Pretrained language models have achieved state-of-the-art performance when adapted to a downstream NLP task. However, theoretical analysis of these models is scarce and challenging since the pretraining and downstream tasks can be very different. We propose an analysis framework that links the pretraining and downstream tasks with an underlying latent variable generative model of text -- the downstream classifier must recover a function of the posterior distribution over the latent variables. We analyze head tuning (learning a classifier on top of the frozen pretrained model) and prompt tuning in this setting. The generative model in our analysis is either a Hidden Markov Model (HMM) or an HMM augmented with a latent memory component, motivated by long-term dependencies in natural language. We show that 1) under certain non-degeneracy conditions on the HMM, simple classification heads can solve the downstream task, 2) prompt tuning obtains downstream guarantees with weaker non-degeneracy conditions, and 3) our recovery guarantees for the memory-augmented HMM are stronger than for the vanilla HMM because task-relevant information is easier to recover from the long-term memory. Experiments on synthetically generated data from HMMs back our theoretical findings.

Our algorithm is spectral in nature, and is easy to implement.

Pretrained language models have achieved state-of-the-art performance when adapted to a downstream NLP task. However, theoretical analysis of these models is scarce and challenging since the pretraining and downstream tasks can be very different. We propose an analysis framework that links the pretraining and downstream tasks with an underlying latent variable generative model of text -- the downstream classifier must recover a function of the posterior distribution over the latent variables. We analyze head tuning (learning a classifier on top of the frozen pretrained model) and prompt tuning in this setting. The generative model in our analysis is either a Hidden Markov Model (HMM) or an HMM augmented with a latent memory component, motivated by long-term dependencies in natural language.

The Certainty Equivalent heuristic (CE) is a widely-used algorithm for various dynamic resource allocation problems in OR and OM. Despite its popularity, existing theoretical guarantees of CE are limited to settings satisfying restrictive fluid regularity conditions, particularly, the non-degeneracy conditions, under the widely held belief that the violation of such conditions leads to performance deterioration and necessitates algorithmic innovation beyond CE. In this work, we conduct a refined performance analysis of CE within the general framework of online linear programming. We show that CE achieves uniformly near-optimal regret (up to a polylogarithmic factor in $T$) under only mild assumptions on the underlying distribution, without relying on any fluid regularity conditions. Our result implies that, contrary to prior belief, CE effectively beats the curse of degeneracy for a wide range of problem instances with continuous conditional reward distributions, highlighting the distinction of the problem's structure between discrete and non-discrete settings. Our explicit regret bound interpolates between the mild $(\log T)^2$ regime and the worst-case $\sqrt{T}$ regime with a parameter $\beta$ quantifying the minimal rate of probability accumulation of the conditional reward distributions, generalizing prior findings in the multisecretary setting. To achieve these results, we develop novel algorithmic analytical techniques. Drawing tools from the empirical processes theory, we establish strong concentration analysis of the solutions to random linear programs, leading to improved regret analysis under significantly relaxed assumptions. These techniques may find potential applications in broader online decision-making contexts.

This paper is concerned with learning Markov random fields (MRF). It is a theoretical paper, which is ultimately focused with proving a particular statement: given a variable X in an MRF, and given some of its Markov blanket variables B, there exists another variable Y that is conditionally dependent on X given the subset of B. In general this statement is not true; so the goal here is to identify some conditions where this is true. Most of this paper is centered around this, from which the ability to learn an MRF follows. The paper is mostly technical; my main complaint is that I do not think it is very intuitive. It appears that central to the results is the assumption on non-degeneracy, which I believe should be explained in higher level terms.

We study the problem of reconstructing a mixture of Markov chains from the trajectories generated by random walks through the state space. Under mild nondegeneracy conditions, we show that we can uniquely reconstruct the underlying chains by only considering trajectories of length three, which represent triples of states. Our algorithm is spectral in nature, and is easy to implement.

We study the problem of reconstructing a mixture of Markov chains from the trajectories generated by random walks through the state space. Under mild non-degeneracy conditions, we show that we can uniquely reconstruct the underlying chains by only considering trajectories of length three, which represent triples of states. Our algorithm is spectral in nature, and is easy to implement.