Jingzhao Zhang

Kwangjun Ahn, Jingzhao Zhang, Suvrit Sra

Most existing analyses of (stochastic) gradient descent rely on the condition that for $L$-smooth costs, the step size is less than $2/L$. However, many works have observed that in machine learning applications step sizes often do not fulfill this condition, yet (stochastic) gradient descent still converges, albeit in an unstable manner. We investigate this unstable convergence phenomenon from first principles, and discuss key causes behind it. We also identify its main characteristics, and how they interrelate based on both theory and experiments, offering a principled view toward understanding the phenomenon.

Lesi Chen, Jing Xu, Jingzhao Zhang

Bilevel optimization reveals the inner structure of otherwise oblique optimization problems, such as hyperparameter tuning, neural architecture search, and meta-learning. A common goal in bilevel optimization is to minimize a hyper-objective that implicitly depends on the solution set of the lower-level function. Although this hyper-objective approach is widely used, its theoretical properties have not been thoroughly investigated in cases where the lower-level functions lack strong convexity. In this work, we first provide hardness results to show that the goal of finding stationary points of the hyper-objective for nonconvex-convex bilevel optimization can be intractable for zero-respecting algorithms. Then we study a class of tractable nonconvex-nonconvex bilevel problems when the lower-level function satisfies the Polyak-Łojasiewicz (PL) condition. We show a simple first-order algorithm can achieve better complexity bounds of $\tilde{\mathcal{O}}(ε^{-2})$, $\tilde{\mathcal{O}}(ε^{-4})$ and $\tilde{\mathcal{O}}(ε^{-6})$ in the deterministic, partially stochastic, and fully stochastic setting respectively. The complexities in the first two cases are optimal up to logarithmic factors.

Wei Fu, Weihua Du, Jingwei Li et al. · cmu

In complex reinforcement learning (RL) problems, policies with similar rewards may have substantially different behaviors. It remains a fundamental challenge to optimize rewards while also discovering as many diverse strategies as possible, which can be crucial in many practical applications. Our study examines two design choices for tackling this challenge, i.e., diversity measure and computation framework. First, we find that with existing diversity measures, visually indistinguishable policies can still yield high diversity scores. To accurately capture the behavioral difference, we propose to incorporate the state-space distance information into the diversity measure. In addition, we examine two common computation frameworks for this problem, i.e., population-based training (PBT) and iterative learning (ITR). We show that although PBT is the precise problem formulation, ITR can achieve comparable diversity scores with higher computation efficiency, leading to improved solution quality in practice. Based on our analysis, we further combine ITR with two tractable realizations of the state-distance-based diversity measures and develop a novel diversity-driven RL algorithm, State-based Intrinsic-reward Policy Optimization (SIPO), with provable convergence properties. We empirically examine SIPO across three domains from robot locomotion to multi-agent games. In all of our testing environments, SIPO consistently produces strategically diverse and human-interpretable policies that cannot be discovered by existing baselines.

Xinran Gu, Kaifeng Lyu, Sanjeev Arora et al. · tsinghua

In distributed deep learning with data parallelism, synchronizing gradients at each training step can cause a huge communication overhead, especially when many nodes work together to train large models. Local gradient methods, such as Local SGD, address this issue by allowing workers to compute locally for $H$ steps without synchronizing with others, hence reducing communication frequency. While $H$ has been viewed as a hyperparameter to trade optimization efficiency for communication cost, recent research indicates that setting a proper $H$ value can lead to generalization improvement. Yet, selecting a proper $H$ is elusive. This work proposes a theory-grounded method for determining $H$, named the Quadratic Synchronization Rule (QSR), which recommends dynamically setting $H$ in proportion to $\frac{1}{η^2}$ as the learning rate $η$ decays over time. Extensive ImageNet experiments on ResNet and ViT show that local gradient methods with QSR consistently improve the test accuracy over other synchronization strategies. Compared with the standard data parallel training, QSR enables Local AdamW on ViT-B to cut the training time on 16 or 64 GPUs down from 26.7 to 20.2 hours or from 8.6 to 5.5 hours and, at the same time, achieves $1.16\%$ or $0.84\%$ higher top-1 validation accuracy.

Lesi Chen, Jing Xu, Jingzhao Zhang

Bilevel optimization reveals the inner structure of otherwise oblique optimization problems, such as hyperparameter tuning, neural architecture search, and meta-learning. A common goal in bilevel optimization is to minimize a hyper-objective that implicitly depends on the solution set of the lower-level function. Although this hyper-objective approach is widely used, its theoretical properties have not been thoroughly investigated in cases where the lower-level functions lack strong convexity. In this work, we first provide hardness results to show that the goal of finding stationary points of the hyper-objective for nonconvex-convex bilevel optimization can be intractable for zero-respecting algorithms. Then we study a class of tractable nonconvex-nonconvex bilevel problems when the lower-level function satisfies the Polyak-Łojasiewicz (PL) condition. We show a simple first-order algorithm can achieve better complexity bounds of $\tilde{\mathcal{O}}(ε^{-2})$, $\tilde{\mathcal{O}}(ε^{-4})$ and $\tilde{\mathcal{O}}(ε^{-6})$ in the deterministic, partially stochastic, and fully stochastic setting respectively. The complexities in the first two cases are optimal up to logarithmic factors.

Jing Dong, Jingwei Li, Baoxiang Wang et al.

Reinforcement learning (RL) has exceeded human performance in many synthetic settings such as video games and Go. However, real-world deployment of end-to-end RL models is less common, as RL models can be very sensitive to slight perturbation of the environment. The robust Markov decision process (MDP) framework -- in which the transition probabilities belong to an uncertainty set around a nominal model -- provides one way to develop robust models. While previous analysis shows RL algorithms are effective assuming access to a generative model, it remains unclear whether RL can be efficient under a more realistic online setting, which requires a careful balance between exploration and exploitation. In this work, we consider online robust MDP by interacting with an unknown nominal system. We propose a robust optimistic policy optimization algorithm that is provably efficient. To address the additional uncertainty caused by an adversarial environment, our model features a new optimistic update rule derived via Fenchel conjugates. Our analysis establishes the first regret bound for online robust MDPs.

Lesi Chen, Yaohua Ma, Jingzhao Zhang

In this work, we consider bilevel optimization when the lower-level problem is strongly convex. Recent works show that with a Hessian-vector product (HVP) oracle, one can provably find an $ε$-stationary point within ${\mathcal{O}}(ε^{-2})$ oracle calls. However, the HVP oracle may be inaccessible or expensive in practice. Kwon et al. (ICML 2023) addressed this issue by proposing a first-order method that can achieve the same goal at a slower rate of $\tilde{\mathcal{O}}(ε^{-3})$. In this paper, we incorporate a two-time-scale update to improve their method to achieve the near-optimal $\tilde {\mathcal{O}}(ε^{-2})$ first-order oracle complexity. Our analysis is highly extensible. In the stochastic setting, our algorithm can achieve the stochastic first-order oracle complexity of $\tilde {\mathcal{O}}(ε^{-4})$ and $\tilde {\mathcal{O}}(ε^{-6})$ when the stochastic noises are only in the upper-level objective and in both level objectives, respectively. When the objectives have higher-order smoothness conditions, our deterministic method can escape saddle points by injecting noise, and can be accelerated to achieve a faster rate of $\tilde {\mathcal{O}}(ε^{-1.75})$ using Nesterov's momentum.

Xiang Cheng, Bohan Wang, Jingzhao Zhang et al.

MCMC algorithms offer empirically efficient tools for sampling from a target distribution $π(x) \propto \exp(-V(x))$. However, on the theory side, MCMC algorithms suffer from slow mixing rate when $π(x)$ is non-log-concave. Our work examines this gap and shows that when Poincaré-style inequality holds on a subset $\mathcal{X}$ of the state space, the conditional distribution of MCMC iterates over $\mathcal{X}$ mixes fast to the true conditional distribution. This fast mixing guarantee can hold in cases when global mixing is provably slow. We formalize the statement and quantify the conditional mixing rate. We further show that conditional mixing can have interesting implications for sampling from mixtures of Gaussians, parameter estimation for Gaussian mixture models and Gibbs-sampling with well-connected local minima.

Zeyang Zhang, Chengwei Liang, Xingyan Chen et al.

Diffusion language models (DLMs) are promising alternatives to autoregressive language models (ARMs), yet the intrinsic differences in their generated text remain underexplored. We first find empirically that off-the-shelf DLMs exhibit lower $n$-gram entropy, higher semantic coherence, and higher semantic diversity. To understand the cause, we conduct controlled experiments that decouple the effects of training objectives and decoding algorithms. Results suggest that the DLM training objective contributes to the increases in semantic coherence and semantic diversity, but has a minor influence on entropy. These differences are primarily driven by the bidirectional context; other components in the training objective, such as input masking, label masking, and the weighting function, have a much weaker influence. Further, our experiments demonstrate that the reduction in entropy stems from DLMs' decoding algorithms, particularly confidence-based remasking strategies. We provide a theoretical understanding for this entropy reduction phenomenon. Together, our work uncovers key mechanisms underlying the differences between DLMs and ARMs in text generation, and informs future design of training objectives and decoding algorithms in DLMs.

Pengkun Yang, Jingzhao Zhang

We study the scaling of classification error rates with respect to the size of the training dataset. In contrast to classical results where rates are minimax optimal for a problem class, this work starts with the empirical observation that, even for a fixed data distribution, the error scaling can have \emph{diverse} rates across different ranges of sample size. To understand when and why the error rate is non-uniform, we theoretically analyze nearest neighbor classifiers. We show that an error scaling law can have fine-grained rates: in the early phase, the test error depends polynomially on the data dimension and decreases fast; whereas in the later phase, the error depends exponentially on the data dimension and decreases slowly. Our analysis highlights the complexity of the data distribution in determining the test error. When the data are distributed benignly, we show that the generalization error of nearest neighbor classifier can depend polynomially, instead of exponentially, on the data dimension.

Huaqing Zhang, Lesi Chen, Jing Xu et al.

This paper studies simple bilevel problems, where a convex upper-level function is minimized over the optimal solutions of a convex lower-level problem. We first show the fundamental difficulty of simple bilevel problems, that the approximate optimal value of such problems is not obtainable by first-order zero-respecting algorithms. Then we follow recent works to pursue the weak approximate solutions. For this goal, we propose a novel method by reformulating them into functionally constrained problems. Our method achieves near-optimal rates for both smooth and nonsmooth problems. To the best of our knowledge, this is the first near-optimal algorithm that works under standard assumptions of smoothness or Lipschitz continuity for the objective functions.

Xiang Cheng, Jingzhao Zhang, Suvrit Sra

We study the task of efficiently sampling from a Gibbs distribution $d π^* = e^{-h} d {vol}_g$ over a Riemannian manifold $M$ via (geometric) Langevin MCMC; this algorithm involves computing exponential maps in random Gaussian directions and is efficiently implementable in practice. The key to our analysis of Langevin MCMC is a bound on the discretization error of the geometric Euler-Murayama scheme, assuming $\nabla h$ is Lipschitz and $M$ has bounded sectional curvature. Our error bound matches the error of Euclidean Euler-Murayama in terms of its stepsize dependence. Combined with a contraction guarantee for the geometric Langevin Diffusion under Kendall-Cranston coupling, we prove that the Langevin MCMC iterates lie within $ε$-Wasserstein distance of $π^*$ after $\tilde{O}(ε^{-2})$ steps, which matches the iteration complexity for Euclidean Langevin MCMC. Our results apply in general settings where $h$ can be nonconvex and $M$ can have negative Ricci curvature. Under additional assumptions that the Riemannian curvature tensor has bounded derivatives, and that $π^*$ satisfies a $CD(\cdot,\infty)$ condition, we analyze the stochastic gradient version of Langevin MCMC, and bound its iteration complexity by $\tilde{O}(ε^{-2})$ as well.

Jing Xu, Jingzhao Zhang

Fine-tuning large language models (LLM) can be costly. Parameter-efficient fine-tuning (PEFT) addresses the problems by training a fraction of the parameters, whose success reveals the expressiveness and flexibility of pretrained models. This paper studies the limit of PEFT, by further simplifying its design and reducing the number of trainable parameters beyond standard setups. To this end, we use Random Masking to fine-tune the pretrained model. Despite its simplicity, we show that Random Masking is surprisingly effective: with a larger-than-expected learning rate, Random Masking can match the performance of standard PEFT algorithms such as LoRA on various tasks, using fewer trainable parameters. We provide both empirical and theoretical explorations into the success of Random Masking. We show that masking induces a flatter loss landscape and more distant solutions, which allows for and necessitates large learning rates.

Jing Xu, Jiazheng Li, Jingzhao Zhang

Model merging offers an effective way to integrate the capabilities of multiple fine-tuned models. However, the performance degradation of the merged model remains a challenge, particularly when none or few data are available. This paper first highlights the necessity of domain-specific data for model merging by proving that data-agnostic algorithms can have arbitrarily bad worst-case performance. Building on this theoretical insight, we explore the relationship between model merging and distillation, introducing a novel few-shot merging algorithm, ProDistill (Progressive Layer-wise Distillation). Unlike common belief that layer wise training hurts performance, we show that layer-wise teacher-student distillation not only enhances the scalability but also improves model merging performance. We conduct extensive experiments to show that compared to existing few-shot merging methods, ProDistill achieves state-of-the-art performance, with up to 6.14% and 6.61% improvements in vision and NLU tasks. Furthermore, we extend the experiments to models with over 10B parameters, showcasing the exceptional scalability of ProDistill.

Amirhesam Abedsoltan, Huaqing Zhang, Kaiyue Wen et al.

Large language models (LLMs) exhibit remarkable task generalization, solving tasks they were never explicitly trained on with only a few demonstrations. This raises a fundamental question: When can learning from a small set of tasks generalize to a large task family? In this paper, we investigate task generalization through the lens of autoregressive compositional structure, where each task is a composition of $T$ operations, and each operation is among a finite family of $D$ subtasks. This yields a total class of size $D^T$. We first show that generalization to all $D^T$ tasks is theoretically achievable by training on only $\widetilde{O}(D)$ tasks. Empirically, we demonstrate that Transformers achieve such exponential task generalization on sparse parity functions via In-context Learning (ICL) and chain-of-thought (CoT) reasoning. We further show generalization in arithmetic and translation, beyond parity functions.

Lesi Chen, Chengchang Liu, Jingzhao Zhang

This paper studies second-order methods for convex-concave minimax optimization. Monteiro and Svaiter (2012) proposed a method to solve the problem with an optimal iteration complexity of $\mathcal{O}(ε^{-3/2})$ to find an $ε$-saddle point. However, it is unclear whether the computational complexity, $\mathcal{O}((N+ d^2) d ε^{-2/3})$, can be improved. In the above, we follow Doikov et al. (2023) and assume the complexity of obtaining a first-order oracle as $N$ and the complexity of obtaining a second-order oracle as $dN$. In this paper, we show that the computation cost can be reduced by reusing Hessian across iterations. Our methods take the overall computational complexity of $ \tilde{\mathcal{O}}( (N+d^2)(d+ d^{2/3}ε^{-2/3}))$, which improves those of previous methods by a factor of $d^{1/3}$. Furthermore, we generalize our method to strongly-convex-strongly-concave minimax problems and establish the complexity of $\tilde{\mathcal{O}}((N+d^2) (d + d^{2/3} κ^{2/3}) )$ when the condition number of the problem is $κ$, enjoying a similar speedup upon the state-of-the-art method. Numerical experiments on both real and synthetic datasets also verify the efficiency of our method.

Lesi Chen, Jingzhao Zhang

Bilevel optimization minimizes an objective function, defined by an upper-level problem whose feasible region is the solution of a lower-level problem. We study the oracle complexity of finding an $ε$-stationary point with first-order methods when the upper-level problem is nonconvex and the lower-level problem is strongly convex. Recent works (Ji et al., ICML 2021; Arbel and Mairal, ICLR 2022; Chen el al., JMLR 2025) achieve a $\tilde{\mathcal{O}}(κ^4 ε^{-2})$ upper bound that is near-optimal in $ε$. However, the optimal dependency on the condition number $κ$ is unknown. In this work, we establish a new $Ω(κ^2 ε^{-2})$ lower bound and $\tilde{\mathcal{O}}(κ^{7/2} ε^{-2})$ upper bound for this problem, establishing the first provable gap between bilevel problems and minimax problems in this setup. Our lower bounds can be extended to various settings, including high-order smooth functions, stochastic oracles, and convex hyper-objectives: (1) For second-order and arbitrarily smooth problems, we show $Ω(κ_y^{13/4} ε^{-12/7})$ and $Ω(κ^{17/10} ε^{-8/5})$ lower bounds, respectively. (2) For convex-strongly-convex problems, we improve the previously best lower bound (Ji and Liang, JMLR 2022) from $Ω(κ/\sqrtε)$ to $Ω(κ^{5/4} / \sqrtε)$. (3) For smooth stochastic problems, we show an $Ω(κ^4 ε^{-4})$ lower bound.

Hengbo Xiao, Jingyuan Fan, Xin Tong et al.

Tasks on complex systems require high-precision numerical computation to support decisions, but current large language models (LLMs) cannot integrate such computations as an intrinsic and interpretable capability with existing architectures. Multi-agent approaches can leverage external experts, but inevitably introduce communication overhead and suffer from inefficiency caused by limited scalability. To this end, we propose Physically-isolated Experts Routing Network (PiERN), an architecture for integrating computation and reasoning. Instead of the tool-use workflows or function-calling, PiERN endogenously integrates computational capabilities into neural networks after separately training experts, a text-to-computation module, and a router. At inference, the router directs computation and reasoning at the token level, thereby enabling iterative alternation within a single chain of thought. We evaluate PiERN on representative linear and nonlinear computation-reasoning tasks against LLM finetuning and the multi-agent system approaches. Results show that the PiERN architecture achieves not only higher accuracy than directly finetuning LLMs but also significant improvements in response latency, token usage, and GPU energy consumption compared with mainstream multi-agent approaches. PiERN offers an efficient, interpretable, and scalable paradigm for interfacing language models with scientific systems.

Hong Lu, Jiali Chen, Jingzhao Zhang et al.

Industrial-scale battery management involves various types of tasks, such as estimation, prediction, and system-level diagnostics. Each task employs distinct data across temporal scales, sensor resolutions, and data channels. Building task-specific methods requires a great deal of data and engineering effort, which limits the scalability of intelligent battery management. Here we present the Flexible Masked Autoencoder (FMAE), a flexible pretraining framework that can learn with missing battery data channels and capture inter-correlations across data snippets. FMAE learns unified battery representations from heterogeneous data and can be adopted by different tasks with minimal data and engineering efforts. Experimentally, FMAE consistently outperforms all task-specific methods across five battery management tasks with eleven battery datasets. On remaining life prediction tasks, FMAE uses 50 times less inference data while maintaining state-of-the-art results. Moreover, when real-world data lack certain information, such as system voltage, FMAE can still be applied with marginal performance impact, achieving comparable results with the best hand-crafted features. FMAE demonstrates a practical route to a flexible, data-efficient model that simplifies real-world multi-task management of dynamical systems.

Weizhi Fei, Hao Shi, Jing Xu et al. · tsinghua

Efficiently editing knowledge stored in large language models (LLMs) enables model updates without large-scale training. One possible solution is Locate-and-Edit (L\&E), allowing simultaneous modifications of a massive number of facts. However, such editing may compromise the general abilities of LLMs and even result in forgetting edited facts when scaling up to thousands of edits. In this paper, we model existing linear L\&E methods as querying a Key-Value (KV) database. From this perspective, we then propose NeuralDB, an editing framework that explicitly represents the edited facts as a neural KV database equipped with a non-linear gated retrieval module, % In particular, our gated module only operates when inference involves the edited facts, effectively preserving the general abilities of LLMs. Comprehensive experiments involving the editing of 10,000 facts were conducted on the ZsRE and CounterFacts datasets, using GPT2-XL, GPT-J (6B) and Llama-3 (8B). The results demonstrate that NeuralDB not only excels in editing efficacy, generalization, specificity, fluency, and consistency, but also preserves overall performance across six representative text understanding and generation tasks. Further experiments indicate that NeuralDB maintains its effectiveness even when scaled to 100,000 facts (\textbf{50x} more than in prior work).

Lesi Chen, Chengchang Liu, Luo Luo et al.

Previous algorithms can solve convex-concave minimax problems $\min_{x \in \mathcal{X}} \max_{y \in \mathcal{Y}} f(x,y)$ with $\mathcal{O}(ε^{-2/3})$ second-order oracle calls using Newton-type methods. This result has been speculated to be optimal because the upper bound is achieved by a natural generalization of the optimal first-order method. In this work, we show an improved upper bound of $\tilde{\mathcal{O}}(ε^{-4/7})$ by generalizing the optimal second-order method for convex optimization to solve the convex-concave minimax problem. We further apply a similar technique to lazy Hessian algorithms and show that our proposed algorithm can also be seen as a second-order ``Catalyst'' framework (Lin et al., JMLR 2018) that could accelerate any globally convergent algorithms for solving minimax problems.

Haowei He, Jingzhao Zhang, Yanan Wang et al.

Electric vehicles (EVs) play an important role in reducing carbon emissions. As EV adoption accelerates, safety issues caused by EV batteries have become an important research topic. In order to benchmark and develop data-driven methods for this task, we introduce a large and comprehensive dataset of EV batteries. Our dataset includes charging records collected from hundreds of EVs from three manufacturers over several years. Our dataset is the first large-scale public dataset on real-world battery data, as existing data either include only several vehicles or is collected in the lab environment. Meanwhile, our dataset features two types of labels, corresponding to two key tasks - battery health estimation and battery capacity estimation. In addition to demonstrating how existing deep learning algorithms can be applied to this task, we further develop an algorithm that exploits the data structure of battery systems. Our algorithm achieves better results and shows that a customized method can improve model performances. We hope that this public dataset provides valuable resources for researchers, policymakers, and industry professionals to better understand the dynamics of EV battery aging and support the transition toward a sustainable transportation system.

Jingzhao Zhang, Haochuan Li, Suvrit Sra et al.

This work examines the deep disconnect between existing theoretical analyses of gradient-based algorithms and the practice of training deep neural networks. Specifically, we provide numerical evidence that in large-scale neural network training (e.g., ImageNet + ResNet101, and WT103 + TransformerXL models), the neural network's weights do not converge to stationary points where the gradient of the loss is zero. Remarkably, however, we observe that even though the weights do not converge to stationary points, the progress in minimizing the loss function halts and training loss stabilizes. Inspired by this observation, we propose a new perspective based on ergodic theory of dynamical systems to explain it. Rather than studying the evolution of weights, we study the evolution of the distribution of weights. We prove convergence of the distribution of weights to an approximate invariant measure, thereby explaining how the training loss can stabilize without weights necessarily converging to stationary points. We further discuss how this perspective can better align optimization theory with empirical observations in machine learning practice.

Xinran Gu, Kaixuan Huang, Jingzhao Zhang et al.

Federated Learning (FL) coordinates with numerous heterogeneous devices to collaboratively train a shared model while preserving user privacy. Despite its multiple advantages, FL faces new challenges. One challenge arises when devices drop out of the training process beyond the control of the central server. In this case, the convergence of popular FL algorithms such as FedAvg is severely influenced by the straggling devices. To tackle this challenge, we study federated learning algorithms under arbitrary device unavailability and propose an algorithm named Memory-augmented Impatient Federated Averaging (MIFA). Our algorithm efficiently avoids excessive latency induced by inactive devices, and corrects the gradient bias using the memorized latest updates from the devices. We prove that MIFA achieves minimax optimal convergence rates on non-i.i.d. data for both strongly convex and non-convex smooth functions. We also provide an explicit characterization of the improvement over baseline algorithms through a case study, and validate the results by numerical experiments on real-world datasets.

Haochuan Li, Yi Tian, Jingzhao Zhang et al.

We provide a first-order oracle complexity lower bound for finding stationary points of min-max optimization problems where the objective function is smooth, nonconvex in the minimization variable, and strongly concave in the maximization variable. We establish a lower bound of $Ω\left(\sqrtκε^{-2}\right)$ for deterministic oracles, where $ε$ defines the level of approximate stationarity and $κ$ is the condition number. Our analysis shows that the upper bound achieved in (Lin et al., 2020b) is optimal in the $ε$ and $κ$ dependence up to logarithmic factors. For stochastic oracles, we provide a lower bound of $Ω\left(\sqrtκε^{-2} + κ^{1/3}ε^{-4}\right)$. It suggests that there is a significant gap between the upper bound $\mathcal{O}(κ^3 ε^{-4})$ in (Lin et al., 2020a) and our lower bound in the condition number dependence.

Tiancheng Yu, Yi Tian, Jingzhao Zhang et al.

We study multi-objective reinforcement learning (RL) where an agent's reward is represented as a vector. In settings where an agent competes against opponents, its performance is measured by the distance of its average return vector to a target set. We develop statistically and computationally efficient algorithms to approach the associated target set. Our results extend Blackwell's approachability theorem (Blackwell, 1956) to tabular RL, where strategic exploration becomes essential. The algorithms presented are adaptive; their guarantees hold even without Blackwell's approachability condition. If the opponents use fixed policies, we give an improved rate of approaching the target set while also tackling the more ambitious goal of simultaneously minimizing a scalar cost function. We discuss our analysis for this special case by relating our results to previous works on constrained RL. To our knowledge, this work provides the first provably efficient algorithms for vector-valued Markov games and our theoretical guarantees are near-optimal.

Zeke Xie, Zhiqiang Xu, Jingzhao Zhang et al.

Weight decay is a simple yet powerful regularization technique that has been very widely used in training of deep neural networks (DNNs). While weight decay has attracted much attention, previous studies fail to discover some overlooked pitfalls on large gradient norms resulted by weight decay. In this paper, we discover that, weight decay can unfortunately lead to large gradient norms at the final phase (or the terminated solution) of training, which often indicates bad convergence and poor generalization. To mitigate the gradient-norm-centered pitfalls, we present the first practical scheduler for weight decay, called the Scheduled Weight Decay (SWD) method that can dynamically adjust the weight decay strength according to the gradient norm and significantly penalize large gradient norms during training. Our experiments also support that SWD indeed mitigates large gradient norms and often significantly outperforms the conventional constant weight decay strategy for Adaptive Moment Estimation (Adam).

Jingzhao Zhang, Aditya Menon, Andreas Veit et al.

The label shift problem refers to the supervised learning setting where the train and test label distributions do not match. Existing work addressing label shift usually assumes access to an \emph{unlabelled} test sample. This sample may be used to estimate the test label distribution, and to then train a suitably re-weighted classifier. While approaches using this idea have proven effective, their scope is limited as it is not always feasible to access the target domain; further, they require repeated retraining if the model is to be deployed in \emph{multiple} test environments. Can one instead learn a \emph{single} classifier that is robust to arbitrary label shifts from a broad family? In this paper, we answer this question by proposing a model that minimises an objective based on distributionally robust optimisation (DRO). We then design and analyse a gradient descent-proximal mirror ascent algorithm tailored for large-scale problems to optimise the proposed objective. %, and establish its convergence. Finally, through experiments on CIFAR-100 and ImageNet, we show that our technique can significantly improve performance over a number of baselines in settings where label shift is present.

Jingzhao Zhang, Hongzhou Lin, Subhro Das et al.

We study oracle complexity of gradient based methods for stochastic approximation problems. Though in many settings optimal algorithms and tight lower bounds are known for such problems, these optimal algorithms do not achieve the best performance when used in practice. We address this theory-practice gap by focusing on instance-dependent complexity instead of worst case complexity. In particular, we first summarize known instance-dependent complexity results and categorize them into three levels. We identify the domination relation between different levels and propose a fourth instance-dependent bound that dominates existing ones. We then provide a sufficient condition according to which an adaptive algorithm with moment estimation can achieve the proposed bound without knowledge of noise levels. Our proposed algorithm and its analysis provide a theoretical justification for the success of moment estimation as it achieves improved instance complexity.

Jingzhao Zhang, Hongzhou Lin, Stefanie Jegelka et al.

We provide the first non-asymptotic analysis for finding stationary points of nonsmooth, nonconvex functions. In particular, we study the class of Hadamard semi-differentiable functions, perhaps the largest class of nonsmooth functions for which the chain rule of calculus holds. This class contains examples such as ReLU neural networks and others with non-differentiable activation functions. We first show that finding an $ε$-stationary point with first-order methods is impossible in finite time. We then introduce the notion of $(δ, ε)$-stationarity, which allows for an $ε$-approximate gradient to be the convex combination of generalized gradients evaluated at points within distance $δ$ to the solution. We propose a series of randomized first-order methods and analyze their complexity of finding a $(δ, ε)$-stationary point. Furthermore, we provide a lower bound and show that our stochastic algorithm has min-max optimal dependence on $δ$. Empirically, our methods perform well for training ReLU neural networks.

Jingzhao Zhang, Sai Praneeth Karimireddy, Andreas Veit et al.

While stochastic gradient descent (SGD) is still the \emph{de facto} algorithm in deep learning, adaptive methods like Clipped SGD/Adam have been observed to outperform SGD across important tasks, such as attention models. The settings under which SGD performs poorly in comparison to adaptive methods are not well understood yet. In this paper, we provide empirical and theoretical evidence that a heavy-tailed distribution of the noise in stochastic gradients is one cause of SGD's poor performance. We provide the first tight upper and lower convergence bounds for adaptive gradient methods under heavy-tailed noise. Further, we demonstrate how gradient clipping plays a key role in addressing heavy-tailed gradient noise. Subsequently, we show how clipping can be applied in practice by developing an \emph{adaptive} coordinate-wise clipping algorithm (ACClip) and demonstrate its superior performance on BERT pretraining and finetuning tasks.

Jingzhao Zhang, Tianxing He, Suvrit Sra et al.

We provide a theoretical explanation for the effectiveness of gradient clipping in training deep neural networks. The key ingredient is a new smoothness condition derived from practical neural network training examples. We observe that gradient smoothness, a concept central to the analysis of first-order optimization algorithms that is often assumed to be a constant, demonstrates significant variability along the training trajectory of deep neural networks. Further, this smoothness positively correlates with the gradient norm, and contrary to standard assumptions in the literature, it can grow with the norm of the gradient. These empirical observations limit the applicability of existing theoretical analyses of algorithms that rely on a fixed bound on smoothness. These observations motivate us to introduce a novel relaxation of gradient smoothness that is weaker than the commonly used Lipschitz smoothness assumption. Under the new condition, we prove that two popular methods, namely, \emph{gradient clipping} and \emph{normalized gradient}, converge arbitrarily faster than gradient descent with fixed stepsize. We further explain why such adaptively scaled gradient methods can accelerate empirical convergence and verify our results empirically in popular neural network training settings.

Tianxing He, Jingzhao Zhang, Zhiming Zhou et al.

Exposure bias has been regarded as a central problem for auto-regressive language models (LM). It claims that teacher forcing would cause the test-time generation to be incrementally distorted due to the training-generation discrepancy. Although a lot of algorithms have been proposed to avoid teacher forcing and therefore alleviate exposure bias, there is little work showing how serious the exposure bias problem actually is. In this work, we focus on the task of open-ended language generation, propose metrics to quantify the impact of exposure bias in the aspects of quality, diversity, and consistency. Our key intuition is that if we feed ground-truth data prefixes (instead of prefixes generated by the model itself) into the model and ask it to continue the generation, the performance should become much better because the training-generation discrepancy in the prefix is removed. Both automatic and human evaluations are conducted in our experiments. On the contrary to the popular belief in exposure bias, we find that the the distortion induced by the prefix discrepancy is limited, and does not seem to be incremental during the generation. Moreover, our analysis reveals an interesting self-recovery ability of the LM, which we hypothesize to be countering the harmful effects from exposure bias.

Jingzhao Zhang, Hongyi Zhang, Suvrit Sra

We study smooth stochastic optimization problems on Riemannian manifolds. Via adapting the recently proposed SPIDER algorithm \citep{fang2018spider} (a variance reduced stochastic method) to Riemannian manifold, we can achieve faster rate than known algorithms in both the finite sum and stochastic settings. Unlike previous works, by \emph{not} resorting to bounding iterate distances, our analysis yields curvature independent convergence rates for both the nonconvex and strongly convex cases.

Jingzhao Zhang, Aryan Mokhtari, Suvrit Sra et al.

We study gradient-based optimization methods obtained by directly discretizing a second-order ordinary differential equation (ODE) related to the continuous limit of Nesterov's accelerated gradient method. When the function is smooth enough, we show that acceleration can be achieved by a stable discretization of this ODE using standard Runge-Kutta integrators. Specifically, we prove that under Lipschitz-gradient, convexity and order-$(s+2)$ differentiability assumptions, the sequence of iterates generated by discretizing the proposed second-order ODE converges to the optimal solution at a rate of $\mathcal{O}({N^{-2\frac{s}{s+1}}})$, where $s$ is the order of the Runge-Kutta numerical integrator. Furthermore, we introduce a new local flatness condition on the objective, under which rates even faster than $\mathcal{O}(N^{-2})$ can be achieved with low-order integrators and only gradient information. Notably, this flatness condition is satisfied by several standard loss functions used in machine learning. We provide numerical experiments that verify the theoretical rates predicted by our results.