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1. A High Probability Analysis of Adaptive SGD with Momentum

   Li, Xiaoyu; Orabona, Francesco (January 2020, Workshop on Beyond First Order Methods in ML Systems at ICML'20)

   null (Ed.)

   Stochastic Gradient Descent (SGD) and its variants are the most used algorithms in machine learning applications. In particular, SGD with adaptive learning rates and momentum is the industry standard to train deep networks. Despite the enormous success of these methods, our theoretical understanding of these variants in the non-convex setting is not complete, with most of the results only proving convergence in expectation and with strong assumptions on the stochastic gradients. In this paper, we present a high probability analysis for adaptive and momentum algorithms, under weak assumptions on the function, stochastic gradients, and learning rates. We use it to prove for the first time the convergence of the gradients to zero in high probability in the smooth nonconvex setting for Delayed AdaGrad with momentum. 

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2. Accelerated Over-Relaxation Heavy-Ball Method: Achieving Global Accelerated Convergence with Broad Generalization

   Wei, Jingrong; Chen, Long (January 2025, International Conference on Learning Representations (ICLR))

   The heavy-ball momentum method accelerates gradient descent with a momentum term but lacks accelerated convergence for general smooth strongly convex problems. This work introduces the Accelerated Over-Relaxation Heavy-Ball (AOR-HB) method, the first variant with provable global and accelerated convergence for such problems. AOR-HB closes a long-standing theoretical gap, extends to composite convex optimization and min-max problems, and achieves optimal complexity bounds. It offers three key advantages: (1) broad generalization ability, (2) potential to reshape acceleration techniques, and (3) conceptual clarity and elegance compared to existing methods. 

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3. Provable Complexity Improvement of AdaGrad over SGD: Upper and Lower Bounds in Stochastic Non-Convex Optimization

   Jiang, R; Maladkar, D; Mokhtari, A (June 2025, https://doi.org/10.48550/arXiv.2406.04592)

   Adaptive gradient methods, such as AdaGrad, are among the most successful optimization algorithms for neural network training. While these methods are known to achieve better dimensional dependence than stochastic gradient descent (SGD) for stochastic convex optimization under favorable geometry, the theoretical justification for their success in stochastic non-convex optimization remains elusive. In fact, under standard assumptions of Lipschitz gradients and bounded noise variance, it is known that SGD is worst-case optimal in terms of finding a near-stationary point with respect to the l2-norm, making further improvements impossible. Motivated by this limitation, we introduce refined assumptions on the smoothness structure of the objective and the gradient noise variance, which better suit the coordinate-wise nature of adaptive gradient methods. Moreover, we adopt the l1-norm of the gradient as the stationarity measure, as opposed to the standard l2-norm, to align with the coordinate-wise analysis and obtain tighter convergence guarantees for AdaGrad. Under these new assumptions and the l1-norm stationarity measure, we establish an upper bound on the convergence rate of AdaGrad and a corresponding lower bound for SGD. In particular, we identify non-convex settings in which the iteration complexity of AdaGrad is favorable over SGD and show that, for certain configurations of problem parameters, it outperforms SGD by a factor of d, where d is the problem dimension. To the best of our knowledge, this is the first result to demonstrate a provable gain of adaptive gradient methods over SGD in a non-convex setting. We also present supporting lower bounds, including one specific to AdaGrad and one applicable to general deterministic first-order methods, showing that our upper bound for AdaGrad is tight and unimprovable up to a logarithmic factor under certain conditions. 

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4. Improved Convergence in High Probability of Clipped Gradient Methods with Heavy Tailed Noise

   Nguyen, Ta Duy; Nguyen, Thien Hang; Ene, Alina; Nguyen, Huy (September 2023, Advances in Neural Information Processing Systems)

   In this work, we study the convergence \emph{in high probability} of clipped gradient methods when the noise distribution has heavy tails, i.e., with bounded $$p$$th moments, for some $$1 

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5. Improved Convergence in High Probability of Clipped Gradient Methods with Heavy Tailed Noise

   Nguyen, Ta Duy; Nguyen, Thien Hang; Ene, Alina; Nguyen, Huy L (December 2023, Advances in neural information processing systems)

   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 $$1< p \leq 2$$. 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 existing 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. 

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