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Optimization · Convex

Paul Mangold, Alain Oliviero Durmus, Aymeric Dieuleveut, Eric Moulines

This paper proposes a novel analysis for the Scaffold algorithm, a popular method for dealing with data heterogeneity in federated learning. While its convergence in deterministic settings—where local control variates mitigate client drift—is well established, the impact of stochastic gradient updates on its performance is less understood. To address this problem, we first show that its global parameters and control variates define a Markov chain that converges to a stationary distribution in the Wasserstein distance. Leveraging this result, we prove that Scaffold achieves linear speed-up in the number of clients up to higher-order terms in the step size. Nevertheless, our analysis reveals that Scaffold retains a higher-order bias, similar to FedAvg, that does not decrease as the number of clients increases. This highlights opportunities for developing improved stochastic federated learning algorithms.

Social Aspects · Security

Yu-Wei Chen, Raghu Pasupathy, Jordan A Awan

This work identifies the first privacy-aware stratified sampling scheme that minimizes the variance for general private mean estimation under the Laplace, Discrete Laplace (DLap) and Truncated-Uniform-Laplace (TuLap) mechanisms within the framework of differential privacy (DP). We view stratified sampling as a subsampling operation, which amplifies the privacy guarantee; however, to have the same final privacy guarantee for each group, different nominal privacy budgets need to be used depending on the subsampling rate. Ignoring the effect of DP, traditional stratified sampling strategies risk significant variance inflation. We phrase our optimal survey design as an optimization problem, where we determine the optimal subsampling sizes for each group with the goal of minimizing the variance of the resulting estimator. We establish strong convexity of the variance objective, propose an efficient algorithm to identify the integer-optimal design, and offer insights on the structure of the optimal design.

Optimization

Lisa Jin, Jianhao Ma, Zechun Liu, Andrey Gromov, Aaron Defazio, Lin Xiao

We develop a novel optimization method for quantization-aware training (QAT). Specifically, we show that convex, piecewise-affine regularization (PAR) can effectively induce neural network weights to cluster towards discrete values. We minimize PAR-regularized loss functions using an aggregate proximal stochastic gradient method (AProx) and prove that it enjoys last-iterate convergence. Our approach provides an interpretation of the straight-through estimator (STE), a widely used heuristic for QAT, as the asymptotic form of PARQ. We conduct experiments to demonstrate that PARQ obtains competitive performance on convolution- and transformer-based vision tasks.

Optimization · Everything Else

Yan Liu, Mingjie Chen, Chaojie Ji, Hao Zhang, Ruxin Wang

Recently, the expansion of Variance Reduction (VR) to Riemannian stochastic non-convex optimization has attracted increasing interest. Inspired by recursive momentum, we first introduce Stochastic Recursive Variance Reduced Gradient (SRVRG) algorithm and further present Stochastic Recursive Gradient Estimator (SRGE) in Euclidean spaces, which unifies the prevailing variance reduction estimators. We then extend SRGE to Riemannian spaces, resulting in a unified Stochastic rEcursive vaRiance reducEd gradieNt frAmework (SERENA) for Riemannian non-convex optimization. This framework includes the proposed R-SRVRG, R-SVRRM, and R-Hybrid-SGD methods, as well as other existing Riemannian VR methods. Furthermore, we establish a unified theoretical analysis for Riemannian non-convex optimization under retraction and vector transport. The IFO complexity of our proposed R-SRVRG and R-SVRRM to converge to $\varepsilon$-accurate solution is $\mathcal{O}\left(\min \{n^{1/2}{\varepsilon^{-2}}, \varepsilon^{-3}\}\right)$ in the finite-sum setting and ${\mathcal{O}\left( \varepsilon^{-3}\right)}$ for the online case, both of which align with the lower IFO complexity bound. Experimental results indicate that the proposed algorithms surpass other existing Riemannian optimization methods.

Theory · Reinforcement Learning and Planning

Uri Sherman, Tomer Koren, Yishay Mansour

Modern policy optimization methods roughly follow the policy mirror descent (PMD) algorithmic template, for which there are by now numerous theoretical convergence results. However, most of these either target tabular environments, or can be applied effectively only when the class of policies being optimized over satisfies strong closure conditions, which is typically not the case when working with parametric policy classes in large-scale environments. In this work, we develop a theoretical framework for PMD for general policy classes where we replace the closure conditions with a generally weaker variational gradient dominance assumption, and obtain upper bounds on the rate of convergence to the best-in-class policy. Our main result leverages a novel notion of smoothness with respect to a local norm induced by the occupancy measure of the current policy, and casts PMD as a particular instance of smooth non-convex optimization in non-Euclidean space.

Optimization · Convex

Abdel-Rahim Mezidi, Jordan Patracone, Saverio Salzo, Amaury Habrard, Massimiliano Pontil, Rémi Emonet, Marc Sebban

We present several advances on neural operators by viewing the action of operator layers as the minimizers of Bregman regularized optimization problems over Banach function spaces. The proposed framework allows interpreting the activation operators as Bregman proximity operators from dual to primal space. This novel viewpoint is general enough to recover classical neural operators as well as a new variant, coined Bregman neural operators, which includes the inverse activation operator and features the same expressivity of standard neural operators. Numerical experiments support the added benefits of the Bregman variant of Fourier neural operators for training deeper and more accurate models.

Theory · Optimization

Leello Dadi, Volkan Cevher

We study generalization of iterative noisy gradient schemes on smooth non-convex losses. Formally, we establish time-independent information theoretic generalization bounds for Stochastic Gradient Langevin Dynamics (SGLD) that do not diverge as the iteration count increases. Our bounds are obtained through a stability argument: we analyze the difference between two SGLD sequences ran in parallel on two datasets sampled from the same distribution. Our result only requires an isoperimetric inequality to hold, which is merely a restriction on the tails of the loss. Our work relaxes the assumptions of prior work to establish that the iterates stay within a bounded KL divergence from each other. Under an additional dissipativity assumption, we show that the stronger Renyi divergence also stays bounded by establishing a uniform log-Sobolev constant of the iterates. Without dissipativity, we sidestep the need for local log-Sobolev inequalities and instead exploit the regularizing properties of Gaussian convolution. These techniques allow us to show that strong convexity is not necessary for finite stability bounds. Our work shows that noisy SGD can have finite, iteration-independent, generalization and differential privacy bounds in unbounded non-convex settings.

Optimization

William de Vazelhes, Xiaotong Yuan, Bin Gu

In sparse optimization, enforcing hard constraints using the $\ell_0$ pseudo-norm offers advantages like controlled sparsity compared to convex relaxations. However, many real-world applications demand not only sparsity constraints but also some extra constraints. While prior algorithms have been developed to address this complex scenario with mixed combinatorial and convex constraints, they typically require the closed form projection onto the mixed constraints which might not exist, and/or only provide local guarantees of convergence which is different from the global guarantees commonly sought in sparse optimization. To fill this gap, in this paper, we study the problem of sparse optimization with extra *support-preserving* constraints commonly encountered in the literature. We present a new variant of iterative hard-thresholding algorithm equipped with a two-step consecutive projection operator customized for these mixed constraints, serving as a simple alternative to the Euclidean projection onto the mixed constraint. By introducing a novel trade-off between sparsity relaxation and sub-optimality, we provide global guarantees in objective value for the output of our algorithm, in the deterministic, stochastic, and zeroth-order settings, under the conventional restricted strong-convexity/smoothness assumptions. As a fundamental contribution in proof techniques, we develop a novel extension of the classic three-point lemma to the considered two-step non-convex projection operator, which allows us to analyze the convergence in objective value in an elegant way that has not been possible with existing techniques. In the zeroth-order case, such technique also improves upon the state-of-the-art result from de Vazelhes et. al. (2022), even in the case without additional constraints, by allowing us to remove a non-vanishing system error present in their work.

Optimization · Stochastic

Waïss Azizian, Franck Iutzeler, Jérôme Malick, Panayotis Mertikopoulos

In this paper, we examine the time it takes for stochastic gradient descent (SGD) to reach the global minimum of a general, non-convex loss function. We approach this question through the lens of large deviations theory and randomly perturbed dynamical systems, and we provide a tight characterization of the associated hitting times of SGD with matching upper and lower bounds. Our analysis reveals that the global convergence time of SGD is dominated by the most "costly" set of obstacles that the algorithm may need to overcome in order to reach a global minimizer, coupling in this way the geometry of the underlying loss landscape with the statistics of the noise entering the process. Finally, motivated by applications to the training of deep neural networks, we provide a series of refinements and extensions of our analysis to, among others, loss functions with no spurious local minima or ones with bounded depths.

Optimization · Convex

Zijian Liu, Zhengyuan Zhou

We study the convergence of the shuffling gradient method, a popular algorithm employed to minimize the finite-sum function with regularization, in which functions are passed to apply (Proximal) Gradient Descent (GD) one by one whose order is determined by a permutation on the indices of functions. In contrast to its easy implementation and effective performance in practice, the theoretical understanding remains limited. A recent advance by (Liu & Zhou, 2024b) establishes the first last-iterate convergence results under various settings, especially proving the optimal rates for smooth (strongly) convex optimization. However, their bounds for nonsmooth (strongly) convex functions are only as fast as Proximal GD. In this work, we provide the first improved last-iterate analysis for the nonsmooth case demonstrating that the widely used Random Reshuffle ($\textsf{RR}$) and Single Shuffle ($\textsf{SS}$) strategies are both provably faster than Proximal GD, reflecting the benefit of randomness. As an important implication, we give the first (nearly) optimal convergence result for the suffix average under the $\textsf{RR}$ sampling scheme in the general convex case, matching the lower bound shown by (Koren et al., 2022).

Theory · Online Learning and Bandits

Aditya Gangrade, Aldo Pacchiano, Clay Scott, Venkatesh Saligrama

We study the 'feasible action search' (FAS) problem for linear bandits, wherein a learner attempts to discover a feasible point for a set of linear constraints $\Phi_* a \ge 0,$ without knowledge of the matrix $\Phi_* \in \mathbb{R}^{m \times d}$. A FAS learner selects a sequence of actions $a_t,$ and uses observations of the form $\Phi_* a_t + \mathrm{noise}$ to either find a point with nearly optimal 'safety margin', or detect that the constraints are infeasible, where the safety margin of an action measures its (signed) distance from the constraint boundary. While of interest in its own right, the FAS problem also directly addresses a key deficiency in the extant theory of 'safe linear bandits' (SLBs), by discovering a safe initialisation for low-regret SLB methods.We propose and analyse a novel efficient FAS-learner. Our method, FAST, is based on Thompson Sampling. It applies a _coupled_ random perturbation to an estimate of $\Phi_*,$ and plays a maximin point of a game induced by this perturbed matrix. We prove that FAST stops in $\tilde{O}(d^3/\varepsilon^2 M_*^2)$ steps, and incurs $\tilde{O}(d^3/|M_*|)$ safety costs, to either correctly detect infeasibility, or output a point that is at least $(1-\varepsilon) M_*$-safe, where $M_*$ is the _optimal safety margin_ of $\Phi_*$. Further, instantiating prior SLB methods with the output of FAS yields the first SLB methods that incur $\tilde{O}(\sqrt{d^3 T/M_*^2})$ regret and $O(1)$ risk without a priori knowledge of a safe action. The main technical novelty lies in the extension of Thompson Sampling to this multiobjective setting, for which we both propose a coupled noise design, and provide an analysis that avoids convexity considerations.

Optimization

Shuo Xie, Tianhao Wang, Sashank J. Reddi, Sanjiv Kumar, Zhiyuan Li

We present a novel unified analysis for a broad class of adaptive optimization algorithms with structured (e.g., layerwise, diagonal, and kronecker-factored) preconditioners for both online regret minimization and offline convex optimization. Our analysis not only provides matching rate to several important structured preconditioned algorithms including diagonal AdaGrad, full-matrix AdaGrad, and AdaGrad-Norm, but also gives an improved convergence rate for a one-sided variant of Shampoo over that of original Shampoo. Interestingly, more structured preconditioners (e.g., diagonal Adagrad, AdaGrad-Norm which use less space and compute) are often presented as computationally efficient approximations to full-matrix Adagrad, aiming for improved optimization performance through better approximations. Our unified analysis challenges this prevailing view and reveals, perhaps surprisingly, that more structured preconditioners, despite using less space and computation per step, can outperform their less structured counterparts. To demonstrate this, we show that one-sided Shampoo, which is relatively much cheaper than full-matrix AdaGrad could outperform it both theoretically and experimentally.

Optimization · Non-Convex

Arsalan Sharifnassab, Saber Salehkaleybar, Rich Sutton

We address the challenge of optimizing meta-parameters (hyperparameters) in machine learning, a key factor for efficient training and high model performance. Rather than relying on expensive meta-parameter search methods, we introduce MetaOptimize: a dynamic approach that adjusts meta-parameters, particularly step sizes (also known as learning rates), during training. More specifically, MetaOptimize can wrap around any first-order optimization algorithm, tuning step sizes on the fly to minimize a specific form of regret that considers the long-term impact of step sizes on training, through a discounted sum of future losses. We also introduce lower-complexity variants of MetaOptimize that, in conjunction with its adaptability to various optimization algorithms, achieve performance comparable to those of the best hand-crafted learning rate schedules across diverse machine learning tasks.

General Machine Learning · Everything Else

Enming Liang, Minghua Chen

Neural networks (NNs) have emerged as promising tools for solving constrained optimization problems in real-time. However, ensuring constraint satisfaction for NN-generated solutions remains challenging due to prediction errors. Existing methods to ensure NN feasibility either suffer from high computational complexity or are limited to specific constraint types.We present Bisection Projection, an efficient approach to ensure NN solution feasibility for optimization over general compact sets with non-empty interiors.Our method comprises two key components:(i) a dedicated NN (called IPNN) that predicts interior points (IPs) with low eccentricity, which naturally accounts for approximation errors;(ii) a bisection algorithm that leverages these IPs to recover solution feasibility when initial NN solutions violate constraints.We establish theoretical guarantees by providing sufficient conditions for IPNN feasibility and proving bounded optimality loss of the bisection operation under IP predictions. Extensive evaluations on real-world non-convex problems demonstrate that Bisection Projection achieves superior feasibility and computational efficiency compared to existing methods, while maintaining comparable optimality gaps.

Theory · Game Theory

Linjian Meng, Wubing Chen, Wenbin Li, Tianpei Yang, Youzhi Zhang, Yang Gao

Nash equilibrium (NE) plays an important role in game theory. How to efficiently compute an NE in NFGs is challenging due to its complexity and non-convex optimization property. Machine Learning (ML), the cornerstone of modern artificial intelligence, has demonstrated remarkable empirical performance across various applications including non-convex optimization. To leverage non-convex stochastic optimization techniques from ML for approximating an NE, various loss functions have been proposed. Among these, only one loss function is unbiased, allowing for unbiased estimation under the sampled play. Unfortunately, this loss function suffers from high variance, which degrades the convergence rate. To improve the convergence rate by mitigating the high variance associated with the existing unbiased loss function, we propose a novel surrogate loss function named Nash Advantage Loss (NAL). NAL is theoretically proved unbiased and exhibits significantly lower variance than the existing unbiased loss function. Experimental results demonstrate that the algorithm minimizing NAL achieves a significantly faster empirical convergence rates compared to other algorithms, while also reducing the variance of estimated loss value by several orders of magnitude.

General Machine Learning · Causality

Gabriele DAcunto, Fabio Massimo Zennaro, Yorgos Felekis, Paolo Di Lorenzo

Structural causal models (SCMs) allow us to investigate complex systems at multiple levels of resolution.The causal abstraction (CA) framework formalizes the mapping between high- and low-level SCMs. We address CA learning in a challenging and realistic setting, where SCMs are inaccessible, interventional data is unavailable, and sample data is misaligned.A key principle of our framework is *semantic embedding*, formalized as the high-level distribution lying on a subspace of the low-level one. This principle naturally links linear CA to the geometry of the *Stiefel manifold*.We present a category-theoretic approach to SCMs that enables the learning of a CA by finding a morphism between the low- and high-level probability measures, adhering to the semantic embedding principle.Consequently, we formulate a general CA learning problem.As an application, we solve the latter problem for linear CA; considering Gaussian measures and the Kullback-Leibler divergence as an objective.Given the nonconvexity of the learning task, we develop three algorithms building upon existing paradigms for Riemannian optimization.We demonstrate that the proposed methods succeed on both synthetic and real-world brain data with different degrees of prior information about the structure of CA.

Theory · Optimization

Kunjie Ren, Luo Luo

This paper studies zeroth-order optimization for stochastic convex minimization problems. We propose a parameter-free stochastic zeroth-order method (POEM), which introduces a step-size scheme based on the distance over finite difference and an adaptive smoothingparameter. Our theoretical analysis shows that POEM achieves near-optimal stochastic zeroth-order oracle complexity. Furthermore, numerical experiments demonstrate that POEM outperforms existing zeroth-order methods in practice.

Social Aspects · Fairness

Francesco Tonin, Alex Lambert, Johan Suykens, Volkan Cevher

Fairness of decision-making algorithms is an increasingly important issue. In this paper, we focus on spectral clustering with group fairness constraints, where every demographic group is represented in each cluster proportionally as in the general population. We present a new efficient method for fair spectral clustering (Fair SC) by casting the Fair SC problem within the difference of convex functions (DC) framework. To this end, we introduce a novel variable augmentation strategy and employ an alternating direction method of multipliers type of algorithm adapted to DC problems. We show that each associated subproblem can be solved efficiently, resulting in higher computational efficiency compared to prior work, which required a computationally expensive eigendecomposition. Numerical experimentsdemonstrate the effectiveness of our approach on both synthetic and real-world benchmarks, showing significant speedups in computation time over prior art, especially as the problem size grows. This work thus represents a considerable step forward towards the adoption of fair clustering in real-world applications.

Theory · Learning Theory

Xin Yu, Zelin He, Ying Sun, Lingzhou Xue, Runze Li

Personalized federated learning (PFL) offers a flexible framework for aggregating information across distributed clients with heterogeneous data. This work considers a personalized federated learning setting that simultaneously learns global and local models. While purely local training has no communication cost, collaborative learning among the clients can leverage shared knowledge to improve statistical accuracy, presenting an accuracy-communication trade-off in personalized federated learning. However, the theoretical analysis of how personalization quantitatively influences sample and algorithmic efficiency and their inherent trade-off is largely unexplored. This paper makes a contribution towards filling this gap, by providing a quantitative characterization of the personalization degree on the tradeoff. The results further offer theoretical insights for choosing the personalization degree. As a side contribution, we establish the minimax optimality in terms of statistical accuracy for a widely studied PFL formulation. The theoretical result is validated on both synthetic and real-world datasets and its generalizability is verified in a non-convex setting.

Theory · Online Learning and Bandits

Christophe Roux, David Martinez-Rubio, Sebastian Pokutta

We introduce a Riemannian optimistic online learning algorithm for Hadamard manifolds based on inexact implicit updates. Unlike prior work, our method can handle in-manifold constraints, and matches the best known regret bounds in the Euclidean setting with no dependence on geometric constants, like the minimum curvature. Building on this, we develop algorithms for g-convex, g-concave smooth min-max problems on Hadamard manifolds. Notably, one method nearly matches the gradient oracle complexity of the lower bound for Euclidean problems, for the first time.