Catastrophic forgetting in deep neural networks occurs when learning new tasks degrades performance on previously learned tasks due to knowledge overwriting. Among the approaches to mitigate this issue, regularization techniques aim to identify and constrain "important" parameters to preserve previous knowledge. In the highly nonconvex optimization landscape of deep learning, we propose a novel perspective: tracking parameters during the final training plateau is more effective than monitoring them throughout the entire training process. We argue that parameters that exhibit higher activity (movement and variability) during this plateau reveal directions in the loss landscape that are relatively flat, making them suitable for adaptation to new tasks while preserving knowledge from previous ones. Our comprehensive experiments demonstrate that this approach achieves superior performance in balancing catastrophic forgetting mitigation with strong performance on newly learned tasks.

The labor-intensive nature of medical data annotation presents a significant challenge for respiratory disease diagnosis, resulting in a scarcity of high-quality labeled datasets in resource-constrained settings. Moreover, patient privacy concerns complicate the direct sharing of local medical data across institutions, and existing centralized data-driven approaches, which rely on amounts of available data, often compromise data privacy. This study proposes a federated few-shot learning framework with privacy-preserving mechanisms to address the issues of limited labeled data and privacy protection in diagnosing respiratory diseases. In particular, a meta-stochastic gradient descent algorithm is proposed to mitigate the overfitting problem that arises from insufficient data when employing traditional gradient descent methods for neural network training. Furthermore, to ensure data privacy against gradient leakage, differential privacy noise from a standard Gaussian distribution is integrated into the gradients during the training of private models with local data, thereby preventing the reconstruction of medical images. Given the impracticality of centralizing respiratory disease data dispersed across various medical institutions, a weighted average algorithm is employed to aggregate local diagnostic models from different clients, enhancing the adaptability of a model across diverse scenarios. Experimental results show that the proposed method yields compelling results with the implementation of differential privacy, while effectively diagnosing respiratory diseases using data from different structures, categories, and distributions.

Variational inference with natural-gradient descent often shows fast convergence in practice, but its theoretical convergence guarantees have been challenging to establish. This is true even for the simplest cases that involve concave log-likelihoods and use a Gaussian approximation. We show that the challenge can be circumvented for such cases using a square-root parameterization for the Gaussian covariance. This approach establishes novel convergence guarantees for natural-gradient variational-Gaussian inference and its continuous-time gradient flow. Our experiments demonstrate the effectiveness of natural gradient methods and highlight their advantages over algorithms that use Euclidean or Wasserstein geometries.

The fine-tuning of pretrained large language models (LLMs) has demonstrated state-of-the-art performance across a range of downstream applications. However, two main challenges hinder the wide adoption of these models: the substantial memory requirements of gradient-based optimizers used for fine-tuning and the critical need to protect the privacy of domain-specific fine-tuning data. As fine-tuning LLMs grows increasingly memory-intensive, a range of strategies has emerged to address this issue. In particular, zeroth-order (ZO) optimization methods recently have gained traction due to their memory efficiency, as they do not require explicit gradient computations. Instead, the zeroth-order gradients can be computed using forward step only, significantly reducing memory use compared to gradient computation. In a pioneering approach, Malladi et al. (2023) introduced a memory-efficient technique for fine-tuning LLMs using two-point Simultaneous Perturbation Stochastic Approximation (SPSA) estimators (Spall, 1992), enabling large model fine-tuning on memory-limited devices. Since then, zeroth-order methods have gained popularity in dealing with large machine learning models due to their memory efficiency and favorable upper bounds on gap from optimality under certain conditions on the Hessian of the objective function (Zhang et al., 2024b,a; Guo et al., 2024). Another major concern in training LLMs is privacy . As large parameterized models are increasingly used in sensitive data applications, these models must protect sensitive information, especially given privacy regulations like the E.U.

Differentially Private Stochastic Gradient Descent (DP-SGD) has been widely used for solving optimization problems with privacy guarantees in machine learning and statistics. Despite this, a systematic non-asymptotic convergence analysis for DP-SGD, particularly in the context of online problems and local differential privacy (LDP) models, remains largely elusive. Existing non-asymptotic analyses have focused on non-private optimization methods, and hence are not applicable to privacy-preserving optimization problems. This work initiates the analysis to bridge this gap and opens the door to non-asymptotic convergence analysis of private optimization problems. A general framework is investigated for the online LDP model in stochastic optimization problems. We assume that sensitive information from individuals is collected sequentially and aim to estimate, in real-time, a static parameter that pertains to the population of interest. Most importantly, we conduct a comprehensive non-asymptotic convergence analysis of the proposed estimators in finite-sample situations, which gives their users practical guidelines regarding the effect of various hyperparameters, such as step size, parameter dimensions, and privacy budgets, on convergence rates. Our proposed estimators are validated in the theoretical and practical realms by rigorous mathematical derivations and carefully constructed numerical experiments.

Differential privacy has been proven effective for stochastic gradient descent; however, existing methods often suffer from performance degradation in high-dimensional settings, as the scale of injected noise increases with dimensionality. To tackle this challenge, we propose AdaDPIGU--a new differentially private SGD framework with importance-based gradient updates tailored for deep neural networks. In the pretraining stage, we apply a differentially private Gaussian mechanism to estimate the importance of each parameter while preserving privacy. During the gradient update phase, we prune low-importance coordinates and introduce a coordinate-wise adaptive clipping mechanism, enabling sparse and noise-efficient gradient updates. Theoretically, we prove that AdaDPIGU satisfies $(\varepsilon, δ)$-differential privacy and retains convergence guarantees. Extensive experiments on standard benchmarks validate the effectiveness of AdaDPIGU. All results are reported under a fixed retention ratio of 60%. On MNIST, our method achieves a test accuracy of 99.12% under a privacy budget of $ε= 8$, nearly matching the non-private model. Remarkably, on CIFAR-10, it attains 73.21% accuracy at $ε= 4$, outperforming the non-private baseline of 71.12%, demonstrating that adaptive sparsification can enhance both privacy and utility.

Adam is widely recognized as one of the most effective optimizers for training deep neural networks (DNNs). Despite its remarkable empirical success, its theoretical convergence analysis remains unsatisfactory. Existing works predominantly interpret Adam as a preconditioned stochastic gradient descent with momentum (SGDM), formulated as $\bm{x}_{t+1} = \bm{x}_t - \frac{γ_t}{{\sqrt{\bm{v}_t}+ε}} \circ \bm{m}_t$. This perspective necessitates strong assumptions and intricate techniques, resulting in lengthy and opaque convergence proofs that are difficult to verify and extend. In contrast, we propose a novel interpretation by treating Adam as a sign-like optimizer, expressed as $\bm{x}_{t+1} = \bm{x}_t - γ_t \frac{|\bm{m}_t|}{{\sqrt{\bm{v}_t}+ε}} \circ {\rm Sign}(\bm{m}_t)$. This reformulation significantly simplifies the convergence analysis. For the first time, with some mild conditions, we prove that Adam achieves the optimal rate of ${\cal O}(\frac{1}{T^{\sfrac{1}{4}}})$ rather than the previous ${\cal O} \left(\frac{\ln T}{T^{\sfrac{1}{4}}}\right)$ under weak assumptions of the generalized $p$-affine variance and $(L_0, L_1, q)$-smoothness, without dependence on the model dimensionality or the numerical stability parameter $ε$. Additionally, our theoretical analysis provides new insights into the role of momentum as a key factor ensuring convergence and offers practical guidelines for tuning learning rates in Adam, further bridging the gap between theory and practice.

Federated learning with differential privacy suffers from two major costs: each client must transmit $d$-dimensional gradients every round, and the magnitude of DP noise grows with $d$. Yet empirical studies show that gradient updates exhibit strong temporal correlations and lie in a $k$-dimensional subspace with $k \ll d$. Motivated by this, we introduce DOME, a decentralized DP optimization framework in which each client maintains a compact sketch to project gradients into $\mathbb{R}^k$ before privatization and Secure Aggregation. This reduces per-round communication from order $d$ to order $k$ and moves towards a gradient approximation mean-squared error of $σ^2 k$. To allow the sketch to span new directions and prevent it from collapsing onto historical gradients, we augment it with random probes orthogonal to historical directions. We prove that our overall protocol satisfies $(ε,δ)$-Differential Privacy.

This monograph, spanning three chapters, explores Inverse Reinforcement Learning (IRL). The first two chapters view inverse reinforcement learning (IRL) through the lens of revealed preferences from microeconomics while the third chapter studies adaptive IRL via Langevin dynamics stochastic gradient algorithms. Chapter uses classical revealed preference theory (Afriat's theorem and extensions) to identify constrained utility maximizers based on observed agent actions. This allows for the reconstruction of set-valued estimates of an agent's utility. We illustrate this procedure by identifying the presence of a cognitive radar and reconstructing its utility function. The chapter also addresses the construction of a statistical detector for utility maximization behavior when agent actions are corrupted by noise. Chapter 2 studies Bayesian IRL. It investigates how an analyst can determine if an observed agent is a rationally inattentive Bayesian utility maximizer (i.e., simultaneously optimizing its utility and observation likelihood). The chapter discusses inverse stopping-time problems, focusing on reconstructing the continuation and stopping costs of a Bayesian agent operating over a random horizon. We then apply this IRL methodology to identify the presence of a Bayes-optimal sequential detector. Additionally, Chapter 2 provides a concise overview of discrete choice models, inverse Bayesian filtering, and inverse stochastic gradient algorithms for adaptive IRL. Finally, Chapter 3 introduces an adaptive IRL approach utilizing passive Langevin dynamics. This method aims to track time-varying utility functions given noisy and misspecified gradients. In essence, the adaptive IRL algorithms presented in Chapter 3 can be conceptualized as inverse stochastic gradient algorithms, as they learn the utility function in real-time while a stochastic gradient algorithm is in operation.

The rapid advancement of Embodied AI has led to an increasing demand for large-scale, high-quality real-world data. However, collecting such embodied data remains costly and inefficient. As a result, simulation environments have become a crucial surrogate for training robot policies. Yet, the significant Real2Sim2Real gap remains a critical bottleneck, particularly in terms of physical dynamics and visual appearance. To address this challenge, we propose EmbodieDreamer, a novel framework that reduces the Real2Sim2Real gap from both the physics and appearance perspectives. Specifically, we propose PhysAligner, a differentiable physics module designed to reduce the Real2Sim physical gap. It jointly optimizes robot-specific parameters such as control gains and friction coefficients to better align simulated dynamics with real-world observations. In addition, we introduce VisAligner, which incorporates a conditional video diffusion model to bridge the Sim2Real appearance gap by translating low-fidelity simulated renderings into photorealistic videos conditioned on simulation states, enabling high-fidelity visual transfer. Extensive experiments validate the effectiveness of EmbodieDreamer. The proposed PhysAligner reduces physical parameter estimation error by 3.74% compared to simulated annealing methods while improving optimization speed by 89.91\%. Moreover, training robot policies in the generated photorealistic environment leads to a 29.17% improvement in the average task success rate across real-world tasks after reinforcement learning. Code, model and data will be publicly available.