Machine Learning

In April 2025, Anthropic CEO Dario Amodei published “The Urgency of Interpretability,” sounding an alarm that rippled through the AI research community. His message was stark: we’re building systems of unprecedented capability while remaining fundamentally unable to understand how they arrive at their outputs. The timing was deliberate—after years of incremental progress, a technique called Sparse Autoencoders (SAEs) had finally cracked open the black box, revealing millions of interpretable concepts hidden inside large language models. ...

The most valuable resource in training large language models isn’t compute, parameters, or architecture—it’s data. Yet high-quality training data has become increasingly scarce, expensive, and in some domains, simply unavailable. This constraint has pushed researchers toward an elegant paradox: using AI to train AI. Synthetic data generation, once considered a last resort for data-starved applications, has evolved into a sophisticated discipline that powers some of today’s most capable models. Microsoft’s Phi-4, a 14-billion parameter model that rivals models five times its size, was trained primarily on synthetic data. Meta’s Llama models use synthetic data generation for fine-tuning and reasoning capabilities. The question is no longer whether synthetic data works, but how to generate it without triggering model collapse—the degenerative process that turns capable models into noise generators. ...

The mathematics of neural networks has long been considered settled: gradients flow through continuous-valued weights, optimized via backpropagation through floating-point arithmetic. Yet in February 2024, Microsoft Research challenged this orthodoxy with a question that seemed absurd: what if every weight in a large language model could be expressed using only three values—{-1, 0, 1}? The answer, it turns out, rewrites everything we thought we knew about the efficiency-accuracy trade-off. BitNet b1.58, trained natively with ternary weights, matches full-precision LLaMA models in perplexity while consuming 90% less memory. QuEST demonstrates that LLMs can be trained stably at 1-bit precision. NanoQuant pushes further, achieving sub-1-bit compression that runs a 70B model on a consumer 8GB GPU. ...

For years, the next-token prediction (NTP) paradigm has been the unquestioned foundation of large language model training. Given a sequence of tokens $x_{1:t}$, the model learns to maximize $P(x_{t+1} | x_{1:t})$. Simple, elegant, and remarkably effective—until you realize the fundamental inefficiency baked into this approach. The problem is that transformers spend the same computational budget predicting filler words (“the”, “and”, “is”) as they do on information-carrying tokens (“quantum”, “entanglement”, “superposition”). Research from Apple and EPFL reveals that over 50% of English text consists of function words—linguistic glue that carries minimal semantic weight. Yet models trained on NTP treat every token with equal reverence, creating a massive computational inefficiency. ...

Traditional approaches to controlling Large Language Model behavior have followed two well-worn paths: prompt engineering at the input level, and fine-tuning or RLHF at the weight level. But what if we could modify how a model “thinks” in real-time, without changing its weights or crafting the perfect prompt? Representation Engineering (RepE) offers exactly this capability—a paradigm that treats internal activations, rather than neurons or circuits, as the fundamental unit of analysis and control. ...

The emergence of reasoning models like DeepSeek-R1, OpenAI’s o3, and Google’s Gemini thinking mode has fundamentally shifted how we think about LLM inference. These models don’t just generate—they search. The question is no longer “what should the model output?” but “how should the model search for the answer?” This shift from generation to search has spawned an entire taxonomy of inference-time algorithms, each with distinct trade-offs between computational cost and output quality. Understanding these methods—their mathematical foundations, implementation details, and practical performance—is essential for anyone deploying reasoning models in production. ...

The standard RLHF pipeline was never elegant. Train a reward model from human preferences, then use Proximal Policy Optimization (PPO) to maximize that reward while staying close to your original model—requiring four separate models in memory during training, sampling from the policy during optimization, and navigating a landscape of hyperparameter sensitivity that could turn a week of training into a costly failure. Direct Preference Optimization (DPO) changed everything. By recognizing that the optimal policy under a KL-constrained reward maximization objective could be derived in closed form, DPO eliminated reinforcement learning entirely. What followed was an explosion of variants—KTO, ORPO, SimPO, IPO, AlphaDPO—each addressing different limitations with different inductive biases. Understanding when to use which method requires understanding not just their formulas, but the assumptions they encode about human preferences and the trade-offs they make between data requirements, computational efficiency, and alignment quality. ...

Fine-tuning a 7-billion parameter model used to demand 100+ GB of VRAM—roughly the memory of four A100 GPUs. Today, the same task runs on a consumer RTX 4090 with 24 GB. This 4× reduction didn’t come from better hardware; it came from a mathematical insight about the structure of neural network adaptations. Low-Rank Adaptation (LoRA), introduced by Microsoft in 2021, fundamentally changed how we think about model fine-tuning. The core idea is deceptively simple: instead of updating all parameters, inject small trainable matrices that modify the model’s behavior. But behind this simplicity lies deep connections to linear algebra, information theory, and the geometry of neural network weight spaces. ...

A math student solves a complex integration problem. Her final answer is correct, but halfway through, she made a sign error that accidentally canceled out in the next step. The teacher gives full marks—after all, the answer is right. But should it count? This scenario captures the fundamental flaw in how we’ve traditionally evaluated Large Language Model (LLM) reasoning: Outcome Reward Models (ORMs) only check the final destination, ignoring whether the path was sound. Process Reward Models (PRMs) represent a paradigm shift—verifying every step of reasoning, catching those hidden errors that coincidentally produce correct answers, and enabling the test-time scaling that powers reasoning models like OpenAI’s o1 and DeepSeek-R1. ...

For years, the path to better LLMs seemed straightforward: more parameters, more training data, more compute. The scaling laws articulated by Kaplan et al. and refined by Chinchilla painted a clear picture—performance improved predictably with model size. Then OpenAI released o1, and suddenly the rules changed. A model that “thinks longer” at inference time was solving problems that eluded models 10x its size. The breakthrough wasn’t just engineering—it was a fundamental shift in how we think about compute allocation. The question flipped from “how big should we train?” to “how long should we let it think?” ...

On January 20, 2025, DeepSeek released R1—a 671B parameter Mixture-of-Experts model that achieved something remarkable: matching OpenAI’s o1 on reasoning benchmarks while being fully open-source. The breakthrough wasn’t just in scale or architecture, but in a fundamentally different approach to training reasoning capabilities: Group Relative Policy Optimization (GRPO), a reinforcement learning algorithm that eliminates the need for reward models while enabling sophisticated reasoning behaviors to emerge naturally. The Problem with Traditional LLM Training Standard large language models excel at pattern matching and next-token prediction, but struggle with tasks requiring multi-step logical deduction, self-correction, and complex problem decomposition. Chain-of-thought prompting helped, but it required extensive human-annotated demonstrations and still couldn’t match the systematic reasoning humans employ. ...

In 1986, David Rumelhart, Geoffrey Hinton, and Ronald Williams published a paper in Nature that would transform artificial intelligence. The paper, “Learning representations by back-propagating errors,” demonstrated that a mathematical technique from the 1970s could train neural networks orders of magnitude faster than existing methods. The speedup wasn’t incremental—it was the difference between a model taking a week to train and taking 200,000 years. But backpropagation wasn’t invented in 1986. Its modern form was first published in 1970 by Finnish master’s student Seppo Linnainmaa, who described it as “reverse mode automatic differentiation.” Even earlier, Henry J. Kelley derived the foundational concepts in 1960 for optimal flight path calculations. What the 1986 paper achieved wasn’t invention—it was recognition. The authors demonstrated that this obscure numerical technique was exactly what neural networks needed. ...