Sarah Sachs

Sarah Sachs, Tim van Erven, Liam Hodgkinson et al.

Algorithm- and data-dependent generalization bounds are required to explain the generalization behavior of modern machine learning algorithms. In this context, there exists information theoretic generalization bounds that involve (various forms of) mutual information, as well as bounds based on hypothesis set stability. We propose a conceptually related, but technically distinct complexity measure to control generalization error, which is the empirical Rademacher complexity of an algorithm- and data-dependent hypothesis class. Combining standard properties of Rademacher complexity with the convenient structure of this class, we are able to (i) obtain novel bounds based on the finite fractal dimension, which (a) extend previous fractal dimension-type bounds from continuous to finite hypothesis classes, and (b) avoid a mutual information term that was required in prior work; (ii) we greatly simplify the proof of a recent dimension-independent generalization bound for stochastic gradient descent; and (iii) we easily recover results for VC classes and compression schemes, similar to approaches based on conditional mutual information.

Sarah Sachs, Hedi Hadiji, Tim van Erven et al.

Stochastic and adversarial data are two widely studied settings in online learning. But many optimization tasks are neither i.i.d. nor fully adversarial, which makes it of fundamental interest to get a better theoretical understanding of the world between these extremes. In this work we establish novel regret bounds for online convex optimization in a setting that interpolates between stochastic i.i.d. and fully adversarial losses. By exploiting smoothness of the expected losses, these bounds replace a dependence on the maximum gradient length by the variance of the gradients, which was previously known only for linear losses. In addition, they weaken the i.i.d. assumption by allowing, for example, adversarially poisoned rounds, which were previously considered in the related expert and bandit settings. In the fully i.i.d. case, our regret bounds match the rates one would expect from results in stochastic acceleration, and we also recover the optimal stochastically accelerated rates via online-to-batch conversion. In the fully adversarial case our bounds gracefully deteriorate to match the minimax regret. We further provide lower bounds showing that our regret upper bounds are tight for all intermediate regimes in terms of the stochastic variance and the adversarial variation of the loss gradients.

Hédi Hadiji, Sarah Sachs, Cristóbal Guzmán

Tracking the solution of time-varying variational inequalities is an important problem with applications in game theory, optimization, and machine learning. Existing work considers time-varying games or time-varying optimization problems. For strongly convex optimization problems or strongly monotone games, these results provide tracking guarantees under the assumption that the variation of the time-varying problem is restrained, that is, problems with a sublinear solution path. In this work we extend existing results in two ways: In our first result, we provide tracking bounds for (1) variational inequalities with a sublinear solution path but not necessarily monotone functions, and (2) for periodic time-varying variational inequalities that do not necessarily have a sublinear solution path-length. Our second main contribution is an extensive study of the convergence behavior and trajectory of discrete dynamical systems of periodic time-varying VI. We show that these systems can exhibit provably chaotic behavior or can converge to the solution. Finally, we illustrate our theoretical results with experiments.

Sarah Sachs, Hédi Hadiji, Tim van Erven et al.

Stochastic and adversarial data are two widely studied settings in online learning. But many optimization tasks are neither i.i.d. nor fully adversarial, which makes it of fundamental interest to get a better theoretical understanding of the world between these extremes. In this work we establish novel regret bounds for online convex optimization in a setting that interpolates between stochastic i.i.d. and fully adversarial losses. By exploiting smoothness of the expected losses, these bounds replace a dependence on the maximum gradient length by the variance of the gradients, which was previously known only for linear losses. In addition, they weaken the i.i.d. assumption by allowing, for example, adversarially poisoned rounds, which were previously considered in the expert and bandit setting. Our results extend this to the online convex optimization framework. In the fully i.i.d. case, our bounds match the rates one would expect from results in stochastic acceleration, and in the fully adversarial case they gracefully deteriorate to match the minimax regret. We further provide lower bounds showing that our regret upper bounds are tight for all intermediate regimes in terms of the stochastic variance and the adversarial variation of the loss gradients.