Density estimation is essential for generative modeling, particularly with the rise of modern neural networks. While existing methods capture complex data distributions, they often lack interpretability and uncertainty quantification.

We utilise a sampler originating from nonequilibrium statistical mechanics, termed here Jarzynski-adjusted Langevin algorithm (JALA), to build statistical estimation methods in latent variable models. We achieve this by leveraging Jarzynski's equality and developing algorithms based on a weighted version of the unadjusted Langevin algorithm (ULA) with recursively updated weights. Adapting this for latent variable models, we develop a sequential Monte Carlo (SMC) method that provides the maximum marginal likelihood estimate of the parameters, termed JALA-EM. Under suitable regularity assumptions on the marginal likelihood, we provide a nonasymptotic analysis of the JALA-EM scheme implemented with stochastic gradient descent and show that it provably converges to the maximum marginal likelihood estimate. We demonstrate the performance of JALA-EM on a variety of latent variable models and show that it performs comparably to existing methods in terms of accuracy and computational efficiency. Importantly, the ability to recursively estimate marginal likelihoods--an uncommon feature among scalable methods--makes our approach particularly suited for model selection, which we validate through dedicated experiments.

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.

This paper investigates the computational complexity of reinforcement learning within a novel linear function approximation regime, termed partial $q^{\pi}$-realizability. In this framework, the objective is to learn an $\epsilon$-optimal policy with respect to a predefined policy set $\Pi$, under the assumption that all value functions corresponding to policies in $\Pi$ are linearly realizable. This framework adopts assumptions that are weaker than those in the $q^{\pi}$-realizability setting yet stronger than those in the q*-realizability setup. As a result, it provides a more practical model for reinforcement learning scenarios where function approximation naturally arise. We prove that learning an $\epsilon$-optimal policy in this newly defined setting is computationally hard. More specifically, we establish NP-hardness under a parameterized greedy policy set (i.e., argmax) and, further, show that--unless NP = RP--an exponential lower bound (exponential in feature vector dimension) holds when the policy set contains softmax policies, under the Randomized Exponential Time Hypothesis. Our hardness results mirror those obtained in the $q^*$-realizability settings, and suggest that computational difficulty persists even when the policy class $ \Pi $ is expanded beyond the optimal policy, reinforcing the unbreakable nature of the computational hardness result regarding partial $ q^{\pi} $-realizability under two important policy sets. To establish our negative result, our primary technical contribution is a reduction from two complexity problems, $\delta$-Max-3SAT and $\delta$-Max-3SAT($b$), to instances of our problem settings: GLinear-$\kappa$-RL (under the greedy policy set) and SLinear-$\kappa$-RL (under the softmax policy set), respectively. Our findings indicate that positive computational results are generally unattainable in the context of partial $ q^{\pi} $-realizability, in sharp contrast to the $ q^{\pi} $-realizability setting under a generative access model.

Scaling laws have emerged as a unifying lens for understanding and guiding the training of large language models (LLMs). However, existing studies predominantly focus on the final-step loss, leaving open whether the entire $\textit{loss dynamics}$ obey similar laws and, crucially, how the $\textit{learning rate schedule}$ (LRS) shapes them. We address these gaps in a controlled theoretical setting by analyzing stochastic gradient descent (SGD) on a power-law kernel regression model. The key insight is a novel $\textbf{intrinsic-time}$ viewpoint, which captures the training progress more faithfully than iteration count. We then establish a $\textbf{Functional Scaling Law (FSL)}$ that captures the full loss trajectory under arbitrary LRSs, with the schedule's influence entering through a simple convolutional functional. We further instantiate the theory for three representative LRSs---constant, exponential decay, and warmup-stable-decay (WSD)---and derive explicit scaling relations in both data-and compute-limited regimes. These comparisons explain key empirical phenomena: (i) higher-capacity models are more data-and compute-efficient; (ii) learning-rate decay improves training efficiency; and (iii) WSD-type schedules outperform pure decay. Finally, experiments on LLMs ranging from 0.1B to 1B parameters demonstrate the practical relevance of FSL as a surrogate model for fitting and predicting loss trajectories in large-scale pre-training.

Modern machine learning often requires training with large batch size, distributed data, and massively parallel compute hardware (like mobile and other edge devices or distributed data centers). Communication becomes a major bottleneck in such settings but methods like Local Stochastic Gradient Descent (Local SGD) show great promise to reduce the global communication need. Local SGD consists of three parts: a local optimization processes, an aggregation mechanism, and an outer optimizer that uses the aggregated updates from the nodes to produce a new model. While there exists an extensive literature on understanding the impact of hyperparameters in the local optimization process, the choice of outer optimizer and its hyperparameters is less clear. We study the role of the outer learning in Local SGD, and prove new convergence guarantees for the algorithm. In particular, we show that tuning the outer learning rate allows us to (a) trade off between optimization error and stochastic gradient noise variance, and (b) make up for ill-tuning of the inner learning rate. Our theory suggests that the outer learning rate should sometimes be set to values greater than $1$. We extend our results to apply to when we use momentum in the outer optimizer, and also introduce a novel data-dependent analysis of Local SGD that yields further insights on outer learning rate tuning. We conduct comprehensive experiments with standard language models and various outer optimizers to validate our theory.

We investigate language generation in the limit - a model by Kleinberg and Mullainathan and extended by Li, Raman, and Tewari. While Kleinberg and Mullainathan proved generation is possible for all countable collections, Li, Raman, and Tewari defined a hierarchy of generation notions (uniform, non-uniform, and generatable) and explored their feasibility for uncountable collections. Our first set of results resolve two open questions of Li et al. by proving finite unions of generatable or non-uniformly generatable classes need not be generatable. These follow from a stronger result: there is non-uniformly generatable class and a uniformly generatable class whose union is non-generatable. This adds to the aspects along which language generation in the limit is different from traditional tasks in statistical learning theory like classification, which are closed under finite unions. In particular, it implies that given two generators for different collections, one cannot combine them to obtain a single more powerful generator, prohibiting this notion of boosting. Our construction also addresses a third of Li et al.'s open questions on whether there are uncountable classes that are non-uniformly generatable and do not satisfy the eventually unbounded closure (EUC) condition introduced by Li et al. Our approach utilizes carefully constructed classes along with a novel diagonalization argument that could be of independent interest in the growing area of language generation.

Time series forecasting plays a pivotal role in critical domains such as energy management and financial markets. Although deep learning-based approaches (e.g., MLP, RNN, Transformer) have achieved remarkable progress, the prevailing long-sequence information gain hypothesis exhibits inherent limitations. Through systematic experimentation, this study reveals a counterintuitive phenomenon: appropriately truncating historical data can paradoxically enhance prediction accuracy, indicating that existing models learn substantial redundant features (e.g., noise or irrelevant fluctuations) during training, thereby compromising effective signal extraction. Building upon information bottleneck theory, we propose an innovative solution termed Adaptive Masking Loss with Representation Consistency (AMRC), which features two core components: 1) Dynamic masking loss, which adaptively identified highly discriminative temporal segments to guide gradient descent during model training; 2) Representation consistency constraint, which stabilized the mapping relationships among inputs, labels, and predictions. Experimental results demonstrate that AMRC effectively suppresses redundant feature learning while significantly improving model performance. This work not only challenges conventional assumptions in temporal modeling but also provides novel theoretical insights and methodological breakthroughs for developing efficient and robust forecasting models. We have made our code available at \url{https://github.com/MazelTovy/AMRC}.

Stochastic Gradient Descent (SGD) and its Ruppert-Polyak averaged variant (ASGD) lie at the heart of modern large-scale learning, yet their theoretical properties in high-dimensional settings are rarely understood. In this paper, we provide rigorous statistical guarantees for constant learning-rate SGD and ASGD in high-dimensional regimes. Our key innovation is to transfer powerful tools from high-dimensional time series to online learning. Specifically, by viewing SGD as a nonlinear autoregressive process and adapting existing coupling techniques, we prove the geometric-moment contraction of high-dimensional SGD for constant learning rates, thereby establishing asymptotic stationarity of the iterates. Building on this, we derive the $q$-th moment convergence of SGD and ASGD for any $q\ge2$ in general $\ell^s$-norms, and, in particular, the $\ell^{\infty}$-norm that is frequently adopted in high-dimensional sparse or structured models. Furthermore, we provide sharp high-probability concentration analysis which entails the probabilistic bound of high-dimensional ASGD. Beyond closing a critical gap in SGD theory, our proposed framework offers a novel toolkit for analyzing a broad class of high-dimensional learning algorithms.

We introduce a novel framework for differentially private (DP) statistical estimation via data truncation, addressing a key challenge in DP estimation when the data support is unbounded. Traditional approaches rely on problem-specific sensitivity analysis, limiting their applicability. By leveraging techniques from truncated statistics, we develop computationally efficient DP estimators for exponential family distributions, including Gaussian mean and covariance estimation, achieving near-optimal sample complexity. Previous works on exponential families only consider bounded or one-dimensional families. Our approach mitigates sensitivity through truncation while carefully correcting for the introduced bias using maximum likelihood estimation and DP stochastic gradient descent. Along the way, we establish improved uniform convergence guarantees for the log-likelihood function of exponential families, which may be of independent interest. Our results provide a general blueprint for DP algorithm design via truncated statistics.