Swap regret, a generic performance measure of online decision-making algorithms, plays an important role in the theory of repeated games, along with a close connection to correlated equilibria in strategic games. This paper shows an $\Omega( \sqrt{T N\log{N}})$-lower bound for swap regret, where $T$ and $N$ denote the numbers of time steps and available actions, respectively. Our lower bound is tight up to a constant, and resolves an open problem mentioned, e.g., in the book by Nisan et al. (2007). Besides, we present a computationally efficient reduction method that converts no-external-regret algorithms to no-swap-regret algorithms. This method can be applied not only to the full-information setting but also to the bandit setting and provides a better regret bound than previous results.

We develop a rigorous mathematical analysis of zero-shot learning with attributes. In this setting, the goal is to label novel classes with no training data, only detectors for attributes and a description of how those attributes are correlated with the target classes, called the class-attribute matrix. We develop the first non-trivial lower bound on the worst-case error of the best map from attributes to classes for this setting, even with perfect attribute detectors. The lower bound characterizes the theoretical intrinsic difficulty of the zero-shot problem based on the available information---the class-attribute matrix---and the bound is practically computable from it. Our lower bound is tight, as we show that we can always find a randomized map from attributes to classes whose expected error is upper bounded by the value of the lower bound. We show that our analysis can be predictive of how standard zero-shot methods behave in practice, including which classes will likely be confused with others.

We develop a rigorous mathematical analysis of zero-shot learning with attributes. In this setting, the goal is to label novel classes with no training data, only detectors for attributes and a description of how those attributes are correlated with the target classes, called the class-attribute matrix. We develop the first non-trivial lower bound on the worst-case error of the best map from attributes to classes for this setting, even with perfect attribute detectors. The lower bound characterizes the theoretical intrinsic difficulty of the zero-shot problem based on the available information---the class-attribute matrix---and the bound is practically computable from it. Our lower bound is tight, as we show that we can always find a randomized map from attributes to classes whose expected error is upper bounded by the value of the lower bound.

Performative prediction is a framework accounting for the shift in the data distribution induced by the prediction of a model deployed in the real world. Ensuring rapid convergence to a stable solution where the data distribution remains the same after the model deployment is crucial, especially in evolving environments. This paper extends the Repeated Risk Minimization (RRM) framework by utilizing historical datasets from previous retraining snapshots, yielding a class of algorithms that we call Affine Risk Minimizers and enabling convergence to a performatively stable point for a broader class of problems. We introduce a new upper bound for methods that use only the final iteration of the dataset and prove for the first time the tightness of both this new bound and the previous existing bounds within the same regime. We also prove that utilizing historical datasets can surpass the lower bound for last iterate RRM, and empirically observe faster convergence to the stable point on various perfor-mative prediction benchmarks. We offer at the same time the first lower bound analysis for RRM within the class of Affine Risk Min-imizers, quantifying the potential improvements in convergence speed that could be achieved with other variants in our framework.

Swap regret, a generic performance measure of online decision-making algorithms, plays an important role in the theory of repeated games, along with a close connection to correlated equilibria in strategic games. This paper shows an \Omega( \sqrt{T N\log{N}}) -lower bound for swap regret, where T and N denote the numbers of time steps and available actions, respectively. Our lower bound is tight up to a constant, and resolves an open problem mentioned, e.g., in the book by Nisan et al. (2007). Besides, we present a computationally efficient reduction method that converts no-external-regret algorithms to no-swap-regret algorithms. This method can be applied not only to the full-information setting but also to the bandit setting and provides a better regret bound than previous results.

Leveraging algorithmic stability to derive sharp generalization bounds is a classic and powerful approach in learning theory. Since Vapnik and Chervonenkis [1974] first formalized the idea for analyzing SVMs, it has been utilized to study many fundamental learning algorithms (e.g., $k$-nearest neighbors [Rogers and Wagner, 1978], stochastic gradient method [Hardt et al., 2016], linear regression [Maurer, 2017], etc). In a recent line of great works by Feldman and Vondrak [2018, 2019] as well as Bousquet et al. [2020b], they prove a high probability generalization upper bound of order $\widetilde{\mathcal{O}}(\gamma +\frac{L}{\sqrt{n}})$ for any uniformly $\gamma$-stable algorithm and $L$-bounded loss function. Although much progress was achieved in proving generalization upper bounds for stable algorithms, our knowledge of lower bounds is rather limited. In fact, there is no nontrivial lower bound known ever since the study on uniform stability began [Bousquet and Elisseeff, 2002], to the best of our knowledge. In this paper we fill the gap by proving a tight generalization lower bound of order $\Omega(\gamma+\frac{L}{\sqrt{n}})$, which matches the best known upper bound up to logarithmic factors

The Combinatorial Multi-Armed Bandit problem is a sequential decision-making problem in which an agent selects a set of arms on each round, observes feedback for each of these arms and aims to maximize a known reward function of the arms it chose. While previous work proved regret upper bounds in this setting for general reward functions, only a few works provided matching lower bounds, all for specific reward functions. In this work, we prove regret lower bounds for combinatorial bandits that hold under mild assumptions for all smooth reward functions. We derive both problem-dependent and problem-independent bounds and show that the recently proposed Giniweighted smoothness parameter (Merlis and Mannor, 2019) also determines the lower bounds for monotone reward functions. Notably, this implies that our lower bounds are tight up to log-factors.

The homology groups of a manifold are important topological invariants that provide an algebraic summary of the manifold. These groups contain rich topological information, for instance, about the connected components, holes, tunnels and sometimes the dimension of the manifold. In earlier work, we have considered the statistical problem of estimating the homology of a manifold from noiseless samples and from noisy samples under several different noise models. We derived upper and lower bounds on the minimax risk for this problem. In this note we revisit the noiseless case. In previous work we used Le Cam's lemma to establish a lower bound that differed from the upper bound of Niyogi, Smale and Weinberger by a polynomial factor in the condition number. In this note we use a different construction based on the direct analysis of the likelihood ratio test to show that the upper bound of Niyogi, Smale and Weinberger is in fact tight, thus establishing rate optimal asymptotic minimax bounds for the problem. The techniques we use here extend in a straightforward way to the noisy settings considered in our earlier work.