Originally published on Towards AI the World's Leading AI and Technology News and Media Company. If you are building an AI-related product or service, we invite you to consider becoming an AI sponsor. At Towards AI, we help scale AI and technology startups. Let us help you unleash your technology to the masses. Most deep learning algorithms train by optimizing some kind of objective (or loss) function. We typically try to either maximize or minimize these functions.

Optimizing expensive-to-evaluate black-box functions of discrete (and potentially continuous) design parameters is a ubiquitous problem in scientific and engineering applications. Bayesian optimization (BO) is a popular, sample-efficient method that leverages a probabilistic surrogate model and an acquisition function (AF) to select promising designs to evaluate. However, maximizing the AF over mixed or high-cardinality discrete search spaces is challenging standard gradient-based methods cannot be used directly or evaluating the AF at every point in the search space would be computationally prohibitive. To address this issue, we propose using probabilistic reparameterization (PR). Instead of directly optimizing the AF over the search space containing discrete parameters, we instead maximize the expectation of the AF over a probability distribution defined by continuous parameters. We prove that under suitable reparameterizations, the BO policy that maximizes the probabilistic objective is the same as that which maximizes the AF, and therefore, PR enjoys the same regret bounds as the original BO policy using the underlying AF. Moreover, our approach provably converges to a stationary point of the probabilistic objective under gradient ascent using scalable, unbiased estimators of both the probabilistic objective and its gradient. Therefore, as the number of starting points and gradient steps increase, our approach will recover of a maximizer of the AF (an often-neglected requisite for commonly used BO regret bounds). We validate our approach empirically and demonstrate state-of-the-art optimization performance on a wide range of real-world applications. PR is complementary to (and benefits) recent work and naturally generalizes to settings with multiple objectives and black-box constraints.

Variance-reduced stochastic gradient methods have gained popularity in recent times. Several variants exist with different strategies for the storing and sampling of gradients and this work concerns the interactions between these two aspects. We present a general proximal variance-reduced gradient method and analyze it under strong convexity assumptions. Special cases of the algorithm include SAGA, L-SVRG and their proximal variants. Our analysis sheds light on epoch-length selection and the need to balance the convergence of the iterates with how often gradients are stored. The analysis improves on other convergence rates found in the literature and produces a new and faster converging sampling strategy for SAGA. Problem instances for which the predicted rates are the same as the practical rates are presented together with problems based on real world data.

In response to the growing demand, differential-private (DP) machine learning [10] provides a computational framework for privacy protection and has been widely studied in various settings, including both convex and non-convex optimization [15, 33, 34]. One widely used procedure for privacy-preserving learning is the (Differentially) Private Gradient Descent (PGD) [1, 3]. A typical gradient descent procedure updates its model by gradients of the loss evaluated on a training dataset. When the data is sensitive, the gradients should be privatized to prevent excess privacy leakage. The PGD privatizes a gradient by adding controlled noise. As such, the models from PGD is expected to have a lower utility as compared to those from unprotected algorithms. In the cases where strict privacy control is exercised, or equivalently, a tight privacy budget, accumulating effects from highly-noised gradients may lead to unacceptable model performance. It is thus critical to design effective privatization procedures for PGD to maintain a great balance between utility and privacy. Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page.

Despite the established convergence theory of Optimistic Gradient Descent Ascent (OGDA) and Extragradient (EG) methods for the convex-concave minimax problems, little is known about the theoretical guarantees of these methods in nonconvex settings. To bridge this gap, for the first time, this paper establishes the convergence of OGDA and EG methods under the nonconvex-strongly-concave (NC-SC) and nonconvex-concave (NC-C) settings by providing a unified analysis through the lens of single-call extra-gradient methods. We further establish lower bounds on the convergence of GDA/OGDA/EG, shedding light on the tightness of our analysis. We also conduct experiments supporting our theoretical results. We believe our results will advance the theoretical understanding of OGDA and EG methods for solving complicated nonconvex minimax real-world problems, e.g., Generative Adversarial Networks (GANs) or robust neural networks training.

Prior work on differential privacy analysis of randomized SGD algorithms relies on composition theorems, where the implicit (unrealistic) assumption is that the internal state of the iterative algorithm is revealed to the adversary. As a result, the R\'enyi DP bounds derived by such composition-based analyses linearly grow with the number of training epochs. When the internal state of the algorithm is hidden, we prove a converging privacy bound for noisy stochastic gradient descent (on strongly convex smooth loss functions). We show how to take advantage of privacy amplification by sub-sampling and randomized post-processing, and prove the dynamics of privacy bound for "shuffle and partition" and "sample without replacement" stochastic mini-batch gradient descent schemes. We prove that, in these settings, our privacy bound converges exponentially fast and is substantially smaller than the composition bounds, notably after a few number of training epochs. Thus, unless the DP algorithm converges fast, our privacy analysis shows that hidden state analysis can significantly amplify differential privacy.

The phenomenon that stochastic gradient descent (SGD) favors flat minima has played a critical role in understanding the implicit regularization of SGD. In this paper, we provide an explanation of this striking phenomenon by relating the particular noise structure of SGD to its \emph{linear stability} (Wu et al., 2018). Specifically, we consider training over-parameterized models with square loss. We prove that if a global minimum $\theta^*$ is linearly stable for SGD, then it must satisfy $\|H(\theta^*)\|_F\leq O(\sqrt{B}/\eta)$, where $\|H(\theta^*)\|_F, B,\eta$ denote the Frobenius norm of Hessian at $\theta^*$, batch size, and learning rate, respectively. Otherwise, SGD will escape from that minimum \emph{exponentially} fast. Hence, for minima accessible to SGD, the sharpness -- as measured by the Frobenius norm of the Hessian -- is bounded \emph{independently} of the model size and sample size. The key to obtaining these results is exploiting the particular structure of SGD noise: The noise concentrates in sharp directions of local landscape and the magnitude is proportional to loss value. This alignment property of SGD noise provably holds for linear networks and random feature models (RFMs), and is empirically verified for nonlinear networks. Moreover, the validity and practical relevance of our theoretical findings are also justified by extensive experiments on CIFAR-10 dataset.

Modern deep learning has achieved great success in various fields. However, it requires the labeling of huge amounts of data, which is expensive and labor-intensive. Active learning (AL), which identifies the most informative samples to be labeled, is becoming increasingly important to maximize the efficiency of the training process. The existing AL methods mostly use only a single final fixed model for acquiring the samples to be labeled. This strategy may not be good enough in that the structural uncertainty of a model for given training data is not considered to acquire the samples. In this study, we propose a novel acquisition criterion based on temporal self-ensemble generated by conventional stochastic gradient descent (SGD) optimization. These self-ensemble models are obtained by capturing the intermediate network weights obtained through SGD iterations. Our acquisition function relies on a consistency measure between the student and teacher models. The student models are given a fixed number of temporal self-ensemble models, and the teacher model is constructed by averaging the weights of the student models. Using the proposed acquisition criterion, we present an AL algorithm, namely student-teacher consistency-based AL (ST-CoNAL). Experiments conducted for image classification tasks on CIFAR-10, CIFAR-100, Caltech-256, and Tiny ImageNet datasets demonstrate that the proposed ST-CoNAL achieves significantly better performance than the existing acquisition methods. Furthermore, extensive experiments show the robustness and effectiveness of our methods.

Gradient descant is a popular optimization in neural networks. Nowadays, neural network has attracted much attention in any fields, there are many different types of the neural networks that has already been developed(Haykin, 2009). The convolutional neural networks is very popular in the classification field(Huang et al., 2017), even though it is a kind of the feedforward neural network(Haykin, 2009). In recent years, the structure of convolutional neural network has been rapidly improved(Smith and Topin, 2016). However, many improvements develop the structural algorithm of convolutional neural network, but not much improve it from optimization.

Adaptive algorithms like AdaGrad and AMSGrad are successful in nonconvex optimization owing to their parameter-agnostic ability -- requiring no a priori knowledge about problem-specific parameters nor tuning of learning rates. However, when it comes to nonconvex minimax optimization, direct extensions of such adaptive optimizers without proper time-scale separation may fail to work in practice. We provide such an example proving that the simple combination of Gradient Descent Ascent (GDA) with adaptive stepsizes can diverge if the primal-dual stepsize ratio is not carefully chosen; hence, a fortiori, such adaptive extensions are not parameter-agnostic. To address the issue, we formally introduce a Nested Adaptive framework, NeAda for short, that carries an inner loop for adaptively maximizing the dual variable with controllable stopping criteria and an outer loop for adaptively minimizing the primal variable. Such mechanism can be equipped with off-the-shelf adaptive optimizers and automatically balance the progress in the primal and dual variables. Theoretically, for nonconvex-strongly-concave minimax problems, we show that NeAda can achieve the near-optimal $\tilde{O}(\epsilon^{-2})$ and $\tilde{O}(\epsilon^{-4})$ gradient complexities respectively in the deterministic and stochastic settings, without prior information on the problem's smoothness and strong concavity parameters. To the best of our knowledge, this is the first algorithm that simultaneously achieves near-optimal convergence rates and parameter-agnostic adaptation in the nonconvex minimax setting. Numerically, we further illustrate the robustness of the NeAda family with experiments on simple test functions and a real-world application.