Meta-learning owns unique effectiveness and swiftness in tackling emerging tasks with limited data. Its broad applicability is revealed by viewing it as a bi-level optimization problem. The resultant algorithmic viewpoint however, faces scalability issues when the inner-level optimization relies on gradient-based iterations. Implicit differentiation has been considered to alleviate this challenge, but it is restricted to an isotropic Gaussian prior, and only favors deterministic meta-learning approaches. This work markedly mitigates the scalability bottleneck by cross-fertilizing the benefits of implicit differentiation to probabilistic Bayesian meta-learning. The novel implicit Bayesian meta-learning (iBaML) method not only broadens the scope of learnable priors, but also quantifies the associated uncertainty. Furthermore, the ultimate complexity is well controlled regardless of the inner-level optimization trajectory. Analytical error bounds are established to demonstrate the precision and efficiency of the generalized implicit gradient over the explicit one. Extensive numerical tests are also carried out to empirically validate the performance of the proposed method.

The generalized equation (also called the [non]linear inclusion) provides a unified template to model various problems in computational mathematics and related fields su ch as the optimality condition of optimization problems (in both unconstrained and constrained settings), minimax optimization, variational inequality, complementarity, two-person game, and fixed-point problem s, see, e.g., [11, 24, 50, 112, 116, 118, 120]. Theory and numerical methods for this equation and its special case s have been extensively studied for many decades, see, e.g., the following monographs and the references quot ed therein [11, 50, 94, 119]. At the same time, several applications of this mathematical tool in operatio ns research, economics, uncertainty quantification, and transportations have been investigated [14, 52, 61, 50, 72]. In the last few years, there has been a surge of research in minimax problems due to new applications in mach ine learning and robust optimization, especially in generative adversarial networks (GANs), adversarial tr aining, and distributionally robust optimization, see, e.g., [4, 14, 55, 76, 84, 114] as a few examples. Minimax probl ems have also found new applications in online learning and reinforcement learning, among many others, se e, e.g., [4, 9, 15, 55, 67, 76, 78, 84, 114, 139]. Such prominent applications have motivated the research in minimax optimization and variational inequality problems (VIPs). On the one hand, classical algorithms such as gradient descent-ascent, extragradient, and primal-dual methods have been revisited, improved, and ext ended. On the other hand, new variants such as accelerated extragradient and accelerated operator split ting schemes have also been developed and equipped with rigorous convergence guarantees and practical perfor mance evaluation. This new development motivates us to write this survey paper, with the focus on sublinear con vergence rate analysis.

We present an approach for designing swarm-based optimizers for the global optimization of expensive black-box functions. In the proposed approach, the problem of finding efficient optimizers is framed as a reinforcement learning problem, where the goal is to find optimization policies that require a few function evaluations to converge to the global optimum. The state of each agent within the swarm is defined as its current position and function value within a design space and the agents learn to take favorable actions that maximize reward, which is based on the final value of the objective function. The proposed approach is tested on various benchmark optimization functions and compared to the performance of other global optimization strategies. Furthermore, the effect of changing the number of agents, as well as the generalization capabilities of the trained agents are investigated. The results show superior performance compared to the other optimizers, desired scaling when the number of agents is varied, and acceptable performance even when applied to unseen functions. On a broader scale, the results show promise for the rapid development of domain-specific optimizers.

Federated Learning is a popular distributed learning paradigm in machine learning. Meanwhile, composition optimization is an effective hierarchical learning model, which appears in many machine learning applications such as meta learning and robust learning. More recently, although a few federated composition optimization algorithms have been proposed, they still suffer from high sample and communication complexities. In the paper, thus, we propose a class of faster federated compositional optimization algorithms (i.e., MFCGD and AdaMFCGD) to solve the nonconvex distributed composition problems, which builds on the momentum-based variance reduced and local-SGD techniques. In particular, our adaptive algorithm (i.e., AdaMFCGD) uses a unified adaptive matrix to flexibly incorporate various adaptive learning rates. Moreover, we provide a solid theoretical analysis for our algorithms under non-i.i.d. setting, and prove our algorithms obtain a lower sample and communication complexities simultaneously than the existing federated compositional algorithms. Specifically, our algorithms obtain lower sample complexity of $\tilde{O}(\epsilon^{-3})$ with lower communication complexity of $\tilde{O}(\epsilon^{-2})$ in finding an $\epsilon$-stationary solution. We conduct the numerical experiments on robust federated learning and distributed meta learning tasks to demonstrate the efficiency of our algorithms.

Yang et al. (2023) recently addressed the open problem of solving Variational Inequalities (VIs) with equality and inequality constraints through a first-order gradient method. However, the proposed primal-dual method called ACVI is applicable when we can compute analytic solutions of its subproblems; thus, the general case remains an open problem. In this paper, we adopt a warm-starting technique where we solve the subproblems approximately at each iteration and initialize the variables with the approximate solution found at the previous iteration. We prove its convergence and show that the gap function of the last iterate of this inexact-ACVI method decreases at a rate of $\mathcal{O}(\frac{1}{\sqrt{K}})$ when the operator is $L$-Lipschitz and monotone, provided that the errors decrease at appropriate rates. Interestingly, we show that often in numerical experiments, this technique converges faster than its exact counterpart. Furthermore, for the cases when the inequality constraints are simple, we propose a variant of ACVI named P-ACVI and prove its convergence for the same setting. We further demonstrate the efficacy of the proposed methods through numerous experiments. We also relax the assumptions in Yang et al., yielding, to our knowledge, the first convergence result that does not rely on the assumption that the operator is $L$-Lipschitz. Our source code is provided at $\texttt{https://github.com/mpagli/Revisiting-ACVI}$.

This note focuses on a simple approach to the unified analysis of SGD-type methods from (Gorbunov et al., 2020) for strongly convex smooth optimization problems. The similarities in the analyses of different stochastic first-order methods are discussed along with the existing extensions of the framework. The limitations of the analysis and several alternative approaches are mentioned as well.

Conventional optimization methods in machine learning and controls rely heavily on first-order update rules. Selecting the right method and hyperparameters for a particular task often involves trial-and-error or practitioner intuition, motivating the field of meta-learning. We generalize a broad family of preexisting update rules by proposing a meta-learning framework in which the inner loop optimization step involves solving a differentiable convex optimization (DCO). We illustrate the theoretical appeal of this approach by showing that it enables one-step optimization of a family of linear least squares problems, given that the meta-learner has sufficient exposure to similar tasks. Various instantiations of the DCO update rule are compared to conventional optimizers on a range of illustrative experimental settings.

Orthogonality constraints naturally appear in many machine learning problems, from Principal Components Analysis to robust neural network training. They are usually solved using Riemannian optimization algorithms, which minimize the objective function while enforcing the constraint. However, enforcing the orthogonality constraint can be the most time-consuming operation in such algorithms. Recently, Ablin and Peyré (2022) proposed the Landing algorithm, a method with cheap iterations that does not enforce the orthogonality constraint but is attracted towards the manifold in a smooth manner. In this article, we provide new practical and theoretical developments for the landing algorithm. First, the method is extended to the Stiefel manifold, the set of rectangular orthogonal matrices. We also consider stochastic and variance reduction algorithms when the cost function is an average of many functions. We demonstrate that all these methods have the same rate of convergence as their Riemannian counterparts that exactly enforce the constraint. Finally, our experiments demonstrate the promise of our approach to an array of machine-learning problems that involve orthogonality constraints.

A range of complicated real-world problems have inspired the development of several optimization methods. Here, a novel hybrid version of the Ant colony optimization (ACO) method is developed using the sample space reduction technique of the Cohort Intelligence (CI) Algorithm. The algorithm is developed, and accuracy is tested by solving 35 standard benchmark test functions. Furthermore, the constrained version of the algorithm is used to solve two mechanical design problems involving stepped cantilever beams and I-section beams. The effectiveness of the proposed technique of solution is evaluated relative to contemporary algorithmic approaches that are already in use. The results show that our proposed hybrid ACO-CI algorithm will take lesser number of iterations to produce the desired output which means lesser computational time. For the minimization of weight of stepped cantilever beam and deflection in I-section beam a proposed hybrid ACO-CI algorithm yielded best results when compared to other existing algorithms. The proposed work could be investigate for variegated real world applications encompassing domains of engineering, combinatorial and health care problems.

We address the task of controlled generation of small molecules, which entails finding novel molecules with desired properties under certain constraints (e.g., similarity to a reference molecule). Here we introduce MolMIM, a probabilistic auto-encoder for small molecule drug discovery that learns an informative and clustered latent space. MolMIM is trained with Mutual Information Machine (MIM) learning, and provides a fixed length representation of variable length SMILES strings. Since encoder-decoder models can learn representations with ``holes'' of invalid samples, here we propose a novel extension to the training procedure which promotes a dense latent space, and allows the model to sample valid molecules from random perturbations of latent codes. We provide a thorough comparison of MolMIM to several variable-size and fixed-size encoder-decoder models, demonstrating MolMIM's superior generation as measured in terms of validity, uniqueness, and novelty. We then utilize CMA-ES, a naive black-box and gradient free search algorithm, over MolMIM's latent space for the task of property guided molecule optimization. We achieve state-of-the-art results in several constrained single property optimization tasks as well as in the challenging task of multi-objective optimization, improving over previous success rate SOTA by more than 5\% . We attribute the strong results to MolMIM's latent representation which clusters similar molecules in the latent space, whereas CMA-ES is often used as a baseline optimization method. We also demonstrate MolMIM to be favourable in a compute limited regime, making it an attractive model for such cases.