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Social Aspects/Privacy-preserving Statistics and Machine Learning

Arun Ganesh, Mahdi Haghifam, Thomas Steinke, Abhradeep Guha Thakurta

Differentially private (stochastic) gradient descent is the workhorse of DP private machine learning in both the convex and non-convex settings. Without privacy constraints, second-order methods, like Newton's method, converge faster than first-order methods like gradient descent. In this work, we investigate the prospect of using the second-order information from the loss function to accelerate DP convex optimization. We first develop a private variant of the regularized cubic Newton method of Nesterov and Polyak, and show that for the class of strongly convex loss functions, our algorithm has quadratic convergence and achieves the optimal excess loss. We then design a practical second-order DP algorithm for the unconstrained logistic regression problem. We theoretically and empirically study the performance of our algorithm. Empirical results show our algorithm consistently achieves the best excess loss compared to other baselines and is 10-40x faster than DP-GD/DP-SGD for challenging datasets.

Dachao Lin, Yuze Han, Haishan Ye, Zhihua Zhang

We study finite-sum distributed optimization problems involving a master node and $n-1$ local nodes under the popular $\delta$-similarity and $\mu$-strong convexity conditions. We propose two new algorithms, SVRS and AccSVRS, motivated by previous works. The non-accelerated SVRS method combines the techniques of gradient sliding and variance reduction and achieves a better communication complexity of $\tilde{\mathcal{O}}(n {+} \sqrt{n}\delta/\mu)$ compared to existing non-accelerated algorithms. Applying the framework proposed in Katyusha X, we also develop a directly accelerated version named AccSVRS with the $\tilde{\mathcal{O}}(n {+} n^{3/4}\sqrt{\delta/\mu})$ communication complexity. In contrast to existing results, our complexity bounds are entirely smoothness-free and exhibit superiority in ill-conditioned cases. Furthermore, we establish a nearly matched lower bound to verify the tightness of our AccSVRS method.

Optimization/Stochastic

Angela Yuan, Chris Junchi Li, Gauthier Gidel, Michael Jordan, Quanquan Gu, Simon Du

We consider the problem of solving stochastic monotone variational inequalities with a separable structure using a stochastic first-order oracle. Building on standard extragradient for variational inequalities we propose a novel algorithm---stochastic \emph{accelerated gradient-extragradient} (AG-EG)---for strongly monotone variational inequalities (VIs). Our approach combines the strengths of extragradient and Nesterov acceleration. By showing that its iterates remain in a bounded domain and applying scheduled restarting, we prove that AG-EG has an optimal convergence rate for strongly monotone VIs. Furthermore, when specializing to the particular case of bilinearly coupled strongly-convex-strongly-concave saddle-point problems, including bilinear games, our algorithm achieves fine-grained convergence rates that match the respective lower bounds, with the stochasticity being characterized by an additive statistical error term that is optimal up to a constant prefactor.

Optimization/Large Scale, Parallel and Distributed

Wenzhi Gao, Qi Deng

This paper studies delayed stochastic algorithms for weakly convex optimization in a distributed network with workers connected to a master node. Recently, Xu~et~al.~2022 showed that an inertial stochastic subgradient method converges at a rate of $\mathcal{O}(\tau_{\text{max}}/\sqrt{K})$ which depends on the maximum information delay $\tau_{\text{max}}$. In this work, we show that the delayed stochastic subgradient method ($\texttt{DSGD}$) obtains a tighter convergence rate which depends on the expected delay $\bar{\tau}$. Furthermore, for an important class of composition weakly convex problems, we develop a new delayed stochastic prox-linear ($\texttt{DSPL}$) method in which the delays only affect the high-order term in the rate and hence, are negligible after a certain number of $\texttt{DSPL}$ iterations. In addition, we demonstrate the robustness of our proposed algorithms against arbitrary delays. By incorporating a simple safeguarding step in both methods, we achieve convergence rates that depend solely on the number of workers, eliminating the effect of delays. Our numerical experiments further confirm the empirical superiority of our proposed methods.

Optimization

Lam Nguyen, Trang H. Tran

Stochastic gradient descent (SGD) algorithm is the method of choice in many machine learning tasks thanks to its scalability and efficiency in dealing with large-scale problems. In this paper, we focus on the shuffling version of SGD which matches the mainstream practical heuristics. We show the convergence to a global solution of shuffling SGD for a class of non-convex functions under over-parameterized settings. Our analysis employs more relaxed non-convex assumptions than previous literature. Nevertheless, we maintain the desired computational complexity as shuffling SGD has achieved in the general convex setting.

Miscellaneous Aspects of Machine Learning/Online Learning, Active Learning and Bandits

Abishek Sankararaman, Balakrishnan Narayanaswamy

The real-time estimation of time-varying parameters from high-dimensional, heavy-tailed and corrupted data-streams is a common sub-routine in systems ranging from those for network monitoring and anomaly detection to those for traffic scheduling in data-centers. For estimation tasks that can be cast as minimizing a strongly convex loss function, we prove that an appropriately tuned version of the {\ttfamily clipped Stochastic Gradient Descent} (SGD) is simultaneously {\em(i)} adaptive to drift, {\em (ii)} robust to heavy-tailed inliers and arbitrary corruptions, {\em(iii)} requires no distributional knowledge and {\em (iv)} can be implemented in an online streaming fashion. All prior estimation algorithms have only been proven to posses a subset of these practical desiderata. A observation we make is that, neither the $\mathcal{O}\left(\frac{1}{t}\right)$ learning rate for {\ttfamily clipped SGD} known to be optimal for strongly convex loss functions of a \emph{stationary} data-stream, nor the $\mathcal{O}(1)$ learning rate known to be optimal for being adaptive to drift in a \emph{noiseless} environment can be used. Instead, a learning rate of $T^{-\alpha}$ for $ \alpha < 1$ where $T$ is the stream-length is needed to balance adaptivity to potential drift and to combat noise. We develop a new inductive argument and combine it with a martingale concentration result to derive high-probability under \emph{any learning rate} on data-streams exhibiting \emph{arbitrary distribution shift} - a proof strategy that may be of independent interest. Further, using the classical doubling-trick, we relax the knowledge of the stream length $T$. Ours is the first online estimation algorithm that is provably robust to heavy-tails, corruptions and distribution shift simultaneously. We complement our theoretical results empirically on synthetic and real data.

Theory/Learning Theory

Futoshi Futami, Masahiro Fujisawa

We provide novel information-theoretic generalization bounds for stochastic gradient Langevin dynamics (SGLD) under the assumptions of smoothness and dissipativity, which are widely used in sampling and non-convex optimization studies.Our bounds are time-independent and decay to zero as the sample size increases, regardless of the number of iterations and whether the step size is fixed.Unlike previous studies, we derive the generalization error bounds by focusing on the time evolution of the Kullback--Leibler divergence, which is related to the stability of datasets and is the upper bound of the mutual information between output parameters and an input dataset.Additionally, we establish the first information-theoretic generalization bound when the training and test loss are the same by showing that a loss function of SGLD is sub-exponential.This bound is also time-independent and removes the problematic step size dependence in existing work, leading to an improved excess risk bound by combining our analysis with the existing non-convex optimization error bounds.

Optimization/Convex

Aleksandr Beznosikov, Sergey Samsonov, Marina Sheshukova, Alexander Gasnikov, Alexey Naumov, Eric Moulines

This paper delves into stochastic optimization problems that involve Markovian noise. We present a unified approach for the theoretical analysis of first-order gradient methods for stochastic optimization and variational inequalities. Our approach covers scenarios for both non-convex and strongly convex minimization problems. To achieve an optimal (linear) dependence on the mixing time of the underlying noise sequence, we use the randomized batching scheme, which is based on the multilevel Monte Carlo method. Moreover, our technique allows us to eliminate the limiting assumptions of previous research on Markov noise, such as the need for a bounded domain and uniformly bounded stochastic gradients. Our extension to variational inequalities under Markovian noise is original. Additionally, we provide lower bounds that match the oracle complexity of our method in the case of strongly convex optimization problems.

Social Aspects/Privacy-preserving Statistics and Machine Learning

Daogao Liu, Arun Ganesh, Sewoong Oh, Abhradeep Guha Thakurta

We reconsider the challenge of non-convex optimization under differential privacy constraint. Building upon the previous variance-reduced algorithm SpiderBoost, we propose a novel framework that employs two types of gradient oracles: one that estimates the gradient at a single point and a more cost-effective option that calculates the gradient difference between two points. Our framework can ensure continuous accuracy of gradient estimations and subsequently enhances the rates of identifying second-order stationary points.Additionally, we consider a more challenging task by attempting to locate the global minima of a non-convex objective via the exponential mechanism without almost any assumptions. Our preliminary results suggest that the regularized exponential mechanism can effectively emulate previous empirical and population risk bounds, negating the need for smoothness assumptions for algorithms with polynomial running time. Furthermore, with running time factors excluded, the exponential mechanism demonstrates promising population risk bound performance, and we provide a nearly matching lower bound.

Theory/Learning Theory

Xiaotong Yuan, Ping Li

Exponential generalization bounds with near-optimal rates have recently been established for uniformly stable algorithms~\citep{feldman2019high,bousquet2020sharper}. We seek to extend these best known high probability bounds from deterministic learning algorithms to the regime of randomized learning. One simple approach for achieving this goal is to define the stability for the expectation over the algorithm's randomness, which may result in sharper parameter but only leads to guarantees regarding the on-average generalization error. Another natural option is to consider the stability conditioned on the algorithm's randomness, which is way more stringent but may lead to generalization with high probability jointly over the randomness of sample and algorithm. The present paper addresses such a tension between these two alternatives and makes progress towards relaxing it inside a classic framework of confidence-boosting. To this end, we first introduce a novel concept of $L_2$-uniform stability that holds uniformly over data but in second-moment over the algorithm's randomness. Then as a core contribution of this work, we prove a strong exponential bound on the first-moment of generalization error under the notion of $L_2$-uniform stability. As an interesting consequence of the bound, we show that a bagging-based meta algorithm leads to near-optimal generalization with high probability jointly over the randomness of data and algorithm. We further substantialize these generic results to stochastic gradient descent (SGD) to derive sharper exponential bounds for convex or non-convex optimization with natural time-decaying learning rates, which have not been possible to prove with the existing stability-based generalization guarantees.

Optimization/Bilevel optimization

Jincheng Cao, Ruichen Jiang, Nazanin Abolfazli, Erfan Yazdandoost Hamedani, Aryan Mokhtari

In this paper, we study a class of stochastic bilevel optimization problems, also known as stochastic simple bilevel optimization, where we minimize a smooth stochastic objective function over the optimal solution set of another stochastic convex optimization problem. We introduce novel stochastic bilevel optimization methods that locally approximate the solution set of the lower-level problem via a stochastic cutting plane, and then run a conditional gradient update with variance reduction techniques to control the error induced by using stochastic gradients. For the case that the upper-level function is convex, our method requires $\mathcal{O}(\max\\{1/\epsilon_f^{2},1/\epsilon_g^{2}\\}) $ stochastic oracle queries to obtain a solution that is $\epsilon_f$-optimal for the upper-level and $\epsilon_g$-optimal for the lower-level. This guarantee improves the previous best-known complexity of $\mathcal{O}(\max\\{1/\epsilon_f^{4},1/\epsilon_g^{4}\\})$. Moreover, for the case that the upper-level function is non-convex, our method requires at most $\mathcal{O}(\max\\{1/\epsilon_f^{3},1/\epsilon_g^{3}\\}) $ stochastic oracle queries to find an $(\epsilon_f, \epsilon_g)$-stationary point. In the finite-sum setting, we show that the number of stochastic oracle calls required by our method are $\mathcal{O}(\sqrt{n}/\epsilon)$ and $\mathcal{O}(\sqrt{n}/\epsilon^{2})$ for the convex and non-convex settings, respectively, where $\epsilon=\min \\{\epsilon_f,\epsilon_g\\}$.

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/Learning Theory

Roi Livni

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

Optimization/Non-Convex

Ya-Ping Hsieh, Mohammad Reza Karimi Jaghargh, Andreas Krause, Panayotis Mertikopoulos

Many modern machine learning applications - from online principal component analysis to covariance matrix identification and dictionary learning - can be formulated as minimization problems on Riemannian manifolds, typically solved with a Riemannian stochastic gradient method (or some variant thereof). However, in many cases of interest, the resulting minimization problem is _not_ geodesically convex, so the convergence of the chosen solver to a desirable solution - i.e., a local minimizer - is by no means guaranteed. In this paper, we study precisely this question, that is, whether stochastic Riemannian optimization algorithms are guaranteed to avoid saddle points with probability $1$. For generality, we study a family of retraction-based methods which, in addition to having a potentially much lower per-iteration cost relative to Riemannian gradient descent, include other widely used algorithms, such as natural policy gradient methods and mirror descent in ordinary convex spaces. In this general setting, we show that, under mild assumptions for the ambient manifold and the oracle providing gradient information, the policies under study avoid strict saddle points / submanifolds with probability $1$, from any initial condition. This result provides an important sanity check for the use of gradient methods on manifolds as it shows that, almost always, the end state of a stochastic Riemannian algorithm can only be a local minimizer.

Optimization/Sampling and Optimization

Mohammad Reza Karimi Jaghargh, Ya-Ping Hsieh, Andreas Krause

Non-convex sampling is a key challenge in machine learning, central to non-convex optimization in deep learning as well as to approximate probabilistic inference. Despite its significance, theoretically there remain some important challenges: Existing guarantees suffer from the drawback of lacking guarantees for the last-iterates, and little is known beyond the elementary schemes of stochastic gradient Langevin dynamics. To address these issues, we develop a novel framework that lifts the above issues by harnessing several tools from the theory of dynamical systems. Our key result is that, for a large class of state-of-the-art sampling schemes, their last-iterate convergence in Wasserstein distances can be reduced to the study of their continuous-time counterparts, which is much better understood. Coupled with standard assumptions of MCMC sampling, our theory immediately yields the last-iterate Wasserstein convergence of many advanced sampling schemes such as mirror Langevin, proximal, randomized mid-point, and Runge-Kutta methods.

Optimization/Convex

Liang Zhang, Junchi YANG, Amin Karbasi, Niao He

Algorithmic reproducibility measures the deviation in outputs of machine learning algorithms upon minor changes in the training process. Previous work suggests that first-order methods would need to trade-off convergence rate (gradient complexity) for better reproducibility. In this work, we challenge this perception and demonstrate that both optimal reproducibility and near-optimal convergence guarantees can be achieved for smooth convex minimization and smooth convex-concave minimax problems under various error-prone oracle settings. Particularly, given the inexact initialization oracle, our regularization-based algorithms achieve the best of both worlds -- optimal reproducibility and near-optimal gradient complexity -- for minimization and minimax optimization. With the inexact gradient oracle, the near-optimal guarantees also hold for minimax optimization. Additionally, with the stochastic gradient oracle, we show that stochastic gradient descent ascent is optimal in terms of both reproducibility and gradient complexity. We believe our results contribute to an enhanced understanding of the reproducibility-convergence trade-off in the context of convex optimization.

Theory/Learning Theory

Ziqiao Wang, Yongyi Mao

We present new information-theoretic generalization guarantees through the a novel construction of the "neighboring-hypothesis" matrix and a new family of stability notions termed sample-conditioned hypothesis (SCH) stability. Our approach yields sharper bounds that improve upon previous information-theoretic bounds in various learning scenarios. Notably, these bounds address the limitations of existing information-theoretic bounds in the context of stochastic convex optimization (SCO) problems, as explored in the recent work by Haghifam et al. (2023).

Optimization/Stochastic

Ta Duy Nguyen, Thien H Nguyen, Alina Ene, Huy Nguyen

In this work, we study the convergence in high probability of clipped gradient methods when the noise distribution has heavy tails, i.e., with bounded $p$th moments, for some $1<p\le2$. Prior works in this setting follow the same recipe of using concentration inequalities and an inductive argument with union bound to bound the iterates across all iterations. This method results in an increase in the failure probability by a factor of $T$, where $T$ is the number of iterations. We instead propose a new analysis approach based on bounding the moment generating function of a well chosen supermartingale sequence. We improve the dependency on $T$ in the convergence guarantee for a wide range of algorithms with clipped gradients, including stochastic (accelerated) mirror descent for convex objectives and stochastic gradient descent for nonconvex objectives. Our high probability bounds achieve the optimal convergence rates and match the best currently known in-expectation bounds. Our approach naturally allows the algorithms to use time-varying step sizes and clipping parameters when the time horizon is unknown, which appears difficult or even impossible using the techniques from prior works. Furthermore, we show that in the case of clipped stochastic mirror descent, several problem constants, including the initial distance to the optimum, are not required when setting step sizes and clipping parameters.

Theory/Online Learning and Bandits

Hamish Flynn, David Reeb, Melih Kandemir, Jan Peters

We present improved algorithms with worst-case regret guarantees for the stochastic linear bandit problem. The widely used "optimism in the face of uncertainty" principle reduces a stochastic bandit problem to the construction of a confidence sequence for the unknown reward function. The performance of the resulting bandit algorithm depends on the size of the confidence sequence, with smaller confidence sets yielding better empirical performance and stronger regret guarantees. In this work, we use a novel tail bound for adaptive martingale mixtures to construct confidence sequences which are suitable for stochastic bandits. These confidence sequences allow for efficient action selection via convex programming. We prove that a linear bandit algorithm based on our confidence sequences is guaranteed to achieve competitive worst-case regret. We show that our confidence sequences are tighter than competitors, both empirically and theoretically. Finally, we demonstrate that our tighter confidence sequences give improved performance in several hyperparameter tuning tasks.

Deep Learning/Theory

Taiji Suzuki, Denny Wu, Atsushi Nitanda

The mean-field Langevin dynamics (MFLD) is a nonlinear generalization of the Langevin dynamics that incorporates a distribution-dependent drift, and it naturally arises from the optimization of two-layer neural networks via (noisy) gradient descent. Recent works have shown that MFLD globally minimizes an entropy-regularized convex functional in the space of measures. However, all prior analyses assumed the infinite-particle or continuous-time limit, and cannot handle stochastic gradient updates. We provide a general framework to prove a uniform-in-time propagation of chaos for MFLD that takes into account the errors due to finite-particle approximation, time-discretization, and stochastic gradient. To demonstrate the wide applicability of our framework, we establish quantitative convergence rate guarantees to the regularized global optimal solution for $(i)$ a wide range of learning problems such as mean-field neural network and MMD minimization, and $(ii)$ different gradient estimators including SGD and SVRG. Despite the generality of our results, we achieve an improved convergence rate in both the SGD and SVRG settings when specialized to the standard Langevin dynamics.