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We introduce a technique for tuning the learning rate scale factor of any base optimization algorithm and schedule automatically, which we call Mechanic. Our method provides a practical realization of recent theoretical reductions for accomplishing a similar goal in online convex optimization. We rigorously evaluate Mechanic on a range of large scale deep learning tasks with varying batch sizes, schedules, and base optimization algorithms. These experiments demonstrate that depending on the problem, Mechanic either comes very close to, matches or even improves upon manual tuning of learning rates. 

We present new algorithms for optimizing non-smooth, non-convex stochastic objectives based on a novel analysis technique. This improves the current best-known complexity for finding a (δ,ϵ)-stationary point from O(ϵ^(-4),δ^(-1)) stochastic gradient queries to O(ϵ^(-3),δ^(-1)), which we also show to be optimal. Our primary technique is a reduction from non-smooth non-convex optimization to online learning, after which our results follow from standard regret bounds in online learning. For deterministic and second-order smooth objectives, applying more advanced optimistic online learning techniques enables a new complexity of O(ϵ^(-1.5),δ^(-0.5)). Our techniques also recover all optimal or best-known results for finding ϵ stationary points of smooth or second-order smooth objectives in both stochastic and deterministic settings. 

We consider the problem of estimating the mean of a sequence of random elements f (θ, X_1) , . . . , f (θ, X_n) where f is a fixed scalar function, S = (X_1, . . . , X_n) are independent random variables, and θ is a possibly S-dependent parameter. An example of such a problem would be to estimate the generalization error of a neural network trained on n examples where f is a loss function. Classically, this problem is approached through concentration inequalities holding uniformly over compact parameter sets of functions f , for example as in Rademacher or VC type analysis. However, in many problems, such inequalities often yield numerically vacuous estimates. Recently, the PAC-Bayes framework has been proposed as a better alternative for this class of problems for its ability to often give numerically non-vacuous bounds. In this paper, we show that we can do even better: we show how to refine the proof strategy of the PAC-Bayes bounds and achieve even tighter guarantees. Our approach is based on the coin-betting framework that derives the numerically tightest known time-uniform concentration inequalities from the regret guarantees of online gambling algorithms. In particular, we derive the first PAC-Bayes concentration inequality based on the coin-betting approach that holds simultaneously for all sample sizes. We demonstrate its tightness showing that by relaxing it we obtain a number of previous results in a closed form including Bernoulli-KL and empirical Bernstein inequalities. Finally, we propose an efficient algorithm to numerically calculate confidence sequences from our bound, which often generates nonvacuous confidence bounds even with one sample, unlike the state-of-the-art PAC-Bayes bounds. 

Traditional analyses in non-convex optimization typically rely on the smoothness assumption, namely requiring the gradients to be Lipschitz. However, recent evidence shows that this smoothness condition does not capture the properties of some deep learning objective functions, including the ones involving Recurrent Neural Networks and LSTMs. Instead, they satisfy a much more relaxed condition, with potentially unbounded smoothness. Under this relaxed assumption, it has been theoretically and empirically shown that the gradient-clipped SGD has an advantage over the vanilla one. In this paper, we show that clipping is not indispensable for Adam-type algorithms in tackling such scenarios: we theoretically prove that a generalized SignSGD algorithm can obtain similar convergence rates as SGD with clipping but does not need explicit clipping at all. This family of algorithms on one end recovers SignSGD and on the other end closely resembles the popular Adam algorithm. Our analysis underlines the critical role that momentum plays in analyzing SignSGD-type and Adam-type algorithms: it not only reduces the effects of noise, thus removing the need for large mini-batch in previous analyses of SignSGD-type algorithms, but it also substantially reduces the effects of unbounded smoothness and gradient norms. To the best of our knowledge, this work is the first one showing the benefit of Adam-type algorithms compared with non-adaptive gradient algorithms such as gradient descent in the unbounded smoothness setting. We also compare these algorithms with popular optimizers on a set of deep learning tasks, observing that we can match the performance of Adam while beating others. 

SGD with Momentum (SGDM) is a widely used family of algorithms for large-scale optimization of machine learning problems. Yet, when optimizing generic convex functions, no advantage is known for any SGDM algorithm over plain SGD. Moreover, even the most recent results require changes to the SGDM algorithms, like averaging of the iterates and a projection onto a bounded domain, which are rarely used in practice. In this paper, we focus on the convergence rate of the last iterate of SGDM. For the first time, we prove that for any constant momentum factor, there exists a Lipschitz and convex function for which the last iterate of SGDM suffers from a suboptimal convergence rate of $\Omega(\frac{\ln T}{\sqrt{T}})$ after $T$ iterations. Based on this fact, we study a class of (both adaptive and non-adaptive) Follow-The-Regularized-Leader-based SGDM algorithms with \emph{increasing momentum} and \emph{shrinking updates}. For these algorithms, we show that the last iterate has optimal convergence $O(\frac{1}{\sqrt{T}})$ for unconstrained convex stochastic optimization problems without projections onto bounded domains nor knowledge of $T$. Further, we show a variety of results for FTRL-based SGDM when used with adaptive stepsizes. Empirical results are shown as well. 

Stochastic Gradient Descent (SGD) is a popular tool in training large-scale machine learning models. Its performance, however, is highly variable, depending crucially on the choice of the step sizes. Accordingly, a variety of strategies for tuning the step sizes have been proposed, ranging from coordinate-wise approaches (a.k.a. “adaptive” step sizes) to sophisticated heuristics to change the step size in each iteration. In this paper, we study two step size schedules whose power has been repeatedly confirmed in practice: the exponential and the cosine step sizes. For the first time, we provide theoretical support for them proving convergence rates for smooth non-convex functions, with and without the Polyak-Łojasiewicz (PL) condition. Moreover, we show the surprising property that these two strategies are adaptive to the noise level in the stochastic gradients of PL functions. That is, contrary to polynomial step sizes, they achieve almost optimal performance without needing to know the noise level nor tuning their hyperparameters based on it. Finally, we conduct a fair and comprehensive empirical evaluation of real-world datasets with deep learning architectures. Results show that, even if only requiring at most two hyperparameters to tune, these two strategies best or match the performance of various finely-tuned state-of-the-art strategies. 

Inspired by the demands of real-time climate and weather forecasting, we develop optimistic online learning algorithms that require no parameter tuning and have optimal regret guarantees under delayed feedback. Our algorithms—DORM, DORM+, and AdaHedgeD—arise from a novel reduction of delayed online learning to optimistic online learning that reveals how optimistic hints can mitigate the regret penalty caused by delay. We pair this delay-as-optimism perspective with a new analysis of optimistic learning that exposes its robustness to hinting errors and a new meta-algorithm for learning effective hinting strategies in the presence of delay. We conclude by benchmarking our algorithms on four subseasonal climate forecasting tasks, demonstrating low regret relative to state-of-the-art forecasting models.