We study trade-offs between the population risk curvature, geometry of the noise, and preconditioning on the generalisation ability of the multipass Preconditioned Stochastic Gradient Descent (PSGD). Many practical optimisation heuristics implicitly navigate this trade-off in different ways — for instance, some aim to whiten gradient noise, while others aim to align updates with expected loss curvature. When the geometry of the population risk curvature and the geometry of the gradient noise do not match, an aggressive choice that improves one aspect can amplify instability along the other, leading to suboptimal statistical behavior. In this paper we employ \emph{on-average algorithmic stability} to connect generalisation of PSGD to the \emph{effective dimension} that depends on these sources of curvature. While existing techniques for on-average stability of SGD are limited to a single pass, as first contribution we develop a new on-average stability analysis for multipass SGD that handles the correlations induced by data reuse. This allows us to derive excess risk bounds that depend on the effective dimension. In particular, we show that an improperly chosen preconditioner can yield suboptimal effective dimension dependence in both optimisation and generalisation. Finally, we complement our upper bounds with matching, instance-dependent lower bounds.

We study first-order algorithms that are uniformly stable for empirical risk minimization (ERM) problems that are convex and smooth with respect to $p$-norms, $p \geq 1$. We propose a black-box reduction method that, by employing properties of uniformly convex regularizers, turns an optimization algorithm for H{ö}lder smooth convex losses into a uniformly stable learning algorithm with optimal statistical risk bounds on the excess risk, up to a constant factor depending on $p$. Achieving a black-box reduction for uniform stability was posed as an open question by Attia and Koren (2022), which had solved the Euclidean case $p=2$. We explore applications that leverage non-Euclidean geometry in addressing binary classification problems.

Recent works of Mei et al. (2023, 2024) have deepened the theoretical understanding of the *Stochastic Gradient Bandit* (SGB) policy, showing that using a constant learning rate guarantees asymptotic convergence to the optimal policy, and that sufficiently *small* learning rates can yield logarithmic regret. However, whether logarithmic regret holds beyond small learning rates remains unclear. In this work, we take a step towards characterizing the regret *regimes* of SGB as a function of its learning rate. For two--armed bandits, we identify a sharp threshold, scaling with the sub-optimality gap $\Delta$, below which SGB achieves *logarithmic* regret on all instances, and above which it can incur *polynomial* regret on some instances. This result highlights the necessity of knowing (or estimating) $\Delta$ to ensure logarithmic regret with a constant learning rate. For general $K$-armed bandits, we further show the learning rate must scale inversely with $K$ to avoid polynomial regret. We introduce novel techniques to derive regret upper bounds for SGB, laying the groundwork for future advances in the theory of gradient-based bandit algorithms.

Optimization over the set of matrices $X$ that satisfy $X^\top B X = I_p$, referred to as the generalized Stiefel manifold, appears in many applications involving sampled covariance matrices such as the canonical correlation analysis (CCA), independent component analysis (ICA), and the generalized eigenvalue problem (GEVP). Solving these problems is typically done by iterative methods that require a fully formed $B$. We propose a cheap stochastic iterative method that solves the optimization problem while having access only to a random estimates of $B$. Our method does not enforce the constraint in every iteration; instead, it produces iterations that converge to critical points on the generalized Stiefel manifold defined in expectation. The method has lower per-iteration cost, requires only matrix multiplications, and has the same convergence rates as its Riemannian optimization counterparts that require the full matrix $B$. Experiments demonstrate its effectiveness in various machine learning applications involving generalized orthogonality constraints, including CCA, ICA, and the GEVP.

Orthogonality constraints naturally appear in many machine learning problems, from principal component analysis to robust neural network training. They are usually solved using Riemannian optimization algorithms, which minimize the objective function while enforcing the constraint. However, enforcing the orthogonality constraint can be the most time-consuming operation in such algorithms. Recently, Ablin and Peyré (2022) proposed the landing algorithm, a method with cheap iterations that does not enforce the orthogonality constraints but is attracted towards the manifold in a smooth manner. This article provides new practical and theoretical developments for the landing algorithm. First, the method is extended to the Stiefel manifold, the set of rectangular orthogonal matrices. We also consider stochastic and variance reduction algorithms when the cost function is an average of many functions. We demonstrate that all these methods have the same rate of convergence as their Riemannian counterparts that exactly enforce the constraint, and converge to the manifold. Finally, our experiments demonstrate the promise of our approach to an array of machine-learning problems that involve orthogonality constraints.

Effective image post-processing algorithms are vital for the successful direct imaging of exoplanets.Existing algorithms use techniques based on a low-rank approximation to separate the rotating planet signal from the quasi-static speckles.In this paper, we present a novel approach that iteratively finds the planet's flux and the low-rank approximation of quasi-static signals, strengthening the existing model based on lowrank approximations.We implement the algorithm with two different norms and test it on data, showing improvement over classical low-rank approaches.Our results highlight the benefits of iterative refinement of low-rank approximation to enhance planet detection.

We propose a direct imaging method for the detection of exo-planets based on a combined low-rank plus structured sparse model. For this task, we develop a dictionary of possible effective circular trajectories a planet can take during the observation time, elements of which can be efficiently computed using rotation and convolution operation. We design a simple alternating iterative hard-thresholding algorithm that jointly promotes a low-rank background and a sparse exoplanet foreground, to solve the non-convex optimisation problem. The experimental comparison on the β-Pictoris exoplanet benchmark dataset shows that our method has the potential to outperform the widely used Annular PCA for specific planet light intensities in terms of the receiver operating characteristic (ROC) curves.

Direct imaging of exoplanets is a challenging task due to the small angular distance and high contrast relative to their host star, and the presence of quasi-static noise. We propose a new statistical method for direct imaging of exoplanets based on a likelihood ratio detection map, which assumes that the noise after the background subtraction step obeys a Laplacian distribution. We compare the method with two detection approaches based on signal-to-noise ratio (SNR) map after performing the background subtraction by the widely used Annular Principal Component Analysis (AnnPCA). The experimental results on the Beta Pictoris data set show the method outperforms SNR maps in terms of achieving the highest true positive rate (TPR) at zero false positive rate (FPR).