Principal component analysis (PCA) has been widely used as an effective technique for feature extraction and dimension reduction. In the High Dimension Low Sample Size (HDLSS) setting, one may prefer modified principal components, with penalized loadings, and automated penalty selection by implementing model selection among these different models with varying penalties. The earlier work [1, 2] has proposed penalized PCA, indicating the feasibility of model selection in $L_2$- penalized PCA through the solution path of Ridge regression, however, it is extremely time-consuming because of the intensive calculation of matrix inverse. In this paper, we propose a fast model selection method for penalized PCA, named Approximated Gradient Flow (AgFlow), which lowers the computation complexity through incorporating the implicit regularization effect introduced by (stochastic) gradient flow [3, 4] and obtains the complete solution path of $L_2$-penalized PCA under varying $L_2$-regularization. We perform extensive experiments on real-world datasets. AgFlow outperforms existing methods (Oja [5], Power [6], and Shamir [7] and the vanilla Ridge estimators) in terms of computation costs.

Overparametrized Deep Neural Networks (DNNs) often achieve astounding performances, but may potentially result in severe generalization error. Recently, the relation between the sharpness of the loss landscape and the generalization error has been established by Foret et al. (2020), in which the Sharpness Aware Minimizer (SAM) was proposed to mitigate the degradation of the generalization. Unfortunately, SAM's computational cost is roughly double that of base optimizers, such as Stochastic Gradient Descent (SGD). This paper thus proposes Efficient Sharpness Aware Minimizer (ESAM), which boosts SAM's efficiency at no cost to its generalization performance. ESAM includes two novel and efficient training strategies--Stochastic Weight Perturbation and Sharpness-Sensitive Data Selection. In the former, the sharpness measure is approximated by perturbing a stochastically chosen set of weights in each iteration; in the latter, the SAM loss is optimized using only a judiciously selected subset of data that is sensitive to the sharpness. We provide theoretical explanations as to why these strategies perform well. We also show, via extensive experiments on the CIFAR and ImageNet datasets, that ESAM enhances the efficiency over SAM from requiring 100% extra computational overhead to 40% vis-à-vis base optimizers, while test accuracies are preserved or even improved.

Finding the optimal configuration of parameters in ResNet is a nonconvex minimization problem, but first-order methods nevertheless find the global optimum in the overparameterized regime. We study this phenomenon with mean-field analysis, by translating the training process of ResNet to a gradient-flow partial differential equation (PDE) and examining the convergence properties of this limiting process. The activation function is assumed to be $2$-homogeneous or partially $1$-homogeneous; the regularized ReLU satisfies the latter condition. We show that if the ResNet is sufficiently large, with depth and width depending algebraically on the accuracy and confidence levels, first-order optimization methods can find global minimizers that fit the training data.

We present a new lossy compression algorithm for statistical floating-point data through a representation learning with binary variables. The algorithm finds a set of basis vectors and their binary coefficients that precisely reconstruct the original data. The optimization for the basis vectors is performed classically, while binary coefficients are retrieved through both simulated and quantum annealing for comparison. A bias correction procedure is also presented to estimate and eliminate the error and bias introduced from the inexact reconstruction of the lossy compression for statistical data analyses. The compression algorithm is demonstrated on two different datasets of lattice quantum chromodynamics simulations. The results obtained using simulated annealing show 3.5 times better compression performance than the algorithms based on a neural-network autoencoder and principal component analysis. Calculations using quantum annealing also show promising results, but performance is limited by the integrated control error of the quantum processing unit, which yields large uncertainties in the biases and coupling parameters. Hardware comparison is further studied between the previous generation D-Wave 2000Q and the current D-Wave Advantage system. Our study shows that the Advantage system is more likely to obtain low-energy solutions for the problems than the 2000Q.

Before training the model, let's dive into an important (hyper)parameter called learning rate that we will be optimizing in this workflow. Neural networks are trained using an optimization algorithm called stochastic gradient descent. In stochastic gradient descent, the error gradient for the current state of the model are estimated using back propagation, which mathematically means that we now have an intuitive understanding about the influence of changing weights on model performance. Using the error gradient, weights of the model are updated using a pre-determined step-size known as learning rate. In other words, learning rate is a variable that controls how much to change the model in response to the estimated error each time the model weights are updated.

Emerging applications in multi-agent environments such as internet-of-things, networked sensing, autonomous systems and federated learning, call for decentralized algorithms for finite-sum optimizations that are resource-efficient in terms of both computation and communication. In this paper, we consider the prototypical setting where the agents work collaboratively to minimize the sum of local loss functions by only communicating with their neighbors over a predetermined network topology. We develop a new algorithm, called DEcentralized STochastic REcurSive gradient methodS (DESTRESS) for nonconvex finite-sum optimization, which matches the optimal incremental first-order oracle (IFO) complexity of centralized algorithms for finding first-order stationary points, while maintaining communication efficiency. Detailed theoretical and numerical comparisons corroborate that the resource efficiencies of DESTRESS improve upon prior decentralized algorithms over a wide range of parameter regimes. DESTRESS leverages several key algorithm design ideas including stochastic recursive gradient updates with mini-batches for local computation, gradient tracking with extra mixing (i.e., multiple gossiping rounds) for per-iteration communication, together with careful choices of hyper-parameters and new analysis frameworks to provably achieve a desirable computation-communication trade-off.

Understanding the concept of the gradient is useful for understanding the logic of the gradient descent algorithm. Let's take a look at the explanation of the concept of stationary point in Wikipedia. As it can be understood from here, the gradient descent algorithm takes the points in the cost function and continues with the aim of reducing the derivative (slope) of these points in each iteration. The reason for this is to find the value whose slope is zero, in other words, the minimum point. When the coordinate values of this point are substituted in the hypothesis function, the function we obtain becomes the hypothesis function of the model with the least error we can create.

Current deep neural networks are highly overparameterized (up to billions of connection weights) and nonlinear. Yet they can fit data almost perfectly through variants of gradient descent algorithms and achieve unexpected levels of prediction accuracy without overfitting. These are formidable results that escape the bias-variance predictions of statistical learning and pose conceptual challenges for non-convex optimization. In this paper, we use methods from statistical physics of disordered systems to analytically study the computational fallout of overparameterization in nonconvex neural network models. As the number of connection weights increases, we follow the changes of the geometrical structure of different minima of the error loss function and relate them to learning and generalisation performance. We find that there exist a gap between the SAT/UNSAT interpolation transition where solutions begin to exist and the point where algorithms start to find solutions, i.e. where accessible solutions appear. This second phase transition coincides with the discontinuous appearance of atypical solutions that are locally extremely entropic, i.e., flat regions of the weight space that are particularly solution-dense and have good generalization properties. Although exponentially rare compared to typical solutions (which are narrower and extremely difficult to sample), entropic solutions are accessible to the algorithms used in learning. We can characterize the generalization error of different solutions and optimize the Bayesian prediction, for data generated from a structurally different network. Numerical tests on observables suggested by the theory confirm that the scenario extends to realistic deep networks.

We say an algorithm is batch size-invariant if changes to the batch size can largely be compensated for by changes to other hyperparameters. Stochastic gradient descent is well-known to have this property at small batch sizes, via the learning rate. However, some policy optimization algorithms (such as PPO) do not have this property, because of how they control the size of policy updates. In this work we show how to make these algorithms batch size-invariant. Our key insight is to decouple the proximal policy (used for controlling policy updates) from the behavior policy (used for off-policy corrections). Our experiments help explain why these algorithms work, and additionally show how they can make more efficient use of stale data.

Two-level stochastic optimization formulations have become instrumental in a number of machine learning contexts such as neural architecture search, continual learning, adversarial learning, and hyperparameter tuning. Practical stochastic bilevel optimization problems become challenging in optimization or learning scenarios where the number of variables is high or there are constraints. The goal of this paper is twofold. First, we aim at promoting the use of bilevel optimization in large-scale learning and we introduce a practical bilevel stochastic gradient method (BSG-1) that requires neither lower level second-order derivatives nor system solves (and dismisses any matrix-vector products). Our BSG-1 method is close to first-order principles, which allows it to achieve a performance better than those that are not, such as DARTS. Second, we develop bilevel stochastic gradient descent for bilevel problems with lower level constraints, and we introduce a convergence theory that covers the unconstrained and constrained cases and abstracts as much as possible from the specifics of the bilevel gradient calculation.