Normalized Gradients for All

In this short note, I show how to adapt to H\"{o}lder smoothness using normalized gradients in a black-box way. Moreover, the bound will depend on a novel notion of local H\"{o}lder smoothness. The main idea directly comes from Levy [2017].

Implicit Interpretation of Importance Weight Aware Updates

Due to its speed and simplicity, subgradient descent is one of the most used optimization algorithms in convex machine learning algorithms. However, tuning its learning rate is probably its most severe bottleneck to achieve consistent good performance. A common way to reduce the dependency on the learning rate is to use implicit/proximal updates. One such variant is the Importance Weight Aware (IWA) updates, which consist of infinitely many infinitesimal updates on each loss function. However, IWA updates' empirical success is not completely explained by their theory. In this paper, we show for the first time that IWA updates have a strictly better regret upper bound than plain gradient updates in the online learning setting. Our analysis is based on the new framework, generalized implicit Follow-the-Regularized-Leader (FTRL) (Chen and Orabona, 2023), to analyze generalized implicit updates using a dual formulation. In particular, our results imply that IWA updates can be considered as approximate implicit/proximal updates.

Generalized Implicit Follow-The-Regularized-Leader

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, 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.

Tighter PAC-Bayes Bounds Through Coin-Betting

We consider the problem of estimating the mean of a sequence of random elements $f(X_1, \theta)$ $, \ldots, $ $f(X_n, \theta)$ where $f$ is a fixed scalar function, $S=(X_1, \ldots, X_n)$ are independent random variables, and $\theta$ 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 \emph{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 \emph{even tighter} guarantees. Our approach is based on the \emph{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 \emph{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.

Optimal Stochastic Non-smooth Non-convex Optimization through Online-to-Non-convex Conversion

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 $(\delta,\epsilon)$-stationary point from $O(\epsilon^{-4}\delta^{-1})$ stochastic gradient queries to $O(\epsilon^{-3}\delta^{-1})$, 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 $O(\epsilon^{-1.5}\delta^{-0.5})$. Our techniques also recover all optimal or best-known results for finding $\epsilon$ stationary points of smooth or second-order smooth objectives in both stochastic and deterministic settings.

Robustness to Unbounded Smoothness of Generalized SignSGD

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. 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 the others.

Implicit Parameter-free Online Learning with Truncated Linear Models

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.

Understanding AdamW through Proximal Methods and Scale-Freeness

Adam has been widely adopted for training deep neural networks due to less hyperparameter tuning and remarkable performance. To improve generalization, Adam is typically used in tandem with a squared $\ell_2$ regularizer (referred to as Adam-$\ell_2$). However, even better performance can be obtained with AdamW, which decouples the gradient of the regularizer from the update rule of Adam-$\ell_2$. Yet, we are still lacking a complete explanation of the advantages of AdamW. In this paper, we tackle this question from both an optimization and an empirical point of view. First, we show how to re-interpret AdamW as an approximation of a proximal gradient method, which takes advantage of the closed-form proximal mapping of the regularizer instead of only utilizing its gradient information as in Adam-$\ell_2$. Next, we consider the property of "scale-freeness" enjoyed by AdamW and by its proximal counterpart: their updates are invariant to component-wise rescaling of the gradients. We provide empirical evidence across a wide range of deep learning experiments showing a correlation between the problems in which AdamW exhibits an advantage over Adam-$\ell_2$ and the degree to which we expect the gradients of the network to exhibit multiple scales, thus motivating the hypothesis that the advantage of AdamW could be due to the scale-free updates.

Minimax Optimal Quantile and Semi-Adversarial Regret via Root-Logarithmic Regularizers

Quantile (and, more generally, KL) regret bounds, such as those achieved by NormalHedge (Chaudhuri, Freund, and Hsu 2009) and its variants, relax the goal of competing against the best individual expert to only competing against a majority of experts on adversarial data. More recently, the semi-adversarial paradigm (Bilodeau, Negrea, and Roy 2020) provides an alternative relaxation of adversarial online learning by considering data that may be neither fully adversarial nor stochastic (i.i.d.). We achieve the minimax optimal regret in both paradigms using FTRL with separate, novel, root-logarithmic regularizers, both of which can be interpreted as yielding variants of NormalHedge. We extend existing KL regret upper bounds, which hold uniformly over target distributions, to possibly uncountable expert classes with arbitrary priors; provide the first full-information lower bounds for quantile regret on finite expert classes (which are tight); and provide an adaptively minimax optimal algorithm for the semi-adversarial paradigm that adapts to the true, unknown constraint faster, leading to uniformly improved regret bounds over existing methods.