Interactive Combinatorial Bandits: Balancing Competitivity and Complementarity

We study non-modular function maximization in the online interactive bandit setting. We are motivated by applications where there is a natural complementarity between certain elements: e.g., in a movie recommendation system, watching the first movie in a series complements the experience of watching a second (and a third, etc.). This is not expressible using only submodular functions which can represent only competitiveness between elements. We extend the purely submodular approach in two ways. First, we assume that the objective can be decomposed into the sum of monotone suBmodular and suPermodular function, known as a BP objective. Here, complementarity is naturally modeled by the supermodular component. We develop a UCB-style algorithm, where at each round a noisy gain is revealed after an action is taken that balances refining beliefs about the unknown objectives (exploration) and choosing actions that appear promising (exploitation). Defining regret in terms of submodular and supermodular curvature with respect to a full-knowledge greedy baseline, we show that this algorithm achieves at most $O(\sqrt{T})$ regret after $T$ rounds of play. Second, for those functions that do not admit a BP structure, we provide analogous regret guarantees in terms of their submodularity ratio; this is applicable for functions that are almost, but not quite, submodular. We numerically study the tasks of movie recommendation on the MovieLens dataset, and selection of training subsets for classification. Through these examples, we demonstrate the algorithm's performance as well as the shortcomings of viewing these problems as being solely submodular.

Towards Sample-efficient Overparameterized Meta-learning

An overarching goal in machine learning is to build a generalizable model with few samples. To this end, overparameterization has been the subject of immense interest to explain the generalization ability of deep nets even when the size of the dataset is smaller than that of the model. While the prior literature focuses on the classical supervised setting, this paper aims to demystify overparameterization for meta-learning. Here we have a sequence of linear-regression tasks and we ask: (1) Given earlier tasks, what is the optimal linear representation of features for a new downstream task? and (2) How many samples do we need to build this representation? This work shows that surprisingly, overparameterization arises as a natural answer to these fundamental meta-learning questions. Specifically, for (1), we first show that learning the optimal representation coincides with the problem of designing a task-aware regularization to promote inductive bias. We leverage this inductive bias to explain how the downstream task actually benefits from overparameterization, in contrast to prior works on few-shot learning. For (2), we develop a theory to explain how feature covariance can implicitly help reduce the sample complexity well below the degrees of freedom and lead to small estimation error. We then integrate these findings to obtain an overall performance guarantee for our meta-learning algorithm. Numerical experiments on real and synthetic data verify our insights on overparameterized meta-learning.

Multiplayer Performative Prediction: Learning in Decision-Dependent Games

Learning problems commonly exhibit an interesting feedback mechanism wherein the population data reacts to competing decision makers' actions. This paper formulates a new game theoretic framework for this phenomenon, called multi-player performative prediction. We focus on two distinct solution concepts, namely (i) performatively stable equilibria and (ii) Nash equilibria of the game. The latter equilibria are arguably more informative, but can be found efficiently only when the game is monotone. We show that under mild assumptions, the performatively stable equilibria can be found efficiently by a variety of algorithms, including repeated retraining and repeated (stochastic) gradient play. We then establish transparent sufficient conditions for strong monotonicity of the game and use them to develop algorithms for finding Nash equilibria. We investigate derivative free methods and adaptive gradient algorithms wherein each player alternates between learning a parametric description of their distribution and gradient steps on the empirical risk. Synthetic and semi-synthetic numerical experiments illustrate the results.

Classification and Adversarial examples in an Overparameterized Linear Model: A Signal Processing Perspective

State-of-the-art deep learning classifiers are heavily overparameterized with respect to the amount of training examples and observed to generalize well on "clean" data, but be highly susceptible to infinitesmal adversarial perturbations. In this paper, we identify an overparameterized linear ensemble, that uses the "lifted" Fourier feature map, that demonstrates both of these behaviors. The input is one-dimensional, and the adversary is only allowed to perturb these inputs and not the non-linear features directly. We find that the learned model is susceptible to adversaries in an intermediate regime where classification generalizes but regression does not. Notably, the susceptibility arises despite the absence of model mis-specification or label noise, which are commonly cited reasons for adversarial-susceptibility. These results are extended theoretically to a random-Fourier-sum setup that exhibits double-descent behavior. In both feature-setups, the adversarial vulnerability arises because of a phenomenon we term spatial localization: the predictions of the learned model are markedly more sensitive in the vicinity of training points than elsewhere. This sensitivity is a consequence of feature lifting and is reminiscent of Gibb's and Runge's phenomena from signal processing and functional analysis. Despite the adversarial susceptibility, we find that classification with these features can be easier than the more commonly studied "independent feature" models.

Classification vs regression in overparameterized regimes: Does the loss function matter?

We compare classification and regression tasks in the overparameterized linear model with Gaussian features. On the one hand, we show that with sufficient overparameterization all training points are support vectors: solutions obtained by least-squares minimum-norm interpolation, typically used for regression, are identical to those produced by the hard-margin support vector machine (SVM) that minimizes the hinge loss, typically used for training classifiers. On the other hand, we show that there exist regimes where these solutions are near-optimal when evaluated by the 0-1 test loss function, but do not generalize if evaluated by the square loss function, i.e. they achieve the null risk. Our results demonstrate the very different roles and properties of loss functions used at the training phase (optimization) and the testing phase (generalization).