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We propose a new class of online learning algorithms, generalized implicit Follow-The-Regularized-Leader (FTRL), that expands the scope of FTRL framework. Generalized implicit FTRL can recover known algorithms, such as FTRL with linearized losses and implicit FTRL, and it allows the design of new update rules, as extensions of aProx and Mirror-Prox to FTRL. Our theory is constructive in the sense that it provides a simple unifying framework to design updates that directly improve the worst-case upper bound on the regret. The key idea is substituting the linearization of the losses with a Fenchel-Young inequality. We show the flexibility of the framework by proving that some known algorithms, like the Mirror-Prox updates, are instantiations of the generalized implicit FTRL. Finally, the new framework allows us to recover the temporal variation bound of implicit OMD, with the same computational complexity. 

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. 

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 increasing momentum and 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. 

Convex-concave min-max problems are ubiquitous in machine learning, and people usually utilize first-order methods (e.g., gradient descent ascent) to find the optimal solution. One feature which separates convex-concave min-max problems from convex minimization problems is that the best known convergence rates for min-max problems have an explicit dependence on the size of the domain, rather than on the distance between initial point and the optimal solution. This means that the convergence speed does not have any improvement even if the algorithm starts from the optimal solution, and hence, is oblivious to the initialization. Here, we show that strict-convexity-strict-concavity is sufficient to get the convergence rate to depend on the initialization. We also show how different algorithms can asymptotically achieve initialization-dependent convergence rates on this class of functions. Furthermore, we show that the so-called “parameter-free” algorithms allow to achieve improved initialization-dependent asymptotic rates without any learning rate to tune. In addition, we utilize this particular parameter-free algorithm as a subroutine to design a new algorithm, which achieves a novel non-asymptotic fast rate for strictly-convex-strictly-concave min-max problems with a growth condition and Hölder continuous solution mapping. Experiments are conducted to verify our theoretical findings and demonstrate the effectiveness of the proposed algorithms. 

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.