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Register a new userTrain simultaneously, generalize better: Stability of gradient-based minimax learners
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Farzan Farnia
Publication date
2020
fields
Informatics Engineering
and research's language is
English
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The success of minimax learning problems of generative adversarial networks (GANs) has been observed to depend on the minimax optimization algorithm used for their training. This dependence is commonly attributed to the convergence speed and robustness properties of the underlying optimization algorithm. In this paper, we show that the optimization algorithm also plays a key role in the generalization performance of the trained minimax model. To this end, we analyze the generalization properties of standard gradient descent ascent (GDA) and proximal point method (PPM) algorithms through the lens of algorithmic stability under both convex concave and non-convex non-concave minimax settings. While the GDA algorithm is not guaranteed to have a vanishing excess risk in convex concave problems, we show the PPM algorithm enjoys a bounded excess risk in the same setup. For non-convex non-concave problems, we compare the generalization performance of stochastic GDA and GDmax algorithms where the latter fully solves the maximization subproblem at every iteration. Our generalization analysis suggests the superiority of GDA provided that the minimization and maximization subproblems are solved simultaneously with similar learning rates. We discuss several numerical results indicating the role of optimization algorithms in the generalization of the learned minimax models.
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Generalization is a central problem in Machine Learning. Most prediction methods require careful calibration of hyperparameters carried out on a hold-out textit{validation} dataset to achieve generalization. The main goal of this paper is to present a novel approach based on a new measure of risk that allows us to develop novel fully automatic procedures for generalization. We illustrate the pertinence of this new framework in the regression problem. The main advantages of this new approach are: (i) it can simultaneously train the model and perform regularization in a single run of a gradient-based optimizer on all available data without any previous hyperparameter tuning; (ii) this framework can tackle several additional objectives simultaneously (correlation, sparsity,...) $via$ the introduction of regularization parameters. Noticeably, our approach transforms hyperparameter tuning as well as feature selection (a combinatorial discrete optimization problem) into a continuous optimization problem that is solvable via classical gradient-based methods ; (iii) the computational complexity of our methods is $O(npK)$ where $n,p,K$ denote respectively the number of observations, features and iterations of the gradient descent algorithm. We observe in our experiments a significantly smaller runtime for our methods as compared to benchmark methods for equivalent prediction score. Our procedures are implemented in PyTorch (code is available for replication).
Smooth minimax games often proceed by simultaneous or alternating gradient updates. Although algorithms with alternating updates are commonly used in practice for many applications (e.g., GAN training), the majority of existing theoretical analyses focus on simultaneous algorithms for convenience of analysis. In this paper, we study alternating gradient descent-ascent (Alt-GDA) in minimax games and show that Alt-GDA is superior to its simultaneous counterpart (Sim-GDA) in many settings. In particular, we prove that Alt-GDA achieves a near-optimal local convergence rate for strongly convex-strongly concave (SCSC) problems while Sim-GDA converges at a much slower rate. To our knowledge, this is the emph{first} result of any setting showing that Alt-GDA converges faster than Sim-GDA by more than a constant. We further prove that the acceleration effect of alternating updates remains when the minimax problem has only strong concavity in the dual variables. Lastly, we adapt the theory of integral quadratic constraints and show that Alt-GDA attains the same rate emph{globally} for a class of SCSC minimax problems. Numerical experiments on quadratic minimax games validate our claims. Empirically, we demonstrate that alternating updates speed up GAN training significantly and the use of optimism only helps for simultaneous algorithms.
We consider a binary classification problem when the data comes from a mixture of two rotationally symmetric distributions satisfying concentration and anti-concentration properties enjoyed by log-concave distributions among others. We show that there exists a universal constant $C_{mathrm{err}}>0$ such that if a pseudolabeler $boldsymbol{beta}_{mathrm{pl}}$ can achieve classification error at most $C_{mathrm{err}}$, then for any $varepsilon>0$, an iterative self-training algorithm initialized at $boldsymbol{beta}_0 := boldsymbol{beta}_{mathrm{pl}}$ using pseudolabels $hat y = mathrm{sgn}(langle boldsymbol{beta}_t, mathbf{x}rangle)$ and using at most $tilde O(d/varepsilon^2)$ unlabeled examples suffices to learn the Bayes-optimal classifier up to $varepsilon$ error, where $d$ is the ambient dimension. That is, self-training converts weak learners to strong learners using only unlabeled examples. We additionally show that by running gradient descent on the logistic loss one can obtain a pseudolabeler $boldsymbol{beta}_{mathrm{pl}}$ with classification error $C_{mathrm{err}}$ using only $O(d)$ labeled examples (i.e., independent of $varepsilon$). Together our results imply that mixture models can be learned to within $varepsilon$ of the Bayes-optimal accuracy using at most $O(d)$ labeled examples and $tilde O(d/varepsilon^2)$ unlabeled examples by way of a semi-supervised self-training algorithm.
Many machine learning problems can be formulated as minimax problems such as Generative Adversarial Networks (GANs), AUC maximization and robust estimation, to mention but a few. A substantial amount of studies are devoted to studying the convergence behavior of their stochastic gradient-type algorithms. In contrast, there is relatively little work on their generalization, i.e., how the learning models built from training examples would behave on test examples. In this paper, we provide a comprehensive generalization analysis of stochastic gradient methods for minimax problems under both convex-concave and nonconvex-nonconcave cases through the lens of algorithmic stability. We establish a quantitative connection between stability and several generalization measures both in expectation and with high probability. For the convex-concave setting, our stability analysis shows that stochastic gradient descent ascent attains optimal generalization bounds for both smooth and nonsmooth minimax problems. We also establish generalization bounds for both weakly-convex-weakly-concave and gradient-dominated problems.
Several learning problems involve solving min-max problems, e.g., empirical distributional robust learning or learning with non-standard aggregated losses. More specifically, these problems are convex-linear problems where the minimization is carried out over the model parameters $winmathcal{W}$ and the maximization over the empirical distribution $pinmathcal{K}$ of the training set indexes, where $mathcal{K}$ is the simplex or a subset of it. To design efficient methods, we let an online learning algorithm play against a (combinatorial) bandit algorithm. We argue that the efficiency of such approaches critically depends on the structure of $mathcal{K}$ and propose two properties of $mathcal{K}$ that facilitate designing efficient algorithms. We focus on a specific family of sets $mathcal{S}_{n,k}$ encompassing various learning applications and provide high-probability convergence guarantees to the minimax values.
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