In this work, we focus on the study of stochastic zeroth-order (ZO) optimization which does not require first-order gradient information and uses only function evaluations. The problem of ZO optimization has emerged in many recent machine learning applications, where the gradient of the objective function is either unavailable or difficult to compute. In such cases, we can approximate the full gradients or stochastic gradients through function value based gradient estimates. Here, we propose a novel hybrid gradient estimator (HGE), which takes advantage of the query-efficiency of random gradient estimates as well as the variance-reduction of coordinate-wise gradient estimates. We show that with a graceful design in coordinate importance sampling, the proposed HGE-based ZO optimization method is efficient both in terms of iteration complexity as well as function query cost. We provide a thorough theoretical analysis of the convergence of our proposed method for non-convex, convex, and strongly-convex optimization. We show that the convergence rate that we derive generalizes the results for some prominent existing methods in the nonconvex case, and matches the optimal result in the convex case. We also corroborate the theory with a real-world black-box attack generation application to demonstrate the empirical advantage of our method over state-of-the-art ZO optimization approaches.

We propose an information-theoretic technique for analyzing privacy guarantees of online algorithms. Specifically, we demonstrate that differential privacy guarantees of iterative algorithms can be determined by a direct application of contraction coefficients derived from strong data processing inequalities for $f$-divergences. Our technique relies on generalizing the Dobrushin's contraction coefficient for total variation distance to an $f$-divergence known as $E_\gamma$-divergence. $E_\gamma$-divergence, in turn, is equivalent to approximate differential privacy. As an example, we apply our technique to derive the differential privacy parameters of gradient descent. Moreover, we also show that this framework can be tailored to batch learning algorithms that can be implemented with one pass over the training dataset.

We introduce a hybrid "Modified Genetic Algorithm-Multilevel Stochastic Gradient Descent" (MGA-MSGD) training algorithm that considerably improves accuracy and efficiency of solving 3D mechanical problems described, in strong-form, by PDEs via ANNs (Artificial Neural Networks). This presented approach allows the selection of a number of locations of interest at which the state variables are expected to fulfil the governing equations associated with a physical problem. Unlike classical PDE approximation methods such as finite differences or the finite element method, there is no need to establish and reconstruct the physical field quantity throughout the computational domain in order to predict the mechanical response at specific locations of interest. The basic idea of MGA-MSGD is the manipulation of the learnable parameters' components responsible for the error explosion so that we can train the network with relatively larger learning rates which avoids trapping in local minima. The proposed training approach is less sensitive to the learning rate value, training points density and distribution, and the random initial parameters. The distance function to minimise is where we introduce the PDEs including any physical laws and conditions (so-called, Physics Informed ANN). The Genetic algorithm is modified to be suitable for this type of ANN in which a Coarse-level Stochastic Gradient Descent (CSGD) is exploited to make the decision of the offspring qualification. Employing the presented approach, a considerable improvement in both accuracy and efficiency, compared with standard training algorithms such as classical SGD and Adam optimiser, is observed. The local displacement accuracy is studied and ensured by introducing the results of Finite Element Method (FEM) at sufficiently fine mesh as the reference displacements. A slightly more complex problem is solved ensuring its feasibility.

Byzantine robustness has received significant attention recently given its importance for distributed and federated learning. In spite of this, we identify severe flaws in existing algorithms even when the data across the participants is assumed to be identical. First, we show that most existing robust aggregation rules may not converge even in the absence of any Byzantine attackers, because they are overly sensitive to the distribution of the noise in the stochastic gradients. Secondly, we show that even if the aggregation rules may succeed in limiting the influence of the attackers in a single round, the attackers can couple their attacks across time eventually leading to divergence. To address these issues, we present two surprisingly simple strategies: a new iterative clipping procedure, and incorporating worker momentum to overcome time-coupled attacks. This is the first provably robust method for the standard stochastic non-convex optimization setting.

Convex optimization over the spectrahedron, i.e., the set of all real $n\times n$ positive semidefinite matrices with unit trace, has important applications in machine learning, signal processing and statistics, mainly as a convex relaxation for optimization with low-rank matrices. It is also one of the most prominent examples in the theory of first-order methods for convex optimization in which non-Euclidean methods can be significantly preferable to their Euclidean counterparts, and in particular the Matrix Exponentiated Gradient (MEG) method which is based on the Bregman distance induced by the (negative) von Neumann entropy. Unfortunately, implementing MEG requires a full SVD computation on each iteration, which is not scalable to high-dimensional problems. In this work we propose efficient implementations of MEG, both with deterministic and stochastic gradients, which are tailored for optimization with low-rank matrices, and only use a single low-rank SVD computation on each iteration. We also provide efficiently-computable certificates for the correct convergence of our methods. Mainly, we prove that under a strict complementarity condition, the suggested methods converge from a "warm-start" initialization with similar rates to their full-SVD-based counterparts. Finally, we bring empirical experiments which both support our theoretical findings and demonstrate the practical appeal of our methods.

As a simple and efficient optimization method in deep learning, stochastic gradient descent (SGD) has attracted tremendous attention. In the vanishing learning rate regime, SGD is now relatively well understood, and the majority of theoretical approaches to SGD set their assumptions in the continuous-time limit. However, the continuous-time predictions are unlikely to reflect the experimental observations well because the practice often runs in the large learning rate regime, where the training is faster and the generalization of models are often better. In this paper, we propose to study the basic properties of SGD and its variants in the non-vanishing learning rate regime. The focus is on deriving exactly solvable results and relating them to experimental observations. The main contributions of this work are to derive the stable distribution for discrete-time SGD in a quadratic loss function with and without momentum. Examples of applications of the proposed theory considered in this work include the approximation error of variants of SGD, the effect of mini-batch noise, the escape rate from a sharp minimum, and and the stationary distribution of a few second order methods.

Bayesian optimisation presents a sample-efficient methodology for global optimisation. Within this framework, a crucial performance-determining subroutine is the maximisation of the acquisition function, a task complicated by the fact that acquisition functions tend to be non-convex and thus nontrivial to optimise. In this paper, we undertake a comprehensive empirical study of approaches to maximise the acquisition function. Additionally, by deriving novel, yet mathematically equivalent, compositional forms for popular acquisition functions, we recast the maximisation task as a compositional optimisation problem, allowing us to benefit from the extensive literature in this field. We highlight the empirical advantages of the compositional approach to acquisition function maximisation across 3958 individual experiments comprising synthetic optimisation tasks as well as tasks from Bayesmark. Given the generality of the acquisition function maximisation subroutine, we posit that the adoption of compositional optimisers has the potential to yield performance improvements across all domains in which Bayesian optimisation is currently being applied.

Matrix factorization is a simple and natural test-bed to investigate the implicit regularization of gradient descent. Gunasekar et al. (2018) conjectured that Gradient Flow with infinitesimal initialization converges to the solution that minimizes the nuclear norm, but a series of recent papers argued that the language of norm minimization is not sufficient to give a full characterization for the implicit regularization. In this work, we provide theoretical and empirical evidence that for depth-2 matrix factorization, gradient flow with infinitesimal initialization is mathematically equivalent to a simple heuristic rank minimization algorithm, Greedy Low-Rank Learning, under some reasonable assumptions. This generalizes the rank minimization view from previous works to a much broader setting and enables us to construct counter-examples to refute the conjecture from Gunasekar et al. (2018). We also extend the results to the case where depth $\ge 3$, and we show that the benefit of being deeper is that the above convergence has a much weaker dependence over initialization magnitude so that this rank minimization is more likely to take effect for initialization with practical scale.

In this article, we will discuss regularization and optimization techniques that are used by programmers to build a more robust and generalized neural network. We will study the most effective regularization techniques like L1, L2, Early Stopping, and Drop out which help for model generalization. We will take a deeper look at different optimization techniques like Batch Gradient Descent, Stochastic Gradient Descent, AdaGrad, and AdaDelta for better convergence of the neural networks. Overfitting and underfitting are the most common problems that programmers face while working with deep learning models. A model that is well generalized to data is considered to be an optimal fit for the data.

We further research on the acceleration phenomenon on Riemannian manifolds by introducing the first global first-order method that achieves the same rates as accelerated gradient descent in the Euclidean space for the optimization of smooth and geodesically convex (g-convex) or strongly g-convex functions defined on the hyperbolic space or a subset of the sphere, up to constants and log factors. To the best of our knowledge, this is the first method that is proved to achieve these rates globally on functions defined on a Riemannian manifold $\mathcal{M}$ other than the Euclidean space. Additionally, for any Riemannian manifold of bounded sectional curvature, we provide reductions from optimization methods for smooth and g-convex functions to methods for smooth and strongly g-convex functions and vice versa. As a proxy, we solve a constrained non-convex Euclidean problem, under a condition between convexity and quasar-convexity.