arXiv subjects

Kimon Antonakopoulos

Publications and source records attributed to Kimon Antonakopoulos.

At least 19 recordsLinked to original sources

arXiv2026

Constrained Stochastic Spectral Preconditioning Converges for Nonconvex Objectives

In this work, we develop proximal preconditioned gradient methods with a focus on spectral gradient methods providing a proximal extension to the Muon and Scion optimizers. We introduce a family of stochastic algorithms that can handle a wide variety of convex and nonconvex constraints and study its convergence under heavy-tailed noise, through a novel analysis tailored to the geometry of the proposed methods. We further propose a variance-reduced version, which achieves faster convergence under standard noise assumptions. Finally, we show that the polynomial iterations used in Muon are more accurately captured by a nonlinear preconditioner than by the ideal matrix sign, leading to a convergence analysis that more faithfully reflects practical implementations.

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arXiv2026

Generalized Gradient Norm Clipping & Non-Euclidean $(L_0,L_1)$-Smoothness

This work introduces a hybrid non-Euclidean optimization method which generalizes gradient norm clipping by combining steepest descent and conditional gradient approaches. The method achieves the best of both worlds by establishing a descent property under a generalized notion of ($L_0$,$L_1$)-smoothness. Weight decay is incorporated in a principled manner by identifying a connection to the Frank-Wolfe short step. In the stochastic case, we show an order optimal $O(n^{-1/4})$ convergence rate by leveraging a momentum based gradient estimator. We discuss how to instantiate the algorithms for deep learning, which we dub Clipped Scion, and demonstrate their properties on image classification and language modeling. The code is available at https://github.com/LIONS-EPFL/ClippedScion.

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arXiv2025

Training Neural Networks at Any Scale

This article reviews modern optimization methods for training neural networks with an emphasis on efficiency and scale. We present state-of-the-art optimization algorithms under a unified algorithmic template that highlights the importance of adapting to the structures in the problem. We then cover how to make these algorithms agnostic to the scale of the problem. Our exposition is intended as an introduction for both practitioners and researchers who wish to be involved in these exciting new developments.

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arXiv2025

Training Deep Learning Models with Norm-Constrained LMOs

In this work, we study optimization methods that leverage the linear minimization oracle (LMO) over a norm-ball. We propose a new stochastic family of algorithms that uses the LMO to adapt to the geometry of the problem and, perhaps surprisingly, show that they can be applied to unconstrained problems. The resulting update rule unifies several existing optimization methods under a single framework. Furthermore, we propose an explicit choice of norm for deep architectures, which, as a side benefit, leads to the transferability of hyperparameters across model sizes. Experimentally, we demonstrate significant speedups on nanoGPT training using our algorithm, Scion, without any reliance on Adam. The proposed method is memory-efficient, requiring only one set of model weights and one set of gradients, which can be stored in half-precision. The code is available at https://github.com/LIONS-EPFL/scion .

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arXiv2025

Multi-Step Alignment as Markov Games: An Optimistic Online Gradient Descent Approach with Convergence Guarantees

Reinforcement Learning from Human Feedback (RLHF) has been highly successful in aligning large language models with human preferences. While prevalent methods like DPO have demonstrated strong performance, they frame interactions with the language model as a bandit problem, which limits their applicability in real-world scenarios where multi-turn conversations are common. Additionally, DPO relies on the Bradley-Terry model assumption, which does not adequately capture the non-transitive nature of human preferences. In this paper, we address these challenges by modeling the alignment problem as a two-player constant-sum Markov game, where each player seeks to maximize their winning rate against the other across all steps of the conversation. Our approach Optimistic Multi-step Preference Optimization (OMPO) is built upon the optimistic online mirror descent algorithm~\citep{rakhlin2013online,joulani17a}. Theoretically, we provide a rigorous analysis for the convergence of OMPO and show that OMPO requires $\mathcal{O}(ε^{-1})$ policy updates to converge to an $ε$-approximate Nash equilibrium. We also validate the effectiveness of our method on multi-turn conversations dataset and math reasoning dataset.

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arXiv2025

Layer-wise Quantization for Quantized Optimistic Dual Averaging

Modern deep neural networks exhibit heterogeneity across numerous layers of various types such as residuals, multi-head attention, etc., due to varying structures (dimensions, activation functions, etc.), distinct representation characteristics, which impact predictions. We develop a general layer-wise quantization framework with tight variance and code-length bounds, adapting to the heterogeneities over the course of training. We then apply a new layer-wise quantization technique within distributed variational inequalities (VIs), proposing a novel Quantized Optimistic Dual Averaging (QODA) algorithm with adaptive learning rates, which achieves competitive convergence rates for monotone VIs. We empirically show that QODA achieves up to a $150\%$ speedup over the baselines in end-to-end training time for training Wasserstein GAN on $12+$ GPUs.

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arXiv2024

Improving SAM Requires Rethinking its Optimization Formulation

This paper rethinks Sharpness-Aware Minimization (SAM), which is originally formulated as a zero-sum game where the weights of a network and a bounded perturbation try to minimize/maximize, respectively, the same differentiable loss. To fundamentally improve this design, we argue that SAM should instead be reformulated using the 0-1 loss. As a continuous relaxation, we follow the simple conventional approach where the minimizing (maximizing) player uses an upper bound (lower bound) surrogate to the 0-1 loss. This leads to a novel formulation of SAM as a bilevel optimization problem, dubbed as BiSAM. BiSAM with newly designed lower-bound surrogate loss indeed constructs stronger perturbation. Through numerical evidence, we show that BiSAM consistently results in improved performance when compared to the original SAM and variants, while enjoying similar computational complexity. Our code is available at https://github.com/LIONS-EPFL/BiSAM.

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arXiv2024

Universal Gradient Methods for Stochastic Convex Optimization

We develop universal gradient methods for Stochastic Convex Optimization (SCO). Our algorithms automatically adapt not only to the oracle's noise but also to the Hölder smoothness of the objective function without a priori knowledge of the particular setting. The key ingredient is a novel strategy for adjusting step-size coefficients in the Stochastic Gradient Method (SGD). Unlike AdaGrad, which accumulates gradient norms, our Universal Gradient Method accumulates appropriate combinations of gradient- and iterate differences. The resulting algorithm has state-of-the-art worst-case convergence rate guarantees for the entire Hölder class including, in particular, both nonsmooth functions and those with Lipschitz continuous gradient. We also present the Universal Fast Gradient Method for SCO enjoying optimal efficiency estimates.

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arXiv2024

Advancing the lower bounds: An accelerated, stochastic, second-order method with optimal adaptation to inexactness

We present a new accelerated stochastic second-order method that is robust to both gradient and Hessian inexactness, which occurs typically in machine learning. We establish theoretical lower bounds and prove that our algorithm achieves optimal convergence in both gradient and Hessian inexactness in this key setting. We further introduce a tensor generalization for stochastic higher-order derivatives. When the oracles are non-stochastic, the proposed tensor algorithm matches the global convergence of Nesterov Accelerated Tensor method. Both algorithms allow for approximate solutions of their auxiliary subproblems with verifiable conditions on the accuracy of the solution.

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arXiv2024

On the Generalization of Stochastic Gradient Descent with Momentum

While momentum-based accelerated variants of stochastic gradient descent (SGD) are widely used when training machine learning models, there is little theoretical understanding on the generalization error of such methods. In this work, we first show that there exists a convex loss function for which the stability gap for multiple epochs of SGD with standard heavy-ball momentum (SGDM) becomes unbounded. Then, for smooth Lipschitz loss functions, we analyze a modified momentum-based update rule, i.e., SGD with early momentum (SGDEM) under a broad range of step-sizes, and show that it can train machine learning models for multiple epochs with a guarantee for generalization. Finally, for the special case of strongly convex loss functions, we find a range of momentum such that multiple epochs of standard SGDM, as a special form of SGDEM, also generalizes. Extending our results on generalization, we also develop an upper bound on the expected true risk, in terms of the number of training steps, sample size, and momentum. Our experimental evaluations verify the consistency between the numerical results and our theoretical bounds. SGDEM improves the generalization error of SGDM when training ResNet-18 on ImageNet in practical distributed settings.

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arXiv2023

Distributed Extra-gradient with Optimal Complexity and Communication Guarantees

We consider monotone variational inequality (VI) problems in multi-GPU settings where multiple processors/workers/clients have access to local stochastic dual vectors. This setting includes a broad range of important problems from distributed convex minimization to min-max and games. Extra-gradient, which is a de facto algorithm for monotone VI problems, has not been designed to be communication-efficient. To this end, we propose a quantized generalized extra-gradient (Q-GenX), which is an unbiased and adaptive compression method tailored to solve VIs. We provide an adaptive step-size rule, which adapts to the respective noise profiles at hand and achieve a fast rate of ${\mathcal O}(1/T)$ under relative noise, and an order-optimal ${\mathcal O}(1/\sqrt{T})$ under absolute noise and show distributed training accelerates convergence. Finally, we validate our theoretical results by providing real-world experiments and training generative adversarial networks on multiple GPUs.

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arXiv2023

No-Regret Learning in Games with Noisy Feedback: Faster Rates and Adaptivity via Learning Rate Separation

We examine the problem of regret minimization when the learner is involved in a continuous game with other optimizing agents: in this case, if all players follow a no-regret algorithm, it is possible to achieve significantly lower regret relative to fully adversarial environments. We study this problem in the context of variationally stable games (a class of continuous games which includes all convex-concave and monotone games), and when the players only have access to noisy estimates of their individual payoff gradients. If the noise is additive, the game-theoretic and purely adversarial settings enjoy similar regret guarantees; however, if the noise is multiplicative, we show that the learners can, in fact, achieve constant regret. We achieve this faster rate via an optimistic gradient scheme with learning rate separation -- that is, the method's extrapolation and update steps are tuned to different schedules, depending on the noise profile. Subsequently, to eliminate the need for delicate hyperparameter tuning, we propose a fully adaptive method that attains nearly the same guarantees as its non-adapted counterpart, while operating without knowledge of either the game or of the noise profile.

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arXiv2022

Extra-Newton: A First Approach to Noise-Adaptive Accelerated Second-Order Methods

This work proposes a universal and adaptive second-order method for minimizing second-order smooth, convex functions. Our algorithm achieves $O(σ/ \sqrt{T})$ convergence when the oracle feedback is stochastic with variance $σ^2$, and improves its convergence to $O( 1 / T^3)$ with deterministic oracles, where $T$ is the number of iterations. Our method also interpolates these rates without knowing the nature of the oracle apriori, which is enabled by a parameter-free adaptive step-size that is oblivious to the knowledge of smoothness modulus, variance bounds and the diameter of the constrained set. To our knowledge, this is the first universal algorithm with such global guarantees within the second-order optimization literature.

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arXiv2022

Adaptive Stochastic Variance Reduction for Non-convex Finite-Sum Minimization

We propose an adaptive variance-reduction method, called AdaSpider, for minimization of $L$-smooth, non-convex functions with a finite-sum structure. In essence, AdaSpider combines an AdaGrad-inspired [Duchi et al., 2011, McMahan & Streeter, 2010], but a fairly distinct, adaptive step-size schedule with the recursive stochastic path integrated estimator proposed in [Fang et al., 2018]. To our knowledge, Adaspider is the first parameter-free non-convex variance-reduction method in the sense that it does not require the knowledge of problem-dependent parameters, such as smoothness constant $L$, target accuracy $ε$ or any bound on gradient norms. In doing so, we are able to compute an $ε$-stationary point with $\tilde{O}\left(n + \sqrt{n}/ε^2\right)$ oracle-calls, which matches the respective lower bound up to logarithmic factors.

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arXiv2022

A universal black-box optimization method with almost dimension-free convergence rate guarantees

Universal methods for optimization are designed to achieve theoretically optimal convergence rates without any prior knowledge of the problem's regularity parameters or the accurarcy of the gradient oracle employed by the optimizer. In this regard, existing state-of-the-art algorithms achieve an $\mathcal{O}(1/T^2)$ value convergence rate in Lipschitz smooth problems with a perfect gradient oracle, and an $\mathcal{O}(1/\sqrt{T})$ convergence rate when the underlying problem is non-smooth and/or the gradient oracle is stochastic. On the downside, these methods do not take into account the problem's dimensionality, and this can have a catastrophic impact on the achieved convergence rate, in both theory and practice. Our paper aims to bridge this gap by providing a scalable universal gradient method - dubbed UnderGrad - whose oracle complexity is almost dimension-free in problems with a favorable geometry (like the simplex, linearly constrained semidefinite programs and combinatorial bandits), while retaining the order-optimal dependence on $T$ described above. These "best-of-both-worlds" results are achieved via a primal-dual update scheme inspired by the dual exploration method for variational inequalities.

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arXiv2022

Routing in an Uncertain World: Adaptivity, Efficiency, and Equilibrium

We consider the traffic assignment problem in nonatomic routing games where the players' cost functions may be subject to random fluctuations (e.g., weather disturbances, perturbations in the underlying network, etc.). We tackle this problem from the viewpoint of a control interface that makes routing recommendations based solely on observed costs and without any further knowledge of the system's governing dynamics -- such as the network's cost functions, the distribution of any random events affecting the network, etc. In this online setting, learning methods based on the popular exponential weights algorithm converge to equilibrium at an $\mathcal{O}({1/\sqrt{T}})$ rate: this rate is known to be order-optimal in stochastic networks, but it is otherwise suboptimal in static networks. In the latter case, it is possible to achieve an $\mathcal{O}({1/T^{2}})$ equilibrium convergence rate via the use of finely tuned accelerated algorithms; on the other hand, these accelerated algorithms fail to converge altogether in the presence of persistent randomness, so it is not clear how to achieve the "best of both worlds" in terms of convergence speed. Our paper seeks to fill this gap by proposing an adaptive routing algortihm with the following desirable properties: $(i)$ it seamlessly interpolates between the $\mathcal{O}({1/T^{2}})$ and $\mathcal{O}({1/\sqrt{T}})$ rates for static and stochastic environments respectively; $(ii)$ its convergence speed is polylogarithmic in the number of paths in the network; ${(iii)}$ the method's per-iteration complexity and memory requirements are both linear in the number of nodes and edges in the network; and ${(iv)}$ it does not require any prior knowledge of the problem's parameters.

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arXiv2021

Adaptive Learning in Continuous Games: Optimal Regret Bounds and Convergence to Nash Equilibrium

In game-theoretic learning, several agents are simultaneously following their individual interests, so the environment is non-stationary from each player's perspective. In this context, the performance of a learning algorithm is often measured by its regret. However, no-regret algorithms are not created equal in terms of game-theoretic guarantees: depending on how they are tuned, some of them may drive the system to an equilibrium, while others could produce cyclic, chaotic, or otherwise divergent trajectories. To account for this, we propose a range of no-regret policies based on optimistic mirror descent, with the following desirable properties: i) they do not require any prior tuning or knowledge of the game; ii) they all achieve O(\sqrt{T}) regret against arbitrary, adversarial opponents; and iii) they converge to the best response against convergent opponents. Also, if employed by all players, then iv) they guarantee O(1) social regret; while v) the induced sequence of play converges to Nash equilibrium with O(1) individual regret in all variationally stable games (a class of games that includes all monotone and convex-concave zero-sum games).

cs.GT↗

arXiv2021

Adaptive first-order methods revisited: Convex optimization without Lipschitz requirements

We propose a new family of adaptive first-order methods for a class of convex minimization problems that may fail to be Lipschitz continuous or smooth in the standard sense. Specifically, motivated by a recent flurry of activity on non-Lipschitz (NoLips) optimization, we consider problems that are continuous or smooth relative to a reference Bregman function - as opposed to a global, ambient norm (Euclidean or otherwise). These conditions encompass a wide range of problems with singular objectives, such as Fisher markets, Poisson tomography, D-design, and the like. In this setting, the application of existing order-optimal adaptive methods - like UnixGrad or AcceleGrad - is not possible, especially in the presence of randomness and uncertainty. The proposed method - which we call adaptive mirror descent (AdaMir) - aims to close this gap by concurrently achieving min-max optimal rates in problems that are relatively continuous or smooth, including stochastic ones.

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