The success of deep learning is due in large part to our ability to solve certain massive non-convex optimization problems with relative ease. Though non-convex optimization is NP-hard, simple algorithms -- often variants of stochastic gradient descent -- exhibit surprising effectiveness in fitting large neural networks in practice. We argue that neural network loss landscapes often contain (nearly) a single basin after accounting for all possible permutation symmetries of hidden units a la Entezari et al. 2021. We introduce three algorithms to permute the units of one model to bring them into alignment with a reference model in order to merge the two models in weight space. This transformation produces a functionally equivalent set of weights that lie in an approximately convex basin near the reference model. Experimentally, we demonstrate the single basin phenomenon across a variety of model architectures and datasets, including the first (to our knowledge) demonstration of zero-barrier linear mode connectivity between independently trained ResNet models on CIFAR-10. Additionally, we identify intriguing phenomena relating model width and training time to mode connectivity. Finally, we discuss shortcomings of the linear mode connectivity hypothesis, including a counterexample to the single basin theory.

When estimating unknown parameters in a dynamic model the optimum solution to the parameter estimation problem may not remain constant. Specifically, the optimal values of the model parameters may change through time because of the evolution of the underlying process: finding them is, in general, not straightforward. A survey of basic techniques for tracking the time-varying dynamics of a system is provided in [Ljung and Gunnarsson, 1990] where recursive algorithms in non-stationary stochastic optimization are analysed under different assumptions about the true system's variations, see also [Simonetto et al., 2020] for a review in a purely deterministic setting. In [Delyon and Juditsky, 1995] the problem of tracking the random drifting parameters of a linear regression system is tackled, and [Zhu and Spall, 2016] builds a computable tracking error bound for how a stochastic approximation with constant gain keeps up with a non-stationary target. Successively, [Wilson et al., 2019] introduces a framework for sequentially solving convex stochastic minimization problems, where the distance between successive minimizers is bounded. The minimization problems are then solved by sequentially applying an optimization algorithm, such as stochastic gradient descent (SGD). In a similar setting, [Cao et al., 2019] establishes an upper bound on the regret of a projected SGD algorithm with respect to the drift of the dynamic optima, while [Cutler et al., 2021] provides novel non-asymptotic convergence guarantees for stochastic algorithms with iterate averaging.

Emerging distributed applications recently boosted the development of decentralized machine learning, especially in IoT and edge computing fields. In real-world scenarios, the common problems of non-convexity and data heterogeneity result in inefficiency, performance degradation, and development stagnation. The bulk of studies concentrates on one of the issues mentioned above without having a more general framework that has been proven optimal. To this end, we propose a unified paradigm called UMP, which comprises two algorithms, D-SUM and GT-DSUM, based on the momentum technique with decentralized stochastic gradient descent(SGD). The former provides a convergence guarantee for general non-convex objectives. At the same time, the latter is extended by introducing gradient tracking, which estimates the global optimization direction to mitigate data heterogeneity(i.e., distribution drift). We can cover most momentum-based variants based on the classical heavy ball or Nesterov's acceleration with different parameters in UMP. In theory, we rigorously provide the convergence analysis of these two approaches for non-convex objectives and conduct extensive experiments, demonstrating a significant improvement in model accuracy by up to 57.6% compared to other methods in practice.

Many mathematical convergence results for gradient descent (GD) based algorithms employ the assumption that the GD process is (almost surely) bounded and, also in concrete numerical simulations, divergence of the GD process may slow down, or even completely rule out, convergence of the error function. In practical relevant learning problems, it thus seems to be advisable to design the ANN architectures in a way so that GD optimization processes remain bounded. The property of the boundedness of GD processes for a given learning problem seems, however, to be closely related to the existence of minimizers in the optimization landscape and, in particular, GD trajectories may escape to infinity if the infimum of the error function (objective function) is not attained in the optimization landscape. This naturally raises the question of the existence of minimizers in the optimization landscape and, in the situation of shallow residual ANNs with multi-dimensional input layers and multi-dimensional hidden layers with the ReLU activation, the main result of this work answers this question affirmatively for a general class of loss functions and all continuous target functions. In our proof of this statement, we propose a kind of closure of the search space, where the limits are called generalized responses, and, thereafter, we provide sufficient criteria for the loss function and the underlying probability distribution which ensure that all additional artificial generalized responses are suboptimal which finally allows us to conclude the existence of minimizers in the optimization landscape.

A central issue in machine learning is how to train models on sensitive user data. Industry has widely adopted a simple algorithm: Stochastic Gradient Descent with noise (a.k.a. Stochastic Gradient Langevin Dynamics). However, foundational theoretical questions about this algorithm's privacy loss remain open -- even in the seemingly simple setting of smooth convex losses over a bounded domain. Our main result resolves these questions: for a large range of parameters, we characterize the differential privacy up to a constant factor. This result reveals that all previous analyses for this setting have the wrong qualitative behavior. Specifically, while previous privacy analyses increase ad infinitum in the number of iterations, we show that after a small burn-in period, running SGD longer leaks no further privacy. Our analysis departs from previous approaches based on fast mixing, instead using techniques based on optimal transport (namely, Privacy Amplification by Iteration) and the Sampled Gaussian Mechanism (namely, Privacy Amplification by Sampling). Our techniques readily extend to other settings, e.g., strongly convex losses, non-uniform stepsizes, arbitrary batch sizes, and random or cyclic choice of batches.

Based on the weight-sharing mechanism, one-shot NAS methods train a supernet and then inherit the pre-trained weights to evaluate sub-models, largely reducing the search cost. However, several works have pointed out that the shared weights suffer from different gradient descent directions during training. And we further find that large gradient variance occurs during supernet training, which degrades the supernet ranking consistency. To mitigate this issue, we propose to explicitly minimize the gradient variance of the supernet training by jointly optimizing the sampling distributions of PAth and DAta (PA&DA). We theoretically derive the relationship between the gradient variance and the sampling distributions, and reveal that the optimal sampling probability is proportional to the normalized gradient norm of path and training data. Hence, we use the normalized gradient norm as the importance indicator for path and training data, and adopt an importance sampling strategy for the supernet training. Our method only requires negligible computation cost for optimizing the sampling distributions of path and data, but achieves lower gradient variance during supernet training and better generalization performance for the supernet, resulting in a more consistent NAS. We conduct comprehensive comparisons with other improved approaches in various search spaces. Results show that our method surpasses others with more reliable ranking performance and higher accuracy of searched architectures, showing the effectiveness of our method. Code is available at https://github.com/ShunLu91/PA-DA.

Stochastic optimization is a fundamental area with extensive applications in many domains, ranging from machine learning to algorithm design and beyond. The design and analysis of iterative methods for stochastic optimization has been the focus of a long line of work, leading to a rich understanding of the convergence of paradigmatic iterative methods such as stochastic gradient descent, mirror descent, and accelerated methods for both convex and non-convex optimization. However, most of these works only establish convergence guarantees that hold only in expectation. Although very meaningful, these results do not fully capture the convergence behaviors of the algorithms when we perform only a small number of runs of the algorithm, as it is typical in modern machine learning applications where there are significant computational and statistical costs associated with performing multiple runs of the algorithm (Harvey et al., 2019; Madden et al., 2020; Davis et al., 2021). Thus, an important direction is to establish convergence guarantees for a single run of the algorithm that hold not only in expectation but also with high probability. Compared to the guarantees that hold in expectation, high probability guarantees are significantly harder to obtain and they hold in more limited settings with stronger assumptions on the problem settings and the stochastic noise distribution. Most existing works that establish high probability guarantees focus on the setting where the length of the stochastic noise follows a light-tail (sub-Gaussian) distribution (Juditsky et al., 2011; Lan, 2012, 2020; Li and Orabona, 2020; Madden et al., 2020; Kavis et al., 2021). Recent works also study the more challenging heavy-tail setting, notably under a bounded variance (Nazin et al., 2019; Gorbunov et al., 2020; Cutkosky and Mehta, 2021) or bounded p-moment assumption (Cutkosky and Mehta, 2021) on the length of the stochastic noise. Both settings are highly relevant in practice: Zhang et al. (2020) empirically studied the noise distribution for two common tasks, training a

Stochastic models with global parameters $\bm{\theta}$ and latent variables $\bm{z}$ are common, and variational inference (VI) is popular for their estimation. This paper uses a variational approximation (VA) that comprises a Gaussian with factor covariance matrix for the marginal of $\bm{\theta}$, and the exact conditional posterior of $\bm{z}|\bm{\theta}$. Stochastic optimization for learning the VA only requires generation of $\bm{z}$ from its conditional posterior, while $\bm{\theta}$ is updated using the natural gradient, producing a hybrid VI method. We show that this is a well-defined natural gradient optimization algorithm for the joint posterior of $(\bm{z},\bm{\theta})$. Fast to compute expressions for the Tikhonov damped Fisher information matrix required to compute a stable natural gradient update are derived. We use the approach to estimate probabilistic Bayesian neural networks with random output layer coefficients to allow for heterogeneity. Simulations show that using the natural gradient is more efficient than using the ordinary gradient, and that the approach is faster and more accurate than two leading benchmark natural gradient VI methods. In a financial application we show that accounting for industry level heterogeneity using the deep model improves the accuracy of probabilistic prediction of asset pricing models.

Quantum federated learning (QFL) is a quantum extension of the classical federated learning model across multiple local quantum devices. An efficient optimization algorithm is always expected to minimize the communication overhead among different quantum participants. In this work, we propose an efficient optimization algorithm, namely federated quantum natural gradient descent (FQNGD), and further, apply it to a QFL framework that is composed of a variational quantum circuit (VQC)-based quantum neural networks (QNN). Compared with stochastic gradient descent methods like Adam and Adagrad, the FQNGD algorithm admits much fewer training iterations for the QFL to get converged. Moreover, it can significantly reduce the total communication overhead among local quantum devices. Our experiments on a handwritten digit classification dataset justify the effectiveness of the FQNGD for the QFL framework in terms of a faster convergence rate on the training set and higher accuracy on the test set.

We derive and solve an ``Equation of Motion'' (EoM) for deep neural networks (DNNs), a differential equation that precisely describes the discrete learning dynamics of DNNs. Differential equations are continuous but have played a prominent role even in the study of discrete optimization (gradient descent (GD) algorithms). However, there still exist gaps between differential equations and the actual learning dynamics of DNNs due to discretization error. In this paper, we start from gradient flow (GF) and derive a counter term that cancels the discretization error between GF and GD. As a result, we obtain EoM, a continuous differential equation that precisely describes the discrete learning dynamics of GD. We also derive discretization error to show to what extent EoM is precise. In addition, we apply EoM to two specific cases: scale- and translation-invariant layers. EoM highlights differences between continuous-time and discrete-time GD, indicating the importance of the counter term for a better description of the discrete learning dynamics of GD. Our experimental results support our theoretical findings.