We study the linear-quadratic control problem for a class of non-exchangeable mean-field systems, which model large populations of heterogeneous interacting agents. We explicitly characterize the optimal control in terms of a new infinite-dimensional system of Riccati equations, for which we establish existence and uniqueness. To illustrate our results, we apply this framework to a systemic risk model involving heterogeneous banks, demonstrating the impact of agent heterogeneity on optimal risk mitigation strategies.
Generative Modeling via Drifting has recently achieved state-of-the-art one-step image generation through a kernel-based drift operator, yet its success is largely empirical and its theoretical foundations remain poorly understood. We observe that under a Gaussian kernel, the drift operator is exactly a score difference on smoothed distributions. This answers three questions left open in the original work: (1) whether a vanishing drift guarantees equality of distributions (V_p,q=0⇒ p=q), (2) how to choose between kernels, and (3) why the stop-gradient operator is indispensable for stable training. Our observations position drifting within the score-matching family. By linearizing the McKean-Vlasov dynamics and probing them in Fourier space, we reveal frequency-dependent convergence timescales comparable to Landau damping in plasma kinetic theory: the Gaussian kernel suffers an exponential high-frequency bottleneck, potentially explaining the empirical preference for the Laplacian kernel. This suggests a fix: an exponential bandwidth annealing schedule σ(t)=σ_0 e^-rt that reduces convergence time from exp(O(K_max^2)) to O(log K_max). Finally, by formalizing drifting as a Wasserstein gradient flow of the smoothed KL divergence, we prove that the stop-gradient operator is not a heuristic but is derived from the frozen-field discretization mandated by the Jordan-Kinderlehrer-Otto (JKO) scheme, and removing it severs training from any gradient-flow guarantee. This variational perspective further provides a general template for constructing novel drift operators, which we demonstrate with a Sinkhorn divergence drift. We validate our analysis on toy datasets and scale it up to ImageNet.
Curriculum learning-organizing training data from easy to hard-has improved efficiency across machine learning domains, yet remains underexplored for language model pretraining. We present the first systematic investigation of curriculum learning in LLM pretraining, with over 200 models trained on up to 100B tokens across three strategies: vanilla curriculum learning, pacing-based sampling, and interleaved curricula, guided by six difficulty metrics spanning linguistic and information-theoretic properties. We evaluate performance on eight benchmarks under three realistic scenarios: limited data, unlimited data, and continual training. Our experiments show that curriculum learning consistently accelerates convergence in early and mid-training phases,reducing training steps by 18-45% to reach baseline performance. When applied as a warmup strategy before standard random sampling, curriculum learning yields sustained improvements up to 3.5%. We identify compression ratio, lexical diversity (MTLD), and readability (Flesch Reading Ease) as the most effective difficulty signals. Our findings demonstrate that data ordering-orthogonal to existing data selection methods-provides a practical mechanism for more efficient LLM pretraining.
