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Theory/Optimization

Junyu Huang, Qilong Feng, Ziyun Huang, Jinhui Xu, Jianxin Wang

The local search methods have been widely used to solve the clustering problems. In practice, local search algorithms for clustering problems mainly adapt the single-swap strategy, which enables them to handle large-scale datasets and achieve linear running time in the data size. However, compared with multi-swap local search algorithms, there is a considerable gap on the approximation ratios of the single-swap local search algorithms. Although the current multi-swap local search algorithms provide small constant approximation, the proposed algorithms tend to have large polynomial running time, which cannot be used to handle large-scale datasets. In this paper, we propose a multi-swap local search algorithm for the $k$-means problem with linear running time in the data size. Given a swap size $t$, our proposed algorithm can achieve a $(50(1+\frac{1}{t})+\epsilon)$-approximation, which improves the current best result 509 (ICML 2019) with linear running time in the data size. Our proposed method, compared with previous multi-swap local search algorithms, is the first one to achieve linear running time in the data size. To obtain a more practical algorithm for the problem with better clustering quality and running time, we propose a sampling-based method which accelerates the process of clustering cost update during swaps. Besides, a recombination mechanism is proposed to find potentially better solutions. Empirical experiments show that our proposed algorithms achieve better performances compared with branch and bound solver (NeurIPS 2022) and other existing state-of-the-art local search algorithms on both small and large datasets.

Theory/Optimization

Ziyu Chen, Wei Zhu

We study the implicit bias of gradient flow on linear equivariant steerable networks in group-invariant binary classification. Our findings reveal that the parameterized predictor converges in direction to the unique group-invariant classifier with a maximum margin defined by the input group action. Under a unitary assumption on the input representation, we establish the equivalence between steerable networks and data augmentation. Furthermore, we demonstrate the improved margin and generalization bound of steerable networks over their non-invariant counterparts.

Theory/Optimization

Siddharth Gollapudi, Sepideh Mahabadi, Varun Sivashankar

Given a set of $n$ vectors in $\mathbb{R}^d$, the goal of the \emph{determinant maximization} problem is to pick $k$ vectors with the maximum volume. Determinant maximization is the MAP-inference task for determinantal point processes (DPP) and has recently received considerable attention for modeling diversity. As most applications for the problem use large amounts of data, this problem has been studied in the relevant \textit{composable coreset} setting.In particular, [Indyk-Mahabadi-OveisGharan-Rezaei--SODA'20, ICML'19] showed that one can get composable coresets with optimal approximation factor of $\tilde O(k)^k$ for the problem, and that a local search algorithm achieves an almost optimal approximation guarantee of $O(k)^{2k}$.In this work, we show that the widely-used Greedy algorithm also provides composable coresets with an almost optimal approximation factor of $O(k)^{3k}$, which improves over the previously known guarantee of $C^{k^2}$, and supports the prior experimental results showing the practicality of the greedy algorithm as a coreset.Our main result follows by showing a local optimality property for Greedy:swapping a single point from the greedy solution with a vector that was not picked by the greedy algorithm can increase the volume by a factor of at most $(1+\sqrt{k})$. This is tight up to the additive constant $1$. Finally, our experiments show that the local optimality of the greedy algorithm is even lower than the theoretical bound on real data sets.

Theory/Optimization

Matan Schliserman, Tomer Koren

We study the generalization properties of unregularized gradient methods applied to separable linear classification---a setting that has received considerable attention since the pioneering work of Soudry et al. (2018).We establish tight upper and lower (population) risk bounds for gradient descent in this setting, for any smooth loss function, expressed in terms of its tail decay rate.Our bounds take the form $\Theta(r_{\ell,T}^2 / \gamma^2 T + r_{\ell,T}^2 / \gamma^2 n)$, where $T$ is the number of gradient steps, $n$ is size of the training set, $\gamma$ is the data margin, and $r_{\ell,T}$ is a complexity term that depends on the tail decay rate of the loss function (and on $T$).Our upper bound greatly improves the existing risk bounds due to Shamir (2021) and Schliserman and Koren (2022), that either applied to specific loss functions or imposed extraneous technical assumptions, and applies to virtually any convex and smooth loss function.Our risk lower bound is the first in this context and establish the tightness of our general upper bound for any given tail decay rate and in all parameter regimes.The proof technique used to show these results is also markedly simpler compared to previous work, and is straightforward to extend to other gradient methods; we illustrate this by providing analogous results for Stochastic Gradient Descent.

Theory/Optimization

Yun Yue, Zhiling Ye, Jiadi Jiang, Yongchao Liu, Ke Zhang

Adaptive optimizers, such as Adam, have achieved remarkable success in deep learning. A key component of these optimizers is the so-called preconditioning matrix, providing enhanced gradient information and regulating the step size of each gradient direction. In this paper, we propose a novel approach to designing the preconditioning matrix by utilizing the gradient difference between two successive steps as the diagonal elements. These diagonal elements are closely related to the Hessian and can be perceived as an approximation of the inner product between the Hessian row vectors and difference of the adjacent parameter vectors. Additionally, we introduce an auto-switching function that enables the preconditioning matrix to switch dynamically between Stochastic Gradient Descent (SGD) and the adaptive optimizer. Based on these two techniques, we develop a new optimizer named AGD that enhances the generalization performance. We evaluate AGD on public datasets of Natural Language Processing (NLP), Computer Vision (CV), and Recommendation Systems (RecSys). Our experimental results demonstrate that AGD outperforms the state-of-the-art (SOTA) optimizers, achieving highly competitive or significantly better predictive performance. Furthermore, we analyze how AGD is able to switch automatically between SGD and the adaptive optimizer and its actual effects on various scenarios. The code is available at https://github.com/intelligent-machine-learning/dlrover/tree/master/atorch/atorch/optimizers.

Theory/Optimization

Ainesh Bakshi, Piotr Indyk, Rajesh Jayaram, Sandeep Silwal, Erik Waingarten

For any two point sets $A,B \subset \mathbb{R}^d$ of size up to $n$, the Chamfer distance from $A$ to $B$ is defined as $\texttt{CH}(A,B)=\sum_{a \in A} \min_{b \in B} d_X(a,b)$, where $d_X$ is the underlying distance measure (e.g., the Euclidean or Manhattan distance). The Chamfer distance is a popular measure of dissimilarity between point clouds, used in many machine learning, computer vision, and graphics applications, and admits a straightforward $O(d n^2)$-time brute force algorithm. Further, Chamfer distance is often used as a proxy for the more computationally demanding Earth-Mover (Optimal Transport) Distance. However, the \emph{quadratic} dependence on $n$ in the running time makes the naive approach intractable for large datasets.We overcome this bottleneck and present the first $(1+\epsilon)$-approximate algorithm for estimating Chamfer distance with a near-linear running time. Specifically, our algorithm runs in time $O(nd \log (n)/\epsilon^2)$ and is implementable. Our experiments demonstrate that it is both accurate and fast on large high-dimensional datasets. We believe that our algorithm will open new avenues for analyzing large high-dimensional point clouds. We also give evidence that if the goal is to report a $(1+\epsilon)$-approximate mapping from $A$ to $B$ (as opposed to just its value), then any sub-quadratic time algorithm is unlikely to exist.

Theory/Optimization

Junyi Li, Feihu Huang, Heng Huang

Bilevel Optimization has witnessed notable progress recently with new emerging efficient algorithms. However, its application in the Federated Learning setting remains relatively underexplored, and the impact of Federated Learning's inherent challenges on the convergence of bilevel algorithms remain obscure.In this work, we investigate Federated Bilevel Optimization problems and propose a communication-efficient algorithm, named FedBiOAcc. The algorithm leverages an efficient estimation of the hyper-gradient in the distributed setting and utilizes the momentum-based variance-reduction acceleration. Remarkably, FedBiOAcc achieves a communication complexity $O(\epsilon^{-1})$, a sample complexity $O(\epsilon^{-1.5})$ and the linear speed up with respect to the number of clients. We also analyze a special case of the Federated Bilevel Optimization problems, where lower level problems are locally managed by clients. We prove that FedBiOAcc-Local, a modified version of FedBiOAcc, converges at the same rate for this type of problems. Finally, we validate the proposed algorithms through two real-world tasks: Federated Data-cleaning and Federated Hyper-representation Learning. Empirical results show superior performance of our algorithms.

Theory/Optimization

Mingze Wang, Chao Ma

The training process of ReLU neural networks often exhibits complicated nonlinear phenomena. The nonlinearity of models and non-convexity of loss pose significant challenges for theoretical analysis. Therefore, most previous theoretical works on the optimization dynamics of neural networks focus either on local analysis (like the end of training) or approximate linear models (like Neural Tangent Kernel). In this work, we conduct a complete theoretical characterization of the training process of a two-layer ReLU network trained by Gradient Flow on a linearly separable data. In this specific setting, our analysis captures the whole optimization process starting from random initialization to final convergence. Despite the relatively simple model and data that we studied, we reveal four different phases from the whole training process showing a general simplifying-to-complicating learning trend.Specific nonlinear behaviors can also be precisely identified and captured theoretically, such asinitial condensation, saddle-to-plateau dynamics, plateau escape, changes of activation patterns, learning with increasing complexity, etc.

Theory/Optimization

Wenjing YAN, Xuanyu Cao

Performative prediction is a recently proposed framework where predictions guide decision-making and hence influence future data distributions. Such performative phenomena are ubiquitous in various areas, such as transportation, finance, public policy, and recommendation systems. To date, work on performative prediction has only focused on unconstrained problems, neglecting the fact that many real-world learning problems are subject to constraints. This paper bridges this gap by studying performative prediction under inequality constraints. Unlike most existing work that provides only performative stable points, we aim to find the optimal solutions. Anticipating performative gradient is a challenging task, due to the agnostic performative effect on data distributions. To address this issue, we first develop a robust primal-dual framework that requires only approximate gradients up to a certain accuracy, yet delivers the same order of performance as the stationary stochastic primal-dual algorithm without performativity. Based on this framework, we then propose an adaptive primal-dual algorithm for location families. Our analysis demonstrates that the proposed adaptive primal-dual algorithm attains $\mathcal{O}(\sqrt{T})$ regret and constraint violations, using only $\sqrt{T} + 2T$ samples, where $T$ is the time horizon. To our best knowledge, this is the first study and analysis on the optimality of the performative prediction problem under inequality constraints. Finally, we validate the effectiveness of our algorithm and theoretical results through numerical simulations.

Theory/Optimization

Miaoxi Zhu, Li Shen, Bo Du, Dacheng Tao

The growing size of available data has attracted increasing interest in solving minimax problems in a decentralized manner for various machine learning tasks. Previous theoretical research has primarily focused on the convergence rate and communication complexity of decentralized minimax algorithms, with little attention given to their generalization. In this paper, we investigate the primal-dual generalization bound of the decentralized stochastic gradient descent ascent (D-SGDA) algorithm using the approach of algorithmic stability under both convex-concave and nonconvex-nonconcave settings. Our theory refines the algorithmic stability in a decentralized manner and demonstrates that the decentralized structure does not destroy the stability and generalization of D-SGDA, implying that it can generalize as well as the vanilla SGDA in certain situations. Our results analyze the impact of different topologies on the generalization bound of the D-SGDA algorithm beyond trivial factors such as sample sizes, learning rates, and iterations. We also evaluate the optimization error and balance it with the generalization gap to obtain the optimal population risk of D-SGDA in the convex-concave setting. Additionally, we perform several numerical experiments which validate our theoretical findings.

Theory/Optimization

Nived Rajaraman, Fnu Devvrit, Aryan Mokhtari, Kannan Ramchandran

Pruning schemes have been widely used in practice to reduce the complexity of trained models with a massive number of parameters. In fact, several practical studies have shown that if the pruned model is fine-tuned with some gradient-based updates it generalizes well to new samples. Although the above pipeline, which we refer to as pruning + fine-tuning, has been extremely successful in lowering the complexity of trained models, there is very little known about the theory behind this success. In this paper we address this issue by investigating the pruning + fine-tuning framework on the overparameterized matrix sensing problem with the ground truth denoted $U_\star \in \mathbb{R}^{d \times r}$ and the overparameterized model $U \in \mathbb{R}^{d \times k}$ with $k \gg r$. We study the approximate local minima of the mean square error, augmented with a smooth version of a group Lasso regularizer, $\sum_{i=1}^{k} \lVert Ue_i \rVert_2 $. In particular, we provably show that pruning all the columns below a certain explicit $\ell_2$-norm threshold results in a solution $U_{\text{prune}}$ which has the minimum number of columns $r$, yet close to the ground truth in training loss. Moreover, in the subsequent fine-tuning phase, gradient descent initialized at $U_{\text{prune}}$ converges at a linear rate to its limit. While our analysis provides insights into the role of regularization in pruning, we also show that running gradient descent in the absence of regularization results in models which {are not suitable for greedy pruning}, i.e., many columns could have their $\ell_2$ norm comparable to that of the maximum. Lastly, we show that our results also extend for the training and pruning of two-layer neural networks with quadratic activation functions. To the best of our knowledge, our results provide the first rigorous insights on why greedy pruning + fine-tuning leads to smaller models which also generalize well.

Theory/Optimization

Mina Dalirrooyfard, Slobodan Mitrovic, Yuriy Nevmyvaka

Finding min $s$-$t$ cuts in graphs is a basic algorithmic tool, with applications in image segmentation, community detection, reinforcement learning, and data clustering. In this problem, we are given two nodes as terminals and the goal is to remove the smallest number of edges from the graph so that these two terminals are disconnected. We study the complexity of differential privacy for the min $s$-$t$ cut problem and show nearly tight lower and upper bounds where we achieve privacy at no cost for running time efficiency. We also develop a differentially private algorithm for the multiway $k$-cut problem, in which we are given $k$ nodes as terminals that we would like to disconnect. As a function of $k$, we obtain privacy guarantees that are exponentially more efficient than applying the advanced composition theorem to known algorithms for multiway $k$-cut. Finally, we empirically evaluate the approximation of our differentially private min $s$-$t$ cut algorithm and show that it almost matches the quality of the output of non-private ones.

Theory/Optimization

Xingjian Bai, Christian Coester

We explore the fundamental problem of sorting through the lens of learning-augmented algorithms, where algorithms can leverage possibly erroneous predictions to improve their efficiency. We consider two different settings: In the first setting, each item is provided a prediction of its position in the sorted list. In the second setting, we assume there is a ``quick-and-dirty'' way of comparing items, in addition to slow-and-exact comparisons. For both settings, we design new and simple algorithms using only $O(\sum_i \log \eta_i)$ exact comparisons, where $\eta_i$ is a suitably defined prediction error for the $i$th element. In particular, as the quality of predictions deteriorates, the number of comparisons degrades smoothly from $O(n)$ to $O(n\log n)$. We prove that this comparison complexity is theoretically optimal with respect to the examined error measures. An experimental evaluation against existing adaptive and non-adaptive sorting algorithms demonstrates the potential of applying learning-augmented algorithms in sorting tasks.

Theory/Optimization

Daesung Kim, Hye Won Chung

The nonconvex formulation of the matrix completion problem has received significant attention in recent years due to its affordable complexity compared to the convex formulation. Gradient Descent (GD) is a simple yet efficient baseline algorithm for solving nonconvex optimization problems. The success of GD has been witnessed in many different problems in both theory and practice when it is combined with random initialization. However, previous works on matrix completion require either careful initialization or regularizers to prove the convergence of GD. In this paper, we study the rank-1 symmetric matrix completion and prove that GD converges to the ground truth when small random initialization is used. We show that in a logarithmic number of iterations, the trajectory enters the region where local convergence occurs. We provide an upper bound on the initialization size that is sufficient to guarantee the convergence, and show that a larger initialization can be used as more samples are available. We observe that the implicit regularization effect of GD plays a critical role in the analysis, and for the entire trajectory, it prevents each entry from becoming much larger than the others.

Theory/Optimization

Deeparnab Chakrabarty, Andrei Graur, Haotian Jiang, Aaron Sidford

We consider the parallel complexity of submodular function minimization (SFM). We provide a pair of methods which obtain two new query versus depth trade-offs a submodular function defined on subsets of $n$ elements that has integer values between $-M$ and $M$. The first method has depth $2$ and query complexity $n^{O(M)}$ and the second method has depth $\widetilde{O}(n^{1/3} M^{2/3})$ and query complexity $O(\mathrm{poly}(n, M))$. Despite a line of work on improved parallel lower bounds for SFM, prior to our work the only known algorithms for parallel SFM either followed from more general methods for sequential SFM or highly-parallel minimization of convex $\ell_2$-Lipschitz functions. Interestingly, to obtain our second result we provide the first highly-parallel algorithm for minimizing $\ell_\infty$-Lipschitz function over the hypercube which obtains near-optimal depth for obtaining constant accuracy.

Theory/Optimization

Rajat Vadiraj Dwaraknath, Ishani Karmarkar, Aaron Sidford

We provide new algorithms and conditional hardness for the problem of estimating effective resistances in $n$-node $m$-edge undirected, expander graphs. We provide an $\widetilde{O}(m\epsilon^{-1})$-time algorithm that produces with high probability, an $\widetilde{O}(n\epsilon^{-1})$-bit sketch from which the effective resistance between any pair of nodes can be estimated, to $(1 \pm \epsilon)$-multiplicative accuracy, in $\widetilde{O}(1)$-time. Consequently, we obtain an $\widetilde{O}(m\epsilon^{-1})$-time algorithm for estimating the effective resistance of all edges in such graphs, improving (for sparse graphs) on the previous fastest runtimes of $\widetilde{O}(m\epsilon^{-3/2})$ [Chu et. al. 2018] and $\widetilde{O}(n^2\epsilon^{-1})$ [Jambulapati, Sidford, 2018] for general graphs and $\widetilde{O}(m + n\epsilon^{-2})$ for expanders [Li, Sachdeva 2022]. We complement this result by showing a conditional lower bound that a broad set of algorithms for computing such estimates of the effective resistances between all pairs of nodes require $\widetilde{\Omega}(n^2 \epsilon^{-1/2})$-time, improving upon the previous best such lower bound of $\widetilde{\Omega}(n^2 \epsilon^{-1/13})$ [Musco et. al. 2017]. Further, we leverage the tools underlying these results to obtain improved algorithms and conditional hardness for more general problems of sketching the pseudoinverse of positive semidefinite matrices and estimating functions of their eigenvalues.

Theory/Optimization

Jiangtao Zhang, Shunyu Liu, Jie Song, Tongtian Zhu, Zhengqi Xu, Mingli Song

Weight Average (WA) is an active research topic due to its simplicity in ensembling deep networks and the effectiveness in promoting generalization. Existing weight average approaches, however, are often carried out along only one training trajectory in a post-hoc manner (i.e., the weights are averaged after the entire training process is finished), which significantly degrades the diversity between networks and thus impairs the effectiveness. In this paper, inspired by weight average, we propose Lookaround, a straightforward yet effective SGD-based optimizer leading to flatter minima with better generalization. Specifically, Lookaround iterates two steps during the whole training period: the around step and the average step. In each iteration, 1) the around step starts from a common point and trains multiple networks simultaneously, each on transformed data by a different data augmentation, and 2) the average step averages these trained networks to get the averaged network, which serves as the starting point for the next iteration. The around step improves the functionality diversity while the average step guarantees the weight locality of these networks during the whole training, which is essential for WA to work. We theoretically explain the superiority of Lookaround by convergence analysis, and make extensive experiments to evaluate Lookaround on popular benchmarks including CIFAR and ImageNet with both CNNs and ViTs, demonstrating clear superiority over state-of-the-arts. Our code is available at https://github.com/Ardcy/Lookaround.

Theory/Optimization

Syamantak Kumar, Purnamrita Sarkar

Since its inception in 1982, Oja's algorithm has become an established method for streaming principle component analysis (PCA). We study the problem of streaming PCA, where the data-points are sampled from an irreducible, aperiodic, and reversible Markov chain starting in stationarity. Our goal is to estimate the top eigenvector of the unknown covariance matrix of the stationary distribution. This setting has implications in scenarios where data can solely be sampled from a Markov Chain Monte Carlo (MCMC) type algorithm, and the objective is to perform inference on parameters of the stationary distribution. Most convergence guarantees for Oja's algorithm in the literature assume that the data-points are sampled IID. For data streams with Markovian dependence, one typically downsamples the data to get a "nearly" independent data stream. In this paper, we obtain the first near-optimal rate for Oja's algorithm on the entire data, where we remove the logarithmic dependence on the sample size, $n$, resulting from throwing data away in downsampling strategies.

Theory/Optimization

Dongkuk Si, Chulhee Yun

Sharpness-Aware Minimization (SAM) is an optimizer that takes a descent step based on the gradient at a perturbation $y_t = x_t + \rho \frac{\nabla f(x_t)}{\lVert \nabla f(x_t) \rVert}$ of the current point $x_t$. Existing studies prove convergence of SAM for smooth functions, but they do so by assuming decaying perturbation size $\rho$ and/or no gradient normalization in $y_t$, which is detached from practice. To address this gap, we study deterministic/stochastic versions of SAM with practical configurations (i.e., constant $\rho$ and gradient normalization in $y_t$) and explore their convergence properties on smooth functions with (non)convexity assumptions.Perhaps surprisingly, in many scenarios, we find out that SAM has limited capability to converge to global minima or stationary points.For smooth strongly convex functions, we show that while deterministic SAM enjoys tight global convergence rates of $\tilde \Theta(\frac{1}{T^2})$, the convergence bound of stochastic SAM suffers an inevitable additive term $\mathcal O(\rho^2)$, indicating convergence only up to neighborhoods of optima.In fact, such $\mathcal O(\rho^2)$ factors arise for stochastic SAM in all the settings we consider, and also for deterministic SAM in nonconvex cases; importantly, we prove by examples that such terms are unavoidable.Our results highlight vastly different characteristics of SAM with vs. without decaying perturbation size or gradient normalization, and suggest that the intuitions gained from one version may not apply to the other.

Theory/Optimization

Gregory Dexter, Petros Drineas, David Woodruff, Taisuke Yasuda

Sketching algorithms have recently proven to be a powerful approach both for designing low-space streaming algorithms as well as fast polynomial time approximation schemes (PTAS). In this work, we develop new techniques to extend the applicability of sketching-based approaches to the sparse dictionary learning and the Euclidean $k$-means clustering problems. In particular, we initiate the study of the challenging setting where the dictionary/clustering assignment for each of the $n$ input points must be output, which has surprisingly received little attention in prior work. On the fast algorithms front, we obtain a new approach for designing PTAS's for the $k$-means clustering problem, which generalizes to the first PTAS for the sparse dictionary learning problem. On the streaming algorithms front, we obtain new upper bounds and lower bounds for dictionary learning and $k$-means clustering. In particular, given a design matrix $\mathbf A\in\mathbb R^{n\times d}$ in a turnstile stream, we show an $\tilde O(nr/\epsilon^2 + dk/\epsilon)$ space upper bound for $r$-sparse dictionary learning of size $k$, an $\tilde O(n/\epsilon^2 + dk/\epsilon)$ space upper bound for $k$-means clustering, as well as an $\tilde O(n)$ space upper bound for $k$-means clustering on random order row insertion streams with a natural "bounded sensitivity" assumption. On the lower bounds side, we obtain a general $\tilde\Omega(n/\epsilon + dk/\epsilon)$ lower bound for $k$-means clustering, as well as an $\tilde\Omega(n/\epsilon^2)$ lower bound for algorithms which can estimate the cost of a single fixed set of candidate centers.