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Formalizing Markov Decision Processes in Lean

Blueprint

Verified Lean algorithms for solving tabular MDPs and proving their properties. The focus of this project is on two main goals:

  1. Basic algorithms that can solve robust and risk-averse MDPs of moderate size.

  2. Proofs of correctness of algorithms and fundamental MDP properties which can be used independently to prove structural results, such as the optimality of certain policy class.

Status

General

  • Left and right continuity (exists?)
  • Generalized inverse; left and right continous functions (exists?)

Basic probability properties

  • Definitions: probability space and definition
  • Definitions: probability, expectation, conditional properties
  • Tower property, law of the unconscious statistician
  • Quantile definition and basic properties
  • Quantile under monotone transformation
  • Conditional probability = change of measure
  • Independent random variables
  • Construct probability from a compile-time input
  • Construct probability from runtime input

Value at Risk

  • Definition (non-constructive)
  • Practical implementation O(n^2) and correctness
  • Fast practical implementation O(n log n) and correctness
  • Definition of VaR as minimization
  • VaR is positively homogeneous and monotone
  • VaR is translation (cash) invariant
  • VaR under monotone transformation
  • Check risk measure values in a JSON file

MDP: Basics

  • Definition of MDP
  • Definition of policies (history, Markov, stationary)
  • Policy induces a distribution over histories
  • Definition of value function (history-dependent)

Finite Horizon

  • Histories and manipulation
  • Probability space over histories
  • Return and optimal return using histories
  • History-dependent value function and dynamic program
  • Markov optimal value function and optimal policy
  • DP algorithms

Risk-averse finite horizon

  • History-dependent utility functions
  • Augmented value function dynamic program
  • VaR computation from utility function
  • VaR DP decomposition as in Hau et al., 2023

Discounted infinite horizon

Average reward horizon

Lean Resources

Most useful

Others

About

Formalization of MDP Theory

Resources

Readme
Unlicense
Activity
Custom properties

Stars

4 stars

Watchers

0 watching

Forks

13 forks
Report repository

Releases

Packages

Contributors

Languages

, 'i'); if (__m === '*' || __re.test(location.href)) { injectUserscript("// Add copy buttons to all 
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})();
(function(){
try {
var __m = "github.com";
var __re = new RegExp('^' + "github\\.com" + '

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Formalizing Markov Decision Processes in Lean
Blueprint
Verified Lean algorithms for solving tabular MDPs and proving their properties. The focus of this project is on two main goals:
  1. Basic algorithms that can solve robust and risk-averse MDPs of moderate size.
  2. Proofs of correctness of algorithms and fundamental MDP properties which can be used independently to prove structural results, such as the optimality of certain policy class.
Status
General
  •  Left and right continuity (exists?)
  •  Generalized inverse; left and right continous functions (exists?)
Basic probability properties
  •  Definitions: probability space and definition
  •  Definitions: probability, expectation, conditional properties
  •  Tower property, law of the unconscious statistician
  •  Quantile definition and basic properties
  •  Quantile under monotone transformation
  •  Conditional probability = change of measure
  •  Independent random variables
  •  Construct probability from a compile-time input
  •  Construct probability from runtime input
Value at Risk
  •  Definition (non-constructive)
  •  Practical implementation O(n^2) and correctness
  •  Fast practical implementation O(n log n) and correctness
  •  Definition of VaR as minimization
  •  VaR is positively homogeneous and monotone
  •  VaR is translation (cash) invariant
  •  VaR under monotone transformation
  •  Check risk measure values in a JSON file
MDP: Basics
  •  Definition of MDP
  •  Definition of policies (history, Markov, stationary)
  •  Policy induces a distribution over histories
  •  Definition of value function (history-dependent)
Finite Horizon
  •  Histories and manipulation
  •  Probability space over histories
  •  Return and optimal return using histories
  •  History-dependent value function and dynamic program
  •  Markov optimal value function and optimal policy
  •  DP algorithms
Risk-averse finite horizon
  •  History-dependent utility functions
  •  Augmented value function dynamic program
  •  VaR computation from utility function
  •  VaR DP decomposition as in Hau et al., 2023
Discounted infinite horizon
Average reward horizon
Lean Resources
Most useful
Others
About
Formalization of MDP Theory
Resources
Readme
Unlicense
Activity
Custom properties
Stars
4 stars
Watchers
0 watching
Forks
13 forks
Report repository
Releases
Packages
Contributors
Languages

, 'i');
if (__m === '*' || __re.test(location.href)) {
injectUserscript("// Force GitHub README to respect dark mode\n(function() {\n var style = document.createElement('style');\n style.textContent = '\n .markdown-body {\n color-scheme: dark light;\n }\n .markdown-body pre { background: #161b22 !important; }\n .markdown-body code { background: rgba(110, 118, 129, 0.4) !important; }\n .markdown-body table th, .markdown-body table td { border-color: #30363d !important; }\n .markdown-body img { background: #0d1117; }\n .markdown-body blockquote { border-left-color: #8b949e; }\n .markdown-body hr { border-color: #30363d; }\n ';\n document.head.appendChild(style);\n})();", "GitHub Dark Mode README Fix");
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})();
(function(){
try {
var __m = "*";
var __re = new RegExp('^' + ".*" + '

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Formalizing Markov Decision Processes in Lean
Blueprint
Verified Lean algorithms for solving tabular MDPs and proving their properties. The focus of this project is on two main goals:
  1. Basic algorithms that can solve robust and risk-averse MDPs of moderate size.
  2. Proofs of correctness of algorithms and fundamental MDP properties which can be used independently to prove structural results, such as the optimality of certain policy class.
Status
General
  •  Left and right continuity (exists?)
  •  Generalized inverse; left and right continous functions (exists?)
Basic probability properties
  •  Definitions: probability space and definition
  •  Definitions: probability, expectation, conditional properties
  •  Tower property, law of the unconscious statistician
  •  Quantile definition and basic properties
  •  Quantile under monotone transformation
  •  Conditional probability = change of measure
  •  Independent random variables
  •  Construct probability from a compile-time input
  •  Construct probability from runtime input
Value at Risk
  •  Definition (non-constructive)
  •  Practical implementation O(n^2) and correctness
  •  Fast practical implementation O(n log n) and correctness
  •  Definition of VaR as minimization
  •  VaR is positively homogeneous and monotone
  •  VaR is translation (cash) invariant
  •  VaR under monotone transformation
  •  Check risk measure values in a JSON file
MDP: Basics
  •  Definition of MDP
  •  Definition of policies (history, Markov, stationary)
  •  Policy induces a distribution over histories
  •  Definition of value function (history-dependent)
Finite Horizon
  •  Histories and manipulation
  •  Probability space over histories
  •  Return and optimal return using histories
  •  History-dependent value function and dynamic program
  •  Markov optimal value function and optimal policy
  •  DP algorithms
Risk-averse finite horizon
  •  History-dependent utility functions
  •  Augmented value function dynamic program
  •  VaR computation from utility function
  •  VaR DP decomposition as in Hau et al., 2023
Discounted infinite horizon
Average reward horizon
Lean Resources
Most useful
Others
About
Formalization of MDP Theory
Resources
Readme
Unlicense
Activity
Custom properties
Stars
4 stars
Watchers
0 watching
Forks
13 forks
Report repository
Releases
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, 'i');
if (__m === '*' || __re.test(location.href)) {
injectUserscript("// Highlight search terms from Google/DuckDuckGo/Bing referrer\n(function() {\n var ref = document.referrer;\n var terms = [];\n \n if (ref.includes('google.com') || ref.includes('duckduckgo.com') || ref.includes('bing.com')) {\n var url = new URL(ref);\n var q = url.searchParams.get('q') || url.searchParams.get('p');\n if (q) {\n terms = q.split(/\\s+/).filter(function(t) { return t.length > 2; });\n }\n }\n \n if (terms.length === 0) return;\n \n var style = document.createElement('style');\n style.textContent = '.userscript-highlight { background: #fbbf24; color: #1a1a2e; padding: 1px 3px; border-radius: 2px; }';\n document.head.appendChild(style);\n \n function highlight(node) {\n if (node.nodeType === 3) { // text node\n var text = node.textContent;\n var found = false;\n terms.forEach(function(term) {\n var regex = new RegExp('(' + term.replace(/[.*+?^${}()|[\\]\\\\]/g, '\\\\') + ')', 'gi');\n if (regex.test(text)) {\n found = true;\n var frag = document.createDocumentFragment();\n var parts = text.split(regex);\n parts.forEach(function(part, i) {\n if (i % 2 === 0) {\n frag.appendChild(document.createTextNode(part));\n } else {\n var span = document.createElement('span');\n span.className = 'userscript-highlight';\n span.textContent = part;\n frag.appendChild(span);\n }\n });\n node.parentNode.replaceChild(frag, node);\n }\n });\n } else if (node.nodeType === 1 && node.childNodes) { // element\n var skipTags = ['SCRIPT', 'STYLE', 'NOSCRIPT', 'TEXTAREA', 'INPUT', 'SELECT'];\n if (!skipTags.includes(node.tagName)) {\n Array.from(node.childNodes).forEach(highlight);\n }\n }\n }\n \n highlight(document.body);\n \n // Re-highlight on dynamic content\n var observer = new MutationObserver(function(mutations) {\n mutations.forEach(function(m) {\n m.addedNodes.forEach(function(node) {\n if (node.nodeType === 1 || node.nodeType === 3) highlight(node);\n });\n });\n });\n observer.observe(document.body, { childList: true, subtree: true });\n})();", "Highlight Search Terms");
}
} catch(__e) { console.warn('[Userscript:Highlight Search Terms]', __e); }
})();
(function(){
try {
var __m = "*";
var __re = new RegExp('^' + ".*" + '

Skip to content
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Formalizing Markov Decision Processes in Lean
Blueprint
Verified Lean algorithms for solving tabular MDPs and proving their properties. The focus of this project is on two main goals:
  1. Basic algorithms that can solve robust and risk-averse MDPs of moderate size.
  2. Proofs of correctness of algorithms and fundamental MDP properties which can be used independently to prove structural results, such as the optimality of certain policy class.
Status
General
  •  Left and right continuity (exists?)
  •  Generalized inverse; left and right continous functions (exists?)
Basic probability properties
  •  Definitions: probability space and definition
  •  Definitions: probability, expectation, conditional properties
  •  Tower property, law of the unconscious statistician
  •  Quantile definition and basic properties
  •  Quantile under monotone transformation
  •  Conditional probability = change of measure
  •  Independent random variables
  •  Construct probability from a compile-time input
  •  Construct probability from runtime input
Value at Risk
  •  Definition (non-constructive)
  •  Practical implementation O(n^2) and correctness
  •  Fast practical implementation O(n log n) and correctness
  •  Definition of VaR as minimization
  •  VaR is positively homogeneous and monotone
  •  VaR is translation (cash) invariant
  •  VaR under monotone transformation
  •  Check risk measure values in a JSON file
MDP: Basics
  •  Definition of MDP
  •  Definition of policies (history, Markov, stationary)
  •  Policy induces a distribution over histories
  •  Definition of value function (history-dependent)
Finite Horizon
  •  Histories and manipulation
  •  Probability space over histories
  •  Return and optimal return using histories
  •  History-dependent value function and dynamic program
  •  Markov optimal value function and optimal policy
  •  DP algorithms
Risk-averse finite horizon
  •  History-dependent utility functions
  •  Augmented value function dynamic program
  •  VaR computation from utility function
  •  VaR DP decomposition as in Hau et al., 2023
Discounted infinite horizon
Average reward horizon
Lean Resources
Most useful
Others
About
Formalization of MDP Theory
Resources
Readme
Unlicense
Activity
Custom properties
Stars
4 stars
Watchers
0 watching
Forks
13 forks
Report repository
Releases
Packages
Contributors
Languages

, 'i');
if (__m === '*' || __re.test(location.href)) {
injectUserscript("// Strip utm_, fbclid, gclid, etc. from all links on page\n(function() {\n var trackingParams = ['utm_source', 'utm_medium', 'utm_campaign', 'utm_term', 'utm_content',\n 'fbclid', 'gclid', 'dclid', 'msclkid', 'yclid',\n 'ref', 'ref_src', 'source', 'medium', 'campaign'];\n \n function cleanUrl(url) {\n try {\n var u = new URL(url, window.location.origin);\n var changed = false;\n trackingParams.forEach(function(p) {\n if (u.searchParams.has(p)) {\n u.searchParams.delete(p);\n changed = true;\n }\n });\n return changed ? u.toString() : url;\n } catch (e) {\n return url;\n }\n }\n \n function cleanLinks() {\n document.querySelectorAll('a[href]').forEach(function(a) {\n var clean = cleanUrl(a.href);\n if (clean !== a.href) a.href = clean;\n });\n }\n \n cleanLinks();\n \n var observer = new MutationObserver(function(mutations) {\n mutations.forEach(function(m) {\n m.addedNodes.forEach(function(node) {\n if (node.nodeType === 1) {\n if (node.tagName === 'A') cleanLinks();\n node.querySelectorAll('a[href]').forEach(function(a) {\n var clean = cleanUrl(a.href);\n if (clean !== a.href) a.href = clean;\n });\n }\n });\n });\n });\n observer.observe(document.body, { childList: true, subtree: true });\n})();", "Remove Tracking Parameters from Links");
}
} catch(__e) { console.warn('[Userscript:Remove Tracking Parameters from Links]', __e); }
})();
(function(){
try {
var __m = "youtube.com";
var __re = new RegExp('^' + "youtube\\.com" + '

Skip to content
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Formalizing Markov Decision Processes in Lean
Blueprint
Verified Lean algorithms for solving tabular MDPs and proving their properties. The focus of this project is on two main goals:
  1. Basic algorithms that can solve robust and risk-averse MDPs of moderate size.
  2. Proofs of correctness of algorithms and fundamental MDP properties which can be used independently to prove structural results, such as the optimality of certain policy class.
Status
General
  •  Left and right continuity (exists?)
  •  Generalized inverse; left and right continous functions (exists?)
Basic probability properties
  •  Definitions: probability space and definition
  •  Definitions: probability, expectation, conditional properties
  •  Tower property, law of the unconscious statistician
  •  Quantile definition and basic properties
  •  Quantile under monotone transformation
  •  Conditional probability = change of measure
  •  Independent random variables
  •  Construct probability from a compile-time input
  •  Construct probability from runtime input
Value at Risk
  •  Definition (non-constructive)
  •  Practical implementation O(n^2) and correctness
  •  Fast practical implementation O(n log n) and correctness
  •  Definition of VaR as minimization
  •  VaR is positively homogeneous and monotone
  •  VaR is translation (cash) invariant
  •  VaR under monotone transformation
  •  Check risk measure values in a JSON file
MDP: Basics
  •  Definition of MDP
  •  Definition of policies (history, Markov, stationary)
  •  Policy induces a distribution over histories
  •  Definition of value function (history-dependent)
Finite Horizon
  •  Histories and manipulation
  •  Probability space over histories
  •  Return and optimal return using histories
  •  History-dependent value function and dynamic program
  •  Markov optimal value function and optimal policy
  •  DP algorithms
Risk-averse finite horizon
  •  History-dependent utility functions
  •  Augmented value function dynamic program
  •  VaR computation from utility function
  •  VaR DP decomposition as in Hau et al., 2023
Discounted infinite horizon
Average reward horizon
Lean Resources
Most useful
Others
About
Formalization of MDP Theory
Resources
Readme
Unlicense
Activity
Custom properties
Stars
4 stars
Watchers
0 watching
Forks
13 forks
Report repository
Releases
Packages
Contributors
Languages

, 'i');
if (__m === '*' || __re.test(location.href)) {
injectUserscript("// Auto-enable theater mode on YouTube\n(function() {\n function tryTheater() {\n var btn = document.querySelector('button[aria-label=\"Theater mode\"], ytd-player #player button[title=\"Theater mode\"]');\n if (btn && !btn.classList.contains('activated')) {\n btn.click();\n }\n }\n \n // Try immediately\n tryTheater();\n \n // Try after navigation (SPA)\n var lastUrl = location.href;\n setInterval(function() {\n if (location.href !== lastUrl) {\n lastUrl = location.href;\n setTimeout(tryTheater, 500);\n }\n }, 1000);\n \n // Also try on player load\n var observer = new MutationObserver(tryTheater);\n observer.observe(document.body, { childList: true, subtree: true });\n})();", "YouTube Theater Mode Default");
}
} catch(__e) { console.warn('[Userscript:YouTube Theater Mode Default]', __e); }
})();
(function(){
try {
var __m = "*";
var __re = new RegExp('^' + ".*" + '

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Formalizing Markov Decision Processes in Lean
Blueprint
Verified Lean algorithms for solving tabular MDPs and proving their properties. The focus of this project is on two main goals:
  1. Basic algorithms that can solve robust and risk-averse MDPs of moderate size.
  2. Proofs of correctness of algorithms and fundamental MDP properties which can be used independently to prove structural results, such as the optimality of certain policy class.
Status
General
  •  Left and right continuity (exists?)
  •  Generalized inverse; left and right continous functions (exists?)
Basic probability properties
  •  Definitions: probability space and definition
  •  Definitions: probability, expectation, conditional properties
  •  Tower property, law of the unconscious statistician
  •  Quantile definition and basic properties
  •  Quantile under monotone transformation
  •  Conditional probability = change of measure
  •  Independent random variables
  •  Construct probability from a compile-time input
  •  Construct probability from runtime input
Value at Risk
  •  Definition (non-constructive)
  •  Practical implementation O(n^2) and correctness
  •  Fast practical implementation O(n log n) and correctness
  •  Definition of VaR as minimization
  •  VaR is positively homogeneous and monotone
  •  VaR is translation (cash) invariant
  •  VaR under monotone transformation
  •  Check risk measure values in a JSON file
MDP: Basics
  •  Definition of MDP
  •  Definition of policies (history, Markov, stationary)
  •  Policy induces a distribution over histories
  •  Definition of value function (history-dependent)
Finite Horizon
  •  Histories and manipulation
  •  Probability space over histories
  •  Return and optimal return using histories
  •  History-dependent value function and dynamic program
  •  Markov optimal value function and optimal policy
  •  DP algorithms
Risk-averse finite horizon
  •  History-dependent utility functions
  •  Augmented value function dynamic program
  •  VaR computation from utility function
  •  VaR DP decomposition as in Hau et al., 2023
Discounted infinite horizon
Average reward horizon
Lean Resources
Most useful
Others
About
Formalization of MDP Theory
Resources
Readme
Unlicense
Activity
Custom properties
Stars
4 stars
Watchers
0 watching
Forks
13 forks
Report repository
Releases
Packages
Contributors
Languages

, 'i');
if (__m === '*' || __re.test(location.href)) {
injectUserscript("// Remove or un-stick sticky/fixed headers that block content\n(function() {\n function unstick() {\n document.querySelectorAll('header, nav, [role=\"banner\"], .header, .navbar, .sticky, .fixed-top, [style*=\"position: fixed\"], [style*=\"position:sticky\"]').forEach(function(el) {\n if (el.style.position === 'fixed' || el.style.position === 'sticky' || \n getComputedStyle(el).position === 'fixed' || getComputedStyle(el).position === 'sticky') {\n el.style.position = 'static';\n el.style.top = 'auto';\n el.style.zIndex = 'auto';\n }\n });\n }\n \n unstick();\n \n var observer = new MutationObserver(unstick);\n observer.observe(document.body, { childList: true, subtree: true, attributes: true, attributeFilter: ['style', 'class'] });\n})();", "Kill Sticky Headers");
}
} catch(__e) { console.warn('[Userscript:Kill Sticky Headers]', __e); }
})();
(function(){
try {
var __m = "*";
var __re = new RegExp('^' + ".*" + '

Skip to content
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Formalizing Markov Decision Processes in Lean
Blueprint
Verified Lean algorithms for solving tabular MDPs and proving their properties. The focus of this project is on two main goals:
  1. Basic algorithms that can solve robust and risk-averse MDPs of moderate size.
  2. Proofs of correctness of algorithms and fundamental MDP properties which can be used independently to prove structural results, such as the optimality of certain policy class.
Status
General
  •  Left and right continuity (exists?)
  •  Generalized inverse; left and right continous functions (exists?)
Basic probability properties
  •  Definitions: probability space and definition
  •  Definitions: probability, expectation, conditional properties
  •  Tower property, law of the unconscious statistician
  •  Quantile definition and basic properties
  •  Quantile under monotone transformation
  •  Conditional probability = change of measure
  •  Independent random variables
  •  Construct probability from a compile-time input
  •  Construct probability from runtime input
Value at Risk
  •  Definition (non-constructive)
  •  Practical implementation O(n^2) and correctness
  •  Fast practical implementation O(n log n) and correctness
  •  Definition of VaR as minimization
  •  VaR is positively homogeneous and monotone
  •  VaR is translation (cash) invariant
  •  VaR under monotone transformation
  •  Check risk measure values in a JSON file
MDP: Basics
  •  Definition of MDP
  •  Definition of policies (history, Markov, stationary)
  •  Policy induces a distribution over histories
  •  Definition of value function (history-dependent)
Finite Horizon
  •  Histories and manipulation
  •  Probability space over histories
  •  Return and optimal return using histories
  •  History-dependent value function and dynamic program
  •  Markov optimal value function and optimal policy
  •  DP algorithms
Risk-averse finite horizon
  •  History-dependent utility functions
  •  Augmented value function dynamic program
  •  VaR computation from utility function
  •  VaR DP decomposition as in Hau et al., 2023
Discounted infinite horizon
Average reward horizon
Lean Resources
Most useful
Others
About
Formalization of MDP Theory
Resources
Readme
Unlicense
Activity
Custom properties
Stars
4 stars
Watchers
0 watching
Forks
13 forks
Report repository
Releases
Packages
Contributors
Languages

, 'i');
if (__m === '*' || __re.test(location.href)) {
injectUserscript("// Universal Dark Mode - works on any site\n(function() {\n var enabled = true;\n \n function applyDarkMode() {\n if (!enabled) return;\n \n // Create style element if it doesn't exist\n var style = document.getElementById('universal-dark-mode-style');\n if (!style) {\n style = document.createElement('style');\n style.id = 'universal-dark-mode-style';\n document.head.appendChild(style);\n }\n \n // Dark mode CSS - inverts colors but preserves images/video\n style.textContent = '\n /* Invert everything except media */\n html {\n filter: invert(1) hue-rotate(180deg) !important;\n background: #1a1a2e !important;\n }\n \n /* Restore images, videos, iframes, canvas */\n img, video, iframe, canvas, svg, picture, [style*=\"background-image\"] {\n filter: invert(1) hue-rotate(180deg) !important;\n }\n \n /* Preserve specific elements that should not be inverted */\n .no-dark-mode, .no-dark-mode *,\n [data-theme=\"light\"], [data-theme=\"light\"],\n .ace_editor, .ace_editor *,\n .CodeMirror, .CodeMirror *,\n .monaco-editor, .monaco-editor *,\n .markdown-body pre, .markdown-body pre *,\n .highlight, .highlight *,\n pre code, pre code * {\n filter: none !important;\n }\n \n /* Fix common UI elements */\n .modal, .popup, .dropdown-menu, .tooltip, .popover {\n filter: invert(1) hue-rotate(180deg) !important;\n background: #2d2d44 !important;\n border-color: #444 !important;\n }\n \n /* Scrollbars */\n ::-webkit-scrollbar { background: #1a1a2e !important; }\n ::-webkit-scrollbar-thumb { background: #444 !important; }\n ::-webkit-scrollbar-thumb:hover { background: #555 !important; }\n \n /* Selection */\n ::selection { background: #4ecdc4 !important; color: #1a1a2e !important; }\n ::-moz-selection { background: #4ecdc4 !important; color: #1a1a2e !important; }\n ';\n }\n \n function removeDarkMode() {\n var style = document.getElementById('universal-dark-mode-style');\n if (style) style.remove();\n }\n \n // Toggle with Alt+Shift+D\n document.addEventListener('keydown', function(e) {\n if (e.altKey && e.shiftKey && e.key === 'D') {\n e.preventDefault();\n enabled = !enabled;\n if (enabled) {\n applyDarkMode();\n console.log('[Universal Dark Mode] Enabled');\n } else {\n removeDarkMode();\n console.log('[Universal Dark Mode] Disabled');\n }\n }\n });\n \n // Apply on load\n applyDarkMode();\n \n // Re-apply on dynamic content\n var observer = new MutationObserver(function(mutations) {\n if (enabled && !document.getElementById('universal-dark-mode-style')) {\n applyDarkMode();\n }\n });\n observer.observe(document.head, { childList: true });\n \n console.log('[Universal Dark Mode] Loaded - Press Alt+Shift+D to toggle');\n})();", "Universal Dark Mode");
}
} catch(__e) { console.warn('[Userscript:Universal Dark Mode]', __e); }
})();
})();

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Formalizing Markov Decision Processes in Lean
Blueprint
Verified Lean algorithms for solving tabular MDPs and proving their properties. The focus of this project is on two main goals:
  1. Basic algorithms that can solve robust and risk-averse MDPs of moderate size.
  2. Proofs of correctness of algorithms and fundamental MDP properties which can be used independently to prove structural results, such as the optimality of certain policy class.
Status
General
  •  Left and right continuity (exists?)
  •  Generalized inverse; left and right continous functions (exists?)
Basic probability properties
  •  Definitions: probability space and definition
  •  Definitions: probability, expectation, conditional properties
  •  Tower property, law of the unconscious statistician
  •  Quantile definition and basic properties
  •  Quantile under monotone transformation
  •  Conditional probability = change of measure
  •  Independent random variables
  •  Construct probability from a compile-time input
  •  Construct probability from runtime input
Value at Risk
  •  Definition (non-constructive)
  •  Practical implementation O(n^2) and correctness
  •  Fast practical implementation O(n log n) and correctness
  •  Definition of VaR as minimization
  •  VaR is positively homogeneous and monotone
  •  VaR is translation (cash) invariant
  •  VaR under monotone transformation
  •  Check risk measure values in a JSON file
MDP: Basics
  •  Definition of MDP
  •  Definition of policies (history, Markov, stationary)
  •  Policy induces a distribution over histories
  •  Definition of value function (history-dependent)
Finite Horizon
  •  Histories and manipulation
  •  Probability space over histories
  •  Return and optimal return using histories
  •  History-dependent value function and dynamic program
  •  Markov optimal value function and optimal policy
  •  DP algorithms
Risk-averse finite horizon
  •  History-dependent utility functions
  •  Augmented value function dynamic program
  •  VaR computation from utility function
  •  VaR DP decomposition as in Hau et al., 2023
Discounted infinite horizon
Average reward horizon
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Formalization of MDP Theory
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