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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. Adaptive Gradient Methods for Constrained Convex Optimization and Variational Inequalities

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

   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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3. High Probability Convergence of Stochastic Gradient Methods

   Liu, Zijian; Nguyen, Ta Duy; Nguyen, Thien Hang; Ene, Alina; Nguyen, Huy (July 2023, International Conference on Machine Learning)

   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 (Ward et al., 2019) 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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4. 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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5. Surrogate Losses for Online Learning of Stepsizes in Stochastic Non-Convex Optimization

   Zhuang, Zhenxun; Cutkosky, Ashok; Orabona, Francesco (June 2019, Proceedings of the 36th International Conference on Machine Learning)

   Stochastic Gradient Descent (SGD) has played a central role in machine learning. However, it requires a carefully hand-picked stepsize for fast convergence, which is notoriously tedious and time-consuming to tune. Over the last several years, a plethora of adaptive gradient-based algorithms have emerged to ameliorate this problem. In this paper, we propose new surrogate losses to cast the problem of learning the optimal stepsizes for the stochastic optimization of a non-convex smooth objective function onto an online convex optimization problem. This allows the use of no-regret online algorithms to compute optimal stepsizes on the fly. In turn, this results in a SGD algorithm with self-tuned stepsizes that guarantees convergence rates that are automatically adaptive to the level of noise. 

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