Path-Guided Particle-based Sampling

Particle-based Bayesian inference methods by sampling from a partition-free target (posterior) distribution, e.g., Stein variational gradient descent (SVGD), have attracted significant attention. We propose a path-guided particle-based sampling~(PGPS) method based on a novel Log-weighted Shrinkage (LwS) density path linking an initial distribution to the target distribution. We propose to utilize a Neural network to learn a vector field motivated by the Fokker-Planck equation of the designed density path. Particles, initiated from the initial distribution, evolve according to the ordinary differential equation defined by the vector field. The distribution of these particles is guided along a density path from the initial distribution to the target distribution. The proposed LwS density path allows for an efficient search of modes of the target distribution while canonical methods fail. We theoretically analyze the Wasserstein distance of the distribution of the PGPS-generated samples and the target distribution due to approximation and discretization errors. Practically, the proposed PGPS-LwS method demonstrates higher Bayesian inference accuracy and better calibration ability in experiments conducted on both synthetic and real-world Bayesian learning tasks, compared to baselines, such as SVGD and Langevin dynamics, etc.

On the Learn-to-Optimize Capabilities of Transformers in In-Context Sparse Recovery

An intriguing property of the Transformer is its ability to perform in-context learning (ICL), where the Transformer can solve different inference tasks without parameter updating based on the contextual information provided by the corresponding input-output demonstration pairs. It has been theoretically proved that ICL is enabled by the capability of Transformers to perform gradient-descent algorithms (Von Oswald et al., 2023a; Bai et al., 2024). This work takes a step further and shows that Transformers can perform learning-to-optimize (L2O) algorithms. Specifically, for the ICL sparse recovery (formulated as LASSO) tasks, we show that a K-layer Transformer can perform an L2O algorithm with a provable convergence rate linear in K. This provides a new perspective explaining the superior ICL capability of Transformers, even with only a few layers, which cannot be achieved by the standard gradient-descent algorithms. Moreover, unlike the conventional L2O algorithms that require the measurement matrix involved in training to match that in testing, the trained Transformer is able to solve sparse recovery problems generated with different measurement matrices. Besides, Transformers as an L2O algorithm can leverage structural information embedded in the training tasks to accelerate its convergence during ICL, and generalize across different lengths of demonstration pairs, where conventional L2O algorithms typically struggle or fail. Such theoretical findings are supported by our experimental results.

On the Training Convergence of Transformers for In-Context Classification

While transformers have demonstrated impressive capacities for in-context learning (ICL) in practice, theoretical understanding of the underlying mechanism enabling transformers to perform ICL is still in its infant stage. This work aims to theoretically study the training dynamics of transformers for in-context classification tasks. We demonstrate that, for in-context classification of Gaussian mixtures under certain assumptions, a single-layer transformer trained via gradient descent converges to a globally optimal model at a linear rate. We further quantify the impact of the training and testing prompt lengths on the ICL inference error of the trained transformer. We show that when the lengths of training and testing prompts are sufficiently large, the prediction of the trained transformer approaches the Bayes-optimal classifier. Experimental results corroborate the theoretical findings.

Data-adaptive Differentially Private Prompt Synthesis for In-Context Learning

Large Language Models (LLMs) rely on the contextual information embedded in examples/demonstrations to perform in-context learning (ICL). To mitigate the risk of LLMs potentially leaking private information contained in examples in the prompt, we introduce a novel data-adaptive differentially private algorithm called AdaDPSyn to generate synthetic examples from the private dataset and then use these synthetic examples to perform ICL. The objective of AdaDPSyn is to adaptively adjust the noise level in the data synthesis mechanism according to the inherent statistical properties of the data, thereby preserving high ICL accuracy while maintaining formal differential privacy guarantees. A key innovation in AdaDPSyn is the Precision-Focused Iterative Radius Reduction technique, which dynamically refines the aggregation radius - the scope of data grouping for noise addition - based on patterns observed in data clustering, thereby minimizing the amount of additive noise. We conduct extensive experiments on standard benchmarks and compare AdaDPSyn with DP few-shot generation algorithm (Tang et al., 2023). The experiments demonstrate that AdaDPSyn not only outperforms DP few-shot generation, but also maintains high accuracy levels close to those of non-private baselines, providing an effective solution for ICL with privacy protection.

Transformers learn variable-order Markov chains in-context

Large language models have demonstrated impressive in-context learning (ICL) capability. However, it is still unclear how the underlying transformers accomplish it, especially in more complex scenarios. Toward this goal, several recent works studied how transformers learn fixed-order Markov chains (FOMC) in context, yet natural languages are more suitably modeled by variable-order Markov chains (VOMC), i.e., context trees (CTs). In this work, we study the ICL of VOMC by viewing language modeling as a form of data compression and focus on small alphabets and low-order VOMCs. This perspective allows us to leverage mature compression algorithms, such as context-tree weighting (CTW) and prediction by partial matching (PPM) algorithms as baselines, the former of which is Bayesian optimal for a class of CTW priors. We empirically observe a few phenomena: 1) Transformers can indeed learn to compress VOMC in-context, while PPM suffers significantly; 2) The performance of transformers is not very sensitive to the number of layers, and even a two-layer transformer can learn in-context quite well; and 3) Transformers trained and tested on non-CTW priors can significantly outperform the CTW algorithm. To explain these phenomena, we analyze the attention map of the transformers and extract two mechanisms, on which we provide two transformer constructions: 1) A construction with $D+2$ layers that can mimic the CTW algorithm accurately for CTs of maximum order $D$, 2) A 2-layer transformer that utilizes the feed-forward network for probability blending. One distinction from the FOMC setting is that a counting mechanism appears to play an important role. We implement these synthetic transformer layers and show that such hybrid transformers can match the ICL performance of transformers, and more interestingly, some of them can perform even better despite the much-reduced parameter sets.

Reframing Data Value for Large Language Models Through the Lens of Plausability

Data valuation seeks to answer the important question, "How much is this data worth?" Existing data valuation methods have largely focused on discriminative models, primarily examining data value through the lens of its utility in training. However, with the push for ever-larger language models, relying on valuation methods that require training becomes increasingly expensive and dependent on specific techniques. We propose an alternative perspective on the data value problem for language models, centering around the plausibility of the data. We posit that data holds lesser value if it can be plausibly generated by the model itself. Starting from some intuitive criteria that align with our notions of valuable data, we develop a novel value function that is computationally tractable and derived from first principles with provable properties. We conduct a theoretical analysis of our value function and evaluate it across multiple scenarios and datasets.

Understanding Uncertainty-based Active Learning Under Model Mismatch

Instead of randomly acquiring training data points, Uncertainty-based Active Learning (UAL) operates by querying the label(s) of pivotal samples from an unlabeled pool selected based on the prediction uncertainty, thereby aiming at minimizing the labeling cost for model training. The efficacy of UAL critically depends on the model capacity as well as the adopted uncertainty-based acquisition function. Within the context of this study, our analytical focus is directed toward comprehending how the capacity of the machine learning model may affect UAL efficacy. Through theoretical analysis, comprehensive simulations, and empirical studies, we conclusively demonstrate that UAL can lead to worse performance in comparison with random sampling when the machine learning model class has low capacity and is unable to cover the underlying ground truth. In such situations, adopting acquisition functions that directly target estimating the prediction performance may be beneficial for improving the performance of UAL.

Provable Policy Gradient Methods for Average-Reward Markov Potential Games

We study Markov potential games under the infinite horizon average reward criterion. Most previous studies have been for discounted rewards. We prove that both algorithms based on independent policy gradient and independent natural policy gradient converge globally to a Nash equilibrium for the average reward criterion. To set the stage for gradient-based methods, we first establish that the average reward is a smooth function of policies and provide sensitivity bounds for the differential value functions, under certain conditions on ergodicity and the second largest eigenvalue of the underlying Markov decision process (MDP). We prove that three algorithms, policy gradient, proximal-Q, and natural policy gradient (NPG), converge to an $\epsilon$-Nash equilibrium with time complexity $O(\frac{1}{\epsilon^2})$, given a gradient/differential Q function oracle. When policy gradients have to be estimated, we propose an algorithm with $\tilde{O}(\frac{1}{\min_{s,a}\pi(a|s)\delta})$ sample complexity to achieve $\delta$ approximation error w.r.t~the $\ell_2$ norm. Equipped with the estimator, we derive the first sample complexity analysis for a policy gradient ascent algorithm, featuring a sample complexity of $\tilde{O}(1/\epsilon^5)$. Simulation studies are presented.

Hierarchical Bayes Approach to Personalized Federated Unsupervised Learning

Statistical heterogeneity of clients' local data is an important characteristic in federated learning, motivating personalized algorithms tailored to the local data statistics. Though there has been a plethora of algorithms proposed for personalized supervised learning, discovering the structure of local data through personalized unsupervised learning is less explored. We initiate a systematic study of such personalized unsupervised learning by developing algorithms based on optimization criteria inspired by a hierarchical Bayesian statistical framework. We develop adaptive algorithms that discover the balance between using limited local data and collaborative information. We do this in the context of two unsupervised learning tasks: personalized dimensionality reduction and personalized diffusion models. We develop convergence analyses for our adaptive algorithms which illustrate the dependence on problem parameters (e.g., heterogeneity, local sample size). We also develop a theoretical framework for personalized diffusion models, which shows the benefits of collaboration even under heterogeneity. We finally evaluate our proposed algorithms using synthetic and real data, demonstrating the effective sample amplification for personalized tasks, induced through collaboration, despite data heterogeneity.

Correlational Lagrangian Schrödinger Bridge: Learning Dynamics with Population-Level Regularization

Accurate modeling of system dynamics holds intriguing potential in broad scientific fields including cytodynamics and fluid mechanics. This task often presents significant challenges when (i) observations are limited to cross-sectional samples (where individual trajectories are inaccessible for learning), and moreover, (ii) the behaviors of individual particles are heterogeneous (especially in biological systems due to biodiversity). To address them, we introduce a novel framework dubbed correlational Lagrangian Schr\"odinger bridge (CLSB), aiming to seek for the evolution "bridging" among cross-sectional observations, while regularized for the minimal population "cost". In contrast to prior methods relying on \textit{individual}-level regularizers for all particles \textit{homogeneously} (e.g. restraining individual motions), CLSB operates at the population level admitting the heterogeneity nature, resulting in a more generalizable modeling in practice. To this end, our contributions include (1) a new class of population regularizers capturing the temporal variations in multivariate relations, with the tractable formulation derived, (2) three domain-informed instantiations based on genetic co-expression stability, and (3) an integration of population regularizers into data-driven generative models as constrained optimization, and a numerical solution, with further extension to conditional generative models. Empirically, we demonstrate the superiority of CLSB in single-cell sequencing data analyses such as simulating cell development over time and predicting cellular responses to drugs of varied doses.