Machine Learning

Reverse-Process Synthetic Data Generation for Math Reasoning

Training LLMs on mathematical reasoning by inverting easy-to-solve problems: generate derivatives, reverse them into integration exercises with full step-by-step solutions.

Watch: Inductive Biases, What Your Architecture Assumes

The inductive-biases series is now a ten-episode animated playlist: every architecture is a bet about the world, and the bet is the bias. No free lunch first, then each architecture read as its assumptions, graded on one scorecard.

Watch: Next-Token Prediction, from Solomonoff to Transformers

The sequential-prediction series is now an eight-episode animated playlist: what-comes-next as the foundation of the language-model era, traced from Solomonoff induction to the transformer.

On Intelligence: The Gap Where Safety Is Decided

There is a beautiful, settled theory of intelligence, and there are the crude systems we shipped. The distance between them is where safety is actually decided.

On Intelligence: and Its Specifications

We taught sand to think. We did it before we learned to say what we wanted it to do. There is a real, settled, and genuinely beautiful theory of what intelligence is: Bayes' theorem, Solomonoff's universal prior, and expected-utility maximization, …

What You Assume vs. What You Compute

Part 4 of What Your RL Algorithm Actually Assumes — model-based vs. model-free, the assumptions table, AIXI as the incomputable ideal, and the unifying claim: representation is prior is assumption.

The Architecture Is the Prior

Part 3 of What Your RL Algorithm Actually Assumes — the architecture decides what kind of features can be learned, and that decision is a Bayesian prior over value functions.

The Features You Choose Are the Assumptions You Make

Part 2 of What Your RL Algorithm Actually Assumes — how hand-crafted features compress the state space, and what you're betting on when you pick them.

The Infinite Table

Part 1 of What Your RL Algorithm Actually Assumes — tabular Q-learning makes zero assumptions about state similarity and pays for it in sample complexity.

Value Functions Over Reasoning Traces

What if reasoning traces could learn their own usefulness? A simple RL framing for trace memory, and why one reward signal is enough.

From A* to GPT: Rational Agents and the Representation Problem

The classical AI curriculum teaches rational agents as utility maximizers. The progression from search to RL to LLMs is really about one thing: finding representations that make decision-making tractable.

The Incomputability of Simple Learning

Why the simplest forms of learning are incomputable, and what that means for the intelligence we can build.

Bayesian Reasoning and Machine Learning

Machine Learning: A Probabilistic Perspective

Pattern Recognition and Machine Learning

Patterns, Predictions, and Actions: A Story About Machine Learning

Notes

Modern graduate ML text with causal inference, decision making, and ML foundations. Accessible free textbook with strong conceptual framing.

The Elements of Statistical Learning

The Master Algorithm

Infinigram: Corpus-Based Language Modeling via Suffix Arrays with LLM Probability Mixing

MCTS-Reasoning: A Canonical Specification of Monte Carlo Tree Search for LLM Reasoning

The Policy: Q-Learning vs Policy Learning

SIGMA uses Q-learning rather than direct policy learning. This architectural choice makes it both transparent and terrifying. You can read its value function, but what you read is chilling.

DreamLog: Logic Programming That Dreams to Improve Itself

A logic programming system that alternates between wake and sleep phases, using LLMs for knowledge generation during wake and compression-based learning during sleep.

DreamLog: Neural-Symbolic Integration through Compression-Based Learning and Wake-Sleep Cycles

Learning Fuzzy Logic: Automatic Rule Discovery Through Differentiable Circuits

Learning fuzzy membership functions and inference rules automatically through gradient descent on soft circuits, instead of hand-crafting them.

Differentiation: Three Ways

Three approaches to computing derivatives, forward-mode AD, reverse-mode AD, and finite differences, each with different trade-offs for numerical computing and machine learning.

Science as Verifiable Search

Science is search through hypothesis space. Intelligence prunes; testing provides signal. Synthetic worlds could accelerate the loop.

MCTS-Reasoning: Tree Search for LLM Reasoning

Applying Monte Carlo Tree Search to large language model reasoning, with a formal specification of the algorithm.

Cluster-Aware Retrieval for RAG Systems

Using GMM clustering to improve retrieval in topically diverse knowledge bases

Latent Reasoning Traces: Memory as Learned Prior

What if LLMs could remember their own successful reasoning? A simple experiment in trace retrieval, and why 'latent' is the right word.

Fuzzy Soft Circuits: Learning Fuzzy Rules from Data

What if fuzzy logic systems could discover their own rules? An interactive demo of differentiable fuzzy circuits that learn membership functions, rule structure, and rule existence, all via gradient descent.

All Induction Is the Same Induction

Solomonoff induction, MDL, speed priors, and neural networks are all special cases of one Bayesian framework with four knobs.

Fisher Flow: Optimization on the Statistical Manifold

Gradient descent in Euclidean space ignores the geometry of probability distributions. Natural gradient descent uses the Fisher information metric instead. Fisher Flow makes this continuous.

FemtoGrad: A Minimal Automatic Differentiation Library

A tiny autodiff library for understanding how backpropagation actually works.

The AI Course: Everything is Utility Maximization

Intelligence as utility maximization under uncertainty. A unifying framework connecting A* search, reinforcement learning, Bayesian networks, and MDPs.

Uses and limits of abstractions

Abstractions let us reason about complex systems despite our cognitive limits. But some systems resist compression entirely.

Working Memory as an Inductive Bias

How the limited capacity of human working memory acts as regularization, shaping our reasoning and possibly preventing cognitive overfitting.

Reverse-Mode Automatic Differentiation

Reverse-mode automatic differentiation is just the chain rule applied systematically. I built one in C++20 to understand what PyTorch and JAX are actually doing.

Discovering ChatGPT: Reconnecting with AI Research

Encountering ChatGPT during cancer treatment and recognizing the Solomonoff connection. Language models as compression, prediction as intelligence. A personal inflection point reconnecting with AI research after years in survival mode.

Forward-Mode Automatic Differentiation

Dual numbers extend our number system with an infinitesimal epsilon where epsilon^2 = 0. Evaluating f(x + epsilon) yields f(x) + epsilon * f'(x)—the derivative emerges automatically from the algebra.

Introduction to Sequential Prediction

The problem of predicting what comes next, from compression to language models