What makes a classifier have the ability to generalize? There have been a lot of important attempts to address this question, but a clear answer is still elusive. Proponents of complexity theory find that the complexity of the classifier's function space is key to deciding generalization, whereas other recent work reveals that classifiers which extract invariant feature representations are likely to generalize better. Recent theoretical and empirical studies, however, have shown that even within a classifier's function space, there can be significant differences in the ability to generalize. Specifically, empirical studies have shown that among functions which have a good training data fit, functions with lower Kolmogorov complexity (KC) are likely to generalize better, while the opposite is true for functions of higher KC.

The Empirical Revenue Maximization (ERM) is one of the most important price learning algorithms in auction design: as the literature shows it can learn approximately optimal reserve prices for revenue-maximizing auctioneers in both repeated auctions and uniform-price auctions. However, in these applications the agents who provide inputs to ERM have incentives to manipulate the inputs to lower the outputted price. We generalize the definition of an incentive-awareness measure proposed by Lavi et al (2019), to quantify the reduction of ERM's outputted price due to a change of m 1 out of N input samples, and provide specific convergence rates of this measure to zero as N goes to infinity for different types of input distributions. By adopting this measure, we construct an efficient, approximately incentive-compatible, and revenue-optimal learning algorithm using ERM in repeated auctions against non-myopic bidders, and show approximate group incentive-compatibility in uniform-price auctions.

We study the fundamental problem of sequential probability assignment, also known as online learning with logarithmic loss, with respect to an arbitrary, possibly nonparametric hypothesis class. Our goal is to obtain a complexity measure for the hypothesis class that characterizes the minimax regret and to determine a general, minimax optimal algorithm.

The maximum a posteriori (MAP) inference for determinantal point processes (DPPs) is crucial for selecting diverse items in many machine learning applications. Although DPP MAP inference is NP-hard, the greedy algorithm often finds highquality solutions, and many researchers have studied its efficient implementation. One classical and practical method is the lazy greedy algorithm, which is applicable to general submodular function maximization, while a recent fast greedy algorithm based on the Cholesky factorization is more efficient for DPP MAP inference. This paper presents how to combine the ideas of "lazy" and "fast", which have been considered incompatible in the literature. Our lazy and fast greedy algorithm achieves almost the same time complexity as the current best one and runs faster in practice. The idea of "lazy + fast" is extendable to other greedy-type algorithms. We also give a fast version of the double greedy algorithm for unconstrained DPP MAP inference.

Multiple Instance Learning (MIL) has been increasingly adopted to mitigate the high costs and complexity associated with labeling individual instances, learning instead from bags of instances labeled at the bag level and enabling instance-level labeling. While existing research has primarily focused on the learnability of MIL at the bag level, there is an absence of theoretical exploration to check if a given MIL algorithm is learnable at the instance level. This paper proposes a theoretical framework based on probably approximately correct (PAC) learning theory to assess the instance-level learnability of deep multiple instance learning (Deep MIL) algorithms. Our analysis exposes significant gaps between current Deep MIL algorithms, highlighting the theoretical conditions that must be satisfied by MIL algorithms to ensure instance-level learnability. With these conditions, we interpret the learnability of the representative Deep MIL algorithms and validate them through empirical studies.

We address a generalization of the bandit with knapsacks problem, where a learner aims to maximize rewards while satisfying an arbitrary set of long-term constraints. Our goal is to design best-of-both-worlds algorithms that perform optimally under both stochastic and adversarial constraints. Previous works address this problem via primal-dual methods, and require some stringent assumptions, namely the Slater's condition, and in adversarial settings, they either assume knowledge of a lower bound on the Slater's parameter, or impose strong requirements on the primal and dual regret minimizers such as requiring weak adaptivity. We propose an alternative and more natural approach based on optimistic estimations of the constraints. Surprisingly, we show that estimating the constraints with an UCBlike approach guarantees optimal performances.

We study problem-dependent rates, i.e., generalization errors that scale tightly with the variance or the effective loss at the "best hypothesis." Existing uniform convergence and localization frameworks, the most widely used tools to study this problem, often fail to simultaneously provide parameter localization and optimal dependence on the sample size. As a result, existing problem-dependent rates are often rather weak when the hypothesis class is "rich" and the worst-case bound of the loss is large. In this paper we propose a new framework based on a "uniform localized convergence" principle. We provide the first (moment-penalized) estimator that achieves the optimal variance-dependent rate for general "rich" classes; we also establish improved loss-dependent rate for standard empirical risk minimization.

The latter allows our approach to subsume the so-called geometrical compressibility as a special case.

We study a generalization of boosting to the multiclass setting. We introduce a weak learning condition for multiclass classification that captures the original notion of weak learnability as being "slightly better than random guessing". We give a simple and efficient boosting algorithm, that does not require realizability assumptions and its sample and oracle complexity bounds are independent of the number of classes. In addition, we utilize our new boosting technique in several theoretical applications within the context of List PAC Learning. First, we establish an equivalence to weak PAC learning. Furthermore, we present a new result on boosting for list learners, as well as provide a novel proof for the characterization of multiclass PAC learning and List PAC learning. Notably, our technique gives rise to a simplified analysis, and also implies an improved error bound for large list sizes, compared to previous results.

Graph classification is a challenging problem owing to the difficulty in quantifying the similarity between graphs or representing graphs as vectors, though there have been a few methods using graph kernels or graph neural networks (GNNs). Graph kernels often suffer from computational costs and manual feature engineering, while GNNs commonly utilize global pooling operations, risking the loss of structural or semantic information. This work introduces Graph Reference Distribution Learning (GRDL), an efficient and accurate graph classification method. GRDL treats each graph's latent node embeddings given by GNN layers as a discrete distribution, enabling direct classification without global pooling, based on maximum mean discrepancy to adaptively learned reference distributions. To fully understand this new model (the existing theories do not apply) and guide its configuration (e.g., network architecture, references' sizes, number, and regularization) for practical use, we derive generalization error bounds for GRDL and verify them numerically.