A Note on High-Probability Analysis of Algorithms with Exponential, Sub-Gaussian, and General Light Tails

This short note describes a simple technique for analyzing probabilistic algorithms that rely on a light-tailed (but not necessarily bounded) source of randomization. We show that the analysis of such an algorithm can be reduced, in a black-box manner and with only a small loss in logarithmic factors, to an analysis of a simpler variant of the same algorithm that uses bounded random variables and often easier to analyze. This approach simultaneously applies to any light-tailed randomization, including exponential, sub-Gaussian, and more general fast-decaying distributions, without needing to appeal to specialized concentration inequalities. Analyses of a generalized Azuma inequality and stochastic optimization with general light-tailed noise are provided to illustrate the technique.

How Free is Parameter-Free Stochastic Optimization?

We study the problem of parameter-free stochastic optimization, inquiring whether, and under what conditions, do fully parameter-free methods exist: these are methods that achieve convergence rates competitive with optimally tuned methods, without requiring significant knowledge of the true problem parameters. Existing parameter-free methods can only be considered ``partially'' parameter-free, as they require some non-trivial knowledge of the true problem parameters, such as a bound on the stochastic gradient norms, a bound on the distance to a minimizer, etc. In the non-convex setting, we demonstrate that a simple hyperparameter search technique results in a fully parameter-free method that outperforms more sophisticated state-of-the-art algorithms. We also provide a similar result in the convex setting with access to noisy function values under mild noise assumptions. Finally, assuming only access to stochastic gradients, we establish a lower bound that renders fully parameter-free stochastic convex optimization infeasible, and provide a method which is (partially) parameter-free up to the limit indicated by our lower bound.

SGD with AdaGrad Stepsizes: Full Adaptivity with High Probability to Unknown Parameters, Unbounded Gradients and Affine Variance

We study Stochastic Gradient Descent with AdaGrad stepsizes: a popular adaptive (self-tuning) method for first-order stochastic optimization. Despite being well studied, existing analyses of this method suffer from various shortcomings: they either assume some knowledge of the problem parameters, impose strong global Lipschitz conditions, or fail to give bounds that hold with high probability. We provide a comprehensive analysis of this basic method without any of these limitations, in both the convex and non-convex (smooth) cases, that additionally supports a general ``affine variance'' noise model and provides sharp rates of convergence in both the low-noise and high-noise~regimes.

Uniform Stability for First-Order Empirical Risk Minimization

We consider the problem of designing uniformly stable first-order optimization algorithms for empirical risk minimization. Uniform stability is often used to obtain generalization error bounds for optimization algorithms, and we are interested in a general approach to achieve it. For Euclidean geometry, we suggest a black-box conversion which given a smooth optimization algorithm, produces a uniformly stable version of the algorithm while maintaining its convergence rate up to logarithmic factors. Using this reduction we obtain a (nearly) optimal algorithm for smooth optimization with convergence rate $\widetilde{O}(1/T^2)$ and uniform stability $O(T^2/n)$, resolving an open problem of Chen et al. (2018); Attia and Koren (2021). For more general geometries, we develop a variant of Mirror Descent for smooth optimization with convergence rate $\widetilde{O}(1/T)$ and uniform stability $O(T/n)$, leaving open the question of devising a general conversion method as in the Euclidean case.

The Instability of Accelerated Gradient Descent

We study the algorithmic stability of Nesterov's accelerated gradient method. For convex quadratic objectives, \citet{chen2018stability} proved that the uniform stability of the method grows quadratically with the number of optimization steps, and conjectured that the same is true for the general convex and smooth case. We disprove this conjecture and show, for two notions of stability, that the stability of Nesterov's accelerated method in fact deteriorates \emph{exponentially fast} with the number of gradient steps. This stands in sharp contrast to the bounds in the quadratic case, but also to known results for non-accelerated gradient methods where stability typically grows linearly with the number of steps.