We consider the Tree Alternating Optimization (TAO) algorithm to train regression trees with linear predictors in the leaves. Unlike the traditional, greedy recursive partitioning algorithms such as CART, TAO guarantees a monotonic decrease of the objective function and results in smaller trees of much better accuracy. We modify the TAO algorithm so that it produces exactly the same result but is much faster, particularly for high input dimensionality or deep trees. The idea is based on the fact that, at each iteration of TAO, each leaf receives only a subset of the training instances. Thus, the optimization of the leaf model can be done exactly but faster by using the Sherman-Morrison-Woodbury formula. This has the unexpected advantage that, once a tree exceeds a critical depth, then making it deeper makes it faster to train, even though the tree is larger and has more parameters. Indeed, this can make learning a nonlinear model (the tree) asymptotically faster than a regular linear regression model. We analyze the corresponding computational complexity and verify the speedups experimentally in various datasets. The argument can be applied to other types of trees, whenever the optimization of a node can be computed in superlinear time of the number of instances.

Weak-to-strong generalization--where a student model trained on imperfect labels generated by a weaker teacher nonetheless surpasses that teacher--has been widely observed, but the mechanisms that enable it have remained poorly understood. In this paper, through a theoretical analysis of simple models, we uncover three core mechanisms that can drive this phenomenon. First, by analyzing ridge linear regression, we study the interplay between the teacher and student regularization parameters and prove that a student can compensate for a teacher's under-regularization and achieve lower test error. We also analyze the role of the parameterization regime of the models and show that qualitatively different phenomena can happen in different regimes. Second, by analyzing weighted ridge linear regression, we show that a student model with a regularization structure better aligned to the target function, can outperform its teacher. Third, in a nonlinear multi-index learning setting, we demonstrate that a student can learn easy, task-specific features from the teacher while leveraging its own broader pre-training to learn hard-to-learn features that the teacher cannot capture.

Understanding how the test risk scales with model complexity is a central question in machine learning. Classical theory is challenged by the learning curves observed for large over-parametrized deep networks. Capacity measures based on parameter count typically fail to account for these empirical observations. To tackle this challenge, we consider norm-based capacity measures and develop our study for random features based estimators, widely used as simplified theoretical models for more complex networks. In this context, we provide a precise characterization of how the estimator's norm concentrates and how it governs the associated test error. Our results show that the predicted learning curve admits a phase transition from under-to over-parameterization, but no double descent behavior. This confirms that more classical U-shaped behavior is recovered considering appropriate capacity measures based on models norms rather than size. From a technical point of view, we leverage deterministic equivalence as the key tool and further develop new deterministic quantities which are of independent interest.

Transformers have demonstrated remarkable in-context learning (ICL) capabilities, adapting to new tasks by simply conditioning on demonstrations without parameter updates. Compelling empirical and theoretical evidence suggests that ICL, as a general-purpose learner, could outperform task-specific models. However, it remains unclear to what extent the transformers optimally learn in-context compared to principled learning algorithms. To investigate this, we employ a meta ICL framework in which each prompt defines a distinctive regression task whose target function is drawn from a hierarchical distribution, requiring inference over both the latent model class and task-specific parameters.

We study population convergence guarantees of stochastic gradient descent (SGD) for smooth convex objectives in the interpolation regime, where the noise at optimum is zero or near zero. The behavior of the last iterate of SGD in this setting--particularly with large (constant) stepsizes--has received growing attention in recent years due to implications for the training of over-parameterized models, as well as to analyzing forgetting in continual learning and to understanding the convergence of the randomized Kaczmarz method for solving linear systems.

Training deep learning models for point cloud prediction tasks such as shape completion and generation depends critically on loss functions that measure discrepancies between predicted and ground-truth point sets. Commonly used functions such as Chamfer Distance (CD), HyperCD, InfoCD and Density-aware CD rely on nearest-neighbor assignments, which often induce many-to-one correspondences, leading to point congestion in dense regions and poor coverage in sparse regions. These losses also involve non-differentiable operations due to index selection, which may affect gradient-based optimization. Earth Mover Distance (EMD) enforces one-to-one correspondences and captures structural similarity more effectively, but its cubic computational complexity limits its practical use. We propose the Adaptive Probabilistic Matching Loss (APML), a fully differentiable approximation of one-to-one matching that leverages Sinkhorn iterations on a temperature-scaled similarity matrix derived from pairwise distances. We analytically compute the temperature to guarantee a minimum assignment probability, eliminating manual tuning. APML achieves near-quadratic runtime, comparable to Chamfer-based losses, and avoids non-differentiable operations.

We study gradient descent (GD) with a constant stepsize for ℓ2-regularized logistic regression with linearly separable data. Classical theory suggests small stepsizes to ensure monotonic reduction of the optimization objective, achieving exponential convergence in eO(κ) steps with κ being the condition number. Surprisingly, we show that this can be accelerated to eO( κ)by simply using a large stepsize--for which the objective evolves nonmonotonically. The acceleration brought by large stepsizes extends to minimizing the population risk for separable distributions, improving on the best-known upper bounds on the number of steps to reach a nearoptimum. Finally, we characterize the largest stepsize for the local convergence of GD, which also determines the global convergence in special scenarios. Our results extend the analysis of Wu et al. (2024) from convex settings with minimizers at infinity to strongly convex cases with finite minimizers.

In this paper, we explore the asymptotically optimal tuning parameter choice in ridge regression for estimating nuisance functions of a statistical functional that has recently gained prominence in conditional independence testing and causal inference. Given a sample of size n, we study estimators of the Expected Conditional Covariance (ECC) between variables Y and Agiven a high-dimensional covariate X Rp. Under linear regression models for Y and A on X and the proportional asymptotic regime p/n c (0,), we evaluate three existing ECC estimators and two sample splitting strategies for estimating the required nuisance functions. Since no consistent estimator of the nuisance functions exists in the proportional asymptotic regime without imposing further structure on the problem, we first derive debiased versions of the ECC estimators that utilize the ridge regression nuisance function estimators. We show that our bias correction strategy yields n-consistent estimators of the ECC across different sample splitting strategies and estimator choices. We then derive the asymptotic variances of these debiased estimators to illustrate the nuanced interplay between the sample splitting strategy, estimator choice, and tuning parameters of the nuisance function estimators for optimally estimating the ECC. Our analysis reveals that prediction-optimal tuning parameters (i.e., those that optimally estimate the nuisance functions) may not lead to the lowest asymptotic variance of the ECC estimator - thereby demonstrating the need to be careful in selecting tuning parameters based on the final goal of inference. Finally, we verify our theoretical results through extensive numerical experiments.

Anthropic still can't distribute Claude Mythos or Fable 5 after running afoul of the Trump administration. But no one can say exactly what the company did wrong. It's been nearly a week since the Trump administration sent an export control directive to Anthropic, forcing one of the world's leading AI labs to pull its most advanced models offline. After days of negotiations between Anthropic and the White House, the two still remain at odds about how to bring Claude Mythos and Fable 5 back. Well, it depends whom you ask.

The growing use of generative models in daily life calls for efficient mechanisms to control their generation, to e.g., produce safe content or provide users with tools to explore style changes. Ideally, such mechanisms should require low volume of unpaired data (i.e., without explicit preference), and should be cheap, both at train and inference time, while preserving output quality. Recent research has shown that such mechanisms can be obtained by intervening exclusively on model activations, with the goal of correcting distributional differences between activations seen when using prompts from a source vs. a target set (e.g., toxic and non-toxic sentences). While cheap, these fast methods are inherently crude: their maps are tuned locally, not accounting for their impact on downstream layers, resulting in interventions that cause unintended shifts when used out-of-sample. We propose in this work linear end-to-end activation steering (LinEAS), an approach trained with a global loss that accounts simultaneously for all layer-wise distributional shifts. In addition to being more robust, the loss used to train LinEAS can be regularized with sparsifying norms, which can automatically carry out neuron selection. LinEAS only requires a handful of unpaired samples to be effective, and beats similar baselines on toxicity mitigation in language models, becoming competitive with oracle-dependent methods that have access to strong supervision. LinEAS is modality-agnostic and we empirically find that it outperforms existing activation steering methods at mitigating and including new concepts at the output of single-step text-to-image generation models.