February 2026

Federated learning (FL) is a privacy-preserving machine learning paradigm that enables multiple parties to collaboratively train models on privately owned data without sharing raw information. While standard FL typically addresses either horizontal or vertical data partitions, many real-world scenarios exhibit a complex hybrid distribution. This paper proposes Hybrid Federated Learning (HybridFL) to address data split both horizontally across disjoint users and vertically across complementary feature sets. We evaluate HybridFL in a financial crime detection context, where a transaction […]

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I reformulated the Bellman Expectation Equation using vector dot products instead of the usual summation sigma summation notation. g = γ⃗ · r⃗ o⃗ = r⃗ + γv⃗’ q = p⃗ · o⃗ v = π⃗ · q⃗ Together they express the full Bellman Expectation Equation: discounted return (g), one-step Bellman backup (o for outcome), Q-value as expected outcome (q) given dynamics (p), and state value (v) as expected value under policy π. This makes the computational structure […]

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This paper introduces a high-order Markov chain task to investigate how transformers learn to integrate information from multiple past positions with varying statistical significance. We demonstrate that transformers learn this task incrementally: each stage is defined by the acquisition of specific information through sparse attention patterns. Notably, we identify a shift in learning dynamics from competitive, where heads converge on the most statistically dominant pattern, to cooperative, where heads specialize in distinct patterns. We model these dynamics using […]

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Learned optimizers are powerful alternatives to hand-designed update rules like Adam, yet they have seen limited practical adoption since they often fail to meta-generalize beyond their training distribution and incur high meta-training cost. For instance, prior work, VeLO, scaled meta-training to 4,000 TPU months ($sim$10$times$ GPT-3 compute) to meta-train a general-purpose optimizer but it failed to generalize beyond 600M parameters tasks. In this work, we present a surprising finding: by crafting a simple normalized optimizer architecture and augmenting […]

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Supervised causal learning has shown promise in causal discovery, yet it often struggles with generalization across diverse interventional settings, particularly when intervention targets are unknown. To address this, we propose TICL (Test-time Interventional Causal Learning), a novel method that synergizes Test-Time Training with Joint Causal Inference. Specifically, we design a self-augmentation strategy to generate instance-specific training data at test time, effectively avoiding distribution shifts. Furthermore, by integrating joint causal inference, we developed a PC-inspired two-phase supervised learning scheme, […]

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