Natural gradient descent, which preconditions a gradient descent update with the Fisher information matrix of the underlying statistical model, is a way to capture partial second-order information. Several highly visible works have advocated an approximation known as the empirical Fisher, drawing connections between approximate second-order methods and heuristics like Adam. We dispute this argument by showing that the empirical Fisher---unlike the Fisher---does not generally capture second-order information. We further argue that the conditions under which the empirical Fisher approaches the Fisher (and the Hessian) are unlikely to be met in practice, and that, even on simple optimization problems, the pathologies of the empirical Fisher can have undesirable effects.

We study the Stein Variational Gradient Descent (SVGD) algorithm, which optimises a set of particles to approximate a target probability distribution \pi\propto e {-V} on \R d . In the population limit, SVGD performs gradient descent in the space of probability distributions on the KL divergence with respect to \pi, where the gradient is smoothed through a kernel integral operator. In this paper, we provide a novel finite time analysis for the SVGD algorithm. We provide a descent lemma establishing that the algorithm decreases the objective at each iteration, and rates of convergence. We also provide a convergence result of the finite particle system corresponding to the practical implementation of SVGD to its population version.

Stochastic gradient descent (SGD) exhibits strong algorithmic regularization effects in practice, which has been hypothesized to play an important role in the generalization of modern machine learning approaches. In this work, we seek to understand these issues in the simpler setting of linear regression (including both underparameterized and overparameterized regimes), where our goal is to make sharp instance-based comparisons of the implicit regularization afforded by (unregularized) average SGD with the explicit regularization of ridge regression. For a broad class of least squares problem instances (that are natural in high-dimensional settings), we show: (1) for every problem instance and for every ridge parameter, (unregularized) SGD, when provided with \emph{logarithmically} more samples than that provided to the ridge algorithm, generalizes no worse than the ridge solution (provided SGD uses a tuned constant stepsize); (2) conversely, there exist instances (in this wide problem class) where optimally-tuned ridge regression requires \emph{quadratically} more samples than SGD in order to have the same generalization performance. Taken together, our results show that, up to the logarithmic factors, the generalization performance of SGD is always no worse than that of ridge regression in a wide range of overparameterized problems, and, in fact, could be much better for some problem instances. More generally, our results show how algorithmic regularization has important consequences even in simpler (overparameterized) convex settings.

Differentially Private Stochastic Gradient Descent (DP-SGD) forms a fundamental building block in many applications for learning over sensitive data. Two standard approaches, privacy amplification by subsampling, and privacy amplification by shuffling, permit adding lower noise in DP-SGD than via na\"{\i}ve schemes. A key assumption in both these approaches is that the elements in the data set can be uniformly sampled, or be uniformly permuted --- constraints that may become prohibitive when the data is processed in a decentralized or distributed fashion. In this paper, we focus on conducting iterative methods like DP-SGD in the setting of federated learning (FL) wherein the data is distributed among many devices (clients). Our main contribution is the \emph{random check-in} distributed protocol, which crucially relies only on randomized participation decisions made locally and independently by each client.

In this paper, we study a class of stochastic bilevel optimization problems, also known as stochastic simple bilevel optimization, where we minimize a smooth stochastic objective function over the optimal solution set of another stochastic convex optimization problem. We introduce novel stochastic bilevel optimization methods that locally approximate the solution set of the lower-level problem via a stochastic cutting plane, and then run a conditional gradient update with variance reduction techniques to control the error induced by using stochastic gradients. For the case that the upper-level function is convex, our method requires \mathcal{O}(\max\\{1/\epsilon_f {2},1/\epsilon_g {2}\\}) stochastic oracle queries to obtain a solution that is \epsilon_f -optimal for the upper-level and \epsilon_g -optimal for the lower-level. Moreover, for the case that the upper-level function is non-convex, our method requires at most \mathcal{O}(\max\\{1/\epsilon_f {3},1/\epsilon_g {3}\\}) stochastic oracle queries to find an (\epsilon_f, \epsilon_g) -stationary point. In the finite-sum setting, we show that the number of stochastic oracle calls required by our method are \mathcal{O}(\sqrt{n}/\epsilon) and \mathcal{O}(\sqrt{n}/\epsilon {2}) for the convex and non-convex settings, respectively, where \epsilon \min \\{\epsilon_f,\epsilon_g\\} .

Inference, estimation, sampling and likelihood evaluation are four primary goals of probabilistic modeling. Practical considerations often force modeling approaches to make compromises between these objectives. We present a novel probabilistic learning framework, called Fenchel Mini-Max Learning (FML), that accommodates all four desiderata in a flexible and scalable manner. Our derivation is rooted in classical maximum likelihood estimation, and it overcomes a longstanding challenge that prevents unbiased estimation of unnormalized statistical models. By reformulating MLE as a mini-max game, FML enjoys an unbiased training objective that (i) does not explicitly involve the intractable normalizing constant and (ii) is directly amendable to stochastic gradient descent optimization.

Uniform stability is a notion of algorithmic stability that bounds the worst case change in the model output by the algorithm when a single data point in the dataset is replaced. An influential work of Hardt et al. [2016] provides strong upper bounds on the uniform stability of the stochastic gradient descent (SGD) algorithm on sufficiently smooth convex losses. These results led to important progress in understanding of the generalization properties of SGD and several applications to differentially private convex optimization for smooth losses. Our work is the first to address uniform stability of SGD on nonsmooth convex losses. Specifically, we provide sharp upper and lower bounds for several forms of SGD and full-batch GD on arbitrary Lipschitz nonsmooth convex losses.

We show, both theoretically and empirically, that the algorithm naturally adapts to data low-rankness and converges exponentially fast to the ground-truth principal subspace. Notably, our result suggests that despite various recent efforts to accelerate the convergence of stochastic-gradient based methods by adding a O(n)-time variance reduction step, for the k- PCA problem, a truly online SGD variant suffices to achieve exponential convergence on intrinsically low-rank data.

In deep neural nets, lower level embedding layers account for a large portion of the total number of parameters. Tikhonov regularization, graph-based regularization, and hard parameter sharing are approaches that introduce explicit biases into training in a hope to reduce statistical complexity. Alternatively, we propose stochastically shared embeddings (SSE), a data-driven approach to regularizing embedding layers, which stochastically transitions between embeddings during stochastic gradient descent (SGD). Because SSE integrates seamlessly with existing SGD algorithms, it can be used with only minor modifications when training large scale neural networks. We develop two versions of SSE: SSE-Graph using knowledge graphs of embeddings; SSE-SE using no prior information.

In this paper, we propose a new covering technique localized for the trajectories of SGD. This localization provides an algorithm-specific complexity measured by the covering number, which can have dimension-independent cardinality in contrast to standard uniform covering arguments that result in exponential dimension dependency. Based on this localized construction, we show that if the objective function is a finite perturbation of a piecewise strongly convex and smooth function with P pieces, i.e., non-convex and non-smooth in general, the generalization error can be upper bounded by O(\sqrt{(\log n\log(nP))/n}), where n is the number of data samples. In particular, this rate is independent of dimension and does not require early stopping and decaying step size. Finally, we employ these results in various contexts and derive generalization bounds for multi-index linear models, multi-class support vector machines, and K -means clustering for both hard and soft label setups, improving the previously known state-of-the-art rates.