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1. Stochastic Optimization under Distributional Drift

   Cutler, Joshua; Drusvyatskiy, Dmitriy; Harchaoui, Zaid (October 2023, Journal of Machine Learning Research)

   We consider the problem of minimizing a convex function that is evolving according to unknown and possibly stochastic dynamics, which may depend jointly on time and on the decision variable itself. Such problems abound in the machine learning and signal processing literature, under the names of concept drift, stochastic tracking, and performative prediction. We provide novel non-asymptotic convergence guarantees for stochastic algorithms with iterate averaging, focusing on bounds valid both in expectation and with high probability. The efficiency estimates we obtain clearly decouple the contributions of optimization error, gradient noise, and time drift. Notably, we identify a low drift-to-noise regime in which the tracking efficiency of the proximal stochastic gradient method benefits significantly from a step decay schedule. Numerical experiments illustrate our results. 

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2. 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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3. 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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4. The Step Decay Schedule: A Near Optimal, Geometrically Decaying Learning Rate Procedure For Least Squares

   Rong Ge, Sham M. (January 2020, Neurips)

   Minimax optimal convergence rates for numerous classes of stochastic convex optimization problems are well characterized, where the majority of results utilize iterate averaged stochastic gradient descent (SGD) with polynomially decaying step sizes. In contrast, the behavior of SGD’s final iterate has received much less attention despite the widespread use in practice. Motivated by this observation, this work provides a detailed study of the following question: what rate is achievable using the final iterate of SGD for the streaming least squares regression problem with and without strong convexity? First, this work shows that even if the time horizon T (i.e. the number of iterations that SGD is run for) is known in advance, the behavior of SGD’s final iterate with any polynomially decaying learning rate scheme is highly sub-optimal compared to the statistical minimax rate (by a condition number factor in the strongly convex √ case and a factor of shows that Step Decay schedules, which cut the learning rate by a constant factor every constant number of epochs (i.e., the learning rate decays geometrically) offer significant improvements over any polynomially decaying step size schedule. In particular, the behavior of the final iterate with step decay schedules is off from the statistical minimax rate by only log factors (in the condition number for the strongly convex case, and in T in the non-strongly convex case). Finally, in stark contrast to the known horizon case, this paper shows that the anytime (i.e. the limiting) behavior of SGD’s final iterate is poor (in that it queries iterates with highly sub-optimal function value infinitely often, i.e. in a limsup sense) irrespective of the stepsize scheme employed. These results demonstrate the subtlety in establishing optimal learning rate schedules (for the final iterate) for stochastic gradient procedures in fixed time horizon settings. 

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5. The Step Decay Schedule: A Near Optimal, Geometrically Decaying Learning Rate Procedure For Least Squares

   Ge, Rong; Kakade, Sham M.; Kidambi, Rahul; Netrapalli, Praneeth (January 2019, Advances in neural information processing systems)

   Minimax optimal convergence rates for classes of stochastic convex optimization problems are well characterized, where the majority of results utilize iterate averaged stochastic gradient descent (SGD) with polynomially decaying step sizes. In contrast, SGD's final iterate behavior has received much less attention despite their widespread use in practice. Motivated by this observation, this work provides a detailed study of the following question: what rate is achievable using the final iterate of SGD for the streaming least squares regression problem with and without strong convexity? First, this work shows that even if the time horizon T (i.e. the number of iterations SGD is run for) is known in advance, SGD's final iterate behavior with any polynomially decaying learning rate scheme is highly sub-optimal compared to the minimax rate (by a condition number factor in the strongly convex case and a factor of T‾‾√ in the non-strongly convex case). In contrast, this paper shows that Step Decay schedules, which cut the learning rate by a constant factor every constant number of epochs (i.e., the learning rate decays geometrically) offers significant improvements over any polynomially decaying step sizes. In particular, the final iterate behavior with a step decay schedule is off the minimax rate by only log factors (in the condition number for strongly convex case, and in T for the non-strongly convex case). Finally, in stark contrast to the known horizon case, this paper shows that the anytime (i.e. the limiting) behavior of SGD's final iterate is poor (in that it queries iterates with highly sub-optimal function value infinitely often, i.e. in a limsup sense) irrespective of the stepsizes employed. These results demonstrate the subtlety in establishing optimal learning rate schemes (for the final iterate) for stochastic gradient procedures in fixed time horizon settings. 

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