This paper establishes a mathematical foundation for the Adam optimizer, elucidating its connection to natural gradient descent through Riemannian and information geometry. We rigorously analyze the diagonal empirical Fisher information matrix (FIM) in Adam, clarifying all detailed approximations and advocating for the use of log probability functions as loss, which should be based on discrete distributions, due to the limitations of empirical FIM. Our analysis uncovers flaws in the original Adam algorithm, leading to proposed corrections such as enhanced momentum calculations, adjusted bias corrections, adaptive epsilon, and gradient clipping. We refine the weight decay term based on our theoretical framework. Our modified algorithm, Fisher Adam (FAdam), demonstrates superior performance across diverse domains including LLM, ASR, and VQ-VAE, achieving state-of-the-art results in ASR.

We present in this paper novel accelerated fully first-order methods in \emph{Bilevel Optimization} (BLO). Firstly, for BLO under the assumption that the lower-level functions admit the typical strong convexity assumption, the \emph{(Perturbed) Restarted Accelerated Fully First-order methods for Bilevel Approximation} (\texttt{PRAF${}^2$BA}) algorithm leveraging \emph{fully} first-order oracles is proposed, whereas the algorithm for finding approximate first-order and second-order stationary points with state-of-the-art oracle query complexities in solving complex optimization tasks. Secondly, applying as a special case of BLO the \emph{nonconvex-strongly-convex} (NCSC) minimax optimization, \texttt{PRAF${}^2$BA} rediscovers \emph{perturbed restarted accelerated gradient descent ascent} (\texttt{PRAGDA}) that achieves the state-of-the-art complexity for finding approximate second-order stationary points. Additionally, we investigate the challenge of finding stationary points of the hyper-objective function in BLO when lower-level functions lack the typical strong convexity assumption, where we identify several regularity conditions of the lower-level problems that ensure tractability and present hardness results indicating the intractability of BLO for general convex lower-level functions. Under these regularity conditions we propose the \emph{Inexact Gradient-Free Method} (\texttt{IGFM}), utilizing the \emph{Switching Gradient Method} (\texttt{SGM}) as an efficient sub-routine to find an approximate stationary point of the hyper-objective in polynomial time. Empirical studies for real-world problems are provided to further validate the outperformance of our proposed algorithms.

Training such a parameterized optimization model is an Embedding parameterized optimization problems instance of bi-level optimization (Gould et al., 2016), as layers into machine learning architectures which is generally challenging. Whenever it is possible serves as a powerful inductive bias. Training to propagate gradients through the optimization problem such architectures with stochastic gradient via an informative derivative of the solution mapping, descent requires care, as degenerate derivatives the task is typically approached with standard stochastic of the embedded optimization problem often gradient descent (GD) (Amos & Kolter, 2017a; Agrawal render the gradients uninformative. We propose et al., 2019b). However, when the optimization problem has Lagrangian Proximal Gradient Descent (LPGD) discrete solutions, the derivatives are typically degenerate, a flexible framework for training architectures as small perturbations of the input do not affect the optimal with embedded optimization layers that seamlessly solution. Previous works have proposed several methods integrates into automatic differentiation to overcome this challenge, ranging from differentiable libraries. LPGD efficiently computes meaningful relaxations (Wang et al., 2019; Wilder et al., 2019a; Mandi replacements of the degenerate optimization & Guns, 2020; Djolonga & Krause, 2017) and stochastic layer derivatives by re-running the forward solver smoothing (Berthet et al., 2020; Dalle et al., 2022), over oracle on a perturbed input. LPGD captures proxy losses (Paulus et al., 2021), to finite-difference based various previously proposed methods as special techniques (Vlastelica et al., 2020).

We address the challenge of online convex optimization where the objective function's gradient exhibits sparsity, indicating that only a small number of dimensions possess non-zero gradients. Our aim is to leverage this sparsity to obtain useful estimates of the objective function's gradient even when the only information available is a limited number of function samples. Our motivation stems from distributed queueing systems like microservices-based applications, characterized by request-response workloads. Here, each request type proceeds through a sequence of microservices to produce a response, and the resource allocation across the collection of microservices is controlled to balance end-to-end latency with resource costs. While the number of microservices is substantial, the latency function primarily reacts to resource changes in a few, rendering the gradient sparse. Our proposed method, CONGO (Compressive Online Gradient Optimization), combines simultaneous perturbation with compressive sensing to estimate gradients. We establish analytical bounds on the requisite number of compressive sensing samples per iteration to maintain bounded bias of gradient estimates, ensuring sub-linear regret. By exploiting sparsity, we reduce the samples required per iteration to match the gradient's sparsity, rather than the problem's original dimensionality. Numerical experiments and real-world microservices benchmarks demonstrate CONGO's superiority over multiple stochastic gradient descent approaches, as it quickly converges to performance comparable to policies pre-trained with workload awareness.

The goal of this paper is to investigate the complexity of gradient algorithms when learning sparse functions (juntas). We introduce a type of Statistical Queries ($\mathsf{SQ}$), which we call Differentiable Learning Queries ($\mathsf{DLQ}$), to model gradient queries on a specified loss with respect to an arbitrary model. We provide a tight characterization of the query complexity of $\mathsf{DLQ}$ for learning the support of a sparse function over generic product distributions. This complexity crucially depends on the loss function. For the squared loss, $\mathsf{DLQ}$ matches the complexity of Correlation Statistical Queries $(\mathsf{CSQ})$--potentially much worse than $\mathsf{SQ}$. But for other simple loss functions, including the $\ell_1$ loss, $\mathsf{DLQ}$ always achieves the same complexity as $\mathsf{SQ}$. We also provide evidence that $\mathsf{DLQ}$ can indeed capture learning with (stochastic) gradient descent by showing it correctly describes the complexity of learning with a two-layer neural network in the mean field regime and linear scaling.

In the context of ground robot navigation in unstructured hazardous environments, the coupling of efficient path planning with an adequate environment representation is a crucial topic in order to guarantee the robot safety while ensuring the accomplishment of its mission. This paper discusses the exploitation of an environment representation obtained via Gaussian process regression (GPR) for smooth path planning using gradient descent B\'ezier curve optimisation (BCO). A continuous differentiable GPR of the terrain traversability and obstacle distance is used to plan paths with a weighted A* discrete planner, a T-RRT sampling-based planner and BCO using A* or T-RRT computed paths as prior. Numerical experiments in procedurally generated 2D environments allowed to compare the paths planned by the described methods and highlight the benefits of the joint use of the GPR continuous representation and the BCO smooth path planning with these different priors.

Differentially Private Stochastic Gradient Descent (DP-SGD) is a popular iterative algorithm used to train machine learning models while formally guaranteeing the privacy of users. However the privacy analysis of DP-SGD makes the unrealistic assumption that all intermediate iterates (aka internal state) of the algorithm are released since in practice, only the final trained model, i.e., the final iterate of the algorithm is released. In this hidden state setting, prior work has provided tighter analyses, albeit only when the loss function is constrained, e.g., strongly convex and smooth or linear. On the other hand, the privacy leakage observed empirically from hidden state DP-SGD, even when using non-convex loss functions suggest that there is in fact a gap between the theoretical privacy analysis and the privacy guarantees achieved in practice. Therefore, it remains an open question whether privacy amplification for DP-SGD is possible in the hidden state setting for general loss functions. Unfortunately, this work answers the aforementioned research question negatively. By carefully constructing a loss function for DP-SGD, we show that for specific loss functions, the final iterate of DP-SGD alone leaks as much information as the sequence of all iterates combined. Furthermore, we empirically verify this result by evaluating the privacy leakage from the final iterate of DP-SGD with our loss function and show that this matches the theoretical upper bound guaranteed by DP exactly. Therefore, we show that the current privacy analysis fo DP-SGD is tight for general loss functions and conclude that no privacy amplification is possible for DP-SGD in general for all (possibly non-convex) loss functions.

This paper introduces and analyses interacting underdamped Langevin algorithms, termed Kinetic Interacting Particle Langevin Monte Carlo (KIPLMC) methods, for statistical inference in latent variable models. We propose a diffusion process that evolves jointly in the space of parameters and latent variables and exploit the fact that the stationary distribution of this diffusion concentrates around the maximum marginal likelihood estimate of the parameters. We then provide two explicit discretisations of this diffusion as practical algorithms to estimate parameters of statistical models. For each algorithm, we obtain nonasymptotic rates of convergence for the case where the joint log-likelihood is strongly concave with respect to latent variables and parameters. In particular, we provide convergence analysis for the diffusion together with the discretisation error, providing convergence rate estimates for the algorithms in Wasserstein-2 distance. To demonstrate the utility of the introduced methodology, we provide numerical experiments that demonstrate the effectiveness of the proposed diffusion for statistical inference and the stability of the numerical integrators utilised for discretisation. Our setting covers a broad number of applications, including unsupervised learning, statistical inference, and inverse problems.

Probabilistic state estimation is essential for robots navigating uncertain environments. Accurately and efficiently managing uncertainty in estimated states is key to robust robotic operation. However, nonlinearities in robotic platforms pose significant challenges that require advanced estimation techniques. Gaussian variational inference (GVI) offers an optimization perspective on the estimation problem, providing analytically tractable solutions and efficiencies derived from the geometry of Gaussian space. We propose a Sequential Gaussian Variational Inference (S-GVI) method to address nonlinearity and provide efficient sequential inference processes. Our approach integrates sequential Bayesian principles into the GVI framework, which are addressed using statistical approximations and gradient updates on the information geometry. Validations through simulations and real-world experiments demonstrate significant improvements in state estimation over the Maximum A Posteriori (MAP) estimation method.

STOchastic Recursive Momentum (STORM)-based algorithms have been widely developed to solve one to $K$-level ($K \geq 3$) stochastic optimization problems. Specifically, they use estimators to mitigate the biased gradient issue and achieve near-optimal convergence results. However, there is relatively little work on understanding their generalization performance, particularly evident during the transition from one to $K$-level optimization contexts. This paper provides a comprehensive generalization analysis of three representative STORM-based algorithms: STORM, COVER, and SVMR, for one, two, and $K$-level stochastic optimizations under both convex and strongly convex settings based on algorithmic stability. Firstly, we define stability for $K$-level optimizations and link it to generalization. Then, we detail the stability results for three prominent STORM-based algorithms. Finally, we derive their excess risk bounds by balancing stability results with optimization errors. Our theoretical results provide strong evidence to complete STORM-based algorithms: (1) Each estimator may decrease their stability due to variance with its estimation target. (2) Every additional level might escalate the generalization error, influenced by the stability and the variance between its cumulative stochastic gradient and the true gradient. (3) Increasing the batch size for the initial computation of estimators presents a favorable trade-off, enhancing the generalization performance.