In privacy-preserving machine learning, differentially private stochastic gradient descent (DP-SGD) performs worse than SGD due to per-sample gradient clipping and noise addition. A recent focus in private learning research is improving the performance of DP-SGD on private data by incorporating priors that are learned on real-world public data. In this work, we explore how we can improve the privacy-utility tradeoff of DP-SGD by learning priors from images generated by random processes and transferring these priors to private data. We propose DP-RandP, a three-phase approach. We attain new state-of-the-art accuracy when training from scratch on CIFAR10, CIFAR100, MedMNIST and ImageNet for a range of privacy budgets $\varepsilon \in [1, 8]$. In particular, we improve the previous best reported accuracy on CIFAR10 from $60.6 \%$ to $72.3 \%$ for $\varepsilon=1$.

Diffusion processes are a class of stochastic differential equations (SDEs) providing a rich family of expressive models that arise naturally in dynamic modelling tasks. Probabilistic inference and learning under generative models with latent processes endowed with a non-linear diffusion process prior are intractable problems. We build upon work within variational inference, approximating the posterior process as a linear diffusion process, and point out pathologies in the approach. We propose an alternative parameterization of the Gaussian variational process using a site-based exponential family description. This allows us to trade a slow inference algorithm with fixed-point iterations for a fast algorithm for convex optimization akin to natural gradient descent, which also provides a better objective for learning model parameters.

Biological and artificial learning agents face numerous choices about how to learn, ranging from hyperparameter selection to aspects of task distributions like curricula. Understanding how to make these meta-learning choices could offer normative accounts of cognitive control functions in biological learners and improve engineered systems. Yet optimal strategies remain challenging to compute in modern deep networks due to the complexity of optimizing through the entire learning process. Here we theoretically investigate optimal strategies in a tractable setting. We present a learning effort framework capable of efficiently optimizing control signals on a fully normative objective: discounted cumulative performance throughout learning. We obtain computational tractability by using average dynamical equations for gradient descent, available for simple neural network architectures. Our framework accommodates a range of meta-learning and automatic curriculum learning methods in a unified normative setting. We apply this framework to investigate the effect of approximations in common meta-learning algorithms; infer aspects of optimal curricula; and compute optimal neuronal resource allocation in a continual learning setting. Across settings, we find that control effort is most beneficial when applied to easier aspects of a task early in learning; followed by sustained effort on harder aspects. Overall, the learning effort framework provides a tractable theoretical test bed to study normative benefits of interventions in a variety of learning systems, as well as a formal account of optimal cognitive control strategies over learning trajectories posited by established theories in cognitive neuroscience.

The increasing scale of data propels the popularity of leveraging parallelism to speed up the optimization. Minibatch stochastic gradient descent (minibatch SGD) and local SGD are two popular methods for parallel optimization. The existing theoretical studies show a linear speedup of these methods with respect to the number of machines, which, however, is measured by optimization errors. As a comparison, the stability and generalization of these methods are much less studied. In this paper, we study the stability and generalization analysis of minibatch and local SGD to understand their learnability by introducing a novel expectation-variance decomposition. We incorporate training errors into the stability analysis, which shows how small training errors help generalization for overparameterized models. We show both minibatch and local SGD achieve a linear speedup to attain the optimal risk bounds.

To that aim we introduce new g-convex optimization results, of independent interest: we show global linear convergence for metric-projected Riemannian gradient descent and improve existing accelerated methods by reducing geometric constants. Additionally, we complete the analysis of two previous works applying to the Riemannian min-max case by removing an assumption about iterates staying in a pre-specified compact set.

We study first-order optimization algorithms for computing the barycenter of Gaussian distributions with respect to the optimal transport metric. Although the objective is geodesically non-convex, Riemannian GD empirically converges rapidly, in fact faster than off-the-shelf methods such as Euclidean GD and SDP solvers. This stands in stark contrast to the best-known theoretical results for Riemannian GD, which depend exponentially on the dimension. In this work, we prove new geodesic convexity results which provide stronger control of the iterates, yielding a dimension-free convergence rate. Our techniques also enable the analysis of two related notions of averaging, the entropically-regularized barycenter and the geometric median, providing the first convergence guarantees for Riemannian GD for these problems.

Sharpness-aware Minimization (SAM) [Foret et al., 2020] is a new gradient-based neural network training algorithm that advanced the state-of-the-art test accuracy on a number of prominent benchmark datasets. As its name suggests, it explicitly seeks to find a solution that not only fits the training data, but that avoids "sharp" minima, for which nearby parameter vectors perform poorly. SAM is an incremental algorithm that updates its parameters using a gradient computed at a neighbor of the current solution. The neighbor is the point in parameter space found by taking a step of length ρ "uphill" in the gradient direction. The practical success of SAM has motivated theoretical research [Bartlett et al., 2022, Wen et al., 2023, Andriushchenko et al., 2023], including results highlighting senses in which SAM's update may be viewed, under certain conditions, as including a component that performs gradient descent on the operator norm of the Hessian [Bartlett et al., 2022, Wen et al., 2023]. Meanwhile, Cohen et al. [2021], building on the work of Jastrzebski et al. [2020] and others, exposed a striking phenomenon regarding neural network training with the original gradient descent (GD) method: for many initialization schemes and learning rates η, the operator norm of the Hessian eventually settles in the neighborhood of 2/η. This has been termed the "edge of stability", in part because a convex quadratic trained by gradient descent with a learning rate η will only Also affiliated with University of California, Berkeley.

Wasserstein distance (WD) and the associated optimal transport plan have been proven useful in many applications where probability measures are at stake. In this paper, we propose a new proxy of the squared WD, coined min-SWGG, that is based on the transport map induced by an optimal one-dimensional projection of the two input distributions. We draw connections between min-SWGG and Wasserstein generalized geodesics in which the pivot measure is supported on a line. We notably provide a new closed form for the exact Wasserstein distance in the particular case of one of the distributions supported on a line allowing us to derive a fast computational scheme that is amenable to gradient descent optimization. We show that min-SWGG is an upper bound of WD and that it has a complexity similar to as Sliced-Wasserstein, with the additional feature of providing an associated transport plan. We also investigate some theoretical properties such as metricity, weak convergence, computational and topological properties. Empirical evidences support the benefits of min-SWGG in various contexts, from gradient flows, shape matching and image colorization, among others.

Nonconvex-nonconcave minimax optimization has received intense attention over the last decade due to its broad applications in machine learning. Most existing algorithms rely on one-sided information, such as the convexity (resp. concavity) of the primal (resp. dual) functions, or other specific structures, such as the Polyak-\L{}ojasiewicz (P\L{}) and Kurdyka-\L{}ojasiewicz (K\L{}) conditions. However, verifying these regularity conditions is challenging in practice. To meet this challenge, we propose a novel universally applicable single-loop algorithm, the doubly smoothed gradient descent ascent method (DS-GDA), which naturally balances the primal and dual updates. That is, DS-GDA with the same hyperparameters is able to uniformly solve nonconvex-concave, convex-nonconcave, and nonconvex-nonconcave problems with one-sided K\L{} properties, achieving convergence with $\mathcal{O}(\epsilon^{-4})$ complexity. Sharper (even optimal) iteration complexity can be obtained when the K\L{} exponent is known. Specifically, under the one-sided K\L{} condition with exponent $\theta\in(0,1)$, DS-GDA converges with an iteration complexity of $\mathcal{O}(\epsilon^{-2\max\{2\theta,1\}})$. They all match the corresponding best results in the literature. Moreover, we show that DS-GDA is practically applicable to general nonconvex-nonconcave problems even without any regularity conditions, such as the P\L{} condition, K\L{} condition, or weak Minty variational inequalities condition. For various challenging nonconvex-nonconcave examples in the literature, including ``Forsaken'', ``Bilinearly-coupled minimax'', ``Sixth-order polynomial'', and ``PolarGame'', the proposed DS-GDA can all get rid of limit cycles. To the best of our knowledge, this is the first first-order algorithm to achieve convergence on all of these formidable problems.

We focus on gradient descent and its variants, as they are widely adopted and scale well when the model dimensionality d is large (Bottou et al., 2018). The optimization problem (OPT) finds many applications: in solving linear systems, logistic regression, support vector machines, and other areas of machine learning (Boyd and Vandenberghe, 2004). Equally important, methods designed for (stochastic) convex optimization also influence the intuition for and design of methods for nonconvex optimization-for example, momentum (Polyak, 1964), AdaGrad (Duchi et al., 2010), and Adam (Kingma and Ba, 2015) were all first analyzed in the convex optimization framework. As models become larger and more complex, the cost and environmental impact of training have rapidly grown as well (Sharir et al., 2020; Patterson et al., 2021). Therefore, it is vital that we develop more efficient and effective methods of solving machine learning optimization tasks. One of the chief challenges in applying gradient-based methods is that they often require tuning one or more stepsize parameters (Goodfellow et al., 2016), and the choice of stepsize can significantly influence a method's convergence speed as well as the quality of the obtained solutions, especially in deep learning (Wilson et al., 2017). The cost and impact of hyperparameter tuning on the optimization process have led to significant research activity in designing parameter-free and adaptive optimization methods in recent years, see e.g.