Adaptive optimizers such as Adam (Kingma & Ba, 2015) have been central to the success of large language models. However, they often require maintaining optimizer states throughout training, which can result in memory requirements several times greater than the model footprint. This overhead imposes constraints on scalability and computational efficiency. Stochastic Gradient Descent (SGD), in contrast, is a stateless optimizer, as it does not track state variables during training. Consequently, it achieves optimal memory efficiency. However, its capability in LLM training is limited (Zhao et al., 2024b) In this work, we show that pre-processing SGD using normalization and whitening in a stateless manner can achieve similar performance as Adam for LLM training, while maintaining the same memory footprint of SGD. Specifically, we show that normalization stabilizes gradient distributions, and whitening counteracts the local curvature of the loss landscape. This results in SWAN (SGD with Whitening And Normalization), a stochastic optimizer that eliminates the need to store any optimizer states. Empirically, SWAN achieves 50% reduction on total end-to-end memory compared to Adam. Under the memory-efficienct LLaMA training benchmark of (Zhao et al., 2024a), SWAN reaches the same evaluation perplexity using half as many tokens for 350M and 1.3B model.

Training large language models (LLMs) commonly relies on adaptive optimizers such as Adam (Kingma & Ba 2015), which accelerate convergence through moment estimates but incur substantial memory overhead. Recent stateless approaches such as SWAN (Ma et al., 2024) have shown that appropriate preprocessing of instantaneous gradient matrices can match the performance of adaptive methods without storing optimizer states. Building on this insight, we introduce \emph{gradient multi-normalization}, a principled framework for designing stateless optimizers that normalize gradients with respect to multiple norms simultaneously. Whereas standard first-order methods can be viewed as gradient normalization under a single norm (Bernstein & Newhouse, 2024), our formulation generalizes this perspective to a multi-norm setting. We derive an efficient alternating scheme that enforces these normalization constraints and show that our procedure can produce, up to an arbitrary precision, a fixed-point of the problem. This unifies and extends prior stateless optimizers, showing that SWAN arises as a specific instance with particular norm choices. Leveraging this principle, we develop SinkGD, a lightweight matrix optimizer that retains the memory footprint of SGD while substantially reducing computation relative to whitening-based methods. On the memory-efficient LLaMA training benchmark (Zhao et al., 2024), SinkGD achieves state-of-the-art performance, reaching the same evaluation perplexity as Adam using only 40\% of the training tokens.

Discovering the underlying relationships among variables from temporal observations has been a longstanding challenge in numerous scientific disciplines, including biology, finance, and climate science. The dynamics of such systems are often best described using continuous-time stochastic processes. Unfortunately, most existing structure learning approaches assume that the underlying process evolves in discrete-time and/or observations occur at regular time intervals. These mismatched assumptions can often lead to incorrect learned structures and models. In this work, we introduce a novel structure learning method, SCOTCH, which combines neural stochastic differential equations (SDE) with variational inference to infer a posterior distribution over possible structures. This continuous-time approach can naturally handle both learning from and predicting observations at arbitrary time points. Theoretically, we establish sufficient conditions for an SDE and SCOTCH to be structurally identifiable, and prove its consistency under infinite data limits. Empirically, we demonstrate that our approach leads to improved structure learning performance on both synthetic and real-world datasets compared to relevant baselines under regular and irregular sampling intervals.

Bayesian causal discovery aims to infer the posterior distribution over causal models from observed data, quantifying epistemic uncertainty and benefiting downstream tasks. However, computational challenges arise due to joint inference over combinatorial space of Directed Acyclic Graphs (DAGs) and nonlinear functions. Despite recent progress towards efficient posterior inference over DAGs, existing methods are either limited to variational inference on node permutation matrices for linear causal models, leading to compromised inference accuracy, or continuous relaxation of adjacency matrices constrained by a DAG regularizer, which cannot ensure resulting graphs are DAGs. In this work, we introduce a scalable Bayesian causal discovery framework based on a combination of stochastic gradient Markov Chain Monte Carlo (SG-MCMC) and Variational Inference (VI) that overcomes these limitations. Our approach directly samples DAGs from the posterior without requiring any DAG regularization, simultaneously draws function parameter samples and is applicable to both linear and nonlinear causal models. To enable our approach, we derive a novel equivalence to the permutation-based DAG learning, which opens up possibilities of using any relaxed gradient estimator defined over permutations. To our knowledge, this is the first framework applying gradient-based MCMC sampling for causal discovery. Empirical evaluation on synthetic and real-world datasets demonstrate our approach's effectiveness compared to state-of-the-art baselines.

Learning structures between groups of variables from data with missing values is an important task in the real world, yet difficult to solve. One typical scenario is discovering the structure among topics in the education domain to identify learning pathways. Here, the observations are student performances for questions under each topic which contain missing values. However, most existing methods focus on learning structures between a few individual variables from the complete data. In this work, we propose VISL, a novel scalable structure learning approach that can simultaneously infer structures between groups of variables under missing data and perform missing value imputations with deep learning. Particularly, we propose a generative model with a structured latent space and a graph neural network-based architecture, scaling to a large number of variables. Empirically, we conduct extensive experiments on synthetic, semi-synthetic, and real-world education data sets. We show improved performances on both imputation and structure learning accuracy compared to popular and recent approaches.

Sliced Stein discrepancy (SSD) and its kernelized variants have demonstrated promising successes in goodness-of-fit tests and model learning in high dimensions. Despite the theoretical elegance, their empirical performance depends crucially on the search of the optimal slicing directions to discriminate between two distributions. Unfortunately, previous gradient-based optimisation approach returns sub-optimal results for the slicing directions: it is computationally expensive, sensitive to initialization, and it lacks theoretical guarantee for convergence. We address these issues in two steps. First, we show in theory that the requirement of using optimal slicing directions in the kernelized version of SSD can be relaxed, validating the resulting discrepancy with finite random slicing directions. Second, given that good slicing directions are crucial for practical performance, we propose a fast algorithm for finding good slicing directions based on ideas of active sub-space construction and spectral decomposition. Experiments in goodness-of-fit tests and model learning show that our approach achieves both the best performance and the fastest convergence. Especially, we demonstrate 14-80x speed-up in goodness-of-fit tests when compared with the gradient-based approach.

In this paper, we address the ice-start problem, i.e., the challenge of deploying machine learning models when only a little or no training data is initially available, and acquiring each feature element of data is associated with costs. This setting is representative of the real-world machine learning applications. For instance, in the health care domain, obtaining every single measurement comes with a cost. We propose Icebreaker, a principled framework for elementwise training data acquisition. Icebreaker introduces a full Bayesian Deep Latent Gaussian Model (BELGAM) with a novel inference method, which combines recent advances in amortized inference and stochastic gradient MCMC to enable fast and accurate posterior inference. By utilizing BELGAM’s ability to fully quantify model uncertainty, we also propose two information acquisition functions for imputation and active prediction problems. We demonstrate that BELGAM performs significantly better than previous variational autoencoder (VAE) based models, when the data set size is small, using both machine learning benchmarks and real world recommender systems and health-care applications. Moreover, Icebreaker not only demonstrates improved performance compared to baselines, but it is also capable of achieving better test performance with less training data available.