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1. Adaptive Gradient Methods for Constrained Convex Optimization and Variational Inequalities

   Ene, Alina ; Nguyen, Huy L ; Vladu, Adrian ( May 2021 , Proceedings of the AAAI Conference on Artificial Intelligence)

   null (Ed.)

   We provide new adaptive first-order methods for constrained convex optimization. Our main algorithms AdaACSA and AdaAGD+ are accelerated methods, which are universal in the sense that they achieve nearly-optimal convergence rates for both smooth and non-smooth functions, even when they only have access to stochastic gradients. In addition, they do not require any prior knowledge on how the objective function is parametrized, since they automatically adjust their per-coordinate learning rate. These can be seen as truly accelerated Adagrad methods for constrained optimization. We complement them with a simpler algorithm AdaGrad+ which enjoys the same features, and achieves the standard non-accelerated convergence rate. We also present a set of new results involving adaptive methods for unconstrained optimization and variational inequalities arising from monotone operators. 

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2. On the Convergence of AdaGrad(Norm) on R^d: Beyond Convexity, Non-Asymptotic Rate and Acceleration

   Liu, Zijian ; Nguyen, Ta Duy ; Ene, Alina ; Nguyen, Huy ( February 2023 , International Conference on Learning Representations)

   Existing analysis of AdaGrad and other adaptive methods for smooth convex optimization is typically for functions with bounded domain diameter. In unconstrained problems, previous works guarantee an asymptotic convergence rate without an explicit constant factor that holds true for the entire function class. Furthermore, in the stochastic setting, only a modified version of AdaGrad, different from the one commonly used in practice, in which the latest gradient is not used to update the stepsize, has been analyzed. Our paper aims at bridging these gaps and developing a deeper understanding of AdaGrad and its variants in the standard setting of smooth convex functions as well as the more general setting of quasar convex functions. First, we demonstrate new techniques to explicitly bound the convergence rate of the vanilla AdaGrad for unconstrained problems in both deterministic and stochastic settings. Second, we propose a variant of AdaGrad for which we can show the convergence of the last iterate, instead of the average iterate. Finally, we give new accelerated adaptive algorithms and their convergence guarantee in the deterministic setting with explicit dependency on the problem parameters, improving upon the asymptotic rate shown in previous works. 

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3. On the Convergence of AdaGrad(Norm) on R^d: Beyond Convexity, Non-Asymptotic Rate and Acceleration

   Liu, Zijian ; Nguyen, Ta Duy ; Ene, Alina ; Nguyen, Huy ( May 2023 , International Conference on Learning Representations)

   Existing analysis of AdaGrad and other adaptive methods for smooth convex optimization is typically for functions with bounded domain diameter. In unconstrained problems, previous works guarantee an asymptotic convergence rate without an explicit constant factor that holds true for the entire function class. Furthermore, in the stochastic setting, only a modified version of AdaGrad, different from the one commonly used in practice, in which the latest gradient is not used to update the stepsize, has been analyzed. Our paper aims at bridging these gaps and developing a deeper understanding of AdaGrad and its variants in the standard setting of smooth convex functions as well as the more general setting of quasar convex functions. First, we demonstrate new techniques to explicitly bound the convergence rate of the vanilla AdaGrad for unconstrained problems in both deterministic and stochastic settings. Second, we propose a variant of AdaGrad for which we can show the convergence of the last iterate, instead of the average iterate. Finally, we give new accelerated adaptive algorithms and their convergence guarantee in the deterministic setting with explicit dependency on the problem parameters, improving upon the asymptotic rate shown in previous works. 

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4. Randomized sketch descent methods for non-separable linearly constrained optimization

   https://doi.org/10.1093/imanum/draa018  

   Necoara, Ion ; Takáč, Martin ( May 2020 , IMA Journal of Numerical Analysis)

   null (Ed.)

   Abstract In this paper we consider large-scale smooth optimization problems with multiple linear coupled constraints. Due to the non-separability of the constraints, arbitrary random sketching would not be guaranteed to work. Thus, we first investigate necessary and sufficient conditions for the sketch sampling to have well-defined algorithms. Based on these sampling conditions we develop new sketch descent methods for solving general smooth linearly constrained problems, in particular, random sketch descent (RSD) and accelerated random sketch descent (A-RSD) methods. To our knowledge, this is the first convergence analysis of RSD algorithms for optimization problems with multiple non-separable linear constraints. For the general case, when the objective function is smooth and non-convex, we prove for the non-accelerated variant sublinear rate in expectation for an appropriate optimality measure. In the smooth convex case, we derive for both algorithms, non-accelerated and A-RSD, sublinear convergence rates in the expected values of the objective function. Additionally, if the objective function satisfies a strong convexity type condition, both algorithms converge linearly in expectation. In special cases, where complexity bounds are known for some particular sketching algorithms, such as coordinate descent methods for optimization problems with a single linear coupled constraint, our theory recovers the best known bounds. Finally, we present several numerical examples to illustrate the performances of our new algorithms. 

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5. High Probability Convergence of Stochastic Gradient Methods

   Liu, Zijian ; Nguyen, Ta Duy ; Nguyen, Thien Hang ; Nguyen, Huy ( July 2023 , Proceedings of Machine Learning Research)

   In this work, we describe a generic approach to show convergence with high probability for both stochastic convex and non-convex optimization with sub-Gaussian noise. In previous works for convex optimization, either the convergence is only in expectation or the bound depends on the diameter of the domain. Instead, we show high probability convergence with bounds depending on the initial distance to the optimal solution. The algorithms use step sizes analogous to the standard settings and are universal to Lipschitz functions, smooth functions, and their linear combinations. The method can be applied to the non-convex case. We demonstrate an $O((1+\sigma^{2}\log(1/\delta))/T+\sigma/\sqrt{T})$ convergence rate when the number of iterations $T$ is known and an $O((1+\sigma^{2}\log(T/\delta))/\sqrt{T})$ convergence rate when $T$ is unknown for SGD, where $1-\delta$ is the desired success probability. These bounds improve over existing bounds in the literature. We also revisit AdaGrad-Norm \cite{ward2019adagrad} and show a new analysis to obtain a high probability bound that does not require the bounded gradient assumption made in previous works. The full version of our paper contains results for the standard per-coordinate AdaGrad. 

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