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The co-localization problem is a model that simultaneously localizes objects of the same class within a series of images or videos. In \cite{joulin2014efficient}, authors present new variants of the Frank-Wolfe algorithm (aka conditional gradient) that increase the efficiency in solving the image and video co-localization problems. The authors show the efficiency of their methods with the rate of decrease in a value called the Wolfe gap in each iteration of the algorithm. In this project, inspired by the conditional gradient sliding algorithm (CGS) \cite{CGS:Lan}, We propose algorithms for solving such problems and demonstrate the efficiency of the proposed algorithms through numerical experiments. The efficiency of these methods with respect to the Wolfe gap is compared with implementing them on the YouTube-Objects dataset for videos., 20 pages, 7 figures, Future Technologies Conference (FTC) 2023

Sampling conditional distributions is a fundamental task for Bayesian inference and density estimation. Generative models, such as normalizing flows and generative adversarial networks, characterize conditional distributions by learning a transport map that pushes forward a simple reference (e.g., a standard Gaussian) to a target distribution. While these approaches successfully describe many non-Gaussian problems, their performance is often limited by parametric bias and the reliability of gradient-based (adversarial) optimizers to learn these transformations. This work proposes a non-parametric generative model that iteratively maps reference samples to the target. The model uses block-triangular transport maps, whose components are shown to characterize conditionals of the target distribution. These maps arise from solving an optimal transport problem with a weighted $L^2$ cost function, thereby extending the data-driven approach in [Trigila and Tabak, 2016] for conditional sampling. The proposed approach is demonstrated on a two dimensional example and on a parameter inference problem involving nonlinear ODEs., 18 pages, 5 figures

We present a non-convex optimization algorithm metaheuristic, based on the training of a deep generative network, which enables effective searching within continuous, ultra-high dimensional landscapes. During network training, populations of sampled local gradients are utilized within a customized loss function to evolve the network output distribution function towards one peak at high-performing optima. The deep network architecture is tailored to support progressive growth over the course of training, which allows the algorithm to manage the curse of dimensionality characteristic of high-dimensional landscapes. We apply our concept to a range of standard optimization problems with dimensions as high as one thousand and show that our method performs better with fewer function evaluations compared to state-of-the-art algorithm benchmarks. We also discuss the role of deep network over-parameterization, loss function engineering, and proper network architecture selection in optimization, and why the required batch size of sampled local gradients is independent of problem dimension. These concepts form the foundation for a new class of algorithms that utilize customizable and expressive deep generative networks to solve non-convex optimization problems.

We introduce Scylla, a primal heuristic for mixed-integer optimization problems. It exploits approximate solves of the Linear Programming relaxations through the matrix-free Primal-Dual Hybrid Gradient algorithm with specialized termination criteria, and derives integer-feasible solutions via fix-and-propagate procedures and feasibility-pump-like updates to the objective function. Computational experiments show that the method is particularly suited to instances with hard linear relaxations.

Linear-quadratic regulator (LQR) is a landmark problem in the field of optimal control, which is the concern of this paper. Generally, LQR is classified into state-feedback LQR (SLQR) and output-feedback LQR (OLQR) based on whether the full state is obtained. It has been suggested in existing literature that both the SLQR and the OLQR could be viewed as \textit{constrained nonconvex matrix optimization} problems in which the only variable to be optimized is the feedback gain matrix. In this paper, we introduce a first-order accelerated optimization framework of handling the LQR problem, and give its convergence analysis for the cases of SLQR and OLQR, respectively. Specifically, a Lipschiz Hessian property of LQR performance criterion is presented, which turns out to be a crucial property for the application of modern optimization techniques. For the SLQR problem, a continuous-time hybrid dynamic system is introduced, whose solution trajectory is shown to converge exponentially to the optimal feedback gain with Nesterov-optimal order $1-\frac{1}{\sqrt{\kappa}}$ ($\kappa$ the condition number). Then, the symplectic Euler scheme is utilized to discretize the hybrid dynamic system, and a Nesterov-type method with a restarting rule is proposed that preserves the continuous-time convergence rate, i.e., the discretized algorithm admits the Nesterov-optimal convergence order. For the OLQR problem, a Hessian-free accelerated framework is proposed, which is a two-procedure method consisting of semiconvex function optimization and negative curvature exploitation. In a time $\mathcal{O}(\epsilon^{-7/4}\log(1/\epsilon))$, the method can find an $\epsilon$-stationary point of the performance criterion; this entails that the method improves upon the $\mathcal{O}(\epsilon^{-2})$ complexity of vanilla gradient descent. Moreover, our method provides the second-order guarantee of stationary point.

Scalarization is a general technique that can be deployed in any multiobjective setting to reduce multiple objectives into one, such as recently in RLHF for training reward models that align human preferences. Yet some have dismissed this classical approach because linear scalarizations are known to miss concave regions of the Pareto frontier. To that end, we aim to find simple non-linear scalarizations that can explore a diverse set of $k$ objectives on the Pareto frontier, as measured by the dominated hypervolume. We show that hypervolume scalarizations with uniformly random weights are surprisingly optimal for provably minimizing the hypervolume regret, achieving an optimal sublinear regret bound of $O(T^{-1/k})$, with matching lower bounds that preclude any algorithm from doing better asymptotically. As a theoretical case study, we consider the multiobjective stochastic linear bandits problem and demonstrate that by exploiting the sublinear regret bounds of the hypervolume scalarizations, we can derive a novel non-Euclidean analysis that produces improved hypervolume regret bounds of $\tilde{O}( d T^{-1/2} + T^{-1/k})$. We support our theory with strong empirical performance of using simple hypervolume scalarizations that consistently outperforms both the linear and Chebyshev scalarizations, as well as standard multiobjective algorithms in bayesian optimization, such as EHVI., ICML 2023 Workshop

Trajectory optimization under uncertainty underpins a wide range of applications in robotics. However, existing methods are limited in terms of reasoning about sources of epistemic and aleatoric uncertainty, space and time correlations, nonlinear dynamics, and non-convex constraints. In this work, we first introduce a continuous-time planning formulation with an average-value-at-risk constraint over the entire planning horizon. Then, we propose a sample-based approximation that unlocks an efficient, general-purpose, and time-consistent algorithm for risk-averse trajectory optimization. We prove that the method is asymptotically optimal and derive finite-sample error bounds. Simulations demonstrate the high speed and reliability of the approach on problems with stochasticity in nonlinear dynamics, obstacle fields, interactions, and terrain parameters.

We study online convex optimization where the possible actions are trace-one elements in a symmetric cone, generalizing the extensively-studied experts setup and its quantum counterpart. Symmetric cones provide a unifying framework for some of the most important optimization models, including linear, second-order cone, and semidefinite optimization. Using tools from the field of Euclidean Jordan Algebras, we introduce the Symmetric-Cone Multiplicative Weights Update (SCMWU), a projection-free algorithm for online optimization over the trace-one slice of an arbitrary symmetric cone. We show that SCMWU is equivalent to Follow-the-Regularized-Leader and Online Mirror Descent with symmetric-cone negative entropy as regularizer. Using this structural result we show that SCMWU is a no-regret algorithm, and verify our theoretical results with extensive experiments. Our results unify and generalize the analysis for the Multiplicative Weights Update method over the probability simplex and the Matrix Multiplicative Weights Update method over the set of density matrices., 27 pages, 7 figures, 2 tables

Time-series datasets are central in numerous fields of science and engineering, such as biomedicine, Earth observation, and network analysis. Extensive research exists on state-space models (SSMs), which are powerful mathematical tools that allow for probabilistic and interpretable learning on time series. Estimating the model parameters in SSMs is arguably one of the most complicated tasks, and the inclusion of prior knowledge is known to both ease the interpretation but also to complicate the inferential tasks. Very recent works have attempted to incorporate a graphical perspective on some of those model parameters, but they present notable limitations that this work addresses. More generally, existing graphical modeling tools are designed to incorporate either static information, focusing on statistical dependencies among independent random variables (e.g., graphical Lasso approach), or dynamic information, emphasizing causal relationships among time series samples (e.g., graphical Granger approaches). However, there are no joint approaches combining static and dynamic graphical modeling within the context of SSMs. This work proposes a novel approach to fill this gap by introducing a joint graphical modeling framework that bridges the static graphical Lasso model and a causal-based graphical approach for the linear-Gaussian SSM. We present DGLASSO (Dynamic Graphical Lasso), a new inference method within this framework that implements an efficient block alternating majorization-minimization algorithm. The algorithm's convergence is established by departing from modern tools from nonlinear analysis. Experimental validation on synthetic and real weather variability data showcases the effectiveness of the proposed model and inference algorithm.

The efficacy of modern generative models is commonly contingent upon the precision of score estimation along the diffusion path, with a focus on diffusion models and their ability to generate high-quality data samples. This study delves into the potentialities of posterior sampling through reverse diffusion. An examination of the sampling literature reveals that score estimation can be transformed into a mean estimation problem via the decomposition of the transition kernel. By estimating the mean of the auxiliary distribution, the reverse diffusion process can give rise to a novel posterior sampling algorithm, which diverges from traditional gradient-based Markov Chain Monte Carlo (MCMC) methods. We provide the convergence analysis in total variation distance and demonstrate that the isoperimetric dependency of the proposed algorithm is comparatively lower than that observed in conventional MCMC techniques, which justifies the superior performance for high dimensional sampling with error tolerance. Our analytical framework offers fresh perspectives on the complexity of score estimation at various time points, as denoted by the properties of the auxiliary distribution.

This article is a short introduction to AI4OPT, the NSF AI Institute for Advances in Optimization. AI4OPT fuses AI and Optimization, inspired by end-use cases in supply chains, energy systems, chip design and manufacturing, and sustainable food systems. AI4OPT also applies its "teaching the teachers" philosophy to provide longitudinal educational pathways in AI for engineering.

In our work, we build upon the established connection between Residual Neural Networks (ResNets) and continuous-time control systems known as NeurODEs. By construction, NeurODEs have been limited to constant-width layers, making them unsuitable for modeling deep learning architectures with width-varying layers. In this paper, we propose a continuous-time Autoencoder, which we call AutoencODE, and we extend to this case the mean-field control framework already developed for usual NeurODEs. In this setting, we tackle the case of low Tikhonov regularization, resulting in potentially non-convex cost landscapes. While the global results obtained for high Tikhonov regularization may not hold globally, we show that many of them can be recovered in regions where the loss function is locally convex. Inspired by our theoretical findings, we develop a training method tailored to this specific type of Autoencoders with residual connections, and we validate our approach through numerical experiments conducted on various examples., 34 pages, 11 figures

We consider a class of stochastic smooth convex optimization problems under rather general assumptions on the noise in the stochastic gradient observation. As opposed to the classical problem setting in which the variance of noise is assumed to be uniformly bounded, herein we assume that the variance of stochastic gradients is related to the "sub-optimality" of the approximate solutions delivered by the algorithm. Such problems naturally arise in a variety of applications, in particular, in the well-known generalized linear regression problem in statistics. However, to the best of our knowledge, none of the existing stochastic approximation algorithms for solving this class of problems attain optimality in terms of the dependence on accuracy, problem parameters, and mini-batch size. We discuss two non-Euclidean accelerated stochastic approximation routines--stochastic accelerated gradient descent (SAGD) and stochastic gradient extrapolation (SGE)--which carry a particular duality relationship. We show that both SAGD and SGE, under appropriate conditions, achieve the optimal convergence rate, attaining the optimal iteration and sample complexities simultaneously. However, corresponding assumptions for the SGE algorithm are more general; they allow, for instance, for efficient application of the SGE to statistical estimation problems under heavy tail noises and discontinuous score functions. We also discuss the application of the SGE to problems satisfying quadratic growth conditions, and show how it can be used to recover sparse solutions. Finally, we report on some simulation experiments to illustrate numerical performance of our proposed algorithms in high-dimensional settings.

Graph neural networks (GNNs) have become increasingly popular for classification tasks on graph-structured data. Yet, the interplay between graph topology and feature evolution in GNNs is not well understood. In this paper, we focus on node-wise classification, illustrated with community detection on stochastic block model graphs, and explore the feature evolution through the lens of the "Neural Collapse" (NC) phenomenon. When training instance-wise deep classifiers (e.g. for image classification) beyond the zero training error point, NC demonstrates a reduction in the deepest features' within-class variability and an increased alignment of their class means to certain symmetric structures. We start with an empirical study that shows that a decrease in within-class variability is also prevalent in the node-wise classification setting, however, not to the extent observed in the instance-wise case. Then, we theoretically study this distinction. Specifically, we show that even an "optimistic" mathematical model requires that the graphs obey a strict structural condition in order to possess a minimizer with exact collapse. Interestingly, this condition is viable also for heterophilic graphs and relates to recent empirical studies on settings with improved GNNs' generalization. Furthermore, by studying the gradient dynamics of the theoretical model, we provide reasoning for the partial collapse observed empirically. Finally, we present a study on the evolution of within- and between-class feature variability across layers of a well-trained GNN and contrast the behavior with spectral methods., 55 pages, 34 figures

Breaking safety constraints in control systems can lead to potential risks, resulting in unexpected costs or catastrophic damage. Nevertheless, uncertainty is ubiquitous, even among similar tasks. In this paper, we develop a novel adaptive safe control framework that integrates meta learning, Bayesian models, and control barrier function (CBF) method. Specifically, with the help of CBF method, we learn the inherent and external uncertainties by a unified adaptive Bayesian linear regression (ABLR) model, which consists of a forward neural network (NN) and a Bayesian output layer. Meta learning techniques are leveraged to pre-train the NN weights and priors of the ABLR model using data collected from historical similar tasks. For a new control task, we refine the meta-learned models using a few samples, and introduce pessimistic confidence bounds into CBF constraints to ensure safe control. Moreover, we provide theoretical criteria to guarantee probabilistic safety during the control processes. To validate our approach, we conduct comparative experiments in various obstacle avoidance scenarios. The results demonstrate that our algorithm significantly improves the Bayesian model-based CBF method, and is capable for efficient safe exploration even with multiple uncertain constraints.

Given a compact subset of a Banach space, the Chebyshev center problem consists of finding a minimal circumscribing ball containing the set. In this article we establish a numerically tractable algorithm for solving the Chebyshev center problem in the context of optimal learning from a finite set of data points. For a hypothesis space realized as a compact but not necessarily convex subset of a finite-dimensional subspace of some underlying Banach space, this algorithm computes the Chebyshev radius and the Chebyshev center of the hypothesis space, thereby solving the problem of optimal recovery of functions from data. The algorithm itself is based on, and significantly extends, recent results for near-optimal solutions of convex semi-infinite problems by means of targeted sampling, and it is of independent interest. Several examples of numerical computations of Chebyshev centers are included in order to illustrate the effectiveness of the algorithm., 22 pages, 16 figures

Time-dependent scheduling with linear deterioration involves determining when to execute jobs whose processing times degrade as their beginning is delayed. Each job i is associated with a release time r_i and a processing time function p_i(s_i)=alpha_i + beta_i*s_i, where alpha_i, beta_i>0$ are constants and s_i is the job's start time. In this setting, the approximability of both single-machine minimum makespan and total completion time problems remains open. Here, we take a step forward by developing new bounds and approximation results for the interesting special case of the problems with uniform deterioration, i.e.\ beta_i=beta, for each i. The key contribution is a O(1+1/beta)-approximation algorithm for the makespan problem and a O(1+1/beta^2)-approximation algorithm for the total completion time problem. Further, we propose greedy constant-factor approximation algorithms for instances with beta=O(1/n) and beta=Omega(n), where n is the number of jobs. Our analysis is based on a new approach for comparing computed and optimal schedules via bounding pseudomatchings.

Optimizer is an essential component for the success of deep learning, which guides the neural network to update the parameters according to the loss on the training set. SGD and Adam are two classical and effective optimizers on which researchers have proposed many variants, such as SGDM and RAdam. In this paper, we innovatively combine the backward-looking and forward-looking aspects of the optimizer algorithm and propose a novel \textsc{Admeta} (\textbf{A} \textbf{D}ouble exponential \textbf{M}oving averag\textbf{E} \textbf{T}o \textbf{A}daptive and non-adaptive momentum) optimizer framework. For backward-looking part, we propose a DEMA variant scheme, which is motivated by a metric in the stock market, to replace the common exponential moving average scheme. While in the forward-looking part, we present a dynamic lookahead strategy which asymptotically approaches a set value, maintaining its speed at early stage and high convergence performance at final stage. Based on this idea, we provide two optimizer implementations, \textsc{AdmetaR} and \textsc{AdmetaS}, the former based on RAdam and the latter based on SGDM. Through extensive experiments on diverse tasks, we find that the proposed \textsc{Admeta} optimizer outperforms our base optimizers and shows advantages over recently proposed competitive optimizers. We also provide theoretical proof of these two algorithms, which verifies the convergence of our proposed \textsc{Admeta}.

The dynamics of a high-dimensional metastable molecular system can often be characterised by a few features of the system, i.e. collective variables (CVs). Thanks to the rapid advance in the area of machine learning, various deep learning-based CV identification techniques have been developed in recent years, allowing accurate modelling and efficient simulation of complex molecular systems. In this paper, we look at two different categories of deep learning-based approaches for finding CVs, either by computing leading eigenfunctions of infinitesimal generator or transfer operator associated to the underlying dynamics, or by learning an autoencoder via minimisation of reconstruction error. We present a concise overview of the mathematics behind these two approaches and conduct a comparative numerical study of these two approaches on illustrative examples., 16 pages

Obesity is a critical healthcare issue affecting the United States. The least risky treatments available for obesity are behavioral interventions meant to promote diet and exercise. Often these interventions contain a mobile component that allows interventionists to collect participants level data and provide participants with incentives and goals to promote long term behavioral change. Recently, there has been interest in using direct financial incentives to promote behavior change. However, adherence is challenging in these interventions, as each participant will react differently to different incentive structure and amounts, leading researchers to consider personalized interventions. The key challenge for personalization, is that the clinicians do not know a priori how best to administer incentives to participants, and given finite intervention budgets how to disburse costly resources efficiently. In this paper, we consider this challenge of designing personalized weight loss interventions that use direct financial incentives to motivate weight loss while remaining within a budget. We create a machine learning approach that is able to predict how individuals may react to different incentive schedules within the context of a behavioral intervention. We use this predictive model in an adaptive framework that over the course of the intervention computes what incentives to disburse to participants and remain within the study budget. We provide both theoretical guarantees for our modeling and optimization approaches as well as demonstrate their performance in a simulated weight loss study. Our results highlight the cost efficiency and effectiveness of our personalized intervention design for weight loss., 48 pages, 5 figures

This paper introduces a framework that utilizes the Safe Screening technique to accelerate the optimization process of the Unbalanced Optimal Transport (UOT) problem by proactively identifying and eliminating zero elements in the sparse solutions. We demonstrate the feasibility of applying Safe Screening to the UOT problem with $\ell_2$-penalty and KL-penalty by conducting an analysis of the solution's bounds and considering the local strong convexity of the dual problem. Considering the specific structural characteristics of the UOT in comparison to general Lasso problems on the index matrix, we specifically propose a novel approximate projection, an elliptical safe region construction, and a two-hyperplane relaxation method. These enhancements significantly improve the screening efficiency for the UOT's without altering the algorithm's complexity.

Screw and Lie group theory allows for user-friendly modeling of multibody systems (MBS) while at the same they give rise to computationally efficient recursive algorithms. The inherent frame invariance of such formulations allows for use of arbitrary reference frames within the kinematics modeling (rather than obeying modeling conventions such as the Denavit-Hartenberg convention) and to avoid introduction of joint frames. The computational efficiency is owed to a representation of twists, accelerations, and wrenches that minimizes the computational effort. This can be directly carried over to dynamics formulations. In this paper recursive $O\left( n\right) $ Newton-Euler algorithms are derived for the four most frequently used representations of twists, and their specific features are discussed. These formulations are related to the corresponding algorithms that were presented in the literature. The MBS motion equations are derived in closed form using the Lie group formulation. One are the so-called 'Euler-Jourdain' or 'projection' equations, of which Kane's equations are a special case, and the other are the Lagrange equations. The recursive kinematics formulations are readily extended to higher orders in order to compute derivatives of the motions equations. To this end, recursive formulations for the acceleration and jerk are derived. It is briefly discussed how this can be employed for derivation of the linearized motion equations and their time derivatives. The geometric modeling allows for direct application of Lie group integration methods, which is briefly discussed.

We present a method for solving general nonconvex-strongly-convex bilevel optimization problems. Our method -- the \emph{Restarted Accelerated HyperGradient Descent} (\texttt{RAHGD}) method -- finds an $\epsilon$-first-order stationary point of the objective with $\tilde{\mathcal{O}}(\kappa^{3.25}\epsilon^{-1.75})$ oracle complexity, where $\kappa$ is the condition number of the lower-level objective and $\epsilon$ is the desired accuracy. We also propose a perturbed variant of \texttt{RAHGD} for finding an $\big(\epsilon,\mathcal{O}(\kappa^{2.5}\sqrt{\epsilon}\,)\big)$-second-order stationary point within the same order of oracle complexity. Our results achieve the best-known theoretical guarantees for finding stationary points in bilevel optimization and also improve upon the existing upper complexity bound for finding second-order stationary points in nonconvex-strongly-concave minimax optimization problems, setting a new state-of-the-art benchmark. Empirical studies are conducted to validate the theoretical results in this paper.

We propose to enhance the training of physics-informed neural networks (PINNs). To this aim, we introduce nonlinear additive and multiplicative preconditioning strategies for the widely used L-BFGS optimizer. The nonlinear preconditioners are constructed by utilizing the Schwarz domain-decomposition framework, where the parameters of the network are decomposed in a layer-wise manner. Through a series of numerical experiments, we demonstrate that both, additive and multiplicative preconditioners significantly improve the convergence of the standard L-BFGS optimizer, while providing more accurate solutions of the underlying partial differential equations. Moreover, the additive preconditioner is inherently parallel, thus giving rise to a novel approach to model parallelism., 22 pages, 7 figures

Single-use anion-exchange resins can reduce hazardous chromates to safe levels in drinking water. However, since most process control strategies monitor effluent concentrations, detection of any chromate leakage leads to premature resin replacement. Furthermore, variations in the inlet chromate concentration and other process conditions make process control a challenging step. In this work, we capture the uncertainty of the process conditions by applying the Ito process of Brownian motion with drift into a stochastic optimal control strategy. The ion exchange process is modeled using the method of moments which helps capture the process dynamics, later formulated into mathematical objectives representing desired chromate removal. We then solved our developed models as an optimal control problem via Pontryagin's maximum principle. The objectives enabled a successful control via flow rate adjustments leading to higher chromate extraction. Such an approach maximized the capacity of the resin and column efficiency to remove toxic compounds from water while capturing deviations in the process conditions., 43 pages, 19 figures, 3 tables

As artificial intelligence (AI) systems are increasingly embedded in our lives, their presence leads to interactions that shape our behaviour, decision-making, and social interactions. Existing theoretical research has primarily focused on human-to-human interactions, overlooking the unique dynamics triggered by the presence of AI. In this paper, resorting to methods from evolutionary game theory, we study how different forms of AI influence the evolution of cooperation in a human population playing the one-shot Prisoner's Dilemma game in both well-mixed and structured populations. We found that Samaritan AI agents that help everyone unconditionally, including defectors, can promote higher levels of cooperation in humans than Discriminatory AI that only help those considered worthy/cooperative, especially in slow-moving societies where change is viewed with caution or resistance (small intensities of selection). Intuitively, in fast-moving societies (high intensities of selection), Discriminatory AIs promote higher levels of cooperation than Samaritan AIs., This work is the result of the Complexity72h 2023 workshop

Understanding the geometric properties of gradient descent dynamics is a key ingredient in deciphering the recent success of very large machine learning models. A striking observation is that trained over-parameterized models retain some properties of the optimization initialization. This "implicit bias" is believed to be responsible for some favorable properties of the trained models and could explain their good generalization properties. The purpose of this article is threefold. First, we rigorously expose the definition and basic properties of "conservation laws", which are maximal sets of independent quantities conserved during gradient flows of a given model (e.g. of a ReLU network with a given architecture) with any training data and any loss. Then we explain how to find the exact number of these quantities by performing finite-dimensional algebraic manipulations on the Lie algebra generated by the Jacobian of the model. Finally, we provide algorithms (implemented in SageMath) to: a) compute a family of polynomial laws; b) compute the number of (not necessarily polynomial) conservation laws. We provide showcase examples that we fully work out theoretically. Besides, applying the two algorithms confirms for a number of ReLU network architectures that all known laws are recovered by the algorithm, and that there are no other laws. Such computational tools pave the way to understanding desirable properties of optimization initialization in large machine learning models.

A ReLU neural network leads to a finite polyhedral decomposition of input space and a corresponding finite dual graph. We show that while this dual graph is a coarse quantization of input space, it is sufficiently robust that it can be combined with persistent homology to detect homological signals of manifolds in the input space from samples. This property holds for a variety of networks trained for a wide range of purposes that have nothing to do with this topological application. We found this feature to be surprising and interesting; we hope it will also be useful., Accepted by Proceedings of the 2 nd Annual Workshop on Topology, Algebra, and Geometry in Machine Learning (TAG-ML) at the 40 th In- ternational Conference on Machine Learning

Convex optimization is crucial in controlling legged robots, where stability and optimal control are vital. Many control problems can be formulated as convex optimization problems, with a convex cost function and constraints capturing system dynamics. Our review focuses on active balancing problems and presents a general framework for formulating them as second-order cone programming (SOCP) for robustness and efficiency with existing interior point algorithms. We then discuss some prior work around the Zero Moment Point stability criterion, Linear Quadratic Regulator Control, and then the feedback model predictive control (MPC) approach to improve prediction accuracy and reduce computational costs. Finally, these techniques are applied to stabilize the robot for jumping and landing tasks. Further research in convex optimization of legged robots can have a significant societal impact. It can lead to improved gait planning and active balancing which enhances their ability to navigate complex environments, assist in search and rescue operations and perform tasks in hazardous environments. These advancements have the potential to revolutionize industries and help humans in daily life., 12 pages

Recently, Ye et al. (Mathematical Programming 2023) designed an algorithm for solving a specific class of bilevel programs with an emphasis on applications related to hyperparameter selection, utilizing the difference of convex algorithm based on the value function approach reformulation. The proposed algorithm is particularly powerful when the lower level problem is fully convex , such as a support vector machine model or a least absolute shrinkage and selection operator model. In this paper, to suit more applications related to machine learning and statistics, we substantially weaken the underlying assumption from lower level full convexity to weak convexity. Accordingly, we propose a new reformulation using Moreau envelope of the lower level problem and demonstrate that this reformulation is a difference of weakly convex program. Subsequently, we develop a sequentially convergent algorithm for solving this difference of weakly convex program. To evaluate the effectiveness of our approach, we conduct numerical experiments on the bilevel hyperparameter selection problem from elastic net, sparse group lasso, and RBF kernel support vector machine models.

Byte-Pair Encoding (BPE) is a popular algorithm used for tokenizing data in NLP, despite being devised initially as a compression method. BPE appears to be a greedy algorithm at face value, but the underlying optimization problem that BPE seeks to solve has not yet been laid down. We formalize BPE as a combinatorial optimization problem. Via submodular functions, we prove that the iterative greedy version is a $\frac{1}{{\sigma(\boldsymbol{\mu}^\star)}}(1-e^{-{\sigma(\boldsymbol{\mu}^\star)}})$-approximation of an optimal merge sequence, where ${\sigma(\boldsymbol{\mu}^\star)}$ is the total backward curvature with respect to the optimal merge sequence $\boldsymbol{\mu}^\star$. Empirically the lower bound of the approximation is $\approx 0.37$. We provide a faster implementation of BPE which improves the runtime complexity from $\mathcal{O}\left(N M\right)$ to $\mathcal{O}\left(N \log M\right)$, where $N$ is the sequence length and $M$ is the merge count. Finally, we optimize the brute-force algorithm for optimal BPE using memoization., Comment: ACL 2023

We focus on the problem of uncertainty informed allocation of medical resources (vaccines) to heterogeneous populations for managing epidemic spread. We tackle two related questions: (1) For a compartmental ordinary differential equation (ODE) model of epidemic spread, how can we estimate and integrate parameter uncertainty into resource allocation decisions? (2) How can we computationally handle both nonlinear ODE constraints and parameter uncertainties for a generic stochastic optimization problem for resource allocation? To the best of our knowledge current literature does not fully resolve these questions. Here, we develop a data-driven approach to represent parameter uncertainty accurately and tractably in a novel stochastic optimization problem formulation. We first generate a tractable scenario set by estimating the distribution on ODE model parameters using Bayesian inference with Gaussian processes. Next, we develop a parallelized solution algorithm that accounts for scenario-dependent nonlinear ODE constraints. Our scenario-set generation procedure and solution approach are flexible in that they can handle any compartmental epidemiological ODE model. Our computational experiments on two different non-linear ODE models (SEIR and SEPIHR) indicate that accounting for uncertainty in key epidemiological parameters can improve the efficacy of time-critical allocation decisions by 4-8%. This improvement can be attributed to data-driven and optimal (strategic) nature of vaccine allocations, especially in the early stages of the epidemic when the allocation strategy can crucially impact the long-term trajectory of the disease.

In Statistics, log-concave density estimation is a central problem within the field of nonparametric inference under shape constraints. Despite great progress in recent years on the statistical theory of the canonical estimator, namely the log-concave maximum likelihood estimator, adoption of this method has been hampered by the complexities of the non-smooth convex optimization problem that underpins its computation. We provide enhanced understanding of the structural properties of this optimization problem, which motivates the proposal of new algorithms, based on both randomized and Nesterov smoothing, combined with an appropriate integral discretization of increasing accuracy. We prove that these methods enjoy, both with high probability and in expectation, a convergence rate of order $1/T$ up to logarithmic factors on the objective function scale, where $T$ denotes the number of iterations. The benefits of our new computational framework are demonstrated on both synthetic and real data, and our implementation is available in a github repository \texttt{LogConcComp} (Log-Concave Computation).

Numerous applications in machine learning and data analytics can be formulated as equilibrium computation over Riemannian manifolds. Despite the extensive investigation of their Euclidean counterparts, the performance of Riemannian gradient-based algorithms remain opaque and poorly understood. We revisit the original scheme of Riemannian gradient descent (RGD) and analyze it under a geodesic monotonicity assumption, which includes the well-studied geodesically convex-concave min-max optimization problem as a special case. Our main contribution is to show that, despite the phenomenon of distance distortion, the RGD scheme, with a step size that is agnostic to the manifold's curvature, achieves a curvature-independent and linear last-iterate convergence rate in the geodesically strongly monotone setting. To the best of our knowledge, the possibility of curvature-independent rates and/or last-iterate convergence in the Riemannian setting has not been considered before.

Public transit is a popular mode of transit due to its affordability, despite the inconveniences due to the necessity of transfers required to reach most areas. For example, in the bus and metro network of New Delhi, only 30\% of stops can be directly accessed from any starting point, thus requiring transfers for most commutes. Additionally, last-mile services like rickshaws, tuk-tuks or shuttles are commonly used as feeders to the nearest public transit access points, which further adds to the complexity and inefficiency of a journey. Ultimately, users often face a tradeoff between coverage and transfers to reach their destination, regardless of the mode of transit or the use of last-mile services. To address the problem of limited accessibility and inefficiency due to transfers in public transit systems, we propose ``opti-mile," a novel trip planning approach that combines last-mile services with public transit such that no transfers are required. Opti-mile allows users to customise trip parameters such as maximum walking distance, and acceptable fare range. We analyse the transit network of New Delhi, evaluating the efficiency, feasibility and advantages of opti-mile for optimal multi-modal trips between randomly selected source-destination pairs. We demonstrate that opti-mile trips lead to a 10% reduction in distance travelled for 18% increase in price compared to traditional shortest paths. We also show that opti-mile trips provide better coverage of the city than public transit, without a significant fare increase.

This paper studies the q-learning, recently coined as the continuous time counterpart of Q-learning by Jia and Zhou (2023), for continuous time Mckean-Vlasov control problems in the setting of entropy-regularized reinforcement learning. In contrast to the single agent's control problem in Jia and Zhou (2023), the mean-field interaction of agents renders the definition of the q-function more subtle, for which we reveal that two distinct q-functions naturally arise: (i) the integrated q-function (denoted by $q$) as the first-order approximation of the integrated Q-function introduced in Gu, Guo, Wei and Xu (2023), which can be learnt by a weak martingale condition involving test policies; and (ii) the essential q-function (denoted by $q_e$) that is employed in the policy improvement iterations. We show that two q-functions are related via an integral representation under all test policies. Based on the weak martingale condition and our proposed searching method of test policies, some model-free learning algorithms are devised. In two examples, one in LQ control framework and one beyond LQ control framework, we can obtain the exact parameterization of the optimal value function and q-functions and illustrate our algorithms with simulation experiments., Keywords: Continuous-time reinforcement learning, q-function, Mckean-Vlasov control, weak martingale characterization, test policies

The Sum-of-Squares (SOS) approximation method is a technique used in optimization problems to derive lower bounds to the optimal value of an objective function. By representing the objective function as a sum of squares in a feature space, the SOS method transforms non-convex global optimization problems into solvable semidefinite programs. This note presents an overview of the SOS method. We start with its application in finite-dimensional feature spaces and, subsequently, we extend it to infinite-dimensional feature spaces using kernels (k-SOS). Additionally, we highlight the utilization of SOS for estimating some relevant quantities in information theory, including the log-partition function., These are notes from the lecture of Francis Bach given at the summer school "Statistical Physics & Machine Learning", that took place in Les Houches School of Physics in France from 4th to 29th July 2022. The school was organized by Florent Krzakala and Lenka Zdeborov\'a from EPFL. 19 pages, 4 figures

Demand flexibility plays a vital role in maintaining grid balance, reducing peak demand, and saving customers' energy bills. Given their highly shiftable load and significant contribution to a building's energy consumption, Heating, Ventilation, and Air Conditioning (HVAC) systems can provide valuable demand flexibility to the power systems by adjusting their energy consumption in response to electricity price and power system needs. To exploit this flexibility in both operation time and power, it is imperative to accurately model and aggregate the load flexibility of a large population of HVAC systems as well as designing effective control algorithms. In this paper, we tackle the curse of dimensionality issue in modeling and control by utilizing the concept of laxity to quantify the emergency level of each HVAC operation request. We further propose a two-level approach to address energy optimization for a large population of HVAC systems. The lower level involves an aggregator to aggregate HVAC load laxity information and use least-laxity-first (LLF) rule to allocate real-time power for individual HVAC systems based on the controller's total power. Due to the complex and uncertain nature of HVAC systems, we leverage a reinforcement learning (RL)-based controller to schedule the total power based on the aggregated laxity information and electricity price. We evaluate the temperature control and energy cost saving performance of a large-scale group of HVAC systems in both single-zone and multi-zone scenarios, under varying climate and electricity market conditions. The experiment results indicate that proposed approach outperforms the centralized methods in the majority of test scenarios, and performs comparably to model-based method in some scenarios., In Submission

This paper addresses the scheduling problem on two identical parallel machines with a single server in charge of loading and unloading operations of jobs. Each job has to be loaded by the server before being processed on one of the two machines and unloaded by the same server after its processing. No delay is allowed between loading and processing, and between processing and unloading. The objective function involves the minimization of the makespan. This problem referred to as P2, S1|sj , tj |Cmax generalizes the classical parallel machine scheduling problem with a single server which performs only the loading (i.e., setup) operation of each job. For this NP-hard problem, no solution algorithm was proposed in the literature. Therefore, we present two mixedinteger linear programming (MILP) formulations, one with completion-time variables along with two valid inequalities and one with time-indexed variables. In addition, we propose some polynomial-time solvable cases and a tight theoretical lower bound. In addition, we show that the minimization of the makespan is equivalent to the minimization of the total idle times on the machines. To solve large-sized instances of the problem, an efficient General Variable Neighborhood Search (GVNS) metaheuristic with two mechanisms for finding an initial solution is designed. The GVNS is evaluated by comparing its performance with the results provided by the MILPs and another metaheuristic. The results show that the average percentage deviation from the theoretical lower-bound of GVNS is within 0.642%. Some managerial insights are presented and our results are compared with the related literature., 40 pages, 4 figures, 16 tables

In this paper, we consider the network slicing (NS) problem which attempts to map multiple customized virtual network requests to a common shared network infrastructure and allocate network resources to meet diverse service requirements. We propose an efficient decomposition algorithm for solving this NP-hard problem. The proposed algorithm decomposes the large-scale hard NS problem into two relatively easy function placement (FP) and traffic routing (TR) subproblems and iteratively solves them enabling information feedback between each other, which makes it particularly suitable to solve large-scale problems. Specifically, the FP subproblem is to place service functions into cloud nodes in the network, and solving it can return a function placement strategy based on which the TR subproblem is defined; and the TR subproblem is to find paths connecting two nodes hosting two adjacent functions in the network, and solving it can either verify that the solution of the FP subproblem is an optimal solution of the original problem, or return a valid inequality to the FP subproblem that cuts off the current infeasible solution. The proposed algorithm is guaranteed to find the global solution of the NS problem. We demonstrate the effectiveness and efficiency of the proposed algorithm via numerical experiments., 5 pages, 4 figures, accepted for publication in IEEE SPAWC 2023

The focus of this article is on shape and topology optimization of transient vibroacoustic problems. The main contribution is a transient problem formulation that enables optimization over wide ranges of frequencies with complex signals, which are often of interest in industry. The work employs time domain methods to realize wide band optimization in the frequency domain. To this end, the objective function is defined in frequency domain where the frequency response of the system is obtained through a fast Fourier transform (FFT) algorithm on the transient response of the system. The work utilizes a parametric level set approach to implicitly define the geometry in which the zero level describes the interface between acoustic and structural domains. A cut element method is used to capture the geometry on a fixed background mesh through utilization of a special integration scheme that accurately resolves the interface. This allows for accurate solutions to strongly coupled vibroacoustic systems without having to re-mesh at each design update. The present work relies on efficient gradient based optimizers where the discrete adjoint method is used to calculate the sensitivities of objective and constraint functions. A thorough explanation of the consistent sensitivity calculation is given involving the FFT operation needed to define the objective function in frequency domain. Finally, the developed framework is applied to various vibroacoustic filter designs and the optimization results are verified using commercial finite element software with a steady state time-harmonic formulation.

Uncertainty principles present an important theoretical tool in signal processing, as they provide limits on the time-frequency concentration of a signal. In many real-world applications the signal domain has a complicated irregular structure that can be described by a graph. In this paper, we focus on the global uncertainty principle on graphs and propose new connections between the uncertainty bound for graph signals and graph eigenvectors delocalization. We also derive uncertainty bounds for random $d$-regular graphs and provide numerically efficient upper and lower approximations for the uncertainty bound on an arbitrary graph.

In this paper, we present a toolbox for interval analysis in numpy, with an application to formal verification of neural network controlled systems. Using the notion of natural inclusion functions, we systematically construct interval bounds for a general class of mappings. The toolbox offers efficient computation of natural inclusion functions using compiled C code, as well as a familiar interface in numpy with its canonical features, such as n-dimensional arrays, matrix/vector operations, and vectorization. We then use this toolbox in formal verification of dynamical systems with neural network controllers, through the composition of their inclusion functions.

Non-asymptotic convergence analysis of quasi-Newton methods has gained attention with a landmark result establishing an explicit superlinear rate of O$((1/\sqrt{t})^t)$. The methods that obtain this rate, however, exhibit a well-known drawback: they require the storage of the previous Hessian approximation matrix or instead storing all past curvature information to form the current Hessian inverse approximation. Limited-memory variants of quasi-Newton methods such as the celebrated L-BFGS alleviate this issue by leveraging a limited window of past curvature information to construct the Hessian inverse approximation. As a result, their per iteration complexity and storage requirement is O$(\tau d)$ where $\tau \le d$ is the size of the window and $d$ is the problem dimension reducing the O$(d^2)$ computational cost and memory requirement of standard quasi-Newton methods. However, to the best of our knowledge, there is no result showing a non-asymptotic superlinear convergence rate for any limited-memory quasi-Newton method. In this work, we close this gap by presenting a limited-memory greedy BFGS (LG-BFGS) method that achieves an explicit non-asymptotic superlinear rate. We incorporate displacement aggregation, i.e., decorrelating projection, in post-processing gradient variations, together with a basis vector selection scheme on variable variations, which greedily maximizes a progress measure of the Hessian estimate to the true Hessian. Their combination allows past curvature information to remain in a sparse subspace while yielding a valid representation of the full history. Interestingly, our established non-asymptotic superlinear convergence rate demonstrates a trade-off between the convergence speed and memory requirement, which to our knowledge, is the first of its kind. Numerical results corroborate our theoretical findings and demonstrate the effectiveness of our method.

We present a novel approach to non-convex optimization with certificates, which handles smooth functions on the hypercube or on the torus. Unlike traditional methods that rely on algebraic properties, our algorithm exploits the regularity of the target function intrinsic in the decay of its Fourier spectrum. By defining a tractable family of models, we allow at the same time to obtain precise certificates and to leverage the advanced and powerful computational techniques developed to optimize neural networks. In this way the scalability of our approach is naturally enhanced by parallel computing with GPUs. Our approach, when applied to the case of polynomials of moderate dimensions but with thousands of coefficients, outperforms the state-of-the-art optimization methods with certificates, as the ones based on Lasserre's hierarchy, addressing problems intractable for the competitors.

The widely used stochastic gradient methods for minimizing nonconvex composite objective functions require the Lipschitz smoothness of the differentiable part. But the requirement does not hold true for problem classes including quadratic inverse problems and training neural networks. To address this issue, we investigate a family of stochastic Bregman proximal gradient (SBPG) methods, which only require smooth adaptivity of the differentiable part. SBPG replaces the upper quadratic approximation used in SGD with the Bregman proximity measure, resulting in a better approximation model that captures the non-Lipschitz gradients of the nonconvex objective. We formulate the vanilla SBPG and establish its convergence properties under nonconvex setting without finite-sum structure. Experimental results on quadratic inverse problems testify the robustness of SBPG. Moreover, we propose a momentum-based version of SBPG (MSBPG) and prove it has improved convergence properties. We apply MSBPG to the training of deep neural networks with a polynomial kernel function, which ensures the smooth adaptivity of the loss function. Experimental results on representative benchmarks demonstrate the effectiveness and robustness of MSBPG in training neural networks. Since the additional computation cost of MSBPG compared with SGD is negligible in large-scale optimization, MSBPG can potentially be employed as an universal open-source optimizer in the future., 37 pages

In the classical communication setting multiple senders having access to the same source of information and transmitting it over channel(s) to a receiver in general leads to a decrease in estimation error at the receiver as compared with the single sender case. However, if the objectives of the information providers are different from that of the estimator, this might result in interesting strategic interactions and outcomes. In this work, we consider a hierarchical signaling game between multiple senders (information designers) and a single receiver (decision maker) each having their own, possibly misaligned, objectives. The senders lead the game by committing to individual information disclosure policies simultaneously, within the framework of a non-cooperative Nash game among themselves. This is followed by the receiver's action decision. With Gaussian information structure and quadratic objectives (which depend on underlying state and receiver's action) for all the players, we show that in general the equilibrium is not unique. We hence identify a set of equilibria and further show that linear noiseless policies can achieve a minimal element of this set. Additionally, we show that competition among the senders is beneficial to the receiver, as compared with cooperation among the senders. Further, we extend our analysis to a dynamic signaling game of finite horizon with Markovian information evolution. We show that linear memoryless policies can achieve equilibrium in this dynamic game. We also consider an extension to a game with multiple receivers having coupled objectives. We provide algorithms to compute the equilibrium strategies in all these cases. Finally, via extensive simulations, we analyze the effects of multiple senders in varying degrees of alignment among their objectives., This work has been submitted for possible journal publication

Trust-region methods based on Kullback-Leibler divergence are pervasively used to stabilize policy optimization in reinforcement learning. In this paper, we exploit more flexible metrics and examine two natural extensions of policy optimization with Wasserstein and Sinkhorn trust regions, namely Wasserstein policy optimization (WPO) and Sinkhorn policy optimization (SPO). Instead of restricting the policy to a parametric distribution class, we directly optimize the policy distribution and derive their closed-form policy updates based on the Lagrangian duality. Theoretically, we show that WPO guarantees a monotonic performance improvement, and SPO provably converges to WPO as the entropic regularizer diminishes. Moreover, we prove that with a decaying Lagrangian multiplier to the trust region constraint, both methods converge to global optimality. Experiments across tabular domains, robotic locomotion, and continuous control tasks further demonstrate the performance improvement of both approaches, more robustness of WPO to sample insufficiency, and faster convergence of SPO, over state-of-art policy gradient methods.

Support Vector Machine (SVM) has been one of the most successful machine learning techniques for binary classification problems. The key idea is to maximize the margin from the data to the hyperplane subject to correct classification on training samples. The commonly used hinge loss and its variations are sensitive to label noise, and unstable for resampling due to its unboundedness. This paper is concentrated on the kernel SVM with the $\ell_0$-norm hinge loss (referred as $\ell_0$-KSVM), which is a composite function of hinge loss and $\ell_0$-norm and then could overcome the difficulties mentioned above. In consideration of the nonconvexity and nonsmoothness of $\ell_0$-norm hinge loss, we first characterize the limiting subdifferential of the $\ell_0$-norm hinge loss and then derive the equivalent relationship among the proximal stationary point, the Karush-Kuhn-Tucker point, and the local optimal solution of $\ell_0$-KSVM. Secondly, we develop an ADMM algorithm for $\ell_0$-KSVM, and obtain that any limit point of the sequence generated by the proposed algorithm is a locally optimal solution. Lastly, some experiments on the synthetic and real datasets are illuminated to show that $\ell_0$-KSVM can achieve comparable accuracy compared with the standard KSVM while the former generally enjoys fewer support vectors.

Moving horizon estimation (MHE) is a widely studied state estimation approach in several practical applications. In the MHE problem, the state estimates are obtained via the solution of an approximated nonlinear optimization problem. However, this optimization step is known to be computationally complex. Given this limitation, this paper investigates the idea of iteratively preconditioned gradient-descent (IPG) to solve MHE problem with the aim of an improved performance than the existing solution techniques. To our knowledge, the preconditioning technique is used for the first time in this paper to reduce the computational cost and accelerate the crucial optimization step for MHE. The convergence guarantee of the proposed iterative approach for a class of MHE problems is presented. Additionally, sufficient conditions for the MHE problem to be convex are also derived. Finally, the proposed method is implemented on a unicycle localization example. The simulation results demonstrate that the proposed approach can achieve better accuracy with reduced computational costs.