Simulation-based generative models are all the rage. In particular, flow matching now dominates the image and video generation landscape. Because it tolerates few simulation steps well, it has also been widely adopted in applications sensitive to inference time (mostly, robotics). Like we did for denoising diffusion models, this post studies flow matching from first principles.
Some recent work on physical simulation sent me back to solving ordinary differential equations—and how I had little intuition for how non-trivial numerical solvers are born. After covering some basic properties, this post digs into the quadrature viewpoint for synthesising ‘any’ solvers from the Runge-Kutta family. We illustrate the method by rederiving the celebrated RK4 solver from first principles.
For its own sake or as part of a policy iteration scheme, evaluating policies is an important matter. This post is concerned with off-policy policy evaluation, or evaluating policies with data generated by others. The ambition is to provide a smooth progression towards the Retrace estimator and some of its extensions, with a focus on each operator’s properties and stochastic approximation variant.
Coming from a control background, applying RL to text generation is hardly intuitive. Is there an actual, non-degenerate dynamical system at play here? What is the concrete novelty behind the shiny LLM post-training algorithms? This post is an attempt to answer those questions by providing a semiformal derivation of the celebrated GRPO algorithm through a contextual bandit lens.
This post explores the theory of regularised MDPs beyond entropic regularisation (which we covered in an older post). We will introduce convex regularisation of the classical Bellman operators and study the induced regularised policy iteration algorithms. On the way, we will tie some links with several popular algorithms. This post is mostly a good excuse to refresh some convex optimisation classics.
Entropy, divergence, mutual information, etc. are central concepts in statistical machine learning. This post ties them together in a short collection of elementary information theoretic results.
For better or for worse, proximal policy optimisation (PPO) algorithms and its variants are dominating the RL landscape these days. This post aims at retracing their journey, from foundational concepts to LLM-savy innovations. We will start this saga on the theoretical trail, which we will progressively abandon to pay closer attention to algorithmic aspects.
The goal of this post is to explore from first principles the learning of belief models in partially observable MDPs. We will start with a quick refresher on variational inference, and apply it to state estimation in POMDPs. Specifically, we will derive the update rule used to train Dreamer-like models.
In this second post of this series, we take a break from successor measures to focus on successor features. We will first review the use of a generalised policy improvement mechanism that can efficiently leverage the successor features of existing policies to enable zero-shot transfer to new tasks. We will then discuss the generalisation to universal successor features approximations, allowing direct zero-shot control.
Thanks to its relative simplicity and conciseness, the discounted approach to control in MDPs has come to largely prevail in the RL theory and practice landscape. Departing from the myopic nature of discounted control, we study here the average-reward objective which focuses on long-term, steady state rewards. To start gently, we will limit ourselves to establishing Bellman equations for policy evaluation.
The main promise of unsupervised RL is test-time adaptation to newly specified reward functions. This requires a systemic untangling of reward and dynamics in traditional RL tools. In this first post of a short series, we see how this can be done via the concept of successor states and successor representations. We will focus on policy evaluation, leaving control for a follow-up.
This post is interested in state estimation in HMMs: filtering, prediction and smoothing. We will introduce state estimation as the solution of an optimisation problem, and prove the celebrated recursive updates for each inference use-case. A special attention will be given to HMM filters (and how they easily generalise to the celebrated Kalman filters).
Control in MDPs is often focused on the expected return criterion, which has the good taste to follow the Bellman dynamic programing equations we all love and cherish. In this post, we see how those equations generalise beyond the expected return to the full return distribution. We will cover the distributional Bellman equations for policy evaluation, and see them in action in practical distributional dynamic programming algorithm based on quantiles.
We all have heard about diffusion models and played with their generative magic. To me, it felt awkward not properly understanding how they work, from first principles. After all, they are just another latent variable model, right? This post dives into the probabilistic foundations that make them tick, breaking down the tools needed to re-derive variational inference for diffusion models.
This post introduces the Maximum Entropy (MaxEnt) framework for Reinforcement Learning. The ambition is to see how “standard” control algorithms like value or policy iteration translate to the MaxEnt setting, and describe their convergence properties. We will also discuss how said control approach influenced the design of some popular modern algorithms such as the Soft Actor Critic (SAC).
Maximum-Likelihood Estimation (MLE) is ubiquitous in modern (supervised) machine learning, to the point that we sometimes forget it is the very principle behind the criterion we use to train fancy models. Justifying the MLE’s current popularity from a statistical perspective only would be a stretch; nonetheless, it is useful to remember that it enjoys some comfortable (asymptotic) statistical properties. This post provides proof for two such attributes: consistency and asymptotic normality.
This post is the first of a short series, concerned with model-based RL. We will start walking this road via the principled trail: describing and analysing $\texttt{UCB-VI}$, a theoretically grounded algorithm to solve unknown finite MDPs. More precisely, we will see how this model-based approach directly models the unknown MDP and uses optimism for strategic exploration to provably find “good” policies in finite-time. Concentration inequalities and regret bounds incoming, fun!
The goal of this blog post is to provide a self-contained proof of the so-called fundamental theorem of Markov chains. It states that provided some basic properties (spoiler: ergodicity) any Markov chain random walk converges to the same limiting distribution—irrespective of the starting state. A hidden goal is to introduce some important tools for the analysis of the average-reward criterion in MDPs.
The goal of this blog post is walk the path that saw the Value Iteration algorithm mature into a variety of modern value-based reinforcement learning methods empowered by deep learning. In particular, we will first see how VI gave birth to the Q-learning algorithm. We will then see how this tabular algorithm was gradually modified to handle function approximation and give rise to the watershed DQN algorithm.
This blog-post is the first of a short series on the modern algorithmic Reinforcement Learning landscape. The series’ objective is 1) to lay out the borders of some “boxes” where one can fit most modern RL algorithms, 2) define said boxes’ content in an almost self-contained fashion and 3) discuss each algorithmic family’s strengths and weaknesses. This first post provides a very brief description of what those boxes are.
We slightly switch gear in this third blog-post of the multi-agent series. Letting go of game theoretic concepts, we instead discuss some training paradigms for multi-agent reinforcement learning. We will cover the main limitations behind centralised and independent learning, to land at the centralised training with decentralised execution (CTDE), arguably the more established framework to train autonomous agents in decentralised multiplayer games.
This second blog-post of the multi-agent series is dedicated to solution concepts: how we can characterize and compare different joint policies. We will cover Pareto optimality, the definition and existence of Nash equilibrium, as well as the minimax theorem in two-agents zero-sum games.
This blog-post is the first of a short serie on Multi-Agent control and RL. The objective here is to cover the different models of multi-agent interactions, from repeated matrix games to partially observable Markov games. Going up the game hierarchy (from less to more general) we will detail what constitutes valid policies and how they differ from their usual fully and partially observed MDPs cousin.
This post covers the Expectation Maximization (EM) algorithm, a popular heuristic to (approximately) compute maximum likelihood estimates when dealing with unobserved / latent data. We will motivate and derive the EM’s inner mechanisms before making them explicit on two classical examples (Gaussian Mixture parameter estimation and Hidden Markov Model identification).
This post is concerned with the Linear Quadratic Regulator (LQR) in discrete-time. The LQR stands as somewhat of a singularity in optimal control theory: the (only?) non-trivial control problem in continuous state and action space for which a closed-form solution is known.
Partially Observable MDPs allow to model sensor noise, data occlusion, .. in Markov Decision Processes. Partial observability is unquestionably practically relevant, but at first glance this richer interaction setting seems to invalidate the fundamental theorem of MDPs, a.k.a the sufficiency of stationary policies. The goal of this post is to nuance this last statement: we will detail how POMDPs really are just “large” MDPs and therefore enjoy many of the same features.
This post covers the derivation of several policy gradients expressions. It browses through the vanilla policy gradient and the policy gradient theorem to finish at deterministic policy gradients. No attention is given to optimisation aspects (whether and how policy gradients can find locally optimal policies): the focus is the gradient’s (semi) formal derivation.
This blog-post is concerned with computational methods for finding optimal policies. We will cover Value Iteration (VI) and Policy Iteration (PI), which serve as fundamental building blocks for many modern RL methods. We will also quickly cover the Generalized Policy Iteration approach, on top of which stand all fancy modern actor-critic methods.
In this second blog post of the series, we cover the problem of value prediction and control in finite discounted MDPs. This will lead us in particular to review the Bellman prediction and control equations, and to a fundamental theorem of MDPs: stationary policies are enough for optimal control!
This blog post is the first of a short series on control in discounted discrete MDPs. We will cover here the very basics: the definition of an MDP, the different notions of policies (history-dependent, Markovian, stationary,…), as well as the discounted objective for MDPs.