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1. Convergence Rate Analysis of a Stochastic Trust Region Method via Submartingales

   Blanchet, J. (June 2019, INFORMS journal on optimization)

   We propose a novel framework for analyzing convergence rates of stochastic optimization algorithms with adaptive step sizes. This framework is based on analyzing properties of an underlying generic stochastic process; in particular, we derive a bound on the expected stopping time of this process. We utilize this framework to analyze the expected global convergence rates of a stochastic variant of a traditional trust-region method. Although traditional trust-region methods rely on exact computations of the gradient, Hessian, and values of the objective function, this method assumes that these values are available only up to some dynamically adjusted accuracy. Moreover, this accuracy is assumed to hold only with some sufficiently large—but fixed—probability without any additional restrictions on the variance of the errors. This setting applies, for example, to standard stochastic optimization and machine learning formulations. Improving upon prior analysis, we show that the stochastic process defined by the trust-region method satisfies the assumptions of our proposed general framework. The stopping time in this setting is defined by an iterate satisfying a first-order accuracy condition. We demonstrate the first global complexity bound for a stochastic trust-region method under the assumption of sufficiently accurate stochastic gradients. Finally, we apply the same framework to derive second-order complexity bounds under additional assumptions. Previous 

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2. Convergence Rate Analysis of a Stochastic Trust Region Method via Submartin- gales

   Blanchet, J. (June 2019, INFORMS journal on optimization)

   We propose a novel framework for analyzing convergence rates of stochastic optimization algorithms with adaptive step sizes. This framework is based on analyzing properties of an underlying generic stochastic process; in particular, we derive a bound on the expected stopping time of this process. We utilize this framework to analyze the expected global convergence rates of a stochastic variant of a traditional trust-region method. Although traditional trust-region methods rely on exact computations of the gradient, Hessian, and values of the objective function, this method assumes that these values are available only up to some dynamically adjusted accuracy. Moreover, this accuracy is assumed to hold only with some sufficiently large—but fixed—probability without any additional restrictions on the variance of the errors. This setting applies, for example, to standard stochastic optimization and machine learning formulations. Improving upon prior analysis, we show that the stochastic process defined by the trust-region method satisfies the assumptions of our proposed general framework. The stopping time in this setting is defined by an iterate satisfying a first-order accuracy condition. We demonstrate the first global complexity bound for a stochastic trust-region method under the assumption of sufficiently accurate stochastic gradients. Finally, we apply the same framework to derive second-order complexity bounds under additional assumptions. 

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3. High Probability Complexity Bounds for Line Search Based on Stochastic Oracles

   Billy Jin, Katya Scheinberg (January 2021, Advances in Neural Information Processing Systems 34 (NeurIPS 2021))

   We consider a line-search method for continuous optimization under a stochastic setting where the function values and gradients are available only through inexact probabilistic zeroth and first-order oracles. These oracles capture multiple standard settings including expected loss minimization and zeroth-order optimization. Moreover, our framework is very general and allows the function and gradient estimates to be biased. The proposed algorithm is simple to describe, easy to implement, and uses these oracles in a similar way as the standard deterministic line search uses exact function and gradient values. Under fairly general conditions on the oracles, we derive a high probability tail bound on the iteration complexity of the algorithm when applied to non-convex smooth functions. These results are stronger than those for other existing stochastic line search methods and apply in more general settings. 

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4. High Probability Complexity Bounds for Line Search Based on Stochastic Oracles

   Jin, B. (October 2021, Advances in neural information processing systems)

   Ranzato, M.:; Dauphin, Y.; Liang, P.S.; Wortman Vaughan, J. (Ed.)

   We consider a line-search method for continuous optimization under a stochastic setting where the function values and gradients are available only through inexact probabilistic zeroth and first-order oracles. These oracles capture multiple stan- dard settings including expected loss minimization and zeroth-order optimization. Moreover, our framework is very general and allows the function and gradient estimates to be biased. The proposed algorithm is simple to describe, easy to im- plement, and uses these oracles in a similar way as the standard deterministic line search uses exact function and gradient values. Under fairly general conditions on the oracles, we derive a high probability tail bound on the iteration complexity of the algorithm when applied to non-convex smooth functions. These results are stronger than those for other existing stochastic line search methods and apply in more general settings. 

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5. SGD and Hogwild! Convergence Without the Bounded Gradients Assumption

   Nguyen, Lam M.; Nguyen, Phuong Ha; Dijk, Marten van; Richtárik, Peter; Scheinberg, Katya; Takáč, Martin (January 2018, Proceedings of Machine Learning Research)

   Stochastic gradient descent (SGD) is the optimization algorithm of choice in many machine learning applications such as regularized empirical risk minimization and training deep neural networks. The classical analysis of convergence of SGD is carried out under the assumption that the norm of the stochastic gradient is uniformly bounded. While this might hold for some loss functions, it is always violated for cases where the objective function is strongly convex. In (Bottou et al.,2016) a new analysis of convergence of SGD is performed under the assumption that stochastic gradients are bounded with respect to the true gradient norm. Here we show that for stochastic problems arising in machine learning such bound always holds. Moreover, we propose an alternative convergence analysis of SGD with diminishing learning rate regime, which is results in more relaxed conditions that those in (Bottou et al.,2016). We then move on the asynchronous parallel setting, and prove convergence of the Hogwild! algorithm in the same regime, obtaining the first convergence results for this method in the case of diminished learning rate. 

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