":"x = rand(3)\nimplicit_byproduct(x)","category":"page"},{"location":"examples/3_tricks/","page":"Tricks","title":"Tricks","text":"And it works with both ForwardDiff.jl and Zygote.jl","category":"page"},{"location":"examples/3_tricks/","page":"Tricks","title":"Tricks","text":"ForwardDiff.jacobian(first ∘ implicit_byproduct, x)","category":"page"},{"location":"examples/3_tricks/","page":"Tricks","title":"Tricks","text":"Zygote.jacobian(first ∘ implicit_byproduct, x)[1]","category":"page"},{"location":"examples/3_tricks/","page":"Tricks","title":"Tricks","text":"","category":"page"},{"location":"examples/3_tricks/","page":"Tricks","title":"Tricks","text":"This page was generated using Literate.jl.","category":"page"},{"location":"api/#API-reference","page":"API reference","title":"API reference","text":"","category":"section"},{"location":"api/#Index","page":"API reference","title":"Index","text":"","category":"section"},{"location":"api/","page":"API reference","title":"API reference","text":"","category":"page"},{"location":"api/#Docstrings","page":"API reference","title":"Docstrings","text":"","category":"section"},{"location":"api/","page":"API reference","title":"API reference","text":"Modules = [ImplicitDifferentiation]","category":"page"},{"location":"api/#ImplicitDifferentiation.ImplicitDifferentiation","page":"API reference","title":"ImplicitDifferentiation.ImplicitDifferentiation","text":"ImplicitDifferentiation\n\nA Julia package for automatic differentiation of implicit functions.\n\nIts main export is the type ImplicitFunction.\n\n\n\n\n\n","category":"module"},{"location":"api/#ImplicitDifferentiation.AbstractLinearSolver","page":"API reference","title":"ImplicitDifferentiation.AbstractLinearSolver","text":"AbstractLinearSolver\n\nAll linear solvers used within an ImplicitFunction must satisfy this interface.\n\nIt can be useful to roll out your own solver if you need more fine-grained control on convergence / speed / behavior in case of singularity. Check out the source code of IterativeLinearSolver and DirectLinearSolver for implementation examples. \n\nRequired methods\n\npresolve(linear_solver, A, y): Returns a matrix-like object A for which it is cheaper to solve several linear systems with different vectors b (a typical example would be to perform LU factorization).\nsolve(linear_solver, A, b): Returns a vector x satisfying Ax = b. If the linear system has not been solved to satisfaction, every element of x should be a NaN of the appropriate floating point type.\n\n\n\n\n\n","category":"type"},{"location":"api/#ImplicitDifferentiation.DirectLinearSolver","page":"API reference","title":"ImplicitDifferentiation.DirectLinearSolver","text":"DirectLinearSolver\n\nAn implementation of AbstractLinearSolver using the built-in backslash operator.\n\nFields\n\nverbose::Bool: Whether to throw a warning when the solver fails (defaults to true)\n\n\n\n\n\n","category":"type"},{"location":"api/#ImplicitDifferentiation.ImplicitFunction","page":"API reference","title":"ImplicitDifferentiation.ImplicitFunction","text":"ImplicitFunction{F,C,L,B}\n\nWrapper for an implicit function defined by a forward mapping y and a set of conditions c.\n\nAn ImplicitFunction object behaves like a function, and every call is differentiable with respect to the first argument x. When a derivative is queried, the Jacobian of y is computed using the implicit function theorem:\n\n∂/∂y c(x, y(x)) * ∂/∂x y(x) = -∂/∂x c(x, y(x))\n\nThis requires solving a linear system A * J = -B where A = ∂c/∂y, B = ∂c/∂x and J = ∂y/∂x.\n\nFields\n\nforward::F: a callable, does not need to be compatible with automatic differentiation\nconditions::C: a callable, must be compatible with automatic differentiation\nlinear_solver::L: a subtype of AbstractLinearSolver, defines how the linear system will be solved\nconditions_backend::B: either nothing or a subtype of AbstractDifferentiation.AbstractBackend, defines how the conditions will be differentiated within the implicit function theorem \n\nThere are two possible signatures for forward and conditions, which must be consistent with one another:\n\nStandard: forward(x, args...; kwargs...) = y and conditions(x, y, args...; kwargs...) = c\nByproduct: forward(x, args...; kwargs...) = (y, z) and conditions(x, y, z, args...; kwargs...) = c.\n\nIn both cases, x, y and c must be arrays, with size(y) = size(c). In the second case, the byproduct z can be an arbitrary object generated by forward. The positional arguments args... and keyword arguments kwargs... must be the same for both forward and conditions.\n\nwarning: Warning\nThe byproduct z and the other positional arguments args... beyond x are considered constant for differentiation purposes.\n\n\n\n\n\n","category":"type"},{"location":"api/#ImplicitDifferentiation.ImplicitFunction-Tuple{AbstractArray, Vararg{Any}}","page":"API reference","title":"ImplicitDifferentiation.ImplicitFunction","text":"(implicit::ImplicitFunction)(x::AbstractArray, args...; kwargs...)\n\nReturn implicit.forward(x, args...; kwargs...), which can be either an array y or a tuple (y, z).\n\nThis call is differentiable.\n\n\n\n\n\n","category":"method"},{"location":"api/#ImplicitDifferentiation.ImplicitFunction-Tuple{Any, Any}","page":"API reference","title":"ImplicitDifferentiation.ImplicitFunction","text":"ImplicitFunction(\n forward,\n conditions;\n linear_solver=IterativeLinearSolver(),\n conditions_backend=nothing,\n)\n\nConstruct an ImplicitFunction with default parameters.\n\n\n\n\n\n","category":"method"},{"location":"api/#ImplicitDifferentiation.IterativeLinearSolver","page":"API reference","title":"ImplicitDifferentiation.IterativeLinearSolver","text":"IterativeLinearSolver\n\nAn implementation of AbstractLinearSolver using Krylov.gmres.\n\nFields\n\nverbose::Bool: Whether to throw a warning when the solver fails (defaults to true)\n\n\n\n\n\n","category":"type"},{"location":"api/","page":"API reference","title":"API reference","text":"ChainRulesCore.rrule","category":"page"},{"location":"api/#ChainRulesCore.rrule","page":"API reference","title":"ChainRulesCore.rrule","text":"rrule(rc, implicit, x, args...; kwargs...)\n\nCustom reverse rule for an ImplicitFunction, to ensure compatibility with reverse mode autodiff.\n\nThis is only available if ChainRulesCore.jl is loaded (extension), except on Julia < 1.9 where it is always available.\n\nWe compute the vector-Jacobian product Jᵀv by solving Aᵀu = v and setting Jᵀv = -Bᵀu. Positional and keyword arguments are passed to both implicit.forward and implicit.conditions.\n\n\n\n\n\n","category":"function"},{"location":"examples/2_advanced/","page":"Advanced use cases","title":"Advanced use cases","text":"EditURL = \"../../../examples/2_advanced.jl\"","category":"page"},{"location":"examples/2_advanced/#Advanced-use-cases","page":"Advanced use cases","title":"Advanced use cases","text":"","category":"section"},{"location":"examples/2_advanced/","page":"Advanced use cases","title":"Advanced use cases","text":"We dive into more advanced applications of implicit differentiation:","category":"page"},{"location":"examples/2_advanced/","page":"Advanced use cases","title":"Advanced use cases","text":"constrained optimization problems","category":"page"},{"location":"examples/2_advanced/","page":"Advanced use cases","title":"Advanced use cases","text":"using ForwardDiff\nusing ImplicitDifferentiation\nusing LinearAlgebra\nusing Optim\nusing Random\nusing Zygote\n\nRandom.seed!(63);\nnothing #hide","category":"page"},{"location":"examples/2_advanced/#Constrained-optimization","page":"Advanced use cases","title":"Constrained optimization","text":"","category":"section"},{"location":"examples/2_advanced/","page":"Advanced use cases","title":"Advanced use cases","text":"First, we show how to differentiate through the solution of a constrained optimization problem:","category":"page"},{"location":"examples/2_advanced/","page":"Advanced use cases","title":"Advanced use cases","text":"y(x) = undersety in mathbbR^mmathrmargmin f(x y) quad textsubject to quad g(x y) leq 0","category":"page"},{"location":"examples/2_advanced/","page":"Advanced use cases","title":"Advanced use cases","text":"The optimality conditions are a bit trickier than in the previous cases. We can projection on the feasible set mathcalC(x) = y g(x y) leq 0 and exploit the convergence of projected gradient descent with step size eta:","category":"page"},{"location":"examples/2_advanced/","page":"Advanced use cases","title":"Advanced use cases","text":"y = mathrmproj_mathcalC(x) (y - eta nabla_2 f(x y))","category":"page"},{"location":"examples/2_advanced/","page":"Advanced use cases","title":"Advanced use cases","text":"To make verification easy, we minimize the following objective:","category":"page"},{"location":"examples/2_advanced/","page":"Advanced use cases","title":"Advanced use cases","text":"f(x y) = lVert y odot y - x rVert^2","category":"page"},{"location":"examples/2_advanced/","page":"Advanced use cases","title":"Advanced use cases","text":"on the hypercube mathcalC(x) = 0 1^n. In this case, the optimization problem boils down to a thresholded componentwise square root function, but we implement it using a black box solver from Optim.jl.","category":"page"},{"location":"examples/2_advanced/","page":"Advanced use cases","title":"Advanced use cases","text":"function forward_cstr_optim(x)\n f(y) = sum(abs2, y .^ 2 - x)\n lower = zeros(size(x))\n upper = ones(size(x))\n y0 = ones(eltype(x), size(x)) ./ 2\n res = optimize(f, lower, upper, y0, Fminbox(GradientDescent()))\n y = Optim.minimizer(res)\n return y\nend","category":"page"},{"location":"examples/2_advanced/","page":"Advanced use cases","title":"Advanced use cases","text":"function proj_hypercube(p)\n return max.(0, min.(1, p))\nend\n\nfunction conditions_cstr_optim(x, y)\n ∇₂f = @. 4 * (y^2 - x) * y\n η = 0.1\n return y .- proj_hypercube(y .- η .* ∇₂f)\nend","category":"page"},{"location":"examples/2_advanced/","page":"Advanced use cases","title":"Advanced use cases","text":"We now have all the ingredients to construct our implicit function.","category":"page"},{"location":"examples/2_advanced/","page":"Advanced use cases","title":"Advanced use cases","text":"implicit_cstr_optim = ImplicitFunction(forward_cstr_optim, conditions_cstr_optim)","category":"page"},{"location":"examples/2_advanced/","page":"Advanced use cases","title":"Advanced use cases","text":"And indeed, it behaves as it should when we call it:","category":"page"},{"location":"examples/2_advanced/","page":"Advanced use cases","title":"Advanced use cases","text":"x = rand(2) .+ [0, 1]","category":"page"},{"location":"examples/2_advanced/","page":"Advanced use cases","title":"Advanced use cases","text":"The second component of x is 1, so its square root will be thresholded to one, and the corresponding derivative will be 0.","category":"page"},{"location":"examples/2_advanced/","page":"Advanced use cases","title":"Advanced use cases","text":"implicit_cstr_optim(x) .^ 2","category":"page"},{"location":"examples/2_advanced/","page":"Advanced use cases","title":"Advanced use cases","text":"J_thres = Diagonal([0.5 / sqrt(x[1]), 0])","category":"page"},{"location":"examples/2_advanced/","page":"Advanced use cases","title":"Advanced use cases","text":"Forward mode autodiff","category":"page"},{"location":"examples/2_advanced/","page":"Advanced use cases","title":"Advanced use cases","text":"ForwardDiff.jacobian(implicit_cstr_optim, x)","category":"page"},{"location":"examples/2_advanced/","page":"Advanced use cases","title":"Advanced use cases","text":"ForwardDiff.jacobian(forward_cstr_optim, x)","category":"page"},{"location":"examples/2_advanced/","page":"Advanced use cases","title":"Advanced use cases","text":"Reverse mode autodiff","category":"page"},{"location":"examples/2_advanced/","page":"Advanced use cases","title":"Advanced use cases","text":"Zygote.jacobian(implicit_cstr_optim, x)[1]","category":"page"},{"location":"examples/2_advanced/","page":"Advanced use cases","title":"Advanced use cases","text":"try\n Zygote.jacobian(forward_cstr_optim, x)[1]\ncatch e\n e\nend","category":"page"},{"location":"examples/2_advanced/","page":"Advanced use cases","title":"Advanced use cases","text":"","category":"page"},{"location":"examples/2_advanced/","page":"Advanced use cases","title":"Advanced use cases","text":"This page was generated using Literate.jl.","category":"page"},{"location":"","page":"Home","title":"Home","text":"EditURL = \"https://github.com/gdalle/ImplicitDifferentiation.jl/blob/main/README.md\"","category":"page"},{"location":"#ImplicitDifferentiation.jl","page":"Home","title":"ImplicitDifferentiation.jl","text":"","category":"section"},{"location":"","page":"Home","title":"Home","text":"(Image: Stable) (Image: Dev) (Image: Build Status) (Image: Coverage) (Image: Code Style: Blue) (Image: Aqua QA)","category":"page"},{"location":"","page":"Home","title":"Home","text":"ImplicitDifferentiation.jl is a package for automatic differentiation of functions defined implicitly, i.e., forward mappings","category":"page"},{"location":"","page":"Home","title":"Home","text":"x in mathbbR^n longmapsto y(x) in mathbbR^m","category":"page"},{"location":"","page":"Home","title":"Home","text":"whose output is defined by conditions","category":"page"},{"location":"","page":"Home","title":"Home","text":"c(xy(x)) = 0 in mathbbR^m","category":"page"},{"location":"#Background","page":"Home","title":"Background","text":"","category":"section"},{"location":"","page":"Home","title":"Home","text":"Implicit differentiation is useful to differentiate through two types of functions:","category":"page"},{"location":"","page":"Home","title":"Home","text":"Those for which automatic differentiation fails. Reasons can vary depending on your backend, but the most common include calls to external solvers, mutating operations or type restrictions.\nThose for which automatic differentiation is very slow. A common example is iterative procedures like fixed point equations or optimization algorithms.","category":"page"},{"location":"","page":"Home","title":"Home","text":"If you just need a quick overview, check out our JuliaCon 2022 talk. If you want a deeper dive into the theory, you can refer to the paper Efficient and modular implicit differentiation by Blondel et al. (2022).","category":"page"},{"location":"#Getting-started","page":"Home","title":"Getting started","text":"","category":"section"},{"location":"","page":"Home","title":"Home","text":"To install the stable version, open a Julia REPL and run:","category":"page"},{"location":"","page":"Home","title":"Home","text":"julia> using Pkg; Pkg.add(\"ImplicitDifferentiation\")","category":"page"},{"location":"","page":"Home","title":"Home","text":"For the latest version, run this instead:","category":"page"},{"location":"","page":"Home","title":"Home","text":"julia> using Pkg; Pkg.add(url=\"https://github.com/gdalle/ImplicitDifferentiation.jl\")","category":"page"},{"location":"","page":"Home","title":"Home","text":"Please read the documentation, especially the examples and FAQ.","category":"page"},{"location":"#Related-projects","page":"Home","title":"Related projects","text":"","category":"section"},{"location":"","page":"Home","title":"Home","text":"In Julia:","category":"page"},{"location":"","page":"Home","title":"Home","text":"jump-dev/DiffOpt.jl: differentiation of convex optimization problems\naxelparmentier/InferOpt.jl: approximate differentiation of combinatorial optimization problems\nJuliaNonconvex/NonconvexUtils.jl: contains the original implementation from which this package drew inspiration","category":"page"},{"location":"","page":"Home","title":"Home","text":"In Python:","category":"page"},{"location":"","page":"Home","title":"Home","text":"google/jaxopt: hardware accelerated, batchable and differentiable optimizers in JAX","category":"page"},{"location":"faq/#Frequently-Asked-Questions","page":"FAQ","title":"Frequently Asked Questions","text":"","category":"section"},{"location":"faq/#Supported-autodiff-backends","page":"FAQ","title":"Supported autodiff backends","text":"","category":"section"},{"location":"faq/","page":"FAQ","title":"FAQ","text":"To differentiate an ImplicitFunction, the following backends are supported.","category":"page"},{"location":"faq/","page":"FAQ","title":"FAQ","text":"Backend Forward mode Reverse mode\nForwardDiff.jl yes -\nChainRules.jl-compatible yes soon\nEnzyme.jl someday someday","category":"page"},{"location":"faq/","page":"FAQ","title":"FAQ","text":"By default, the conditions are differentiated with the same backend as the ImplicitFunction that contains them. However, this can be switched to any backend compatible with AbstractDifferentiation.jl (i.e. a subtype of AD.AbstractBackend). You can specify it with the conditions_backend keyword argument when constructing an ImplicitFunction.","category":"page"},{"location":"faq/","page":"FAQ","title":"FAQ","text":"warning: Warning\nAt the moment, conditions_backend can only be nothing or AD.ForwardDiffBackend(). We are investigating why the other backends fail.","category":"page"},{"location":"faq/#Input-and-output-types","page":"FAQ","title":"Input and output types","text":"","category":"section"},{"location":"faq/#Arrays","page":"FAQ","title":"Arrays","text":"","category":"section"},{"location":"faq/","page":"FAQ","title":"FAQ","text":"Functions that eat or spit out arbitrary arrays are supported, as long as the forward mapping and conditions return arrays of the same size.","category":"page"},{"location":"faq/","page":"FAQ","title":"FAQ","text":"If the output is a small array (say, less than 100 elements), consider using StaticArrays.jl for increased performance.","category":"page"},{"location":"faq/#Scalars","page":"FAQ","title":"Scalars","text":"","category":"section"},{"location":"faq/","page":"FAQ","title":"FAQ","text":"Functions that eat or spit out a single number are not supported. The forward mapping and conditions need arrays: instead of returning val you should return [val] (a 1-element Vector). Or better yet, wrap it in a static vector: SVector(val).","category":"page"},{"location":"faq/#Sparse-arrays","page":"FAQ","title":"Sparse arrays","text":"","category":"section"},{"location":"faq/","page":"FAQ","title":"FAQ","text":"danger: Danger\nSparse arrays are not supported and might give incorrect values or NaNs!","category":"page"},{"location":"faq/#Number-of-inputs-and-outputs","page":"FAQ","title":"Number of inputs and outputs","text":"","category":"section"},{"location":"faq/","page":"FAQ","title":"FAQ","text":"Most of the documentation is written for the simple case where the forward mapping is x -> y, i.e. one input and one output. What can you do to handle multiple inputs or outputs? Well, it depends whether you want their derivatives or not.","category":"page"},{"location":"faq/","page":"FAQ","title":"FAQ","text":" Derivatives needed Derivatives not needed\nMultiple inputs Make x a ComponentVector Supply args and kwargs to forward\nMultiple outputs Make y and c two ComponentVectors Let forward return a byproduct","category":"page"},{"location":"faq/","page":"FAQ","title":"FAQ","text":"We now detail each of these options.","category":"page"},{"location":"faq/#Multiple-inputs-or-outputs-Derivatives-needed","page":"FAQ","title":"Multiple inputs or outputs | Derivatives needed","text":"","category":"section"},{"location":"faq/","page":"FAQ","title":"FAQ","text":"Say your forward mapping takes multiple input arrays and returns multiple output arrays, such that you want derivatives for all of them.","category":"page"},{"location":"faq/","page":"FAQ","title":"FAQ","text":"The trick is to leverage ComponentArrays.jl to wrap all the inputs inside a single a ComponentVector, and do the same for all the outputs. See the examples for a demonstration.","category":"page"},{"location":"faq/","page":"FAQ","title":"FAQ","text":"warning: Warning\nYou may run into issues trying to differentiate through the ComponentVector constructor. For instance, Zygote.jl will throw ERROR: Mutating arrays is not supported. Check out this issue for a dirty workaround involving custom chain rules for the constructor.","category":"page"},{"location":"faq/#Multiple-inputs-Derivatives-not-needed","page":"FAQ","title":"Multiple inputs | Derivatives not needed","text":"","category":"section"},{"location":"faq/","page":"FAQ","title":"FAQ","text":"If your forward mapping (or conditions) takes multiple inputs but you don't care about derivatives, then you can add further positional and keyword arguments beyond x. It is important to make sure that the forward mapping and conditions accept the same set of arguments, even if each of these functions only uses a subset of them.","category":"page"},{"location":"faq/","page":"FAQ","title":"FAQ","text":"forward(x, arg1, arg2; kwarg1, kwarg2) = y\nconditions(x, arg1, arg2; kwarg1, kwarg2) = c","category":"page"},{"location":"faq/","page":"FAQ","title":"FAQ","text":"All of the positional and keyword arguments apart from x will get zero tangents during differentiation of the implicit function.","category":"page"},{"location":"faq/#Multiple-outputs-Derivatives-not-needed","page":"FAQ","title":"Multiple outputs | Derivatives not needed","text":"","category":"section"},{"location":"faq/","page":"FAQ","title":"FAQ","text":"The last and most tricky situation is when your forward mapping returns multiple outputs, but you only care about some of their derivatives. Then, you need to group the objects you don't want to differentiate into a \"byproduct\" z, returned alongside the actual output y. This way, derivatives of z will not be computed: the byproduct is considered constant during differentiation.","category":"page"},{"location":"faq/","page":"FAQ","title":"FAQ","text":"The signatures of your functions will need to be be slightly different from the previous cases:","category":"page"},{"location":"faq/","page":"FAQ","title":"FAQ","text":"forward(x, arg1, arg2; kwarg1, kwarg2) = (y, z)\nconditions(x, y, z, arg1, arg2; kwarg1, kwarg2) = c","category":"page"},{"location":"faq/","page":"FAQ","title":"FAQ","text":"See the examples for a demonstration.","category":"page"},{"location":"faq/","page":"FAQ","title":"FAQ","text":"This is mainly useful when the solution procedure creates objects such as Jacobians, which we want to reuse when computing or differentiating the conditions. In that case, you may want to write the conditions differentiation rules yourself. A more advanced application is given by DifferentiableFrankWolfe.jl.","category":"page"},{"location":"faq/#Modeling-tips","page":"FAQ","title":"Modeling tips","text":"","category":"section"},{"location":"faq/#Writing-conditions","page":"FAQ","title":"Writing conditions","text":"","category":"section"},{"location":"faq/","page":"FAQ","title":"FAQ","text":"We recommend that the conditions themselves do not involve calls to autodiff, even when they describe a gradient. Otherwise, you will need to make sure that nested autodiff works well in your case. For instance, if you're differentiating your implicit function (and your conditions) in reverse mode with Zygote.jl, you may want to use ForwardDiff.jl mode to compute gradients inside the conditions.","category":"page"},{"location":"faq/#Dealing-with-constraints","page":"FAQ","title":"Dealing with constraints","text":"","category":"section"},{"location":"faq/","page":"FAQ","title":"FAQ","text":"To express constrained optimization problems as implicit functions, you might need differentiable projections or proximal operators to write the optimality conditions. See Efficient and modular implicit differentiation for precise formulations.","category":"page"},{"location":"faq/","page":"FAQ","title":"FAQ","text":"In case these operators are too complicated to code them yourself, here are a few places you can look:","category":"page"},{"location":"faq/","page":"FAQ","title":"FAQ","text":"MathOptSetDistances.jl\nProximalOperators.jl","category":"page"},{"location":"faq/","page":"FAQ","title":"FAQ","text":"An alternative is differentiating through the KKT conditions, which is exactly what DiffOpt.jl does for JuMP models.","category":"page"},{"location":"examples/1_basic/","page":"Basic use cases","title":"Basic use cases","text":"EditURL = \"../../../examples/1_basic.jl\"","category":"page"},{"location":"examples/1_basic/#Basic-use-cases","page":"Basic use cases","title":"Basic use cases","text":"","category":"section"},{"location":"examples/1_basic/","page":"Basic use cases","title":"Basic use cases","text":"We show how to differentiate through very common routines:","category":"page"},{"location":"examples/1_basic/","page":"Basic use cases","title":"Basic use cases","text":"an unconstrained optimization problem\na nonlinear system of equations\na fixed point iteration","category":"page"},{"location":"examples/1_basic/","page":"Basic use cases","title":"Basic use cases","text":"using ForwardDiff\nusing ImplicitDifferentiation\nusing LinearAlgebra\nusing NLsolve\nusing Optim\nusing Random\nusing Zygote\n\nRandom.seed!(63);\nnothing #hide","category":"page"},{"location":"examples/1_basic/","page":"Basic use cases","title":"Basic use cases","text":"In all three cases, we will use the square root as our forward mapping, but expressed in three different ways. Here's our heroic test vector:","category":"page"},{"location":"examples/1_basic/","page":"Basic use cases","title":"Basic use cases","text":"x = rand(2);\nnothing #hide","category":"page"},{"location":"examples/1_basic/","page":"Basic use cases","title":"Basic use cases","text":"Since we already know the mathematical expression of the Jacobian, we will be able to compare it with our numerical results.","category":"page"},{"location":"examples/1_basic/","page":"Basic use cases","title":"Basic use cases","text":"J = Diagonal(0.5 ./ sqrt.(x))","category":"page"},{"location":"examples/1_basic/#Unconstrained-optimization","page":"Basic use cases","title":"Unconstrained optimization","text":"","category":"section"},{"location":"examples/1_basic/","page":"Basic use cases","title":"Basic use cases","text":"First, we show how to differentiate through the solution of an unconstrained optimization problem:","category":"page"},{"location":"examples/1_basic/","page":"Basic use cases","title":"Basic use cases","text":"y(x) = undersety in mathbbR^mmathrmargmin f(x y)","category":"page"},{"location":"examples/1_basic/","page":"Basic use cases","title":"Basic use cases","text":"The optimality conditions are given by gradient stationarity:","category":"page"},{"location":"examples/1_basic/","page":"Basic use cases","title":"Basic use cases","text":"nabla_2 f(x y) = 0","category":"page"},{"location":"examples/1_basic/","page":"Basic use cases","title":"Basic use cases","text":"To make verification easy, we minimize the following objective:","category":"page"},{"location":"examples/1_basic/","page":"Basic use cases","title":"Basic use cases","text":"f(x y) = lVert y odot y - x rVert^2","category":"page"},{"location":"examples/1_basic/","page":"Basic use cases","title":"Basic use cases","text":"In this case, the optimization problem boils down to the componentwise square root function, but we implement it using a black box solver from Optim.jl. Note the presence of a keyword argument.","category":"page"},{"location":"examples/1_basic/","page":"Basic use cases","title":"Basic use cases","text":"function forward_optim(x; method)\n f(y) = sum(abs2, y .^ 2 .- x)\n y0 = ones(eltype(x), size(x))\n result = optimize(f, y0, method)\n return Optim.minimizer(result)\nend","category":"page"},{"location":"examples/1_basic/","page":"Basic use cases","title":"Basic use cases","text":"Even though they are defined as a gradient, it is better to provide optimality conditions explicitly: that way we avoid nesting autodiff calls. By default, the conditions should accept two arguments as input. The forward mapping and the conditions should accept the same set of keyword arguments.","category":"page"},{"location":"examples/1_basic/","page":"Basic use cases","title":"Basic use cases","text":"function conditions_optim(x, y; method)\n ∇₂f = @. 4 * (y^2 - x) * y\n return ∇₂f\nend","category":"page"},{"location":"examples/1_basic/","page":"Basic use cases","title":"Basic use cases","text":"We now have all the ingredients to construct our implicit function.","category":"page"},{"location":"examples/1_basic/","page":"Basic use cases","title":"Basic use cases","text":"implicit_optim = ImplicitFunction(forward_optim, conditions_optim)","category":"page"},{"location":"examples/1_basic/","page":"Basic use cases","title":"Basic use cases","text":"And indeed, it behaves as it should when we call it:","category":"page"},{"location":"examples/1_basic/","page":"Basic use cases","title":"Basic use cases","text":"implicit_optim(x; method=LBFGS()) .^ 2","category":"page"},{"location":"examples/1_basic/","page":"Basic use cases","title":"Basic use cases","text":"Forward mode autodiff","category":"page"},{"location":"examples/1_basic/","page":"Basic use cases","title":"Basic use cases","text":"ForwardDiff.jacobian(_x -> implicit_optim(_x; method=LBFGS()), x)","category":"page"},{"location":"examples/1_basic/","page":"Basic use cases","title":"Basic use cases","text":"In this instance, we could use ForwardDiff.jl directly on the solver, but it returns the wrong result (not sure why).","category":"page"},{"location":"examples/1_basic/","page":"Basic use cases","title":"Basic use cases","text":"ForwardDiff.jacobian(_x -> forward_optim(x; method=LBFGS()), x)","category":"page"},{"location":"examples/1_basic/","page":"Basic use cases","title":"Basic use cases","text":"Reverse mode autodiff","category":"page"},{"location":"examples/1_basic/","page":"Basic use cases","title":"Basic use cases","text":"Zygote.jacobian(_x -> implicit_optim(_x; method=LBFGS()), x)[1]","category":"page"},{"location":"examples/1_basic/","page":"Basic use cases","title":"Basic use cases","text":"In this instance, we cannot use Zygote.jl directly on the solver (due to unsupported try/catch statements).","category":"page"},{"location":"examples/1_basic/","page":"Basic use cases","title":"Basic use cases","text":"try\n Zygote.jacobian(_x -> forward_optim(x; method=LBFGS()), x)[1]\ncatch e\n e\nend","category":"page"},{"location":"examples/1_basic/#Nonlinear-system","page":"Basic use cases","title":"Nonlinear system","text":"","category":"section"},{"location":"examples/1_basic/","page":"Basic use cases","title":"Basic use cases","text":"Next, we show how to differentiate through the solution of a nonlinear system of equations:","category":"page"},{"location":"examples/1_basic/","page":"Basic use cases","title":"Basic use cases","text":"textfind quad y(x) quad textsuch that quad F(x y(x)) = 0","category":"page"},{"location":"examples/1_basic/","page":"Basic use cases","title":"Basic use cases","text":"The optimality conditions are pretty obvious:","category":"page"},{"location":"examples/1_basic/","page":"Basic use cases","title":"Basic use cases","text":"F(x y) = 0","category":"page"},{"location":"examples/1_basic/","page":"Basic use cases","title":"Basic use cases","text":"To make verification easy, we solve the following system:","category":"page"},{"location":"examples/1_basic/","page":"Basic use cases","title":"Basic use cases","text":"F(x y) = y odot y - x = 0","category":"page"},{"location":"examples/1_basic/","page":"Basic use cases","title":"Basic use cases","text":"In this case, the optimization problem boils down to the componentwise square root function, but we implement it using a black box solver from NLsolve.jl.","category":"page"},{"location":"examples/1_basic/","page":"Basic use cases","title":"Basic use cases","text":"function forward_nlsolve(x; method)\n F!(storage, y) = (storage .= y .^ 2 .- x)\n initial_y = similar(x)\n initial_y .= 1\n result = nlsolve(F!, initial_y; method)\n return result.zero\nend","category":"page"},{"location":"examples/1_basic/","page":"Basic use cases","title":"Basic use cases","text":"function conditions_nlsolve(x, y; method)\n c = y .^ 2 .- x\n return c\nend","category":"page"},{"location":"examples/1_basic/","page":"Basic use cases","title":"Basic use cases","text":"implicit_nlsolve = ImplicitFunction(forward_nlsolve, conditions_nlsolve)","category":"page"},{"location":"examples/1_basic/","page":"Basic use cases","title":"Basic use cases","text":"implicit_nlsolve(x; method=:newton) .^ 2","category":"page"},{"location":"examples/1_basic/","page":"Basic use cases","title":"Basic use cases","text":"Forward mode autodiff","category":"page"},{"location":"examples/1_basic/","page":"Basic use cases","title":"Basic use cases","text":"ForwardDiff.jacobian(_x -> implicit_nlsolve(_x; method=:newton), x)","category":"page"},{"location":"examples/1_basic/","page":"Basic use cases","title":"Basic use cases","text":"ForwardDiff.jacobian(_x -> forward_nlsolve(_x; method=:newton), x)","category":"page"},{"location":"examples/1_basic/","page":"Basic use cases","title":"Basic use cases","text":"Reverse mode autodiff","category":"page"},{"location":"examples/1_basic/","page":"Basic use cases","title":"Basic use cases","text":"Zygote.jacobian(_x -> implicit_nlsolve(_x; method=:newton), x)[1]","category":"page"},{"location":"examples/1_basic/","page":"Basic use cases","title":"Basic use cases","text":"try\n Zygote.jacobian(_x -> forward_nlsolve(_x; method=:newton), x)[1]\ncatch e\n e\nend","category":"page"},{"location":"examples/1_basic/#Fixed-point","page":"Basic use cases","title":"Fixed point","text":"","category":"section"},{"location":"examples/1_basic/","page":"Basic use cases","title":"Basic use cases","text":"Finally, we show how to differentiate through the limit of a fixed point iteration:","category":"page"},{"location":"examples/1_basic/","page":"Basic use cases","title":"Basic use cases","text":"y longmapsto T(x y)","category":"page"},{"location":"examples/1_basic/","page":"Basic use cases","title":"Basic use cases","text":"The optimality conditions are pretty obvious:","category":"page"},{"location":"examples/1_basic/","page":"Basic use cases","title":"Basic use cases","text":"y = T(x y)","category":"page"},{"location":"examples/1_basic/","page":"Basic use cases","title":"Basic use cases","text":"To make verification easy, we consider Heron's method:","category":"page"},{"location":"examples/1_basic/","page":"Basic use cases","title":"Basic use cases","text":"T(x y) = frac12 left(y + fracxyright)","category":"page"},{"location":"examples/1_basic/","page":"Basic use cases","title":"Basic use cases","text":"In this case, the fixed point algorithm boils down to the componentwise square root function, but we implement it manually.","category":"page"},{"location":"examples/1_basic/","page":"Basic use cases","title":"Basic use cases","text":"function forward_fixedpoint(x; iterations)\n y = ones(eltype(x), size(x))\n for _ in 1:iterations\n y .= 0.5 .* (y .+ x ./ y)\n end\n return y\nend","category":"page"},{"location":"examples/1_basic/","page":"Basic use cases","title":"Basic use cases","text":"function conditions_fixedpoint(x, y; iterations)\n T = 0.5 .* (y .+ x ./ y)\n return T .- y\nend","category":"page"},{"location":"examples/1_basic/","page":"Basic use cases","title":"Basic use cases","text":"implicit_fixedpoint = ImplicitFunction(forward_fixedpoint, conditions_fixedpoint)","category":"page"},{"location":"examples/1_basic/","page":"Basic use cases","title":"Basic use cases","text":"implicit_fixedpoint(x; iterations=10) .^ 2","category":"page"},{"location":"examples/1_basic/","page":"Basic use cases","title":"Basic use cases","text":"Forward mode autodiff","category":"page"},{"location":"examples/1_basic/","page":"Basic use cases","title":"Basic use cases","text":"ForwardDiff.jacobian(_x -> implicit_fixedpoint(_x; iterations=10), x)","category":"page"},{"location":"examples/1_basic/","page":"Basic use cases","title":"Basic use cases","text":"ForwardDiff.jacobian(_x -> forward_fixedpoint(_x; iterations=10), x)","category":"page"},{"location":"examples/1_basic/","page":"Basic use cases","title":"Basic use cases","text":"Reverse mode autodiff","category":"page"},{"location":"examples/1_basic/","page":"Basic use cases","title":"Basic use cases","text":"Zygote.jacobian(_x -> implicit_fixedpoint(_x; iterations=10), x)[1]","category":"page"},{"location":"examples/1_basic/","page":"Basic use cases","title":"Basic use cases","text":"try\n Zygote.jacobian(_x -> forward_fixedpoint(_x; iterations=10), x)[1]\ncatch e\n e\nend","category":"page"},{"location":"examples/1_basic/","page":"Basic use cases","title":"Basic use cases","text":"","category":"page"},{"location":"examples/1_basic/","page":"Basic use cases","title":"Basic use cases","text":"This page was generated using Literate.jl.","category":"page"}] +} diff --git a/previews/PR109/siteinfo.js b/previews/PR109/siteinfo.js new file mode 100644 index 0000000..3c9d417 --- /dev/null +++ b/previews/PR109/siteinfo.js @@ -0,0 +1 @@ +var DOCUMENTER_CURRENT_VERSION = "previews/PR109";