Optimal Multi-Distribution Learning

Multi-distribution learning (MDL), which seeks to learn a shared model that minimizes the worst-case risk across $k$ distinct data distributions, has emerged as a unified framework in response to the evolving demand for robustness, fairness, multi-group collaboration, etc. Achieving data-efficient MDL necessitates adaptive sampling, also called on-demand sampling, throughout the learning process. However, there exist substantial gaps between the state-of-the-art upper and lower bounds on the optimal sample complexity. Focusing on a hypothesis class of Vapnik-Chervonenkis (VC) dimension $d$, we propose a novel algorithm that yields an $varepsilon$-optimal randomized hypothesis with a sample complexity on the order of $(d+k)/\varepsilon^2$ (modulo some logarithmic factor), matching the best-known lower bound. Our algorithmic ideas and theory have been further extended to accommodate Rademacher classes. The proposed algorithms are oracle-efficient, which access the hypothesis class solely through an empirical risk minimization oracle. Additionally, we establish the necessity of randomization, unveiling a large sample size barrier when only deterministic hypotheses are permitted. These findings successfully resolve three open problems presented in COLT 2023 (i.e., Awasthi et al., (2023, Problem 1, 3 and 4)).

Dichotomy of Early and Late Phase Implicit Biases Can Provably Induce Grokking

Recent work by Power et al. (2022) highlighted a surprising "grokking" phenomenon in learning arithmetic tasks: a neural net first "memorizes" the training set, resulting in perfect training accuracy but near-random test accuracy, and after training for sufficiently longer, it suddenly transitions to perfect test accuracy. This paper studies the grokking phenomenon in theoretical setups and shows that it can be induced by a dichotomy of early and late phase implicit biases. Specifically, when training homogeneous neural nets with large initialization and small weight decay on both classification and regression tasks, we prove that the training process gets trapped at a solution corresponding to a kernel predictor for a long time, and then a very sharp transition to min-norm/max-margin predictors occurs, leading to a dramatic change in test accuracy.

Learning Hierarchical Polynomials with Three-Layer Neural Networks

We study the problem of learning hierarchical polynomials over the standard Gaussian distribution with three-layer neural networks. We specifically consider target functions of the form $h = g \circ p$ where $p : \mathbb{R}^d \rightarrow \mathbb{R}$ is a degree $k$ polynomial and $g: \mathbb{R} \rightarrow \mathbb{R}$ is a degree $q$ polynomial. This function class generalizes the single-index model, which corresponds to $k=1$, and is a natural class of functions possessing an underlying hierarchical structure. Our main result shows that for a large subclass of degree $k$ polynomials $p$, a three-layer neural network trained via layerwise gradient descent on the square loss learns the target $h$ up to vanishing test error in $\widetilde{\mathcal{O}}(d^k)$ samples and polynomial time. This is a strict improvement over kernel methods, which require $\widetilde \Theta(d^{kq})$ samples, as well as existing guarantees for two-layer networks, which require the target function to be low-rank. Our result also generalizes prior works on three-layer neural networks, which were restricted to the case of $p$ being a quadratic. When $p$ is indeed a quadratic, we achieve the information-theoretically optimal sample complexity $\widetilde{\mathcal{O}}(d^2)$, which is an improvement over prior work~\citep{nichani2023provable} requiring a sample size of $\widetilde\Theta(d^4)$. Our proof proceeds by showing that during the initial stage of training the network performs feature learning to recover the feature $p$ with $\widetilde{\mathcal{O}}(d^k)$ samples. This work demonstrates the ability of three-layer neural networks to learn complex features and as a result, learn a broad class of hierarchical functions.

Provably Efficient CVaR RL in Low-rank MDPs

We study risk-sensitive Reinforcement Learning (RL), where we aim to maximize the Conditional Value at Risk (CVaR) with a fixed risk tolerance $\tau$. Prior theoretical work studying risk-sensitive RL focuses on the tabular Markov Decision Processes (MDPs) setting. To extend CVaR RL to settings where state space is large, function approximation must be deployed. We study CVaR RL in low-rank MDPs with nonlinear function approximation. Low-rank MDPs assume the underlying transition kernel admits a low-rank decomposition, but unlike prior linear models, low-rank MDPs do not assume the feature or state-action representation is known. We propose a novel Upper Confidence Bound (UCB) bonus-driven algorithm to carefully balance the interplay between exploration, exploitation, and representation learning in CVaR RL. We prove that our algorithm achieves a sample complexity of $\tilde{O}\left(\frac{H^7 A^2 d^4}{\tau^2 \epsilon^2}\right)$ to yield an $\epsilon$-optimal CVaR, where $H$ is the length of each episode, $A$ is the capacity of action space, and $d$ is the dimension of representations. Computational-wise, we design a novel discretized Least-Squares Value Iteration (LSVI) algorithm for the CVaR objective as the planning oracle and show that we can find the near-optimal policy in a polynomial running time with a Maximum Likelihood Estimation oracle. To our knowledge, this is the first provably efficient CVaR RL algorithm in low-rank MDPs.

Settling the Sample Complexity of Online Reinforcement Learning

A central issue lying at the heart of online reinforcement learning (RL) is data efficiency. While a number of recent works achieved asymptotically minimal regret in online RL, the optimality of these results is only guaranteed in a ``large-sample'' regime, imposing enormous burn-in cost in order for their algorithms to operate optimally. How to achieve minimax-optimal regret without incurring any burn-in cost has been an open problem in RL theory. We settle this problem for the context of finite-horizon inhomogeneous Markov decision processes. Specifically, we prove that a modified version of Monotonic Value Propagation (MVP), a model-based algorithm proposed by \cite{zhang2020reinforcement}, achieves a regret on the order of (modulo log factors) \begin{equation*} \min\big\{ \sqrt{SAH^3K}, \,HK \big\}, \end{equation*} where $S$ is the number of states, $A$ is the number of actions, $H$ is the planning horizon, and $K$ is the total number of episodes. This regret matches the minimax lower bound for the entire range of sample size $K\geq 1$, essentially eliminating any burn-in requirement. It also translates to a PAC sample complexity (i.e., the number of episodes needed to yield $\varepsilon$-accuracy) of $\frac{SAH^3}{\varepsilon^2}$ up to log factor, which is minimax-optimal for the full $\varepsilon$-range. Further, we extend our theory to unveil the influences of problem-dependent quantities like the optimal value/cost and certain variances. The key technical innovation lies in the development of a new regret decomposition strategy and a novel analysis paradigm to decouple complicated statistical dependency -- a long-standing challenge facing the analysis of online RL in the sample-hungry regime.

Teaching Arithmetic to Small Transformers

Large language models like GPT-4 exhibit emergent capabilities across general-purpose tasks, such as basic arithmetic, when trained on extensive text data, even though these tasks are not explicitly encoded by the unsupervised, next-token prediction objective. This study investigates how small transformers, trained from random initialization, can efficiently learn arithmetic operations such as addition, multiplication, and elementary functions like square root, using the next-token prediction objective. We first demonstrate that conventional training data is not the most effective for arithmetic learning, and simple formatting changes can significantly improve accuracy. This leads to sharp phase transitions as a function of training data scale, which, in some cases, can be explained through connections to low-rank matrix completion. Building on prior work, we then train on chain-of-thought style data that includes intermediate step results. Even in the complete absence of pretraining, this approach significantly and simultaneously improves accuracy, sample complexity, and convergence speed. We also study the interplay between arithmetic and text data during training and examine the effects of few-shot prompting, pretraining, and model scale. Additionally, we discuss length generalization challenges. Our work highlights the importance of high-quality, instructive data that considers the particular characteristics of the next-word prediction objective for rapidly eliciting arithmetic capabilities.

Scaling In-Context Demonstrations with Structured Attention

The recent surge of large language models (LLMs) highlights their ability to perform in-context learning, i.e., "learning" to perform a task from a few demonstrations in the context without any parameter updates. However, their capabilities of in-context learning are limited by the model architecture: 1) the use of demonstrations is constrained by a maximum sentence length due to positional embeddings; 2) the quadratic complexity of attention hinders users from using more demonstrations efficiently; 3) LLMs are shown to be sensitive to the order of the demonstrations. In this work, we tackle these challenges by proposing a better architectural design for in-context learning. We propose SAICL (Structured Attention for In-Context Learning), which replaces the full-attention by a structured attention mechanism designed for in-context learning, and removes unnecessary dependencies between individual demonstrations, while making the model invariant to the permutation of demonstrations. We evaluate SAICL in a meta-training framework and show that SAICL achieves comparable or better performance than full attention while obtaining up to 3.4x inference speed-up. SAICL also consistently outperforms a strong Fusion-in-Decoder (FiD) baseline which processes each demonstration independently. Finally, thanks to its linear nature, we demonstrate that SAICL can easily scale to hundreds of demonstrations with continuous performance gains with scaling.

Sample Complexity for Quadratic Bandits: Hessian Dependent Bounds and Optimal Algorithms

In stochastic zeroth-order optimization, a problem of practical relevance is understanding how to fully exploit the local geometry of the underlying objective function. We consider a fundamental setting in which the objective function is quadratic, and provide the first tight characterization of the optimal Hessian-dependent sample complexity. Our contribution is twofold. First, from an information-theoretic point of view, we prove tight lower bounds on Hessian-dependent complexities by introducing a concept called energy allocation, which captures the interaction between the searching algorithm and the geometry of objective functions. A matching upper bound is obtained by solving the optimal energy spectrum. Then, algorithmically, we show the existence of a Hessian-independent algorithm that universally achieves the asymptotic optimal sample complexities for all Hessian instances. The optimal sample complexities achieved by our algorithm remain valid for heavy-tailed noise distributions, which are enabled by a truncation method.

Solving Robust MDPs through No-Regret Dynamics

Reinforcement Learning is a powerful framework for training agents to navigate different situations, but it is susceptible to changes in environmental dynamics. However, solving Markov Decision Processes that are robust to changes is difficult due to nonconvexity and size of action or state spaces. While most works have analyzed this problem by taking different assumptions on the problem, a general and efficient theoretical analysis is still missing. However, we generate a simple framework for improving robustness by solving a minimax iterative optimization problem where a policy player and an environmental dynamics player are playing against each other. Leveraging recent results in online nonconvex learning and techniques from improving policy gradient methods, we yield an algorithm that maximizes the robustness of the Value Function on the order of $\mathcal{O}\left(\frac{1}{T^{\frac{1}{2}}}\right)$ where $T$ is the number of iterations of the algorithm.

How to Query Human Feedback Efficiently in RL?

Reinforcement Learning with Human Feedback (RLHF) is a paradigm in which an RL agent learns to optimize a task using pair-wise preference-based feedback over trajectories, rather than explicit reward signals. While RLHF has demonstrated practical success in fine-tuning language models, existing empirical work does not address the challenge of how to efficiently sample trajectory pairs for querying human feedback. In this study, we propose an efficient sampling approach to acquiring exploratory trajectories that enable accurate learning of hidden reward functions before collecting any human feedback. Theoretical analysis demonstrates that our algorithm requires less human feedback for learning the optimal policy under preference-based models with linear parameterization and unknown transitions, compared to the existing literature. Specifically, our framework can incorporate linear and low-rank MDPs. Additionally, we investigate RLHF with action-based comparison feedback and introduce an efficient querying algorithm tailored to this scenario.