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Machine Learning

Articles

  • Variational Autoencoders: Mathematical Derivation of the ELBO, the Reparameterization Trick, and Mitigating Posterior Collapse

    Variational Autoencoders: Mathematical Derivation of the ELBO, the Reparameterization Trick, and Mitigating Posterior Collapse Traditional autoencoders map high-dimensional data into deterministic latent vectors. While effective for dimensionality reduction and non-linear feature compression, deterministic autoencoders fail as generative models because their latent representations lack continuous probabilistic structure. Unregularized latent spaces contain wide regions of empty space and severe

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  • Non-Contrastive Representation Learning: How Barlow Twins and VICReg Prevent Feature Collapse via Redundancy Reduction

    Self-supervised representation learning provides the foundation for modern foundation models across computer vision, audio, and multimodal systems. By training deep neural networks to produce compact vector embeddings without human annotations, self-supervised pre-training enables models to capture rich semantic structures directly from raw data. Historically, the dominant approach to self-supervised learning was contrastive learning, popularized by architectures such as SimCLR (Chen et al., 20

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  • The Platonic Representation Hypothesis: How Foundation Models Across Modalities Converge on a Shared Geometry of Reality

    In 375 BC, Plato introduced the Allegory of the Cave, describing prisoners who could only perceive reality through shadows projected onto a stone wall. In modern machine learning, deep neural networks ingest distinct sensory shadows of the physical world: vision models process pixel arrays, large language models ingest tokenized text, and speech models parse raw acoustic waveforms. Historically, the machine learning literature assumed that models trained on different modalities, objectives, and

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  • Vector Quantization and VQ-VAEs: How Discrete Codebooks, Straight-Through Estimators, and Commitment Losses Power Multimodal Tokenization

    Autoregressive sequence models excel at discrete token prediction. In natural language processing, words and subwords map onto categorical vocabularies where token identity is exact and cross-entropy loss provides direct likelihood optimization. Continuous multi-dimensional signals—such as images, video frames, raw audio waveforms, and robotic sensorimotor trajectories—present a fundamental mismatch for standard transformer architectures. Historically, variational autoencoders (VAEs) bridged ra

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  • Federated LLM Fine-Tuning in Production: FedLoRA, Differential Privacy, and Cross-Silo Aggregation Architectures

    Fine-tuning foundation large language models on proprietary data is standard enterprise practice, but centralizing sensitive tokens into a single data lake is frequently prohibited. Regulatory frameworks such as HIPAA in healthcare, GDPR and Article 10 of the EU AI Act in Europe, and regional data residency mandates across APAC and North America prevent cross-border or cross-institutional data aggregation. Federated Learning (FL) resolves this bottleneck by decoupling model training from data c

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  • Discrete Diffusion in Large Language Models: How Continuous-Time Markov Chains, Absorbing States, and Score Entropy Challenge Autoregressive Generation

    The dominance of autoregressive architectures in large language models rests on a fundamental mathematical formulation: the chain rule of probability. By factoring the joint distribution of a sequence into a product of conditional probabilities, $p(x) = \prod_{i=1}^N p(x_i \mid x_{<i})$, autoregressive models reduce text generation to sequential next-token prediction. While this left-to-right causal factorization has scaled effectively across compute regimes, it imposes rigid operational constr

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  • Continuous Pre-Training in Production: Domain Adaptation, Replay Buffers, Learning Rate Restarts, and Catastrophic Forgetting Mitigation

    Continuous Pre-Training in Production: Domain Adaptation, Replay Buffers, Learning Rate Restarts, and Catastrophic Forgetting Mitigation Adapting general-purpose foundation models to specialized enterprise domains (such as clinical medicine, corporate law, quantitative finance, and proprietary software codebases) presents a fundamental architectural challenge. While Retrieval-Augmented Generation (RAG) and Supervised Fine-Tuning (SFT) remain standard first-line approaches, both exhibit severe s

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  • Noise-Contrastive Estimation and InfoNCE: How Partition Function Estimation and Mutual Information Lower Bounds Power Modern AI

    Noise-Contrastive Estimation and InfoNCE: How Partition Function Elimination and Mutual Information Lower Bounds Power Modern AI In statistical machine learning and generative modeling, evaluating the exact probability of an observed event frequently requires calculating a normalizing constant known as the partition function. For continuous spaces or discrete spaces with high cardinality, such as a natural language vocabulary spanning over one hundred thousand tokens or high-dimensional pixel d

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  • Distributed RLHF Frameworks in Production: Comparing OpenRLHF, verl, and TRL Architecture, Ray Resource Scheduling, and Rollout-Training Co-Location

    Post-training alignment has shifted from offline preference tuning to large-scale, online reinforcement learning. Modern post-training loops for reasoning models, agentic workflows, and conversational alignment require coordinating multiple distinct neural network roles simultaneously. Under standard Proximal Policy Optimization (PPO), an RL infrastructure pipeline must manage up to four distinct model instances: the Actor (the active policy undergoing gradient updates), the Critic (the value mo

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  • The Gumbel-Softmax Trick: How Continuous Relaxations Enable Differentiable Discrete Sampling

    The Gumbel-Softmax Trick: How Continuous Relaxations Enable Differentiable Discrete Sampling In modern deep learning, end-to-end training depends on reverse-mode automatic differentiation. When an architecture operates on continuous tensors, computing gradients via the chain rule is straightforward. However, many foundational artificial intelligence problems involve discrete choices: selecting tokens from a fixed vocabulary, routing tokens to expert networks in a Mixture-of-Experts (MoE) archit

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  • Flow Matching for Generative Modeling: How Continuous Normalizing Flows and Optimal Transport Paths Replace Diffusion SDEs

    Flow Matching for Generative Modeling: How Continuous Normalizing Flows and Optimal Transport Paths Replace Diffusion SDEs Generative modeling underwent a structural shift with the introduction of Flow Matching (FM), formulated independently by Lipman et al. (2022), Albergo and Vanden-Eijnden (2022), and Liu et al. (2022). While Denoising Diffusion Probabilistic Models (DDPM) and score-based Stochastic Differential Equations (SDEs) established state-of-the-art sample quality across vision and a

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  • The Information Bottleneck in Deep Learning: How Mutual Information Compression Shapes Generalization and Neural Representations

    The Information Bottleneck in Deep Learning: How Mutual Information Compression Shapes Generalization and Neural Representations Classical statistical learning theory struggles to explain why overparameterized deep neural networks generalize well to unseen test data. Traditional complexity measures such as Vapnik-Chervonenkis (VC) dimension and Rademacher complexity scale with the raw count of tunable weights, predicting severe overfitting when parameter counts exceed dataset sample sizes. Yet

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  • Score-Based Generative Modeling via Stochastic Differential Equations: How Continuous SDEs and Score Matching Unify Diffusion Models

    Diffusion and score-based models represent one of the foundational paradigms of modern generative artificial intelligence, underpinning systems across image synthesis, video generation, audio modeling, and continuous multimodal representations. For years, generative diffusion was approached from two distinct perspectives: discrete-step denoising diffusion probabilistic models (DDPM) pioneered by Sohl-Dickstein et al. and Ho et al., and score matching with Langevin dynamics (SMLD / NCSN) introduc

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  • The Lottery Ticket Hypothesis in Large Language Models: How Sparse Subnetworks and Iterative Magnitude Pruning Retain Transformer Capacity

    The Lottery Ticket Hypothesis in Large Language Models: How Sparse Subnetworks and Iterative Magnitude Pruning Retain Transformer Capacity Modern large language models operate under extreme overparameterization. Frontier architectures allocate tens or hundreds of billions of parameters to achieve low perplexity and robust generalization across reasoning, code generation, and factual retrieval. Yet empirical pruning consistently demonstrates that post-training models can lose 30% to 50% of their

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  • Sharpness-Aware Minimization in Large Language Models: How Adversarial Weight Perturbations and Flat Minima Boost Generalization

    In overparameterized deep neural networks, minimizing empirical training loss is insufficient to guarantee optimal generalization on unseen distributions. Modern deep architectures, including vision models and autoregressive Large Language Models (LLMs), operate in regimes where parameter counts far exceed training token counts, producing highly non-convex loss surfaces populated by infinite global minima. Standard optimization via Stochastic Gradient Descent (SGD) or AdamW often converges to sh

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  • Linear Mode Connectivity in Deep Neural Networks: How Permutation Symmetries, Git Re-Basin, and the Single-Basin Hypothesis Unify Model Checkpoints

    title: "Linear Mode Connectivity in Deep Neural Networks: How Permutation Symmetries, Git Re-Basin, and the Single-Basin Hypothesis Unify Model Checkpoints" slug: "linear-mode-connectivity-in-deep-neural-networks-how-permutation-symmetries-git-re-basin-and-the-single-basin-hypothesis-unify-model-checkpoints" feature_image: "https://cms.llms.blog/content/images/2026/08/linear-mode-connectivity-cover.png" excerpt: "Linear Mode Connectivity reveals how neural network checkpoints connect along flat

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  • Deep Double Descent: Why Overparameterization Defies the Classical Bias-Variance Trade-Off

    For decades, statistical learning theory rested on a foundational tenet: the bias-variance trade-off. According to classical machine learning textbooks, increasing model capacity reduces bias on the training set but inevitably inflates variance on unseen test data. The resulting risk curve forms a familiar U-shape: underfitting on the left, an optimal capacity in the center, and severe overfitting on the right. Modern deep learning and large language models (LLMs) fundamentally contradicted thi

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  • Minimum Bayes Risk Decoding: How Utility Consensus and Distributional Reranking Outperform Mode-Seeking in Large Language Models

    Autoregressive large language models generate text by estimating conditional probability distributions over token sequences. In conventional inference pipelines, generating the final output sequence relies almost entirely on search algorithms designed to locate high-probability trajectories: greedy decoding, beam search, or stochastic sampling with temperature and top-p filtering. However, theoretical and empirical analyses demonstrate that the most probable sequence under a model's distributio

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  • In-Context Learning as Implicit Gradient Descent: How Transformers Optimize Models in Forward Activations

    When large language models (LLMs) adapt to new tasks from a handful of prompt demonstrations, their static weights remain completely untouched. No backpropagation runs through the network, no optimizer updates parameters, and no gradients are calculated. Yet, the model’s predictions improve steadily as more input-output examples are added to the prompt context. For years, this phenomenon (known as in-context learning, or ICL) was treated as an empirical black box. Recent theoretical and mechani

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  • Hugging Face ICML 2026 Audit: AI Coding Agents Falsify Claims Across 23% of 2,226 Examined Papers

    Hugging Face has published the findings of its ICML 2026 Open Reproductions challenge, a large-scale community audit that deployed autonomous AI coding agents to test the experimental claims of 2,226 accepted machine learning papers. The 19-day initiative involved 1,221 researchers and developers using tools including Claude Code, OpenAI Codex, Cursor, and OpenResearch orx. Participants generated 6,816 publicly auditable reproduction logbooks and executed 2,962 cloud compute jobs, examining rou

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