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1. Training Deep Networks without Learning Rates Through Coin Betting

   Orabona, Francesco; Tommasi, Tatiana (January 2017, Advances in Neural Information Processing Systems 30)

   Deep learning methods achieve state-of-the-art performance in many application scenarios. Yet, these methods require a significant amount of hyperparameters tuning in order to achieve the best results. In particular, tuning the learning rates in the stochastic optimization process is still one of the main bottlenecks. In this paper, we propose a new stochastic gradient descent procedure for deep networks that does not require any learning rate setting. Contrary to previous methods, we do not adapt the learning rates nor we make use of the assumed curvature of the objective function. Instead, we reduce the optimization process to a game of betting on a coin and propose a learning rate free optimal algorithm for this scenario. Theoretical convergence is proven for convex and quasi-convex functions and empirical evidence shows the advantage of our algorithm over popular stochastic gradient algorithms. 

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2. 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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3. Implicit Parameter-free Online Learning with Truncated Linear Models

   Chen, Keyi; Cutkosky, Ashok; Orabona, Francesco (January 2022, International Conference on Algorithmic Learning Theory)

   Dasgupta, Sanjoy; Haghtalab, Nika (Ed.)

   Parameter-free algorithms are online learning algorithms that do not require setting learning rates. They achieve optimal regret with respect to the distance between the initial point and any competitor. Yet, parameter-free algorithms do not take into account the geometry of the losses. Recently, in the stochastic optimization literature, it has been proposed to instead use truncated linear lower bounds, which produce better performance by more closely modeling the losses. In particular, truncated linear models greatly reduce the problem of overshooting the minimum of the loss function. Unfortunately, truncated linear models cannot be used with parameter-free algorithms because the updates become very expensive to compute. In this paper, we propose new parameter-free algorithms that can take advantage of truncated linear models through a new update that has an “implicit” flavor. Based on a novel decomposition of the regret, the new update is efficient, requires only one gradient at each step, never overshoots the minimum of the truncated model, and retains the favorable parameter-free properties. We also conduct an empirical study demonstrating the practical utility of our algorithms. 

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4. Momentum-Based Variance Reduction in Non-Convex SGD

   Cutkosky, Ashok; Orabona, Francesco (January 2019, Advances in neural information processing systems)

   Wallach, H.; Larochelle, H.; Beygelzimer, A.; d'Alché-Buc, F.; Fox, E.; Garnett, R. (Ed.)

   Variance reduction has emerged in recent years as a strong competitor to stochastic gradient descent in non-convex problems, providing the first algorithms to improve upon the converge rate of stochastic gradient descent for finding first-order critical points. However, variance reduction techniques typically require carefully tuned learning rates and willingness to use excessively large "mega-batches" in order to achieve their improved results. We present a new algorithm, STORM, that does not require any batches and makes use of adaptive learning rates, enabling simpler implementation and less hyperparameter tuning. Our technique for removing the batches uses a variant of momentum to achieve variance reduction in non-convex optimization. On smooth losses $$F$$, STORM finds a point $$\boldsymbol{x}$$ with $$E[\|\nabla F(\boldsymbol{x})\|]\le O(1/\sqrt{T}+\sigma^{1/3}/T^{1/3})$$ in $$T$$ iterations with $$\sigma^2$$ variance in the gradients, matching the optimal rate and without requiring knowledge of $$\sigma$$. 

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5. DoG is SGD’s Best Friend: A Parameter-Free Dynamic Step Size Schedule

   Ivgi, Maor; Hinder, Oliver; Carmon, Yair (July 2023, Proceedings of the 40th International Conference on Machine Learning)

   We propose a tuning-free dynamic SGD step size formula, which we call Distance over Gradients (DoG). The DoG step sizes depend on simple empirical quantities (distance from the initial point and norms of gradients) and have no “learning rate” parameter. Theoretically, we show that, for stochastic convex optimization, a slight variation of the DoG formula enjoys strong, high-probability parameter-free convergence guarantees and iterate movement bounds. Empirically, we consider a broad range of vision and language transfer learning tasks, and show that DoG’s performance is close to that of SGD with tuned learning rate. We also propose a per-layer variant of DoG that generally outperforms tuned SGD, approaching the performance of tuned Adam. A PyTorch implementation of our algorithms is available at https://github.com/formll/dog. 

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