Kefan Dong

Kefan Dong, Tengyu Ma · stanford

Real-world machine learning applications often involve deploying neural networks to domains that are not seen in the training time. Hence, we need to understand the extrapolation of nonlinear models -- under what conditions on the distributions and function class, models can be guaranteed to extrapolate to new test distributions. The question is very challenging because even two-layer neural networks cannot be guaranteed to extrapolate outside the support of the training distribution without further assumptions on the domain shift. This paper makes some initial steps toward analyzing the extrapolation of nonlinear models for structured domain shift. We primarily consider settings where the marginal distribution of each coordinate of the data (or subset of coordinates) does not shift significantly across the training and test distributions, but the joint distribution may have a much bigger shift. We prove that the family of nonlinear models of the form $f(x)=\sum f_i(x_i)$, where $f_i$ is an arbitrary function on the subset of features $x_i$, can extrapolate to unseen distributions, if the covariance of the features is well-conditioned. To the best of our knowledge, this is the first result that goes beyond linear models and the bounded density ratio assumption, even though the assumptions on the distribution shift and function class are stylized.

Arvind Mahankali, Jeff Z. Haochen, Kefan Dong et al. · stanford

Despite recent theoretical progress on the non-convex optimization of two-layer neural networks, it is still an open question whether gradient descent on neural networks without unnatural modifications can achieve better sample complexity than kernel methods. This paper provides a clean mean-field analysis of projected gradient flow on polynomial-width two-layer neural networks. Different from prior works, our analysis does not require unnatural modifications of the optimization algorithm. We prove that with sample size $n = O(d^{3.1})$ where $d$ is the dimension of the inputs, the network trained with projected gradient flow converges in $\text{poly}(d)$ time to a non-trivial error that is not achievable by kernel methods using $n \ll d^4$ samples, hence demonstrating a clear separation between unmodified gradient descent and NTK. As a corollary, we show that projected gradient descent with a positive learning rate and a polynomial number of iterations converges to low error with the same sample complexity.

Kefan Dong, Tengyu Ma · stanford

Past research on interactive decision making problems (bandits, reinforcement learning, etc.) mostly focuses on the minimax regret that measures the algorithm's performance on the hardest instance. However, an ideal algorithm should adapt to the complexity of a particular problem instance and incur smaller regrets on easy instances than worst-case instances. In this paper, we design the first asymptotic instance-optimal algorithm for general interactive decision making problems with finite number of decisions under mild conditions. On every instance $f$, our algorithm outperforms all consistent algorithms (those achieving non-trivial regrets on all instances), and has asymptotic regret $\mathcal{C}(f) \ln n$, where $\mathcal{C}(f)$ is an exact characterization of the complexity of $f$. The key step of the algorithm involves hypothesis testing with active data collection. It computes the most economical decisions with which the algorithm collects observations to test whether an estimated instance is indeed correct; thus, the complexity $\mathcal{C}(f)$ is the minimum cost to test the instance $f$ against other instances. Our results, instantiated on concrete problems, recover the classical gap-dependent bounds for multi-armed bandits [Lai and Robbins, 1985] and prior works on linear bandits [Lattimore and Szepesvari, 2017], and improve upon the previous best instance-dependent upper bound [Xu et al., 2021] for reinforcement learning.

Kefan Dong, Yannis Flet-Berliac, Allen Nie et al. · stanford

We present a model-based offline reinforcement learning policy performance lower bound that explicitly captures dynamics model misspecification and distribution mismatch and we propose an empirical algorithm for optimal offline policy selection. Theoretically, we prove a novel safe policy improvement theorem by establishing pessimism approximations to the value function. Our key insight is to jointly consider selecting over dynamics models and policies: as long as a dynamics model can accurately represent the dynamics of the state-action pairs visited by a given policy, it is possible to approximate the value of that particular policy. We analyze our lower bound in the LQR setting and also show competitive performance to previous lower bounds on policy selection across a set of D4RL tasks.

Kefan Dong, Tengyu Ma

Many machine learning applications require learning a function with a small worst-case error over the entire input domain, that is, the $L_\infty$-error, whereas most existing theoretical works only guarantee recovery in average errors such as the $L_2$-error. $L_\infty$-recovery from polynomial samples is even impossible for seemingly simple function classes such as constant-norm infinite-width two-layer neural nets. This paper makes some initial steps beyond the impossibility results by leveraging the randomness in the ground-truth functions. We prove a polynomial sample complexity bound for random ground-truth functions drawn from Gaussian random fields. Our key technical novelty is to prove that the degree-$k$ spherical harmonics components of a function from Gaussian random field cannot be spiky in that their $L_\infty$/$L_2$ ratios are upperbounded by $O(d \sqrt{\ln k})$ with high probability. In contrast, the worst-case $L_\infty$/$L_2$ ratio for degree-$k$ spherical harmonics is on the order of $Ω(\min\{d^{k/2},k^{d/2}\})$.

Kefan Dong, Tengyu Ma

A fundamental challenge in formal theorem proving by LLMs is the lack of high-quality training data. Although reinforcement learning or expert iteration partially mitigates this issue by alternating between LLM generating proofs and finetuning them on correctly generated ones, performance quickly plateaus due to the scarcity of correct proofs (sparse rewards). To keep improving the models with limited data, we draw inspiration from mathematicians, who continuously develop new results, partly by proposing novel conjectures or exercises (which are often variants of known results) and attempting to solve them. We design the Self-play Theorem Prover (STP) that simultaneously takes on two roles, conjecturer and prover, each providing training signals to the other. The conjecturer is trained iteratively on previously generated conjectures that are barely provable by the current prover, which incentivizes it to generate increasingly challenging conjectures over time. The prover attempts to prove the conjectures with standard expert iteration. We evaluate STP with both Lean and Isabelle formal versifiers. With 51.3 billion tokens generated during the training in Lean, STP proves 28.5% of the statements in the LeanWorkbook dataset, doubling the previous best result of 13.2% achieved through expert iteration. The final model achieves state-of-the-art performance among whole-proof generation methods on miniF2F-test (65.0%, pass@3200), Proofnet-test (23.9%, pass@3200) and PutnamBench (8/644, pass@3200). We release our code, model, and dataset in this URL: https://github.com/kfdong/STP.

Luke Bailey, Kaiyue Wen, Kefan Dong et al.

LLM self-play algorithms are notable in that, in principle, nothing bounds their learning: a Conjecturer model creates problems for a Solver, and both improve together. However, in practice, existing LLM self-play methods do not scale well with large amounts of compute, instead hitting learning plateaus. We argue this is because over long training runs, the Conjecturer learns to hack its reward, collapsing to artificially complex problems that do not help the Solver improve. To overcome this, we introduce Self-Guided Self-Play (SGS), a self-play algorithm in which the language model itself guides the Conjecturer away from degeneracy. In SGS, the model takes on three roles: Solver, Conjecturer, and a Guide that scores synthetic problems by their relevance to unsolved target problems and how clean and natural they are, providing supervision against Conjecturer collapse. Our core hypothesis is that language models can assess whether a subproblem is useful for achieving a goal. We evaluate the scaling properties of SGS by running training for significantly longer than prior works and by fitting scaling laws to cumulative solve rate curves. Applying SGS to formal theorem proving in Lean4, we find that it surpasses the asymptotic solve rate of our strongest RL baseline in fewer than 80 rounds of self-play and enables a 7B parameter model, after 200 rounds of self-play, to solve more problems than a 671B parameter model pass@4.

Kefan Dong, Arvind Mahankali, Tengyu Ma

Mathematical theorem proving is an important testbed for large language models' deep and abstract reasoning capability. This paper focuses on improving LLMs' ability to write proofs in formal languages that permit automated proof verification/evaluation. Most previous results provide human-written lemmas to the theorem prover, which is an arguably oversimplified setting that does not sufficiently test the provers' planning and decomposition capabilities. Instead, we work in a more natural setup where the lemmas that are directly relevant to the theorem are not given to the theorem prover at test time. We design an RL-based training algorithm that encourages the model to decompose a theorem into lemmas, prove the lemmas, and then prove the theorem by using the lemmas. Our reward mechanism is inspired by how mathematicians train themselves: even if a theorem is too challenging to be proved by the current model, a positive reward is still given to the model for any correct and novel lemmas that are proposed and proved in this process. During training, our model proposes and proves lemmas that are not in the training dataset. In fact, these newly-proposed correct lemmas consist of 37.7% of the training replay buffer when we train on the dataset extracted from Archive of Formal Proofs (AFP). The model trained by our RL algorithm outperforms that trained by supervised finetuning, improving the pass rate from 40.8% to 45.5% on AFP test set, and from 36.5% to 39.5% on an out-of-distribution test set.

Andrea Zanette, Kefan Dong, Jonathan Lee et al.

In the stochastic linear contextual bandit setting there exist several minimax procedures for exploration with policies that are reactive to the data being acquired. In practice, there can be a significant engineering overhead to deploy these algorithms, especially when the dataset is collected in a distributed fashion or when a human in the loop is needed to implement a different policy. Exploring with a single non-reactive policy is beneficial in such cases. Assuming some batch contexts are available, we design a single stochastic policy to collect a good dataset from which a near-optimal policy can be extracted. We present a theoretical analysis as well as numerical experiments on both synthetic and real-world datasets.

Kefan Dong, Jiaqi Yang, Tengyu Ma

This paper studies model-based bandit and reinforcement learning (RL) with nonlinear function approximations. We propose to study convergence to approximate local maxima because we show that global convergence is statistically intractable even for one-layer neural net bandit with a deterministic reward. For both nonlinear bandit and RL, the paper presents a model-based algorithm, Virtual Ascent with Online Model Learner (ViOlin), which provably converges to a local maximum with sample complexity that only depends on the sequential Rademacher complexity of the model class. Our results imply novel global or local regret bounds on several concrete settings such as linear bandit with finite or sparse model class, and two-layer neural net bandit. A key algorithmic insight is that optimism may lead to over-exploration even for two-layer neural net model class. On the other hand, for convergence to local maxima, it suffices to maximize the virtual return if the model can also reasonably predict the size of the gradient and Hessian of the real return.

Yuanhao Wang, Kefan Dong

We consider the adversarial Markov Decision Process (MDP) problem, where the rewards for the MDP can be adversarially chosen, and the transition function can be either known or unknown. In both settings, Follow-the-PerturbedLeader (FPL) based algorithms have been proposed in previous literature. However, the established regret bounds for FPL based algorithms are worse than algorithms based on mirrordescent. We improve the analysis of FPL based algorithms in both settings, matching the current best regret bounds using faster and simpler algorithms.

Kefan Dong, Yingkai Li, Qin Zhang et al.

We study multinomial logit bandit with limited adaptivity, where the algorithms change their exploration actions as infrequently as possible when achieving almost optimal minimax regret. We propose two measures of adaptivity: the assortment switching cost and the more fine-grained item switching cost. We present an anytime algorithm (AT-DUCB) with $O(N \log T)$ assortment switches, almost matching the lower bound $Ω(\frac{N \log T}{ \log \log T})$. In the fixed-horizon setting, our algorithm FH-DUCB incurs $O(N \log \log T)$ assortment switches, matching the asymptotic lower bound. We also present the ESUCB algorithm with item switching cost $O(N \log^2 T)$.

Kefan Dong, Yuping Luo, Tengyu Ma

We compare the model-free reinforcement learning with the model-based approaches through the lens of the expressive power of neural networks for policies, $Q$-functions, and dynamics. We show, theoretically and empirically, that even for one-dimensional continuous state space, there are many MDPs whose optimal $Q$-functions and policies are much more complex than the dynamics. We hypothesize many real-world MDPs also have a similar property. For these MDPs, model-based planning is a favorable algorithm, because the resulting policies can approximate the optimal policy significantly better than a neural network parameterization can, and model-free or model-based policy optimization rely on policy parameterization. Motivated by the theory, we apply a simple multi-step model-based bootstrapping planner (BOOTS) to bootstrap a weak $Q$-function into a stronger policy. Empirical results show that applying BOOTS on top of model-based or model-free policy optimization algorithms at the test time improves the performance on MuJoCo benchmark tasks.

Kefan Dong, Jian Peng, Yining Wang et al.

In this paper, we consider the problem of online learning of Markov decision processes (MDPs) with very large state spaces. Under the assumptions of realizable function approximation and low Bellman ranks, we develop an online learning algorithm that learns the optimal value function while at the same time achieving very low cumulative regret during the learning process. Our learning algorithm, Adaptive Value-function Elimination (AVE), is inspired by the policy elimination algorithm proposed in (Jiang et al., 2017), known as OLIVE. One of our key technical contributions in AVE is to formulate the elimination steps in OLIVE as contextual bandit problems. This technique enables us to apply the active elimination and expert weighting methods from (Dudik et al., 2011), instead of the random action exploration scheme used in the original OLIVE algorithm, for more efficient exploration and better control of the regret incurred in each policy elimination step. To the best of our knowledge, this is the first $\sqrt{n}$-regret result for reinforcement learning in stochastic MDPs with general value function approximation.

Zhizhou Ren, Kefan Dong, Yuan Zhou et al.

Goal-oriented reinforcement learning has recently been a practical framework for robotic manipulation tasks, in which an agent is required to reach a certain goal defined by a function on the state space. However, the sparsity of such reward definition makes traditional reinforcement learning algorithms very inefficient. Hindsight Experience Replay (HER), a recent advance, has greatly improved sample efficiency and practical applicability for such problems. It exploits previous replays by constructing imaginary goals in a simple heuristic way, acting like an implicit curriculum to alleviate the challenge of sparse reward signal. In this paper, we introduce Hindsight Goal Generation (HGG), a novel algorithmic framework that generates valuable hindsight goals which are easy for an agent to achieve in the short term and are also potential for guiding the agent to reach the actual goal in the long term. We have extensively evaluated our goal generation algorithm on a number of robotic manipulation tasks and demonstrated substantially improvement over the original HER in terms of sample efficiency.

Kefan Dong, Yuanhao Wang, Xiaoyu Chen et al.

A fundamental question in reinforcement learning is whether model-free algorithms are sample efficient. Recently, Jin et al. \cite{jin2018q} proposed a Q-learning algorithm with UCB exploration policy, and proved it has nearly optimal regret bound for finite-horizon episodic MDP. In this paper, we adapt Q-learning with UCB-exploration bonus to infinite-horizon MDP with discounted rewards \emph{without} accessing a generative model. We show that the \textit{sample complexity of exploration} of our algorithm is bounded by $\tilde{O}({\frac{SA}{ε^2(1-γ)^7}})$. This improves the previously best known result of $\tilde{O}({\frac{SA}{ε^4(1-γ)^8}})$ in this setting achieved by delayed Q-learning \cite{strehl2006pac}, and matches the lower bound in terms of $ε$ as well as $S$ and $A$ except for logarithmic factors.