This poster has been moved from Monday #86 to Thursday #101. Stochastic convex optimization is a basic and well studied primitive in machine learning. It is well known that convex and Lipschitz functions can be minimized efficiently using Stochastic Gradient Descent (SGD).The Normalized Gradient Descent (NGD) algorithm, is an adaptation of Gradient Descent, which updates according to the direction of the gradients, rather than the gradients themselves. In this paper we analyze a stochastic version of NGD and prove its convergence to a global minimum for a wider class of functions: we require the functions to be quasi-convex and locally-Lipschitz. Quasi-convexity broadens the concept of unimodality to multidimensions and allows for certain types of saddle points, which are a known hurdle for first-order optimization methods such as gradient descent.

Distributed-memory implementations of numerical optimization algorithm, such as stochastic gradient descent (SGD), require interprocessor communication at every iteration of the algorithm. On modern distributed-memory clusters where communication is more expensive than computation, the scalability and performance of these algorithms are limited by communication cost. This work generalizes prior work on 1D $s$-step SGD and 1D Federated SGD with Averaging (FedAvg) to yield a 2D parallel SGD method (HybridSGD) which attains a continuous performance trade off between the two baseline algorithms. We present theoretical analysis which show the convergence, computation, communication, and memory trade offs between $s$-step SGD, FedAvg, 2D parallel SGD, and other parallel SGD variants. We implement all algorithms in C++ and MPI and evaluate their performance on a Cray EX supercomputing system. Our empirical results show that HybridSGD achieves better convergence than FedAvg at similar processor scales while attaining speedups of $5.3\times$ over $s$-step SGD and speedups up to $121\times$ over FedAvg when used to solve binary classification tasks using the convex, logistic regression model on datasets obtained from the LIBSVM repository.

The holy grail of machine learning is to enable Continual Federated Learning (CFL) to enhance the efficiency, privacy, and scalability of AI systems while learning from streaming data. The primary challenge of a CFL system is to overcome global catastrophic forgetting, wherein the accuracy of the global model trained on new tasks declines on the old tasks. In this work, we propose Continual Federated Learning with Aggregated Gradients (C-FLAG), a novel replay-memory based federated strategy consisting of edge-based gradient updates on memory and aggregated gradients on the current data. We provide convergence analysis of the C-FLAG approach which addresses forgetting and bias while converging at a rate of $O(1/\sqrt{T})$ over $T$ communication rounds. We formulate an optimization sub-problem that minimizes catastrophic forgetting, translating CFL into an iterative algorithm with adaptive learning rates that ensure seamless learning across tasks. We empirically show that C-FLAG outperforms several state-of-the-art baselines on both task and class-incremental settings with respect to metrics such as accuracy and forgetting.

We show via a combination of mathematical arguments and empirical evidence that data distributions sampled from diffusion models satisfy a Concentration of Measure Property saying that any Lipschitz $1$-dimensional projection of a random vector is not too far from its mean with high probability. This implies that such models are quite restrictive and gives an explanation for a fact previously observed in arXiv:2410.14171 that conventional diffusion models cannot capture "heavy-tailed" data (i.e. data $\mathbf{x}$ for which the norm $\|\mathbf{x}\|_2$ does not possess a subgaussian tail) well. We then proceed to train a generalized linear model using stochastic gradient descent (SGD) on the diffusion-generated data for a multiclass classification task and observe empirically that a Gaussian universality result holds for the test error. In other words, the test error depends only on the first and second order statistics of the diffusion-generated data in the linear setting. Results of such forms are desirable because they allow one to assume the data itself is Gaussian for analyzing performance of the trained classifier. Finally, we note that current approaches to proving universality do not apply to this case as the covariance matrices of the data tend to have vanishing minimum singular values for the diffusion-generated data, while the current proofs assume that this is not the case (see Subsection 3.4 for more details). This leaves extending previous mathematical universality results as an intriguing open question.

The seminal work of Jordan, Kinderlehrer, and Otto [33] developed what is now widely known as the JKO scheme, a foundational method for generating iterative algorithms to compute distributions and reshaping our understanding of sampling algorithms. The JKO scheme can be interpreted as a gradient flow of the free energy with respect to the Wasserstein metric, often referred to as the Wasserstein gradient flow. This interpretation has led to significant advancements in machine learning, including applications in reinforcement learning to solve policy-distribution optimization problems [55]. While the JKO scheme traditionally assumes that the underlying model is fully known, in this paper, we relax this assumption by allowing models with unknown parameters. We develop statistical approaches to estimate these parameters and adapt the JKO scheme to work with the estimated values. Specifically, Langevin equations--stochastic differential equations--play a key role in describing the evolution of physical systems, facilitating stochastic gradient descent in machine learning, and enabling Markov chain Monte Carlo (MCMC) simulations in numerical computing. For examples and detailed discussions, see [11, 8, 51, 22, 19, 39, 43]. Solutions to Langevin equations, known as Langevin diffusions, are stochastic processes whose distributions evolve according to the Fokker-Planck equations [27, 48].

Merging multiple expert models offers a promising approach for performing multi-task learning without accessing their original data. Existing methods attempt to alleviate task conflicts by sparsifying task vectors or promoting orthogonality among them. However, they overlook the fundamental requirement of model merging: ensuring the merged model performs comparably to task-specific models on respective tasks. We find these methods inevitably discard task-specific information that, while causing conflicts, is crucial for performance. Based on our findings, we frame model merging as a constrained optimization problem ($\textit{i.e.}$, minimizing the gap between the merged model and individual models, subject to the constraint of retaining shared knowledge) and solve it via adaptive projective gradient descent. Specifically, we align the merged model with individual models by decomposing and reconstituting the loss function, alleviating conflicts through $\textit{data-free}$ optimization of task vectors. To retain shared knowledge, we optimize this objective by projecting gradients within a $\textit{shared subspace}$ spanning all tasks. Moreover, we view merging coefficients as adaptive learning rates and propose a task-aware, training-free strategy. Experiments show that our plug-and-play approach consistently outperforms previous methods, achieving state-of-the-art results across diverse architectures and tasks in both vision and NLP domains.

Bandit optimization is a difficult problem, especially if the reward model is high-dimensional. When rewards are modeled by neural networks, sublinear regret has only been shown under strong assumptions, usually when the network is extremely wide. In this thesis, we investigate how pre-training can help us in the regime of smaller models. We consider a stochastic contextual bandit with the rewards modeled by a multi-layer neural network. The last layer is a linear predictor, and the layers before it are a black box neural architecture, which we call a representation network. We model pre-training as an initial guess of the weights of the representation network provided to the learner. To leverage the pre-trained weights, we introduce a novel algorithm we call Explore Twice then Commit (E2TC). During its two stages of exploration, the algorithm first estimates the last layer's weights using Ridge regression, and then runs Stochastic Gradient Decent jointly on all the weights. For a locally convex loss function, we provide conditions on the pre-trained weights under which the algorithm can learn efficiently. Under these conditions, we show sublinear regret of E2TC when the dimension of the last layer and number of actions $K$ are much smaller than the horizon $T$. In the weak training regime, when only the last layer is learned, the problem reduces to a misspecified linear bandit. We introduce a measure of misspecification $\epsilon_0$ for this bandit and use it to provide bounds $O(\epsilon_0\sqrt{d}KT+(KT)^{4 /5})$ or $\tilde{O}(\epsilon_0\sqrt{d}KT+d^{1 /3}(KT)^{2 /3})$ on the regret, depending on regularization strength. The first of these bounds has a dimension-independent sublinear term, made possible by the stochasticity of contexts. We also run experiments to evaluate the regret of E2TC and sample complexity of its exploration in practice.

Online learning and model reference adaptive control have many interesting intersections. One area where they differ however is in how the algorithms are analyzed and what objective or metric is used to discriminate "good" algorithms from "bad" algorithms. In adaptive control there are usually two objectives: 1) prove that all time varying parameters/states of the system are bounded, and 2) that the instantaneous error between the adaptively controlled system and a reference system converges to zero over time (or at least a compact set). For online learning the performance of algorithms is often characterized by the regret the algorithm incurs. Regret is defined as the cumulative loss (cost) over time from the online algorithm minus the cumulative loss (cost) of the single optimal fixed parameter choice in hindsight. Another significant difference between the two areas of research is with regard to the assumptions made in order to obtain said results. Adaptive control makes assumptions about the input-output properties of the control problem and derives solutions for a fixed error model or optimization task. In the online learning literature results are derived for classes of loss functions (i.e. convex) while a priori assuming that all time varying parameters are bounded, which for many optimization tasks is not unrealistic, but is a non starter in control applications. In this work we discuss these differences in detail through the regret based analysis of gradient descent for convex functions and the control based analysis of a streaming regression problem. We close with a discussion about the newly defined paradigm of online adaptive control and ask the following question "Are regret optimal control strategies deployable?"

We provide a general and malleable heuristic for the air conflict resolution problem. This heuristic is based on a new neighborhood structure for searching the solution space of trajectories and flight-levels. Using unsupervised learning, the core idea of our heuristic is to cluster the conflict points and disperse them in various flight levels. Our first algorithm is called Cluster & Disperse and in each iteration it assigns the most problematic flights in each cluster to another flight-level. In effect, we shuffle them between the flight-levels until we achieve a well-balanced configuration. The Cluster & Disperse algorithm then uses any horizontal plane conflict resolution algorithm as a subroutine to solve these well-balanced instances. Nevertheless, we develop a novel algorithm for the horizontal plane based on a similar idea. That is we cluster and disperse the conflict points spatially in the same flight level using the gradient descent and a social force. We use a novel maneuver making flights travel on an arc instead of a straight path which is based on the aviation routine of the Radius to Fix legs. Our algorithms can handle a high density of flights within a reasonable computation time. We put their performance in context with some notable algorithms from the literature. Being a general framework, a particular strength of the Cluster & Disperse is its malleability in allowing various constraints regarding the aircraft or the environment to be integrated with ease. This is in contrast to the models for instance based on mixed integer programming.

Optimal Transport (OT) has established itself as a robust framework for quantifying differences between distributions, with applications that span fields such as machine learning, data science, and computer vision. This paper offers a detailed examination of the OT problem, beginning with its theoretical foundations, including the classical formulations of Monge and Kantorovich and their extensions to modern computational techniques. It explores cutting-edge algorithms, including Sinkhorn iterations, primal-dual strategies, and reduction-based approaches, emphasizing their efficiency and scalability in addressing high-dimensional problems. The paper also highlights emerging trends, such as integrating OT into machine learning frameworks, the development of novel problem variants, and ongoing theoretical advancements. Applications of OT are presented across a range of domains, with particular attention to its innovative application in time series data analysis via Optimal Transport Warping (OTW), a robust alternative to methods like Dynamic Time Warping. Despite the significant progress made, challenges related to scalability, robustness, and ethical considerations remain, necessitating further research. The paper underscores OT's potential to bridge theoretical depth and practical utility, fostering impactful advancements across diverse disciplines.