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Stochastic approximation (SA) is a classical approach for stochastic convex optimization. Previous studies have demonstrated that the convergence rate of SA can be improved by introducing either smoothness or strong convexity condition. In…
Although there is an extensive literature on the maxima of Gaussian processes, there are relatively few non-asymptotic bounds on their lower-tail probabilities. The aim of this paper is to develop such a bound, while also allowing for many…
In recent years, non-convex optimization problems are more often described by generalized $(L_0, L_1)$-smoothness assumption rather than standard one. Meanwhile, severely corrupted data used in these problems has increased the demand for…
This work is concerned with stochastic consensus conditions of multi-agent systems with both time-delays and measurement noises. For the case of additive noises, we develop some necessary conditions and sufficient conditions for stochastic…
We consider a finite element approximation of a general semi-linear stochastic partial differential equation (SPDE) driven by space-time multiplicative and additive noise. We examine the full weak convergence rate of the exponential Euler…
Multi-task learning (MTL) considers learning a joint model for multiple tasks by optimizing a convex combination of all task losses. To solve the optimization problem, existing methods use an adaptive weight updating scheme, where task…
In this paper, we propose practical normalized stochastic first-order methods with Polyak momentum, multi-extrapolated momentum, and recursive momentum for solving unconstrained optimization problems. These methods employ dynamically…
The study of tail behaviour of SGD-induced processes has been attracting a lot of interest, due to offering strong guarantees with respect to individual runs of an algorithm. While many works provide high-probability guarantees, quantifying…
We consider the stochastic optimization problem with smooth but not necessarily convex objectives in the heavy-tailed noise regime, where the stochastic gradient's noise is assumed to have bounded $p$th moment ($p\in(1,2]$). Zhang et al.…
We investigate the optimality of perturbation based algorithms in the stochastic and adversarial multi-armed bandit problems. For the stochastic case, we provide a unified regret analysis for both sub-Weibull and bounded perturbations when…
This paper aims to investigate the numerical approximation of a general second order parabolic stochastic partial differential equation(SPDE) driven by multiplicative and additive noise under more relaxed conditions. The SPDE is discretized…
Stochastic approximation (SA) that involves multiple coupled sequences, known as multiple-sequence SA (MSSA), finds diverse applications in the fields of signal processing and machine learning. However, existing theoretical understandings…
Evolutionary strategies have recently been shown to achieve competing levels of performance for complex optimization problems in reinforcement learning. In such problems, one often needs to optimize an objective function subject to a set of…
We study a stochastically perturbed version of the well-known Krasnoselski--Mann iteration for computing fixed points of nonexpansive maps in finite dimensional normed spaces. We discuss sufficient conditions on the stochastic noise and…
This paper concerns quasi-stochastic approximation (QSA) to solve root finding problems commonly found in applications to optimization and reinforcement learning. The general constant gain algorithm may be expressed as the…
This paper establishes a stochastic maximum principle for optimal control problems governed by time-changed forward-backward stochastic differential equations with L\'evy noise. The system incorporates a random, non-decreasing operational…
Stochastic heavy ball momentum (SHB) is commonly used to train machine learning models, and often provides empirical improvements over stochastic gradient descent. By primarily focusing on strongly-convex quadratics, we aim to better…
Understanding the generalization properties of optimization algorithms under heavy-tailed noise has gained growing attention. However, the existing theoretical results mainly focus on stochastic gradient descent (SGD) and the analysis of…
The computational burden of attention in long-context language models has motivated two largely independent lines of work: sparse attention mechanisms that reduce complexity by attending to selected tokens, and gated attention variants that…
In this paper, we prove convergence rates for time discretisation schemes for semi-linear stochastic evolution equations with additive or multiplicative Gaussian noise, where the leading operator $A$ is the generator of a strongly…