We provide a first-order oracle complexity lower bound for finding stationary points of min-max optimization problems where the objective function is smooth, nonconvex in the minimization variable, and strongly concave in the maximization variable. W
e establish a lower bound of $Omegaleft(sqrt{kappa}epsilon^{-2}right)$ for deterministic oracles, where $epsilon$ defines the level of approximate stationarity and $kappa$ is the condition number. Our analysis shows that the upper bound achieved in (Lin et al., 2020b) is optimal in the $epsilon$ and $kappa$ dependence up to logarithmic factors. For stochastic oracles, we provide a lower bound of $Omegaleft(sqrt{kappa}epsilon^{-2} + kappa^{1/3}epsilon^{-4}right)$. It suggests that there is a significant gap between the upper bound $mathcal{O}(kappa^3 epsilon^{-4})$ in (Lin et al., 2020a) and our lower bound in the condition number dependence.
The label shift problem refers to the supervised learning setting where the train and test label distributions do not match. Existing work addressing label shift usually assumes access to an emph{unlabelled} test sample. This sample may be used to es
timate the test label distribution, and to then train a suitably re-weighted classifier. While approaches using this idea have proven effective, their scope is limited as it is not always feasible to access the target domain; further, they require repeated retraining if the model is to be deployed in emph{multiple} test environments. Can one instead learn a emph{single} classifier that is robust to arbitrary label shifts from a broad family? In this paper, we answer this question by proposing a model that minimises an objective based on distributionally robust optimisation (DRO). We then design and analyse a gradient descent-proximal mirror ascent algorithm tailored for large-scale problems to optimise the proposed objective. %, and establish its convergence. Finally, through experiments on CIFAR-100 and ImageNet, we show that our technique can significantly improve performance over a number of baselines in settings where label shift is present.
We investigate stochastic optimization problems under relaxed assumptions on the distribution of noise that are motivated by empirical observations in neural network training. Standard results on optimal convergence rates for stochastic optimization
assume either there exists a uniform bound on the moments of the gradient noise, or that the noise decays as the algorithm progresses. These assumptions do not match the empirical behavior of optimization algorithms used in neural network training where the noise level in stochastic gradients could even increase with time. We address this behavior by studying convergence rates of stochastic gradient methods subject to changing second moment (or variance) of the stochastic oracle as the iterations progress. When the variation in the noise is known, we show that it is always beneficial to adapt the step-size and exploit the noise variability. When the noise statistics are unknown, we obtain similar improvements by developing an online estimator of the noise level, thereby recovering close variants of RMSProp. Consequently, our results reveal an important scenario where adaptive stepsize methods outperform SGD.
We provide the first non-asymptotic analysis for finding stationary points of nonsmooth, nonconvex functions. In particular, we study the class of Hadamard semi-differentiable functions, perhaps the largest class of nonsmooth functions for which the
chain rule of calculus holds. This class contains examples such as ReLU neural networks and others with non-differentiable activation functions. We first show that finding an $epsilon$-stationary point with first-order methods is impossible in finite time. We then introduce the notion of $(delta, epsilon)$-stationarity, which allows for an $epsilon$-approximate gradient to be the convex combination of generalized gradients evaluated at points within distance $delta$ to the solution. We propose a series of randomized first-order methods and analyze their complexity of finding a $(delta, epsilon)$-stationary point. Furthermore, we provide a lower bound and show that our stochastic algorithm has min-max optimal dependence on $delta$. Empirically, our methods perform well for training ReLU neural networks.
While stochastic gradient descent (SGD) is still the emph{de facto} algorithm in deep learning, adaptive methods like Clipped SGD/Adam have been observed to outperform SGD across important tasks, such as attention models. The settings under which SGD
performs poorly in comparison to adaptive methods are not well understood yet. In this paper, we provide empirical and theoretical evidence that a heavy-tailed distribution of the noise in stochastic gradients is one cause of SGDs poor performance. We provide the first tight upper and lower convergence bounds for adaptive gradient methods under heavy-tailed noise. Further, we demonstrate how gradient clipping plays a key role in addressing heavy-tailed gradient noise. Subsequently, we show how clipping can be applied in practice by developing an emph{adaptive} coordinate-wise clipping algorithm (ACClip) and demonstrate its superior performance on BERT pretraining and finetuning tasks.
We study gradient-based optimization methods obtained by direct Runge-Kutta discretization of the ordinary differential equation (ODE) describing the movement of a heavy-ball under constant friction coefficient. When the function is high order smooth
and strongly convex, we show that directly simulating the ODE with known numerical integrators achieve acceleration in a nontrivial neighborhood of the optimal solution. In particular, the neighborhood can grow larger as the condition number of the function increases. Furthermore, our results also hold for nonconvex but quasi-strongly convex objectives. We provide numerical experiments that verify the theoretical rates predicted by our results.
We provide a theoretical explanation for the effectiveness of gradient clipping in training deep neural networks. The key ingredient is a new smoothness condition derived from practical neural network training examples. We observe that gradient smoot
hness, a concept central to the analysis of first-order optimization algorithms that is often assumed to be a constant, demonstrates significant variability along the training trajectory of deep neural networks. Further, this smoothness positively correlates with the gradient norm, and contrary to standard assumptions in the literature, it can grow with the norm of the gradient. These empirical observations limit the applicability of existing theoretical analyses of algorithms that rely on a fixed bound on smoothness. These observations motivate us to introduce a novel relaxation of gradient smoothness that is weaker than the commonly used Lipschitz smoothness assumption. Under the new condition, we prove that two popular methods, namely, emph{gradient clipping} and emph{normalized gradient}, converge arbitrarily faster than gradient descent with fixed stepsize. We further explain why such adaptively scaled gradient methods can accelerate empirical convergence and verify our results empirically in popular neural network training settings.
Exposure bias has been regarded as a central problem for auto-regressive language models (LM). It claims that teacher forcing would cause the test-time generation to be incrementally distorted due to the training-generation discrepancy. Although a lo
t of algorithms have been proposed to avoid teacher forcing and therefore alleviate exposure bias, there is little work showing how serious the exposure bias problem actually is. In this work, we focus on the task of open-ended language generation, propose metrics to quantify the impact of exposure bias in the aspects of quality, diversity, and consistency. Our key intuition is that if we feed ground-truth data prefixes (instead of prefixes generated by the model itself) into the model and ask it to continue the generation, the performance should become much better because the training-generation discrepancy in the prefix is removed. Both automatic and human evaluations are conducted in our experiments. On the contrary to the popular belief in exposure bias, we find that the the distortion induced by the prefix discrepancy is limited, and does not seem to be incremental during the generation. Moreover, our analysis reveals an interesting self-recovery ability of the LM, which we hypothesize to be countering the harmful effects from exposure bias.
We study smooth stochastic optimization problems on Riemannian manifolds. Via adapting the recently proposed SPIDER algorithm citep{fang2018spider} (a variance reduced stochastic method) to Riemannian manifold, we can achieve faster rate than known a
lgorithms in both the finite sum and stochastic settings. Unlike previous works, by emph{not} resorting to bounding iterate distances, our analysis yields curvature independent convergence rates for both the nonconvex and strongly convex cases.
We develop a distributed algorithm for convex Empirical Risk Minimization, the problem of minimizing large but finite sum of convex functions over networks. The proposed algorithm is derived from directly discretizing the second-order heavy-ball diff
erential equation and results in an accelerated convergence rate, i.e, faster than distributed gradient descent-based methods for strongly convex objectives that may not be smooth. Notably, we achieve acceleration without resorting to the well-known Nesterovs momentum approach. We provide numerical experiments and contrast the proposed method with recently proposed optimal distributed optimization algorithms.
