Learning Variational Inequalities from Data: Fast Generalization Rates under Strong Monotonicity

Variational inequalities (VIs) are a broad class of optimization problems encompassing machine learning problems ranging from standard convex minimization to more complex scenarios like min-max optimization and computing the equilibria of multi-player games. In convex optimization, strong convexity allows for fast statistical learning rates requiring only $\Theta(1/\epsilon)$ stochastic first-order oracle calls to find an $\epsilon$-optimal solution, rather than the standard $\Theta(1/\epsilon^2)$ calls. In this paper, we explain how one can similarly obtain fast $\Theta(1/\epsilon)$ rates for learning VIs that satisfy strong monotonicity, a generalization of strong convexity. Specifically, we demonstrate that standard stability-based generalization arguments for convex minimization extend directly to VIs when the domain admits a small covering, or when the operator is integrable and suboptimality is measured by potential functions; such as when finding equilibria in multi-player games.

Learning With Multi-Group Guarantees For Clusterable Subpopulations

A canonical desideratum for prediction problems is that performance guarantees should hold not just on average over the population, but also for meaningful subpopulations within the overall population. But what constitutes a meaningful subpopulation? In this work, we take the perspective that relevant subpopulations should be defined with respect to the clusters that naturally emerge from the distribution of individuals for which predictions are being made. In this view, a population refers to a mixture model whose components constitute the relevant subpopulations. We suggest two formalisms for capturing per-subgroup guarantees: first, by attributing each individual to the component from which they were most likely drawn, given their features; and second, by attributing each individual to all components in proportion to their relative likelihood of having been drawn from each component. Using online calibration as a case study, we study a \variational algorithm that provides guarantees for each of these formalisms by handling all plausible underlying subpopulation structures simultaneously, and achieve an $O(T^{1/2})$ rate even when the subpopulations are not well-separated. In comparison, the more natural cluster-then-predict approach that first recovers the structure of the subpopulations and then makes predictions suffers from a $O(T^{2/3})$ rate and requires the subpopulations to be separable. Along the way, we prove that providing per-subgroup calibration guarantees for underlying clusters can be easier than learning the clusters: separation between median subgroup features is required for the latter but not the former.

Algorithmic Content Selection and the Impact of User Disengagement

The content selection problem of digital services is often modeled as a decision-process where a service chooses, over multiple rounds, an arm to pull from a set of arms that each return a certain reward. This classical model does not account for the possibility that users disengage when dissatisfied and thus fails to capture an important trade-off between choosing content that promotes future engagement versus immediate reward. In this work, we introduce a model for the content selection problem where dissatisfied users may disengage and where the content that maximizes immediate reward does not necessarily maximize the odds of future user engagement. We show that when the relationship between each arm's expected reward and effect on user satisfaction are linearly related, an optimal content selection policy can be computed efficiently with dynamic programming under natural assumptions about the complexity of the users' engagement patterns. Moreover, we show that in an online learning setting where users with unknown engagement patterns arrive, there is a variant of Hedge that attains a $\tfrac 12$-competitive ratio regret bound. We also use our model to identify key primitives that determine how digital services should weigh engagement against revenue. For example, when it is more difficult for users to rejoin a service they are disengaged from, digital services naturally see a reduced payoff but user engagement may -- counterintuitively -- increase.

Truthfulness of Calibration Measures

We initiate the study of the truthfulness of calibration measures in sequential prediction. A calibration measure is said to be truthful if the forecaster (approximately) minimizes the expected penalty by predicting the conditional expectation of the next outcome, given the prior distribution of outcomes. Truthfulness is an important property of calibration measures, ensuring that the forecaster is not incentivized to exploit the system with deliberate poor forecasts. This makes it an essential desideratum for calibration measures, alongside typical requirements, such as soundness and completeness. We conduct a taxonomy of existing calibration measures and their truthfulness. Perhaps surprisingly, we find that all of them are far from being truthful. That is, under existing calibration measures, there are simple distributions on which a polylogarithmic (or even zero) penalty is achievable, while truthful prediction leads to a polynomial penalty. Our main contribution is the introduction of a new calibration measure termed the Subsampled Smooth Calibration Error (SSCE) under which truthful prediction is optimal up to a constant multiplicative factor.

Stacking as Accelerated Gradient Descent

Stacking, a heuristic technique for training deep residual networks by progressively increasing the number of layers and initializing new layers by copying parameters from older layers, has proven quite successful in improving the efficiency of training deep neural networks. In this paper, we propose a theoretical explanation for the efficacy of stacking: viz., stacking implements a form of Nesterov's accelerated gradient descent. The theory also covers simpler models such as the additive ensembles constructed in boosting methods, and provides an explanation for a similar widely-used practical heuristic for initializing the new classifier in each round of boosting. We also prove that for certain deep linear residual networks, stacking does provide accelerated training, via a new potential function analysis of the Nesterov's accelerated gradient method which allows errors in updates. We conduct proof-of-concept experiments to validate our theory as well.

The Sample Complexity of Multi-Distribution Learning for VC Classes

Multi-distribution learning is a natural generalization of PAC learning to settings with multiple data distributions. There remains a significant gap between the known upper and lower bounds for PAC-learnable classes. In particular, though we understand the sample complexity of learning a VC dimension d class on $k$ distributions to be $O(\epsilon^{-2} \ln(k)(d + k) + \min\{\epsilon^{-1} dk, \epsilon^{-4} \ln(k) d\})$, the best lower bound is $\Omega(\epsilon^{-2}(d + k \ln(k)))$. We discuss recent progress on this problem and some hurdles that are fundamental to the use of game dynamics in statistical learning.

A Unifying Perspective on Multi-Calibration: Unleashing Game Dynamics for Multi-Objective Learning

We provide a unifying framework for the design and analysis of multi-calibrated and moment-multi-calibrated predictors. Placing the multi-calibration problem in the general setting of \emph{multi-objective learning} -- where learning guarantees must hold simultaneously over a set of distributions and loss functions -- we exploit connections to game dynamics to obtain state-of-the-art guarantees for a diverse set of multi-calibration learning problems. In addition to shedding light on existing multi-calibration guarantees, and greatly simplifying their analysis, our approach yields a $1/\epsilon^2$ improvement in the number of oracle calls compared to the state-of-the-art algorithm of Jung et al. 2021 for learning deterministic moment-calibrated predictors and an exponential improvement in $k$ compared to the state-of-the-art algorithm of Gopalan et al. 2022 for learning a $k$-class multi-calibrated predictor. Beyond multi-calibration, we use these game dynamics to address existing and emerging considerations in the study of group fairness and multi-distribution learning.

On-Demand Sampling: Learning Optimally from Multiple Distributions

Social and real-world considerations such as robustness, fairness, social welfare and multi-agent tradeoffs have given rise to multi-distribution learning paradigms, such as collaborative, group distributionally robust, and fair federated learning. In each of these settings, a learner seeks to minimize its worst-case loss over a set of $n$ predefined distributions, while using as few samples as possible. In this paper, we establish the optimal sample complexity of these learning paradigms and give algorithms that meet this sample complexity. Importantly, our sample complexity bounds exceed that of the sample complexity of learning a single distribution only by an additive factor of $n \log(n) / \epsilon^2$. These improve upon the best known sample complexity of agnostic federated learning by Mohri et al. by a multiplicative factor of $n$, the sample complexity of collaborative learning by Nguyen and Zakynthinou by a multiplicative factor $\log n / \epsilon^3$, and give the first sample complexity bounds for the group DRO objective of Sagawa et al. To achieve optimal sample complexity, our algorithms learn to sample and learn from distributions on demand. Our algorithm design and analysis is enabled by our extensions of stochastic optimization techniques for solving stochastic zero-sum games. In particular, we contribute variants of Stochastic Mirror Descent that can trade off between players' access to cheap one-off samples or more expensive reusable ones.

ERMAS: Becoming Robust to Reward Function Sim-to-Real Gaps in Multi-Agent Simulations

Multi-agent simulations provide a scalable environment for learning policies that interact with rational agents. However, such policies may fail to generalize to the real-world where agents may differ from simulated counterparts due to unmodeled irrationality and misspecified reward functions. We introduce Epsilon-Robust Multi-Agent Simulation (ERMAS), a robust optimization framework for learning AI policies that are robust to such multiagent sim-to-real gaps. While existing notions of multi-agent robustness concern perturbations in the actions of agents, we address a novel robustness objective concerning perturbations in the reward functions of agents. ERMAS provides this robustness by anticipating suboptimal behaviors from other agents, formalized as the worst-case epsilon-equilibrium. We show empirically that ERMAS yields robust policies for repeated bimatrix games and optimal taxation problems in economic simulations. In particular, in the two-level RL problem posed by the AI Economist (Zheng et al., 2020) ERMAS learns tax policies that are robust to changes in agent risk aversion, improving social welfare by up to 15% in complex spatiotemporal simulations.

A unified Neural Network Approach to E-CommerceRelevance Learning

Result relevance scoring is critical to e-commerce search user experience. Traditional information retrieval methods focus on keyword matching and hand-crafted or counting-based numeric features, with limited understanding of item semantic relevance. We describe a highly-scalable feed-forward neural model to provide relevance score for (query, item) pairs, using only user query and item title as features, and both user click feedback as well as limited human ratings as labels. Several general enhancements were applied to further optimize eval/test metrics, including Siamese pairwise architecture, random batch negative co-training, and point-wise fine-tuning. We found significant improvement over GBDT baseline as well as several off-the-shelf deep-learning baselines on an independently constructed ratings dataset. The GBDT model relies on 10 times more features. We also present metrics for select subset combinations of techniques mentioned above.