NeurIPS 2023 Papers

We consider the learning of a single-index target function $f_*: \mathbb{R}^d\to\mathbb{R}$ under spiked covariance data: $$f_*(\boldsymbol{x}) = \textstyle\sigma_*(\frac{1}{\sqrt{1+\theta}}\langle\boldsymbol{x},\boldsymbol{\mu}\rangle), ~~ \boldsymbol{x}\overset{\small\mathrm{i.i.d.}}{\sim}\mathcal{N}(0,\boldsymbol{I_d} + \theta\boldsymbol{\mu}\boldsymbol{\mu}^\top), ~~ \theta\asymp d^{\beta} \text{ for } \beta\in[0,1), $$ where the link function $\sigma_*:\mathbb{R}\to\mathbb{R}$ is a degree-$p$ polynomial with information exponent $k$ (defined as the lowest degree in the Hermite expansion of $\sigma_*$), and it depends on the projection of input $\boldsymbol{x}$ onto the spike (signal) direction $\boldsymbol{\mu}\in\mathbb{R}^d$. In the proportional asymptotic limit where the number of training examples $n$ and the dimensionality $d$ jointly diverge: $n,d\to\infty, n/d\to\psi\in(0,\infty)$, we ask the following question: how large should the spike magnitude $\theta$ (i.e., the strength of the low-dimensional component) be, in order for $(i)$ kernel methods, $(ii)$ neural networks optimized by gradient descent, to learn $f_*$? We show that for kernel ridge regression, $\beta\ge 1-\frac{1}{p}$ is both sufficient and necessary. Whereas for two-layer neural networks trained with gradient descent, $\beta>1-\frac{1}{k}$ suffices. Our results demonstrate that both kernel methods and neural networks benefit from low-dimensional structures in the data. Further, since $k\le p$ by definition, neural networks can adapt to such structures more effectively.

Hedonic Games (HGs) are a classical framework modeling coalition formation of strategic agents guided by their individual preferences. According to these preferences, it is desirable that a coalition structure (i.e. a partition of agents into coalitions) satisfies some form of stability. The most well-known and natural of such notions is arguably core-stability. Informally, a partition is core-stable if no subset of agents would like to deviate by regrouping in a so-called core-blocking coalition. Unfortunately, core-stable partitions seldom exist and even when they do, it is often computationally intractable to find one. To circumvent these problems, we propose the notion of $\varepsilon$-fractional core-stability, where at most an $\varepsilon$-fraction of all possible coalitions is allowed to core-block. It turns out that such a relaxation may guarantee both existence and polynomial-time computation. Specifically, we design efficient algorithms returning an $\varepsilon$-fractional core-stable partition, with $\varepsilon$ exponentially decreasing in the number of agents, for two fundamental classes of HGs: Simple Fractional and Anonymous. From a probabilistic point of view, being the definition of $\varepsilon$-fractional core equivalent to requiring that uniformly sampled coalitions core-block with probability lower than $\varepsilon$, we further extend the definition to handle more complex sampling distributions. Along this line, when valuations have to be learned from samples in a PAC-learning fashion, we give positive and negative results on which distributions allow the efficient computation of outcomes that are $\varepsilon$-fractional core-stable with arbitrarily high confidence.

We study a two-stage scenario for learning to defer with multiple experts, which is crucial in practice for many applications. In this scenario, a predictor is derived in a first stage by training with a common loss function such as cross-entropy. In the second stage, a deferral function is learned to assign the most suitable expert to each input. We design a new family of surrogate loss functions for this scenario both in the score-based and the predictor-rejector settings and prove that they are supported by $H$-consistency bounds, which implies their Bayes-consistency. Moreover, we show that, for a constant cost function, our two-stage surrogate losses are realizable $H$-consistent. While the main focus of this work is a theoretical analysis, we also report the results of several experiments on CIFAR-10 and SVHN datasets.

We propose a general architecture that combines the coefficient learning scheme with a residual operator layer for learning mappings between continuous functions in the 3D Euclidean space. Our proposed model is guaranteed to achieve SE(3)-equivariance by design. From the graph spectrum view, our method can be interpreted as convolution on graphons (dense graphs with infinitely many nodes), which we term InfGCN. By leveraging both the continuous graphon structure and the discrete graph structure of the input data, our model can effectively capture the geometric information while preserving equivariance. Through extensive experiments on large-scale electron density datasets, we observed that our model significantly outperformed the current state-of-the-art architectures. Multiple ablation studies were also carried out to demonstrate the effectiveness of the proposed architecture.

Neural networks are capable of translating between languages—in some cases even between two languages where there is little or no access to parallel translations, in what is known as Unsupervised Machine Translation (UMT). Given this progress, it is intriguing to ask whether machine learning tools can ultimately enable understanding animal communication, particularly that of highly intelligent animals. We propose a theoretical framework for analyzing UMT when no parallel translations are available and when it cannot be assumed that the source and target corpora address related subject domains or posses similar linguistic structure. We exemplify this theory with two stylized models of language, for which our framework provides bounds on necessary sample complexity; the bounds are formally proven and experimentally verified on synthetic data. These bounds show that the error rates are inversely related to the language complexity and amount of common ground. This suggests that unsupervised translation of animal communication may be feasible if the communication system is sufficiently complex.

We focus on Continual Meta-Learning (CML), which targets accumulating and exploiting meta-knowledge on a sequence of non-i.i.d. tasks. The primary challenge is to strike a balance between stability and plasticity, where a model should be stable to avoid catastrophic forgetting in previous tasks and plastic to learn generalizable concepts from new tasks. To address this, we formulate the CML objective as controlling the average excess risk upper bound of the task sequence, which reflects the trade-off between forgetting and generalization. Based on the objective, we introduce a unified theoretical framework for CML in both static and shifting environments, providing guarantees for various task-specific learning algorithms. Moreover, we first present a rigorous analysis of a bi-level trade-off in shifting environments. To approach the optimal trade-off, we propose a novel algorithm that dynamically adjusts the meta-parameter and its learning rate w.r.t environment change. Empirical evaluations on synthetic and real datasets illustrate the effectiveness of the proposed theory and algorithm.

Collaborative learning techniques have significantly advanced in recent years, enabling private model training across multiple organizations. Despite this opportunity, firms face a dilemma when considering data sharing with competitors—while collaboration can improve a company’s machine learning model, it may also benefit competitors and hence reduce profits. In this work, we introduce a general framework for analyzing this data-sharing trade-off. The framework consists of three components, representing the firms’ production decisions, the effect of additional data on model quality, and the data-sharing negotiation process, respectively. We then study an instantiation of the framework, based on a conventional market model from economic theory, to identify key factors that affect collaboration incentives. Our findings indicate a profound impact of market conditions on the data-sharing incentives. In particular, we find that reduced competition, in terms of the similarities between the firms’ products, and harder learning tasks foster collaboration.

The task of tuning regularization coefficients in regularized regression models with provable guarantees across problem instances still poses a significant challenge in the literature. This paper investigates the sample complexity of tuning regularization parameters in linear and logistic regressions under $\ell_1$ and $\ell_2$-constraints in the data-driven setting. For the linear regression problem, by more carefully exploiting the structure of the dual function class, we provide a new upper bound for the pseudo-dimension of the validation loss function class, which significantly improves the best-known results on the problem. Remarkably, we also instantiate the first matching lower bound, proving our results are tight. For tuning the regularization parameters of logistic regression, we introduce a new approach to studying the learning guarantee via an approximation of the validation loss function class. We examine the pseudo-dimension of the approximation class and construct a uniform error bound between the validation loss function class and its approximation, which allows us to instantiate the first learning guarantee for the problem of tuning logistic regression regularization coefficients.

Human perception of the empirical world involves recognizing the diverse appearances, or 'modalities', of underlying objects. Despite the longstanding consideration of this perspective in philosophy and cognitive science, the study of multimodality remains relatively under-explored within the field of machine learning. Nevertheless, current studies of multimodal machine learning are limited to empirical practices, lacking theoretical foundations beyond heuristic arguments. An intriguing finding from the practice of multimodal learning is that a model trained on multiple modalities can outperform a finely-tuned unimodal model, even on unimodal tasks. This paper provides a theoretical framework that explains this phenomenon, by studying generalization properties of multimodal learning algorithms. We demonstrate that multimodal learning allows for a superior generalization bound compared to unimodal learning, up to a factor of $O(\sqrt{n})$, where $n$ represents the sample size. Such advantage occurs when both connection and heterogeneity exist between the modalities.

We examine the relationship between the mutual information between the output model and the empirical sample and the algorithm's generalization in the context of stochastic convex optimization. Despite increasing interest in information-theoretic generalization bounds, it is uncertain if these bounds can provide insight into the exceptional performance of various learning algorithms. Our study of stochastic convex optimization reveals that, for true risk minimization, dimension-dependent mutual information is necessary. This indicates that existing information-theoretic generalization bounds fall short in capturing the generalization capabilities of algorithms like SGD and regularized ERM, which have dimension-independent sample complexity.

Recently, operator learning, or learning mappings between infinite-dimensional function spaces, has garnered significant attention, notably in relation to learning partial differential equations from data. Conceptually clear when outlined on paper, neural operators necessitate discretization in the transition to computer implementations. This step can compromise their integrity, often causing them to deviate from the underlying operators. This research offers a fresh take on neural operators with a framework Representation equivalent Neural Operators (ReNO) designed to address these issues. At its core is the concept of operator aliasing, which measures inconsistency between neural operators and their discrete representations. We explore this for widely-used operator learning techniques. Our findings detail how aliasing introduces errors when handling different discretizations and grids and loss of crucial continuous structures. More generally, this framework not only sheds light on existing challenges but, given its constructive and broad nature, also potentially offers tools for developing new neural operators.

We initiate the mathematical study of replicability as an algorithmic property in the context of reinforcement learning (RL). We focus on the fundamental setting of discounted tabular MDPs with access to a generative model. Inspired by Impagliazzo et al. [2022], we say that an RL algorithm is replicable if, with high probability, it outputs the exact same policy after two executions on i.i.d. samples drawn from the generator when its internal randomness is the same. We first provide an efficient $\rho$-replicable algorithm for $(\varepsilon, \delta)$-optimal policy estimation with sample and time complexity $\widetilde O\left(\frac{N^3\cdot\log(1/\delta)}{(1-\gamma)^5\cdot\varepsilon^2\cdot\rho^2}\right)$, where $N$ is the number of state-action pairs. Next, for the subclass of deterministic algorithms, we provide a lower bound of order $\Omega\left(\frac{N^3}{(1-\gamma)^3\cdot\varepsilon^2\cdot\rho^2}\right)$. Then, we study a relaxed version of replicability proposed by Kalavasis et al. [2023] called TV indistinguishability. We design a computationally efficient TV indistinguishable algorithm for policy estimation whose sample complexity is $\widetilde O\left(\frac{N^2\cdot\log(1/\delta)}{(1-\gamma)^5\cdot\varepsilon^2\cdot\rho^2}\right)$. At the cost of $\exp(N)$ running time, we transform these TV indistinguishable algorithms to $\rho$-replicable ones without increasing their sample complexity. Finally, we introduce the notion of approximate-replicability where we only require that two outputted policies are close under an appropriate statistical divergence (e.g., Renyi) and show an improved sample complexity of $\widetilde O\left(\frac{N\cdot\log(1/\delta)}{(1-\gamma)^5\cdot\varepsilon^2\cdot\rho^2}\right)$.

We design replicable algorithms in the context of statistical clustering under the recently introduced notion of replicability from Impagliazzo et al. [2022]. According to this definition, a clustering algorithm is replicable if, with high probability, its output induces the exact same partition of the sample space after two executions on different inputs drawn from the same distribution, when its internal randomness is shared across the executions. We propose such algorithms for the statistical $k$-medians, statistical $k$-means, and statistical $k$-centers problems by utilizing approximation routines for their combinatorial counterparts in a black-box manner. In particular, we demonstrate a replicable $O(1)$-approximation algorithm for statistical Euclidean $k$-medians ($k$-means) with $\operatorname{poly}(d)$ sample complexity. We also describe an $O(1)$-approximation algorithm with an additional $O(1)$-additive error for statistical Euclidean $k$-centers, albeit with $\exp(d)$ sample complexity. In addition, we provide experiments on synthetic distributions in 2D using the $k$-means++ implementation from sklearn as a black-box that validate our theoretical results.

We initiate the study of smoothed analysis for the sequential probability assignment problem with contexts. We study information-theoretically optimal minmax rates as well as a framework for algorithmic reduction involving the maximum likelihood estimator oracle. Our approach establishes a general-purpose reduction from minimax rates for sequential probability assignment for smoothed adversaries to minimax rates for transductive learning. This leads to optimal (logarithmic) fast rates for parametric classes and classes with finite VC dimension. On the algorithmic front, we develop an algorithm that efficiently taps into the MLE oracle, for general classes of functions. We show that under general conditions this algorithmic approach yields sublinear regret.

Cooperative communication plays a fundamental role in theories of human-human interaction--cognition, culture, development, language, etc.--as well as human-robot interaction. The core challenge in cooperative communication is the problem of common ground: having enough shared knowledge and understanding to successfully communicate. Prior models of cooperative communication, however, uniformly assume the strongest form of common ground, perfect and complete knowledge sharing, and, therefore, fail to capture the core challenge of cooperative communication. We propose a general theory of cooperative communication that is mathematically principled and explicitly defines a spectrum of common ground possibilities, going well beyond that of perfect and complete knowledge sharing, on spaces that permit arbitrary representations of data and hypotheses. Our framework is a strict generalization of prior models of cooperative communication. After considering a parametric form of common ground and viewing the data selection and hypothesis inference processes of communication as encoding and decoding, we establish a connection to variational autoencoding, a powerful model in modern machine learning. Finally, we carry out a series of empirical simulations to support and elaborate on our theoretical results.

The problem of designing learners that provide guarantees that their predictions are provably correct is of increasing importance in machine learning. However, learning theoretic guarantees have only been considered in very specific settings. In this work, we consider the design and analysis of reliable learners in challenging test-time environments as encountered in modern machine learning problems: namely adversarial test-time attacks (in several variations) and natural distribution shifts. In this work, we provide a reliable learner with provably optimal guarantees in such settings. We discuss computationally feasible implementations of the learner and further show that our algorithm achieves strong positive performance guarantees on several natural examples: for example, linear separators under log-concave distributions or smooth boundary classifiers under smooth probability distributions.