Modern adaptive optimization methods, such as Adam and its variants, have emerged as the most widely used tools in deep learning over recent years. These algorithms offer automatic mechanisms for dynamically adjusting the update step based on estimates of gradient statistics. Compared to traditional algorithms like Stochastic Gradient Descent, these adaptive methods are typically more robust to model scale and hyperparameter tuning. However, the gradient statistics employed by these methods often do not leverage sufficient gradient covariance information, leading to suboptimal updates in certain directions of the parameter space and potentially slower convergence. In this work, we keep track of such covariance statistics in the form of a structured preconditioner matrix. Unlike other works, our approach does not apply direct approximations to estimate this matrix. We instead implement an invertible transformation that maps the preconditioner matrix into a new space where it becomes approximately diagonal. This enables a diagonal approximation of the preconditioner matrix in the transformed space, offering several computational advantages. Empirical results show that our approach can substantially enhance the convergence speed of modern adaptive optimizers. Notably, for large language models like LLaMA, we can achieve a speedup of 2x compared to the baseline Adam. Additionally, our method can be integrated with memory-efficient optimizers like Adafactor to manage computational overhead.

Minimizing the inclusive Kullback-Leibler (KL) divergence with stochastic gradient descent (SGD) is challenging since its gradient is defined as an integral over the posterior. Recently, multiple methods have been proposed to run SGD with biased gradient estimates obtained from a Markov chain. This paper provides the first non-asymptotic convergence analysis of these methods by establishing their mixing rate and gradient variance. To do this, we demonstrate that these methods-which we collectively refer to as Markov chain score ascent (MCSA) methods-can be cast as special cases of the Markov chain gradient descent framework. Furthermore, by leveraging this new understanding, we develop a novel MCSA scheme, parallel MCSA (pMCSA), that achieves a tighter bound on the gradient variance. We demonstrate that this improved theoretical result translates to superior empirical performance.

The task of reconstructing a matrix given a sample of observed entries is known as the matrix completion problem. It arises in a wide range of problems, including recommender systems, collaborative filtering, dimensionality reduction, image processing, quantum physics or multi-class classification to name a few. Most works have focused on recovering an unknown real-valued low-rank matrix from randomly sub-sampling its entries. Here, we investigate the case where the observations take a finite number of values, corresponding for examples to ratings in recommender systems or labels in multi-class classification. We also consider a general sampling scheme (not necessarily uniform) over the matrix entries. The performance of a nuclear-norm penalized estimator is analyzed theoretically. More precisely, we derive bounds for the Kullback-Leibler divergence between the true and estimated distributions. In practice, we have also proposed an efficient algorithm based on lifted coordinate gradient descent in order to tackle potentially high dimensional settings.

We improve a recent guarantee of Bach and Moulines on the linear convergence of SGD for smooth and strongly convex objectives, reducing a quadratic dependence on the strong convexity to a linear dependence. Furthermore, we show how reweighting the sampling distribution (i.e.

We provide a new understanding of the stochastic gradient bandit algorithm by showing that it converges to a globally optimal policy almost surely using \emph{any} constant learning rate. This result demonstrates that the stochastic gradient algorithm continues to balance exploration and exploitation appropriately even in scenarios where standard smoothness and noise control assumptions break down. The proofs are based on novel findings about action sampling rates and the relationship between cumulative progress and noise, and extend the current understanding of how simple stochastic gradient methods behave in bandit settings.

Conditional expectation E(Y | X) often fails This limitation has important implications for downstream to capture the complexity of multimodal conditional tasks, particularly in uncertainty quantification for image distributions L(Y | X). To address this, restoration models used in safety-critical domains such as we propose using n-point conditional quantizations--functional autonomous driving and biological imaging. Many existing mappings of X that are learnable approaches rely on per-pixel estimates such as variance via gradient descent--to approximate L(Y | heatmaps (Kendall and Gal, 2017) or confidence intervals X). This approach adapts Competitive Learning (Angelopoulos et al., 2022) to visualize uncertainty. Vector Quantization (CLVQ), tailored for conditional While these methods provide valuable insights, they can distributions. It goes beyond single-valued struggle to represent structured uncertainty, overlooking predictions by providing multiple representative spatial correlations between neighboring pixels.

Training large language models (LLMs) typically relies on adaptive optimizers like Adam (Kingma & Ba, 2015) which store additional state information to accelerate convergence but incur significant memory overhead. Recent efforts, such as SWAN (Ma et al., 2024) address this by eliminating the need for optimizer states while achieving performance comparable to Adam via a multi-step preprocessing procedure applied to instantaneous gradients. Motivated by the success of SWAN, we introduce a novel framework for designing stateless optimizers that normalizes stochastic gradients according to multiple norms. To achieve this, we propose a simple alternating scheme to enforce the normalization of gradients w.r.t these norms. We show that our procedure can produce, up to an arbitrary precision, a fixed-point of the problem, and that SWAN is a particular instance of our approach with carefully chosen norms, providing a deeper understanding of its design. However, SWAN's computationally expensive whitening/orthogonalization step limit its practicality for large LMs. Using our principled perspective, we develop of a more efficient, scalable, and practical stateless optimizer. Our algorithm relaxes the properties of SWAN, significantly reducing its computational cost while retaining its memory efficiency, making it applicable to training large-scale models. Experiments on pre-training LLaMA models with up to 1 billion parameters demonstrate a 3X speedup over Adam with significantly reduced memory requirements, outperforming other memory-efficient baselines.

Knowledge distillation (KD) is a powerful strategy for training deep neural networks (DNNs). Although it was originally proposed to train a more compact ``student'' model from a large ``teacher'' model, many recent efforts have focused on adapting it to promote generalization of the model itself, such as online KD and self KD. % as an effective way Here, we propose an accessible and compatible strategy named Spaced KD to improve the effectiveness of both online KD and self KD, in which the student model distills knowledge from a teacher model trained with a space interval ahead. This strategy is inspired by a prominent theory named \emph{spacing effect} in biological learning and memory, positing that appropriate intervals between learning trials can significantly enhance learning performance. With both theoretical and empirical analyses, we demonstrate that the benefits of the proposed Spaced KD stem from convergence to a flatter loss landscape during stochastic gradient descent (SGD). We perform extensive experiments to validate the effectiveness of Spaced KD in improving the learning performance of DNNs (e.g., the performance gain is up to 2.31\% and 3.34\% on Tiny-ImageNet over online KD and self KD, respectively).

In this paper, we establish non-asymptotic convergence rates in the central limit theorem for Polyak-Ruppert-averaged iterates of stochastic gradient descent (SGD). Our analysis builds on the result of the Gaussian approximation for nonlinear statistics of independent random variables of Shao and Zhang (2022). Using this result, we prove the non-asymptotic validity of the multiplier bootstrap for constructing the confidence sets for the optimal solution of an optimization problem. In particular, our approach avoids the need to approximate the limiting covariance of Polyak-Ruppert SGD iterates, which allows us to derive approximation rates in convex distance of order up to $1/\sqrt{n}$.

The problem of learning single index and multi index models has gained significant interest as a fundamental task in high-dimensional statistics. Many recent works have analysed gradient-based methods, particularly in the setting of isotropic data distributions, often in the context of neural network training. Such studies have uncovered precise characterisations of algorithmic sample complexity in terms of certain analytic properties of the target function, such as the leap, information, and generative exponents. These properties establish a quantitative separation between low and high complexity learning tasks. In this work, we show that high complexity cases are rare. Specifically, we prove that introducing a small random perturbation to the data distribution--via a random shift in the first moment--renders any Gaussian single index model as easy to learn as a linear function. We further extend this result to a class of multi index models, namely sparse Boolean functions, also known as Juntas.