Self-distillation (SD), a technique where a model refines itself from its own predictions, has garnered attention as a simple yet powerful approach in machine learning. Despite its widespread use, the mechanisms underlying its effectiveness remain unclear. In this study, we investigate the efficacy of hyperparameter-tuned multi-stage SD in binary classification tasks with noisy labeled Gaussian mixture data, utilizing a replica theory. Our findings reveals that the primary driver of SD's performance improvement is denoising through hard pseudo-labels, with the most notable gains observed in moderately sized datasets. We also demonstrate the efficacy of practical heuristics, such as early stopping for extracting meaningful signal and bias fixation for imbalanced data. These results provide both theoretical guarantees and practical insights, advancing our understanding and application of SD in noisy settings.

As digital technologies advance, communication networks face challenges in handling the vast data generated by intelligent devices. Autonomous vehicles, smart sensors, and IoT systems necessitate new paradigms. This thesis addresses these challenges by integrating semantic communication and generative models for optimized image compression and edge network resource allocation. Unlike bit-centric systems, semantic communication prioritizes transmitting meaningful data specifically selected to convey the meaning rather than obtain a faithful representation of the original data. The communication infrastructure can benefit to significant improvements in bandwidth efficiency and latency reduction. Central to this work is the design of semantic-preserving image compression using Generative Adversarial Networks and Denoising Diffusion Probabilistic Models. These models compress images by encoding only semantically relevant features, allowing for high-quality reconstruction with minimal transmission. Additionally, a Goal-Oriented edge network optimization framework is introduced, leveraging the Information Bottleneck principle and stochastic optimization to dynamically allocate resources and enhance efficiency. By integrating semantic communication into edge networks, this approach balances computational efficiency and communication effectiveness, making it suitable for real-time applications. The thesis compares semantic-aware models with conventional image compression techniques using classical and semantic evaluation metrics. Results demonstrate the potential of combining generative AI and semantic communication to create more efficient semantic-goal-oriented communication networks that meet the demands of modern data-driven applications.

Structural causal models (SCMs) allow us to investigate complex systems at multiple levels of resolution. The causal abstraction (CA) framework formalizes the mapping between high- and low-level SCMs. We address CA learning in a challenging and realistic setting, where SCMs are inaccessible, interventional data is unavailable, and sample data is misaligned. A key principle of our framework is $\textit{semantic embedding}$, formalized as the high-level distribution lying on a subspace of the low-level one. This principle naturally links linear CA to the geometry of the $\textit{Stiefel manifold}$. We present a category-theoretic approach to SCMs that enables the learning of a CA by finding a morphism between the low- and high-level probability measures, adhering to the semantic embedding principle. Consequently, we formulate a general CA learning problem. As an application, we solve the latter problem for linear CA; considering Gaussian measures and the Kullback-Leibler divergence as an objective. Given the nonconvexity of the learning task, we develop three algorithms building upon existing paradigms for Riemannian optimization. We demonstrate that the proposed methods succeed on both synthetic and real-world brain data with different degrees of prior information about the structure of CA.

We consider a network inventory problem motivated by one-way, on-demand vehicle sharing services. Due to uncertainties in both demand and returns, as well as a fixed number of rental units across an $n$-location network, the service provider must periodically reposition vehicles to match supply with demand spatially while minimizing costs. The optimal repositioning policy under a general $n$-location network is intractable without knowing the optimal value function. We introduce the best base-stock repositioning policy as a generalization of the classical inventory control policy to $n$ dimensions, and establish its asymptotic optimality in two distinct limiting regimes under general network structures. We present reformulations to efficiently compute this best base-stock policy in an offline setting with pre-collected data. In the online setting, we show that a natural Lipschitz-bandit approach achieves a regret guarantee of $\widetilde{O}(T^{\frac{n}{n+1}})$, which suffers from the exponential dependence on $n$. We illustrate the challenges of learning with censored data in networked systems through a regret lower bound analysis and by demonstrating the suboptimality of alternative algorithmic approaches. Motivated by these challenges, we propose an Online Gradient Repositioning algorithm that relies solely on censored demand. Under a mild cost-structure assumption, we prove that it attains an optimal regret of $O(n^{2.5} \sqrt{T})$, which matches the regret lower bound in $T$ and achieves only polynomial dependence on $n$. The key algorithmic innovation involves proposing surrogate costs to disentangle intertemporal dependencies and leveraging dual solutions to find the gradient of policy change. Numerical experiments demonstrate the effectiveness of our proposed methods.

Solving mathematics problems has been an intriguing capability of large language models, and many efforts have been made to improve reasoning by extending reasoning length, such as through self-correction and extensive long chain-of-thoughts. While promising in problem-solving, advanced long reasoning chain models exhibit an undesired single-modal behavior, where trivial questions require unnecessarily tedious long chains of thought. In this work, we propose a way to allow models to be aware of inference budgets by formulating it as utility maximization with respect to an inference budget constraint, hence naming our algorithm Inference Budget-Constrained Policy Optimization (IBPO). In a nutshell, models fine-tuned through IBPO learn to ``understand'' the difficulty of queries and allocate inference budgets to harder ones. With different inference budgets, our best models are able to have a $4.14$\% and $5.74$\% absolute improvement ($8.08$\% and $11.2$\% relative improvement) on MATH500 using $2.16$x and $4.32$x inference budgets respectively, relative to LLaMA3.1 8B Instruct. These improvements are approximately $2$x those of self-consistency under the same budgets.

In multi-objective black-box optimization, the goal is typically to find solutions that optimize a set of T black-box objective functions, $f_1$, ..., $f_T$, simultaneously. Traditional approaches often seek a single Pareto-optimal set that balances trade-offs among all objectives. In this work, we introduce a novel problem setting that departs from this paradigm: finding a smaller set of K solutions, where K < T, that collectively "covers" the T objectives. A set of solutions is defined as "covering" if, for each objective $f_1$, ..., $f_T$, there is at least one good solution. A motivating example for this problem setting occurs in drug design. For example, we may have T pathogens and aim to identify a set of K < T antibiotics such that at least one antibiotic can be used to treat each pathogen. To address this problem, we propose Multi-Objective Coverage Bayesian Optimization (MOCOBO), a principled algorithm designed to efficiently find a covering set. We validate our approach through extensive experiments on challenging high-dimensional tasks, including applications in peptide and molecular design. Experiments demonstrate MOCOBO's ability to find high-performing covering sets of solutions. Additionally, we show that the small sets of K < T solutions found by MOCOBO can match or nearly match the performance of T individually optimized solutions for the same objectives. Our results highlight MOCOBO's potential to tackle complex multi-objective problems in domains where finding at least one high-performing solution for each objective is critical.

Many real-world problems, such as those with fairness constraints, involve complex expectation constraints and large datasets, necessitating the design of efficient stochastic methods to solve them. Most existing research focuses on cases with no {constraint} or easy-to-project constraints or deterministic constraints. In this paper, we consider nonconvex nonsmooth stochastic optimization problems with expectation constraints, for which we build a novel exact penalty model. We first show the relationship between the penalty model and the original problem. Then on solving the penalty problem, we present a single-loop SPIDER-type stochastic subgradient method, which utilizes the subgradients of both the objective and constraint functions, as well as the constraint function value at each iteration. Under certain regularity conditions (weaker than Slater-type constraint qualification or strong feasibility assumed in existing works), we establish an iteration complexity result of $O(\epsilon^{-4})$ to reach a near-$\epsilon$ stationary point of the penalized problem in expectation, matching the lower bound for such tasks. Building on the exact penalization, an $(\epsilon,\epsilon)$-KKT point of the original problem is obtained. For a few scenarios, our complexity of either the {objective} sample subgradient or the constraint sample function values can be lower than the state-of-the-art results by a factor of $\epsilon^{-2}$. Moreover, on solving two fairness-constrained problems, our method is significantly (up to 466 times) faster than the state-of-the-art algorithms, including switching subgradient method and inexact proximal point methods.

Diffusion models (DMs) have emerged as powerful tools for modeling complex data distributions and generating realistic new samples. Over the years, advanced architectures and sampling methods have been developed to make these models practically usable. However, certain synthesis process decisions still rely on heuristics without a solid theoretical foundation. In our work, we offer a novel analysis of the DM's inference process, introducing a comprehensive frequency response perspective. Specifically, by relying on Gaussianity and shift-invariance assumptions, we present the inference process as a closed-form spectral transfer function, capturing how the generated signal evolves in response to the initial noise. We demonstrate how the proposed analysis can be leveraged for optimizing the noise schedule, ensuring the best alignment with the original dataset's characteristics. Our results lead to scheduling curves that are dependent on the frequency content of the data, offering a theoretical justification for some of the heuristics taken by practitioners.

Transformer models are challenging to optimize with SGD and typically require adaptive optimizers such as Adam. However, the reasons behind the superior performance of Adam over SGD remain unclear. In this study, we investigate the optimization of transformer models by focusing on \emph{gradient heterogeneity}, defined as the disparity in gradient norms among parameters. Our analysis shows that gradient heterogeneity hinders gradient-based optimization, including SGD, while sign-based optimization, a simplified variant of Adam, is less affected. We further examine gradient heterogeneity in transformer models and show that it is influenced by the placement of layer normalization. Additionally, we show that the momentum term in sign-based optimization is important for preventing the excessive growth of linear-head parameters in tasks with many classes. Experimental results from fine-tuning transformer models in both NLP and vision domains validate our theoretical analyses. This study provides insights into the optimization challenges of transformer models and offers guidance for designing future optimization algorithms. Code is available at \url{https://github.com/tom4649/gradient-heterogeneity}.

We consider the inverse problem of multi-armed bandits (IMAB) that are widely used in neuroscience and psychology research for behavior modelling. We first show that the IMAB problem is not convex in general, but can be relaxed to a convex problem via variable transformation. Based on this result, we propose a two-step sequential heuristic for (approximately) solving the IMAB problem. We discuss a condition where our method provides global solution to the IMAB problem with certificate, as well as approximations to further save computing time. Numerical experiments indicate that our heuristic method is more robust than directly solving the IMAB problem via repeated local optimization, and can achieve the performance of Monte Carlo methods within a significantly decreased running time. We provide the implementation of our method based on CVXPY, which allows straightforward application by users not well versed in convex optimization.