ICML 2023 Papers

Neural networks constitute a class of functions that are typically non-surjective, with high-dimensional fibers and complicated image. We prove two main results concerning the geometry of the loss landscape of a neural network. First, we provide an explicit effective bound on the sizes of the hidden layers so that the loss landscape has no spurious valleys, which guarantees the success of gradient descent methods. Second, we present a novel method for analyzing whether a given neural network architecture with monomial activation function can represent a target function of interest. The core of our analysis method is the study of a specific set of error values, and its behavior depending on different training datasets.

We develop an analytical framework to characterize the set of optimal ReLU neural networks by reformulating the non-convex training problem as a convex program. We show that the global optima of the convex parameterization are given by a polyhedral set and then extend this characterization to the optimal set of the non-convex training objective. Since all stationary points of the ReLU training problem can be represented as optima of sub-sampled convex programs, our work provide a general expression for all critical points of the non-convex objective. We then leverage our results to provide an optimal pruning algorithm for computing minimal networks, establish conditions for the regularization path of ReLU networks to be continuous, and develop sensitivity results for minimal ReLU networks.

We study variants of the online linear optimization (OLO) problem with bandit feedback, where the algorithm has access to external information about the unknown cost vector. Our motivation is the recent body of work on using such ``hints'' towards improving regret bounds for OLO problems in the full-information setting. Unlike in the full-information OLO setting, with bandit feedback, we first show that one cannot improve the standard regret bounds of $\tilde{O}(\sqrt{T})$ by using hints, even if they are always well-correlated with the cost vector. In contrast, if the algorithm is empowered to issue queries and if all the responses are correct, then we show $O(\log T)$ regret is achievable. We then show how to make this result more robust---when some of the query responses can be adversarial---by using a little feedback on the quality of the responses.

We consider a deep matrix factorization model of covariance matrices trained with the Bures-Wasserstein distance. While recent works have made advances in the study of the optimization problem for overparametrized low-rank matrix approximation, much emphasis has been placed on discriminative settings and the square loss. In contrast, our model considers another type of loss and connects with the generative setting. We characterize the critical points and minimizers of the Bures-Wasserstein distance over the space of rank-bounded matrices. The Hessian of this loss at low-rank matrices can theoretically blow up, which creates challenges to analyze convergence of gradient optimization methods. We establish convergence results for gradient flow using a smooth perturbative version of the loss as well as convergence results for finite step size gradient descent under certain assumptions on the initial weights.

Understanding the neural implementation of complex human behaviors is one of the major goals in neuroscience. To this end, it is crucial to find a true representation of the neural data, which is challenging due to the high complexity of behaviors and the low signal-to-ratio (SNR) of the signals. Here, we propose a novel unsupervised learning framework, Neural Latent Aligner (NLA), to find well-constrained, behaviorally relevant neural representations of complex behaviors. The key idea is to align representations across repeated trials to learn cross-trial consistent information. Furthermore, we propose a novel, fully differentiable time warping model (TWM) to resolve the temporal misalignment of trials. When applied to intracranial electrocorticography (ECoG) of natural speaking, our model learns better representations for decoding behaviors than the baseline models, especially in lower dimensional space. The TWM is empirically validated by measuring behavioral coherence between aligned trials. The proposed framework learns more cross-trial consistent representations than the baselines, and when visualized, the manifold reveals shared neural trajectories across trials.

In this work, we propose introduce a variant of online stochastic gradient descent and prove it converges to Nash equilibria and simultaneously it has sublinear regret for the class of congestion games in the semi-bandit feedback setting. Our proposed method admits convergence rates depending only polynomially on the number of players and the number of facilities, but not on the size of the action set, which can be exponentially large in terms of the number of facilities. Moreover, the running time of our method has polynomial-time dependence on the implicit description of the game. Our analysis exploits techniques from convex geometry, in particular Caratheodory's theorem and recent advances in non-convex stochastic optimization. This work improves upon and answers an open question from (Cui et al 2022).

Kernelized Stein discrepancy (KSD) is a score-based discrepancy widely used in goodness-of-fit tests. It can be applied even when the target distribution has an unknown normalising factor, such as in Bayesian analysis. We show theoretically and empirically that the KSD test can suffer from low power when the target and the alternative distributions have the same well-separated modes but differ in mixing proportions. We propose to perturb the observed sample via Markov transition kernels, with respect to which the target distribution is invariant. This allows us to then employ the KSD test on the perturbed sample. We provide numerical evidence that with suitably chosen transition kernels the proposed approach can lead to substantially higher power than the KSD test.

The computational study of equilibria involving constraints on players' strategies has been largely neglected. However, in real-world applications, players are usually subject to constraints ruling out the feasibility of some of their strategies, such as, e.g., safety requirements and budget caps. Computational studies on constrained versions of the Nash equilibrium have lead to some results under very stringent assumptions, while finding constrained versions of the correlated equilibrium (CE) is still unexplored. In this paper, we introduce and computationally characterize constrained Phi-equilibria---a more general notion than constrained CEs---in normal-form games. We show that computing such equilibria is in general computationally intractable, and also that the set of the equilibria may not be convex, providing a sharp divide with unconstrained CEs. Nevertheless, we provide a polynomial-time algorithm for computing a constrained (approximate) Phi-equilibrium maximizing a given linear function, when either the number of constraints or that of players' actions is fixed. Moreover, in the special case in which a player's constraints do not depend on other players' strategies, we show that an exact, function-maximizing equilibrium can be computed in polynomial time, while one (approximate) equilibrium can be found with an efficient decentralized no-regret learning algorithm.

In this paper, we consider the problem of Iterative Machine Teaching (IMT), where the teacher provides examples to the learner iteratively such that the learner can achieve fast convergence to a target model. However, existing IMT algorithms are solely based on parameterized families of target models. They mainly focus on convergence in the parameter space, resulting in difficulty when the target models are defined to be functions without dependency on parameters. To address such a limitation, we study a more general task -- Nonparametric Iterative Machine Teaching (NIMT), which aims to teach nonparametric target models to learners in an iterative fashion. Unlike parametric IMT that merely operates in the parameter space, we cast NIMT as a functional optimization problem in the function space. To solve it, we propose both random and greedy functional teaching algorithms. We obtain the iterative teaching dimension (ITD) of the random teaching algorithm under proper assumptions, which serves as a uniform upper bound of ITD in NIMT. Further, the greedy teaching algorithm has a significantly lower ITD, which reaches a tighter upper bound of ITD in NIMT. Finally, we verify the correctness of our theoretical findings with extensive experiments in nonparametric scenarios.

Prior knowledge and symbolic rules in machine learning are often expressed in the form of label constraints, especially in structured prediction problems. In this work, we compare two common strategies for encoding label constraints in a machine learning pipeline, *regularization with constraints* and *constrained inference*, by quantifying their impact on model performance. For regularization, we show that it narrows the generalization gap by precluding models that are inconsistent with the constraints. However, its preference for small violations introduces a bias toward a suboptimal model. For constrained inference, we show that it reduces the population risk by correcting a model's violation, and hence turns the violation into an advantage. Given these differences, we further explore the use of two approaches together and propose conditions for constrained inference to compensate for the bias introduced by regularization, aiming to improve both the model complexity and optimal risk.

Generalization is one of the most fundamental challenges in deep learning, aiming to predict model performances on unseen data. Empirically, such predictions usually rely on a validation set, while recent works showed that an unlabeled validation set also works. Without validation sets, it is extremely difficult to obtain non-vacuous generalization bounds, which leads to a weaker task of finding generalization measures that monotonically relate to generalization error. In this paper, we propose a new generalization measure REF Complexity (RElative Fitting degree between signal and noise), motivated by the intuition that a given model-algorithm pair may generalize well if it fits signal (e.g., true labels) fast while fitting noise (e.g., random labels) slowly. Empirically, REF Complexity monotonically relates to test accuracy in real-world datasets without accessing additional validation sets, achieving -0.988 correlation on CIFAR-10 and -0.960 correlation on CIFAR-100. We further theoretically verify the utility of REF Complexity under three different cases, including convex and smooth regimes with stochastic gradient descent, smooth regimes (not necessarily convex) with stochastic gradient Langevin dynamics, and linear regimes with gradient descent. The code is available at https://github.com/962086838/REF-complexity.

We study single-machine scheduling of jobs, each belonging to a job type that determines its duration distribution. We start by analyzing the scenario where the type characteristics are known and then move to two learning scenarios where the types are unknown: non-preemptive problems, where each started job must be completed before moving to another job; and preemptive problems, where job execution can be paused in the favor of moving to a different job. In both cases, we design algorithms that achieve sublinear excess cost, compared to the performance with known types, and prove lower bounds for the non-preemptive case. Notably, we demonstrate, both theoretically and through simulations, how preemptive algorithms can greatly outperform non-preemptive ones when the durations of different job types are far from one another, a phenomenon that does not occur when the type durations are known.

We consider the neural contextual bandit problem. In contrast to the existing work which primarily focuses on ReLU neural nets, we consider a general set of smooth activation functions. Under this more general setting, (i) we derive non-asymptotic error bounds on the difference between an overparameterized neural net and its corresponding neural tangent kernel, (ii) we propose an algorithm with a provable sublinear regret bound that is also efficient in the finite regime as demonstrated by empirical studies. The non-asymptotic error bounds may be of broader interests as a tool to establish the relation between the smoothness of the activation functions in neural contextual bandits and the smoothness of the kernels in kernel bandits.

We propose $\epsilon$-Exploring Thompson Sampling ($\epsilon$-TS), a modified version of the Thompson Sampling (TS) algorithm for multi-armed bandits. In $\epsilon$-TS, arms are selected greedily based on empirical mean rewards with probability $1-\epsilon$, and based on posterior samples obtained from TS with probability $\epsilon$. Here, $\epsilon\in(0,1)$ is a user-defined constant. By reducing exploration, $\epsilon$-TS improves computational efficiency compared to TS while achieving better regret bounds. We establish that $\epsilon$-TS is both minimax optimal and asymptotically optimal for various popular reward distributions, including Gaussian, Bernoulli, Poisson, and Gamma. A key technical advancement in our analysis is the relaxation of the requirement for a stringent anti-concentration bound of the posterior distribution, which was necessary in recent analyses that achieved similar bounds. As a result, $\epsilon$-TS maintains the posterior update structure of TS while minimizing alterations, such as clipping the sampling distribution or solving the inverse of the Kullback-Leibler (KL) divergence between reward distributions, as done in previous work. Furthermore, our algorithm is as easy to implement as TS, but operates significantly faster due to reduced exploration. Empirical evaluations confirm the efficiency and optimality of $\epsilon$-TS.

This work examines the challenges of training neural networks using vector quantization using straight-through estimation. We find that the main cause of training instability is the discrepancy between the model embedding and the code-vector distribution. We identify the factors that contribute to this issue, including the codebook gradient sparsity and the asymmetric nature of the commitment loss, which leads to misaligned code-vector assignments. We propose to address this issue via affine re-parameterization of the code vectors. Additionally, we introduce an alternating optimization to reduce the gradient error introduced by the straight-through estimation. Moreover, we propose an improvement to the commitment loss to ensure better alignment between the codebook representation and the model embedding. These optimization methods improve the mathematical approximation of the straight-through estimation and, ultimately, the model performance. We demonstrate the effectiveness of our methods on several common model architectures, such as AlexNet, ResNet, and ViT, across various tasks, including image classification and generative modeling.

Previous work demonstrates that the optimal safe reinforcement learning policy in a noise-free environment is vulnerable and could be unsafe under observational attacks. While adversarial training effectively improves robustness and safety, collecting samples by attacking the behavior agent online could be expensive or prohibitively dangerous in many applications. We propose the robuSt vAriational ofF-policy lEaRning (SAFER) approach, which only requires benign training data without attacking the agent. SAFER obtains an optimal non-parametric variational policy distribution via convex optimization and then uses it to improve the parameterized policy robustly via supervised learning. The two-stage policy optimization facilitates robust training, and extensive experiments on multiple robot platforms show the efficiency of SAFER in learning a robust and safe policy: achieving the same reward with much fewer constraint violations during training than on-policy baselines.

In Federated Learning (FL) and many other distributed training frameworks, collaborators can hold their private data locally and only share the network weights trained with the local data after multiple iterations. Gradient inversion is a family of privacy attacks that recovers data from its generated gradients. Seemingly, FL can provide a degree of protection against gradient inversion attacks on weight updates, since the gradient of a single step is concealed by the accumulation of gradients over multiple local iterations. In this work, we propose a principled way to extend gradient inversion attacks to weight updates in FL, thereby better exposing weaknesses in the presumed privacy protection inherent in FL. In particular, we propose a surrogate model method based on the characteristic of two-dimensional gradient flow and low-rank property of local updates. Our method largely boosts the ability of gradient inversion attacks on weight updates containing many iterations and achieves state-of-the-art (SOTA) performance. Additionally, our method runs up to $100\times$ faster than the SOTA baseline in the common FL scenario. Our work re-evaluates and highlights the privacy risk of sharing network weights. Our code is available at https://github.com/JunyiZhu-AI/surrogate_model_extension.

We investigate the expressive power of depth-2 bandlimited random neural networks. A random net is a neural network where the hidden layer parameters are frozen with random assignment, and only the output layer parameters are trained by loss minimization. Using random weights for a hidden layer is an effective method to avoid non-convex optimization in standard gradient descent learning. It has also been adopted in recent deep learning theories. Despite the well-known fact that a neural network is a universal approximator, in this study, we mathematically show that when hidden parameters are distributed in a bounded domain, the network may not achieve zero approximation error. In particular, we derive a new nontrivial approximation error lower bound. The proof utilizes the technique of ridgelet analysis, a harmonic analysis method designed for neural networks. This method is inspired by fundamental principles in classical signal processing, specifically the idea that signals with limited bandwidth may not always be able to perfectly reconstruct the original signal. We corroborate our theoretical results with various simulation studies, and generally, two main take-home messages are offered: (i) Not any distribution for selecting random weights is feasible to build a universal approximator; (ii) A suitable assignment of random weights exists but to some degree is associated with the complexity of the target function.

In recent years, decentralized bilevel optimization has gained significant attention thanks to its versatility in modeling a wide range of multi-agent learning problems, such as multi-agent reinforcement learning and multi-agent meta-learning. However, one unexplored and fundamental problem in this area is how to solve decentralized stochastic bilevel optimization problems with **domain constraints** while achieving low sample and communication complexities. This problem often arises from multi-agent learning problems with safety constraints. As shown in this paper, constrained decentralized bilevel optimization is far more challenging than its unconstrained counterpart due to the complex coupling structure, which necessitates new algorithm design and analysis techniques. Toward this end, we investigate a class of constrained decentralized bilevel optimization problems, where multiple agents collectively solve a nonconvex-strongly-convex bilevel problem with constraints in the upper-level variables. We propose an algorithm called Prometheus (proximal tracked stochastic recursive estimator) that achieves the first $\mathcal{O}(\epsilon^{-1})$ results in both sample and communication complexities for constrained decentralized bilevel optimization, where $\epsilon>0$ is a desired stationarity error. Collectively, the results in this work contribute to a theoretical foundation for low sample- and communication-complexity constrained decentralized bilevel learning.