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Optimization · Large Scale, Parallel and Distributed

Youssef Allouah, Sadegh Farhadkhani, Rachid Guerraoui, Nirupam Gupta, Rafael Pinot, Geovani Rizk, Sasha Voitovych

The possibility of adversarial (a.k.a., Byzantine) clients makes federated learning (FL) prone to arbitrary manipulation. The natural approach to robustify FL against adversarial clients is to replace the simple averaging operation at the server in the standard $\mathsf{FedAvg}$ algorithm by a robust averaging rule. While a significant amount of work has been devoted to studying the convergence of federated robust averaging (which we denote by $\mathsf{FedRo}$), prior work has largely ignored the impact of client subsampling and local steps, two fundamental FL characteristics. While client subsampling increases the effective fraction of Byzantine clients, local steps increase the drift between the local updates computed by honest (i.e., non-Byzantine) clients. Consequently, a careless deployment of $\mathsf{FedRo}$ could yield poor performance. We validate this observation by presenting an in-depth analysis of $\mathsf{FedRo}$ tightly analyzing the impact of client subsampling and local steps. Specifically, we present a sufficient condition on client subsampling for nearly-optimal convergence of $\mathsf{FedRo}$ (for smooth non-convex loss). Also, we show that the rate of improvement in learning accuracy diminishes with respect to the number of clients subsampled, as soon as the sample size exceeds a threshold value. Interestingly, we also observe that under a careful choice of step-sizes, the learning error due to Byzantine clients decreases with the number of local steps. We validate our theory by experiments on the FEMNIST and CIFAR-$10$ image classification tasks.

Optimization · Non-Convex

Kwangjun Ahn, Zhiyu Zhang, Yunbum Kook, Yan Dai

Despite the success of the Adam optimizer in practice, the theoretical understanding of its algorithmic components still remains limited. In particular, most existing analyses of Adam show the convergence rate that can be simply achieved by non-adative algorithms like SGD. In this work, we provide a different perspective based on online learning that underscores the importance of Adam's algorithmic components. Inspired by Cutkosky et al. (2023), we consider the framework called online learning of updates/increments, where we choose the updates/increments of an optimizer based on an online learner. With this framework, the design of a good optimizer is reduced to the design of a good online learner. Our main observation is that Adam corresponds to a principled online learning framework called Follow-the-Regularized-Leader (FTRL). Building on this observation, we study the benefits of its algorithmic components from the online learning perspective.

Optimization · Non-Convex

Kwangjun Ahn, Ali Jadbabaie, Suvrit Sra

Modern machine learning applications have witnessed the remarkable success of optimization algorithms that are designed to find flat minima. Motivated by this design choice, we undertake a formal study that (i) formulates the notion of flat minima, and (ii) studies the complexity of finding them. Specifically, we adopt the trace of the Hessian of the cost function as a measure of flatness, and use it to formally define the notion of approximate flat minima. Under this notion, we then analyze algorithms that find approximate flat minima efficiently. For general cost functions, we discuss a gradient-based algorithm that finds an approximate flat local minimum efficiently. The main component of the algorithm is to use gradients computed from randomly perturbed iterates to estimate a direction that leads to flatter minima. For the setting where the cost function is an empirical risk over training data, we present a faster algorithm that is inspired by a recently proposed practical algorithm called sharpness-aware minimization, supporting its success in practice.

Optimization · Non-Convex

Emre Sahinoglu, Shahin Shahrampour

We investigate the finite-time analysis of finding ($\delta, \epsilon$)-stationary points for nonsmooth nonconvex objectives in decentralized stochastic optimization. A set of agents aim at minimizing a global function using only their local information by interacting over a network. We present a novel algorithm, called Multi Epoch Decentralized Online Learning (ME-DOL), for which we establish the sample complexity in various settings. First, using a recently proposed online-to-nonconvex technique, we show that our algorithm recovers the optimal convergence rate of smooth nonconvex objectives. We then extend our analysis to the nonsmooth setting, building on properties of randomized smoothing and Goldstein-subdifferential sets. We establish the sample complexity of $O(\delta^{-1}\epsilon^{-3})$, which to the best of our knowledge is the first finite-time guarantee for decentralized nonsmooth nonconvex stochastic optimization in the first-order setting (without weak-convexity), matching its optimal centralized counterpart. We further prove the same rate for the zero-order oracle setting without using variance reduction.

Optimization · Non-Convex

Kwangjun Ahn, Xiang Cheng, Minhak Song, Chulhee Yun, Ali Jadbabaie, Suvrit Sra

Transformer training is notoriously difficult, requiring a careful design of optimizers and use of various heuristics. We make progress towards understanding the subtleties of training Transformers by carefully studying a simple yet canonical linearized *shallow* Transformer model. Specifically, we train linear Transformers to solve regression tasks, inspired by J. von Oswald et al. (ICML 2023), and K. Ahn et al. (NeurIPS 2023). Most importantly, we observe that our proposed linearized models can reproduce several prominent aspects of Transformer training dynamics. Consequently, the results obtained in this paper suggest that a simple linearized Transformer model could actually be a valuable, realistic abstraction for understanding Transformer optimization.

Deep Learning · Graph Neural Networks

Mert Kosan, Samidha Verma, Burouj Armgaan, Khushbu Pahwa, Ambuj K Singh, Sourav Medya, Sayan Ranu

Numerous explainability methods have been proposed to shed light on the inner workings of GNNs. Despite the inclusion of empirical evaluations in all the proposed algorithms, the interrogative aspects of these evaluations lack diversity. As a result, various facets of explainability pertaining to GNNs, such as a comparative analysis of counterfactual reasoners, their stability to variational factors such as different GNN architectures, noise, stochasticity in non-convex loss surfaces, feasibility amidst domain constraints, and so forth, have yet to be formally investigated. Motivated by this need, we present a benchmarking study on perturbation-based explainability methods for GNNs, aiming to systematically evaluate and compare a wide range of explainability techniques. Among the key findings of our study, we identify the Pareto-optimal methods that exhibit superior efficacy and stability in the presence of noise. Nonetheless, our study reveals thatall algorithms are affected by stability issues when faced with noisy data. Furthermore, we have established that the current generation of counterfactual explainers often fails to provide feasible recourses due to violations of topological constraints encoded by domain-specific considerations. Overall, this benchmarking study empowers stakeholders in the field of GNNs with a comprehensive understanding of the state-of-the-art explainability methods, potential research problems for further enhancement, and the implications of their application in real-world scenarios.

Anshuman Chhabra, Peizhao Li, Prasant Mohapatra, Hongfu Liu

Classification models are ubiquitously deployed in society and necessitate high utility, fairness, and robustness performance. Current research efforts mainly focus on improving model architectures and learning algorithms on fixed datasets to achieve this goal. In contrast, in this paper, we address an orthogonal yet crucial problem: given a fixed convex learning model (or a convex surrogate for a non-convex model) and a function of interest, we assess what data benefits the model by interpreting the feature space, and then aim to improve performance as measured by this function. To this end, we propose the use of influence estimation models for interpreting the classifier's performance from the perspective of the data feature space. Additionally, we propose data selection approaches based on influence that enhance model utility, fairness, and robustness. Through extensive experiments on synthetic and real-world datasets, we validate and demonstrate the effectiveness of our approaches not only for conventional classification scenarios, but also under more challenging scenarios such as distribution shifts, fairness poisoning attacks, utility evasion attacks, online learning, and active learning.

Ian Gemp, Luke Marris, Georgios Piliouras

We propose the first loss function for approximate Nash equilibria of normal-form games that is amenable to unbiased Monte Carlo estimation. This construction allows us to deploy standard non-convex stochastic optimization techniques for approximating Nash equilibria, resulting in novel algorithms with provable guarantees. We complement our theoretical analysis with experiments demonstrating that stochastic gradient descent can outperform previous state-of-the-art approaches.

Deep Learning · Theory

David Robin, Kevin Scaman, marc lelarge

We present a framework to define a large class of neural networks for which, by construction, training by gradient flow provably reaches arbitrarily low loss when the number of parameters grows. Distinct from the fixed-space global optimality of non-convex optimization, this new form of convergence, and the techniques introduced to prove such convergence, pave the way for a usable deep learning convergence theory in the near future, without overparameterization assumptions relating the number of parameters and training samples. We define these architectures from a simple computation graph and a mechanism to lift it, thus increasing the number of parameters, generalizing the idea of increasing the widths of multi-layer perceptrons. We show that architectures similar to most common deep learning models are present in this class, obtained by sparsifying the weight tensors of usual architectures at initialization. Leveraging tools of algebraic topology and random graph theory, we use the computation graph’s geometry to propagate properties guaranteeing convergence to any precision for these large sparse models.

Hong Liu, Zhiyuan Li, David Hall, Percy Liang, Tengyu Ma

Given the massive cost of language model pre-training, a non-trivial improvement of the optimization algorithm would lead to a material reduction on the time and cost of training. Adam and its variants have been state-of-the-art for years, and more sophisticated second-order (Hessian-based) optimizers often incur too much per-step overhead. In this paper, we propose Sophia, a simple scalable second-order optimizer that uses a light-weight estimate of the diagonal Hessian as the pre-conditioner. The update is the moving average of the gradients divided by the moving average of the estimated Hessian, followed by element-wise clipping. The clipping controls the worst-case update size and tames the negative impact of non-convexity and rapid change of Hessian along the trajectory. Sophia only estimates the diagonal Hessian every handful of iterations, which has negligible average per-step time and memory overhead. On language modeling with GPT models of sizes ranging from 125M to 1.5B, Sophia achieves a 2x speed-up compared to Adam in the number of steps, total compute, and wall-clock time, achieving the same perplexity with 50\% fewer steps, less total compute, and reduced wall-clock time.

Haozhao Wang, Haoran Xu, Yichen Li, Yuan Xu, Ruixuan Li, Tianwei Zhang

In Federated Learning (FL), model aggregation is pivotal. It involves a global server iteratively aggregating client local trained models in successive rounds without accessing private data. Traditional methods typically aggregate the local models from the current round alone. However, due to the statistical heterogeneity across clients, the local models from different clients may be greatly diverse, making the obtained global model incapable of maintaining the specific knowledge of each local model. In this paper, we introduce a novel method, FedCDA, which selectively aggregates cross-round local models, decreasing discrepancies between the global model and local models.The principle behind FedCDA is that due to the different global model parameters received in different rounds and the non-convexity of deep neural networks, the local models from each client may converge to different local optima across rounds. Therefore, for each client, we select a local model from its several recent local models obtained in multiple rounds, where the local model is selected by minimizing its divergence from the local models of other clients. This ensures the aggregated global model remains close to all selected local models to maintain their data knowledge. Extensive experiments conducted on various models and datasets reveal our approach outperforms state-of-the-art aggregation methods.

Optimization · Non-Convex

Zhuqing Liu, Xin Zhang, Jia Liu, Zhengyuan Zhu, Songtao Lu

Learning an accurate value function for a given policy is a critical step in solving reinforcement learning (RL) problems. So far, however, the convergence speed and sample complexity performances of most existing policy evaluation algorithms remain unsatisfactory, particularly with non-linear function approximation. This challenge motivates us to develop a new path-integrated primal-dual stochastic gradient (PILOT) method, that is able to achieve a fast convergence speed for RL policy evaluation with nonlinear function approximation. To further alleviate the periodic full gradient evaluation requirement, we further propose an enhanced method with an adaptive-batch adjustment called PILOT$^+$. The main advantages of our methods include: i) PILOT allows the use of {\em{constant}} step sizes and achieves the $\mathcal{O}(1/K)$ convergence rate to first-order stationary points of non-convex policy evaluation problems; ii) PILOT is a generic {\em{single}}-timescale algorithm that is also applicable for solving a large class of non-convex strongly-concave minimax optimization problems; iii) By adaptively adjusting the batch size via historical stochastic gradient information, PILOT$^+$ is more sample-efficient empirically without loss of theoretical convergence rate. Our extensive numerical experiments verify our theoretical findings and showcase the high efficiency of the proposed PILOT and PILOT$^+$ algorithms compared with the state-of-the-art methods.

Deep Learning · Theory

Linara Adilova, Maksym Andriushchenko, Michael Kamp, Asja Fischer, Martin Jaggi

Averaging neural network parameters is an intuitive method for fusing the knowledge of two independent models. It is most prominently used in federated learning. If models are averaged at the end of training, this can only lead to a good performing model if the loss surface of interest is very particular, i.e., the loss in the midpoint between the two models needs to be sufficiently low. This is impossible to guarantee for the non-convex losses of state-of-the-art networks. For averaging models trained on vastly different datasets, it was proposed to average only the parameters of particular layers or combinations of layers, resulting in better performing models. To get a better understanding of the effect of layer-wise averaging, we analyse the performance of the models that result from averaging single layers, or groups of layers. Based on our empirical and theoretical investigation, we introduce a novel notion of the layer-wise linear connectivity, and show that deep networks do not have layer-wise barriers between them.

Optimization · Non-Convex

Aaron Zweig, Joan Bruna

Few neural architectures lend themselves to provable learning with gradient based methods. One popular model is the single-index model, in which labels are produced by composing an unknown linear projection with a possibly unknown scalar link function. Learning this model with SGD is relatively well-understood, whereby the so-called information exponent of the link function governs a polynomial sample complexity rate. However, extending this analysis to deeper or more complicated architectures remains challenging.In this work, we consider single index learning in the setting of symmetric neural networks. Under analytic assumptions on the activation and maximum degree assumptions on the link function, we prove that gradient flow recovers the hidden planted direction, represented as a finitely supported vector in the feature space of power sum polynomials. We characterize a notion of information exponent adapted to our setting that controls the efficiency of learning.

Social Aspects · Trustworthy Machine Learning

Xiaosen Zheng, Tianyu Pang, Chao Du, Jing Jiang, Min Lin

Data attribution seeks to trace model outputs back to training data. With the recent development of diffusion models, data attribution has become a desired module to properly assign valuations for high-quality or copyrighted training samples, ensuring that data contributors are fairly compensated or credited. Several theoretically motivated methods have been proposed to implement data attribution, in an effort to improve the trade-off between computational scalability and effectiveness. In this work, we conduct extensive experiments and ablation studies on attributing diffusion models, specifically focusing on DDPMs trained on CIFAR-10 and CelebA, as well as a Stable Diffusion model LoRA-finetuned on ArtBench. Intriguingly, we report counter-intuitive observations that theoretically unjustified design choices for attribution empirically outperform previous baselines by a large margin, in terms of both linear datamodeling score and counterfactual evaluation. Our work presents a significantly more efficient approach for attributing diffusion models, while the unexpected findings suggest that at least in non-convex settings, constructions guided by theoretical assumptions may lead to inferior attribution performance. The code is available at https://github.com/sail-sg/D-TRAK.

Optimization · Zero-order and Black-box Optimization

Feiyang YE, YUEMING LYU, Xuehao Wang, Yu Zhang, Ivor Tsang

Multi-objective optimization (MOO) has become an influential framework for various machine learning problems, including reinforcement learning and multi-task learning. In this paper, we study the black-box multi-objective optimization problem, where we aim to optimize multiple potentially conflicting objectives with function queries only. To address this challenging problem and find a Pareto optimal solution or the Pareto stationary solution, we propose a novel adaptive stochastic gradient algorithm for black-box MOO, called ASMG. Specifically, we use the stochastic gradient approximation method to obtain the gradient for the distribution parameters of the Gaussian smoothed MOO with function queries only. Subsequently, an adaptive weight is employed to aggregate all stochastic gradients to optimize all objective functions effectively. Theoretically, we explicitly provide the connection between the original MOO problem and the corresponding Gaussian smoothed MOO problem and prove the convergence rate for the proposed ASMG algorithm in both convex and non-convex scenarios.Empirically, the proposed ASMG method achieves competitive performance on multiple numerical benchmark problems. Additionally, the state-of-the-art performance on the black-box multi-task learning problem demonstrates the effectiveness of the proposed ASMG method.

Theory · Domain Adaptation and Transfer Learning

Jiechao GUAN, Hui Xiong

Online-Within-Online (OWO) meta learning stands for the online multi-task learning paradigm in which both tasks and data within each task become available in a sequential order. In this work, we study the OWO meta learning of the initialization and step size of within-task online algorithms in the non-convex setting, and provide improved regret bounds under mild assumptions of loss functions. Previous work analyzing this scenario has obtained for bounded and piecewise Lipschitz functions an averaged regret bound $O((\frac{\sqrt{m}}{T^{1/4}}+\frac{(\log{m})\log{T}}{\sqrt{T}}+V)\sqrt{m})$ across $T$ tasks, with $m$ iterations per task and $V$ the task similarity. Our first contribution is to modify the existing non-convex OWO meta learning algorithm and improve the regret bound to $O((\frac{1}{T^{1/2-\alpha}}+\frac{(\log{T})^{9/2}}{T}+V)\sqrt{m})$, for any $\alpha \in (0,1/2)$. The derived bound has a faster convergence rate with respect to $T$, and guarantees a vanishing task-averaged regret with respect to $m$ (for any fixed $T$). Then, we propose a new algorithm of regret $O((\frac{\log{T}}{T}+V)\sqrt{m})$ for non-convex OWO meta learning. This regret bound exhibits a better asymptotic performance than previous ones, and holds for any bounded (not necessarily Lipschitz) loss functions. Besides the improved regret bounds, our contributions include investigating how to attain generalization bounds for statistical meta learning via regret analysis. Specifically, by online-to-batch arguments, we achieve a transfer risk bound for batch meta learning that assumes all tasks are drawn from a distribution. Moreover, by connecting multi-task generalization error with task-averaged regret, we develop for statistical multi-task learning a novel PAC-Bayes generalization error bound that involves our regret bound for OWO meta learning.

Optimization · Non-Convex

Zhiyuan Li, Yi Wang, Zhiren Wang

Normalization layers are ubiquitous in deep learning, greatly accelerating optimization. However, they also introduce many unexpected phenomena during training, for example, the Fast Equilibrium conjecture proposed by (Li et al.,2020), which states that the scale-invariant normalized network, when trained by SGD with $\eta$ learning rate and $\lambda$ weight decay, mixes to an equilibrium in $\tilde{O}(1/\eta\lambda)$ steps, as opposed to classical $e^{O(\eta^{-1})}$ mixing time. Recent works by Wang & Wang (2022); Li et al. (2022c) proved this conjecture under different sets of assumptions. This paper aims to answer the fast equilibrium conjecture in full generality by removing the non-generic assumptions of Wang & Wang (2022); Li et al. (2022c) that the minima are isolated, that the region near minima forms a unique basin, and that the set of minima is an analytic set. Our main technical contribution is to show that with probability close to 1, in exponential time trajectories will not escape the attracting basin containing its initial position.

Optimization · Non-Convex

Bo Zhao, Robert M. Gower, Robin Walters, Rose Yu

In many neural networks, different values of the parameters may result in the same loss value. Parameter space symmetries are loss-invariant transformations that change the model parameters. Teleportation applies such transformations to accelerate optimization. However, the exact mechanism behind this algorithm's success is not well understood. In this paper, we show that teleportation not only speeds up optimization in the short-term, but gives overall faster time to convergence. Additionally, teleporting to minima with different curvatures improves generalization, which suggests a connection between the curvature of the minimum and generalization ability. Finally, we show that integrating teleportation into a wide range of optimization algorithms and optimization-based meta-learning improves convergence. Our results showcase the versatility of teleportation and demonstrate the potential of incorporating symmetry in optimization.

Optimization · Large Scale, Parallel and Distributed

Yi-Rui Yang, Chang-Wei Shi, Wu-Jun Li

Byzantine-robust distributed learning (BRDL), in which computing devices are likely to behave abnormally due to accidental failures or malicious attacks, has recently become a hot research topic. However, even in the independent and identically distributed (i.i.d.) case, existing BRDL methods will suffer a significant drop on model accuracy due to the large variance of stochastic gradients. Increasing batch sizes is a simple yet effective way to reduce the variance. However, when the total number of gradient computation is fixed, a too-large batch size will lead to a too-small iteration number (update number), which may also degrade the model accuracy. In view of this challenge, we mainly study the effect of batch size when the total number of gradient computation is fixed in this work. In particular, we show that when the total number of gradient computation is fixed, the optimal batch size corresponding to the tightest theoretical upper bound in BRDL increases with the fraction of Byzantine workers. Therefore, compared to the case without attacks, a larger batch size is preferred when under Byzantine attacks. Motivated by the theoretical finding, we propose a novel method called Byzantine-robust stochastic gradient descent with normalized momentum (ByzSGDnm) in order to further increase model accuracy in BRDL. We theoretically prove the convergence of ByzSGDnm for general non-convex cases under Byzantine attacks. Empirical results show that when under Byzantine attacks, compared to the cases of small batch sizes, setting a relatively large batch size can significantly increase the model accuracy, which is consistent with our theoretical results. Moreover, ByzSGDnm can achieve higher model accuracy than existing BRDL methods when under deliberately crafted attacks. In addition, we empirically show that increasing batch sizes has the bonus of training acceleration.