Large Language Models (LLMs) have demonstrated considerable cross-lingual alignment and generalization ability. Current research primarily focuses on improving LLMs’ cross-lingual generalization capabilities. However, there is still a lack of research on the intrinsic mechanisms of how LLMs achieve cross-lingual alignment. From the perspective of region partitioning, this paper conducts several investigations on the linguistic competence of LLMs. We discover a core region in LLMs that corresponds to linguistic competence, accounting for approximately 1% of the total model parameters. Removing this core region by setting parameters to zero results in a significant performance decrease across 30 different languages. Furthermore, this core region exhibits significant dimensional dependence, perturbations to even a single parameter on specific dimensions leading to a loss of linguistic competence. Moreover, we discover that distinct monolingual regions exist for different languages, and disruption to these specific regions substantially reduces the LLMs’ proficiency in those corresponding languages. Our research also indicates that freezing the core linguistic region during further pre-training can mitigate the issue of catastrophic forgetting (CF), a common phenomenon observed during further pre-training of LLMs. Overall, exploring the LLMs’ functional regions provides insights into the foundation of their intelligence.
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Reinforcement Learning · Everything Else
We consider an average reward reinforcement learning (RL) problem and work with asymptotic variance as a risk measure to model safety-critical applications. We design a temporal-difference (TD) type algorithm tailored for policy evaluation in this context. Our algorithm is based on linear stochastic approximation of an equivalent formulation of the asymptotic variance in terms of the solution of the Poisson equation. We consider both the tabular and linear function approximation settings, and establish $\tilde {O}(1/k)$ finite time convergence rate, where $k$ is the number of steps of the algorithm. Our work paves the way for developing actor-critic style algorithms for variance-constrained RL. To the best of our knowledge, our result provides the first sequential estimator for asymptotic variance of a Markov chain with provable finite sample guarantees, which is of independent interest.
Optimization · Zero-order and Black-box Optimization
We leverage multilevel Monte Carlo (MLMC) to improve the performance of multi-step look- ahead Bayesian optimization (BO) methods that involve nested expectations and maximizations. Often these expectations must be computed by Monte Carlo (MC). The complexity rate of naive MC degrades for nested operations, whereas MLMC is capable of achieving the canonical MC convergence rate for this type of problem, independently of dimension and without any smoothness assumptions. Our theoretical study focuses on the approximation improvements for two- and three-step look-ahead acquisition functions, but, as we discuss, the approach is generalizable in various ways, including beyond the context of BO. Our findings are verified numerically and the benefits of MLMC for BO are illustrated on several benchmark examples. Code is available at https://github.com/Shangda-Yang/MLMCBO.
Reinforcement Learning · Function Approximation
We study regret minimization in online episodic linear Markov Decision Processes, and propose a policy optimization algorithm that is computationally efficient, and obtains rate optimal $\widetilde O (\sqrt K)$ regret where $K$ denotes the number of episodes. Our work is the first to establish the optimal rate (in terms of $K$) of convergence in the stochastic setting with bandit feedback using a policy optimization based approach, and the first to establish the optimal rate in the adversarial setup with full information feedback, for which no algorithm with an optimal rate guarantee was previously known.
Theory · Learning Theory
We obtain the first positive results for bounded sample compression in the agnostic regression setting with the $\ell_p$ loss, where $p\in [1,\infty]$. We construct a generic approximate sample compression scheme for real-valued function classes exhibiting exponential size in the fat-shattering dimension but independent of the sample size. Notably, for linear regression, an approximate compression of size linear in the dimension is constructed. Moreover, for $\ell_1$ and $\ell_\infty$ losses, we can even exhibit an efficient exact sample compression scheme of size linear in the dimension. We further show that for every other $\ell_p$ loss, $p\in (1,\infty)$, there does not exist an exact agnostic compression scheme of bounded size. This refines and generalizes a negative result of David, Moran, and Yehudayoff (2016) for the $\ell_2$ loss. We close by posing general open questions: for agnostic regression with $\ell_1$ loss, does every function class admit an exact compression scheme of polynomial size in the pseudo-dimension? For the $\ell_2$ loss, does every function class admit an approximate compression scheme of polynomial size in the fat-shattering dimension? These questions generalize Warmuth's classic sample compression conjecture for realizable-case classification (Warmuth, 2003).
Optimization
We propose a novel algorithm that extends the methods of ball smoothing and Gaussian smoothing for noisy derivative-free optimization by accounting for the heterogeneous curvature of the objective function. The algorithm dynamically adapts the shape of the smoothing kernel to approximate the Hessian of the objective function around a local optimum. This approach significantly reduces the error in estimating the gradient from noisy evaluations through sampling. We demonstrate the efficacy of our method through numerical experiments on artificial problems. Additionally, we show improved performance when tuning NP-hard combinatorial optimization solvers compared to existing state-ofthe-art heuristic derivative-free and Bayesian optimization methods.
Deep Learning · Theory
In this work, we provide a mathematical proof that diffusion models encode data manifolds by approximating their normal bundles. Based on this observation we propose a novel method for extracting the intrinsic dimension of the data manifold from a trained diffusion model. Our insights are based on the fact that a diffusion model approximates the score function i.e. the gradient of the log density of a noise-corrupted version of the target distribution for varying levels of corruption. We prove that as the level of corruption decreases, the score function points towards the manifold, as this direction becomes the direction of maximal likelihood increase. Therefore, at low noise levels, the diffusion model provides us with an approximation of the manifold's normal bundle, allowing for an estimation of the manifold's intrinsic dimension. To the best of our knowledge our method is the first estimator of intrinsic dimension based on diffusion models and it outperforms well established estimators in controlled experiments on both Euclidean and image data.
Deep Learning · Large Language Models
Recent advances in natural language processing have relied heavily on using Transformer-based language models. However, Transformers often require large parameter sizes and model depth. Existing Transformer-free approaches using state-space models demonstrate superiority over Transformers, yet they still lack a neuro-biologically connection to the human brain. This paper proposes ${\it LasF}$, representing ${\bf L}$anguage tokens ${\bf as}$ ${\bf F}$unctionals of semantic fields, to simulate the neuronal behaviors for better language modeling. The ${\it LasF}$ module is equivalent to a nonlinear approximator tailored for sequential data. By replacing the final layers of pre-trained language models with the ${\it LasF}$ module, we obtain ${\it LasF}$-based models. Experiments conducted for standard reading comprehension and question-answering tasks demonstrate that the ${\it LasF}$-based models consistently improve accuracy with fewer parameters. Besides, we use CommonsenseQA's blind test set to evaluate a full-parameter tuned ${\it LasF}$-based model, which outperforms the prior best ensemble and single models by $0.4\%$ and $3.1\%$, respectively. Furthermore, our ${\it LasF}$-only language model trained from scratch outperforms existing parameter-efficient language models on standard datasets such as WikiText103 and PennTreebank.
Theory · Learning Theory
We study the statistical complexity of offline decision-making with function approximation, establishing (near) minimax-optimal rates for stochastic contextual bandits and Markov decision processes. The performance limits are captured by the pseudo-dimension of the (value) function class and a new characterization of the behavior policy that *strictly* subsumes all the previous notions of data coverage in the offline decision-making literature. In addition, we seek to understand the benefits of using offline data in online decision-making and show nearly minimax-optimal rates in a wide range of regimes.
Deep Learning · Generative Models and Autoencoders
Score function estimation is the cornerstone of both training and sampling from diffusion generative models. Despite this fact, the most commonly used estimators are either biased neural network approximations or high variance Monte Carlo estimators based on the conditional score. We introduce a novel nearest neighbour score function estimator which utilizes multiple samples from the training set to dramatically decrease estimator variance. We leverage our low variance estimator in two compelling applications. Training consistency models with our estimator, we report a significant increase in both convergence speed and sample quality. In diffusion models, we show that our estimator can replace a learned network for probability-flow ODE integration, opening promising new avenues of future research. Code will be released upon paper acceptance.
Deep Learning · Generative Models and Autoencoders
Generative adversarial networks (GANs) usually struggle in learning from highly diverse data, whose underlying manifold is complex. In this work, we revisit the mathematical foundations of GANs, and theoretically reveal that the native adversarial loss for GAN training is insufficient to fix the problem of $\textit{subsets with positive Lebesgue measure of the generated data manifold lying out of the real data manifold}$. Instead, we find that score matching serves as a promising solution to this issue thanks to its capability of persistently pushing the generated data points towards the real data manifold. We thereby propose to improve the optimization of GANs with score matching regularity (SMaRt). Regarding the empirical evidences, we first design a toy example to show that training GANs by the aid of a ground-truth score function can help reproduce the real data distribution more accurately, and then confirm that our approach can consistently boost the synthesis performance of various state-of-the-art GANs on real-world datasets with pre-trained diffusion models acting as the approximate score function. For instance, when training Aurora on the ImageNet $64\times64$ dataset, we manage to improve FID from 8.87 to 7.11, on par with the performance of one-step consistency model. Code is available at https://github.com/thuxmf/SMaRt.
We initiate the study of the submodular cover problem in a dynamic setting where the elements of the ground set are inserted and deleted. In the classical submodular cover problem, we are given a monotone submodular function $f : 2^{V} \to \mathbb{R}^{\ge 0}$ and the goal is to obtain a set $S \subseteq V$ that minimizes the cost subject to the constraint $f(S) = f(V)$. This is a classical problem in computer science and generalizes the Set Cover problem, 2-Set Cover, and dominating set problem among others. We consider this problem in a dynamic setting where there are updates to our set $V$, in the form of insertions and deletions of elements from a ground set $\mathcal{V}$, and the goal is to maintain an approximately optimal solution with low query complexity per update. For this problem, we propose a randomized algorithm that, in expectation, obtains a $(1-O(\epsilon), O(\epsilon^{-1}))$-bicriteria approximation using polylogarithmic query complexity per update.
Target Networks and Over-parameterization Stabilize Off-policy Bootstrapping with Function Approximation
PDF ↗Reinforcement Learning · Batch/Offline
We prove that the combination of a target network and over-parameterized linear function approximation establishes a weaker convergence condition for bootstrapped value estimation in certain cases, even with off-policy data. Our condition is naturally satisfied for expected updates over the entire state-action space or learning with a batch of complete trajectories from episodic Markov decision processes. Notably, using only a target network or an over-parameterized model does not provide such a convergence guarantee. Additionally, we extend our results to learning with truncated trajectories, showing that convergence is achievable for all tasks with minor modifications, akin to value truncation for the final states in trajectories. Our primary result focuses on temporal difference estimation for prediction, providing high-probability value estimation error bounds and empirical analysis on Baird's counterexample and a Four-room task. Furthermore, we explore the control setting, demonstrating that similar convergence conditions apply to Q-learning.
Theory · Reinforcement Learning and Planning
In many real-world applications, it is hard to provide a reward signal in each step of a Reinforcement Learning (RL) process and more natural to give feedback when an episode ends. To this end, we study the recently proposed model of RL with Aggregate Bandit Feedback (RL-ABF), where the agent only observes the sum of rewards at the end of an episode instead of each reward individually. Prior work studied RL-ABF only in tabular settings, where the number of states is assumed to be small. In this paper, we extend ABF to linear function approximation and develop two efficient algorithms with near-optimal regret guarantees: a value-based optimistic algorithm built on a new randomization technique with a Q-functions ensemble, and a policy optimization algorithm that uses a novel hedging scheme over the ensemble.
Optimization · Optimization and Learning under Uncertainty
Hypervolume improvement (HVI) is commonly employed in multi-objective Bayesian optimization algorithms to define acquisition functions due to its Pareto-compliant property. Rather than focusing on specific statistical moments of HVI, this work aims to provide the exact expression of HVI's probability distribution for bi-objective problems. Considering a bi-variate Gaussian random variable resulting from Gaussian process (GP) modeling, we derive the probability distribution of its hypervolume improvement via a cell partition-based method. Our exact expression is superior in numerical accuracy and computation efficiency compared to the Monte Carlo approximation of HVI's distribution. Utilizing this distribution, we propose a novel acquisition function - $\varepsilon$-probability of hypervolume improvement ($\varepsilon$-PoHVI). Experimentally, we show that on many widely-applied bi-objective test problems, $\varepsilon$-PoHVI significantly outperforms other related acquisition functions, e.g., $\varepsilon$-PoI, and expected hypervolume improvement, when the GP model exhibits a large the prediction uncertainty.
Guarantees for Nonlinear Representation Learning: Non-identical Covariates, Dependent Data, Fewer Samples
PDF ↗Theory · Learning Theory
A driving force behind the diverse applicability of modern machine learning is the ability to extract meaningful features across many sources. However, many practical domains involve data that are non-identically distributed across sources, and possibly statistically dependent within its source, violating vital assumptions in existing theoretical studies of representation learning. Toward addressing these issues, we establish statistical guarantees for learning general *nonlinear* representations from multiple data sources that admit different input distributions and possibly dependent data. Specifically, we study the sample-complexity of learning $T+1$ functions $f_\star^{(t)} \circ g_\star$ from a function class $\mathcal{F} \times \mathcal{G}$, where $f_\star^{(t)}$ are task specific linear functions and $g_\star$ is a shared non-linear representation. An approximate representation $\hat g$ is estimated using $N$ samples from each of $T$ source tasks, and a fine-tuning function $\hat f^{(0)}$ is fit using $N'$ samples from a target task passed through $\hat g$. Our results show that the excess risk of the estimate $\hat f^{(0)} \circ \hat g$ on the target task decays as $\tilde{\mathcal{O}}\Big(\frac{\mathrm{C}(\mathcal{G})}{N T} + \frac{\text{dim}(\mathcal{F})}{N'}\Big)$, where $\mathrm{C}(\mathcal{G})$ denotes the complexity of $\mathcal{G}$. Notably, our rates match that of the iid setting, while requiring fewer samples per task than prior analysis and admitting *no dependence on the mixing time*. We support our analysis with numerical experiments performing imitation learning over non-linear dynamical systems.
Applications · Physics
Data-driven weather forecasting models are advancing rapidly, yet they rely on initial states (i.e., analysis states) typically produced by traditional data assimilation algorithms. Four-dimensional variational assimilation (4DVar) is one of the most widely adopted data assimilation algorithms in numerical weather prediction centers; it is accurate but computationally expensive. In this paper, we aim to couple the AI forecasting model, FengWu, with 4DVar to build a self-contained data-driven global weather forecasting framework, FengWu-4DVar. To achieve this, we propose an *AI-embedded* 4DVar algorithm that includes three components: (1) a 4DVar objective function embedded with the FengWu forecasting model and its error representation to enhance efficiency and accuracy; (2) a spherical-harmonic-transform-based (SHT-based) approximation strategy for capturing the horizontal correlation of background error; and (3) an auto-differentiation (AD) scheme for determining the optimal analysis fields. Experimental results show that under the ERA5 simulated observational data with varying proportions and noise levels, FengWu-4DVar can generate accurate analysis fields; remarkably, it has achieved stable self-contained global weather forecasts for an entire year for the first time, demonstrating its potential for real-world applications. Additionally, our framework is approximately 100 times faster than the traditional 4DVar algorithm under similar experimental conditions, highlighting its significant computational efficiency.
Reinforcement Learning · Function Approximation
Value-based reinforcement learning is the current State-Of-The-Art due to high sampling efficiency. However, our study shows it suffers from low exploitation in early training period and bias sensitiveness. To address these issues, we propose to augment the decision-making process with hypothesis, a weak form of environment description. Our approach relies on prompting the learning agent with accurate hypotheses, and designing a ready-to-adapt policy through incremental learning. We propose the ALH algorithm, showing detailed analyses on a typical learning scheme and a diverse set of Mujoco benchmarks. Our algorithm produces a significant improvement over value-based learning algorithms and other strong baselines. Our code is available at [Github URL](https://github.com/nbtpj/ALH).
Reinforcement Learning · Function Approximation
The robust $\phi$-regularized Markov Decision Process (RRMDP) framework focuses on designing control policies that are robust against parameter uncertainties due to mismatches between the simulator (nominal) model and real-world settings. This work makes *two* important contributions. First, we propose a *model-free* algorithm called *Robust $\phi$-regularized fitted Q-iteration* for learning an $\epsilon$-optimal robust policy that uses only the historical data collected by rolling out a behavior policy (with *robust exploratory* requirement) on the nominal model. To the best of our knowledge, we provide the *first* unified analysis for a class of $\phi$-divergences achieving robust optimal policies in high-dimensional systems of arbitrary large state space with general function approximation. Second, we introduce the *hybrid robust $\phi$-regularized reinforcement learning* framework to learn an optimal robust policy using both historical data and online sampling. Towards this framework, we propose a model-free algorithm called *Hybrid robust Total-variation-regularized Q-iteration*. To the best of our knowledge, we provide the *first* improved out-of-data-distribution assumption in large-scale problems of arbitrary large state space with general function approximation under the hybrid robust $\phi$-regularized reinforcement learning framework.
Reinforcement Learning · Function Approximation
This work proposes a solution for the problem of training physics-informed networks under partial integro-differential equations. These equations require an infinite or a large number of neural evaluations to construct a single residual for training. As a result, accurate evaluation may be impractical, and we show that naive approximations at replacing these integrals with unbiased estimates lead to biased loss functions and solutions. To overcome this bias, we investigate three types of potential solutions: the deterministic sampling approaches, the double-sampling trick, and the delayed target method. We consider three classes of PDEs for benchmarking; one defining Poisson problems with singular charges and weak solutions of up to 10 dimensions, another involving weak solutions on electro-magnetic fields and a Maxwell equation, and a third one defining a Smoluchowski coagulation problem. Our numerical results confirm the existence of the aforementioned bias in practice and also show that our proposed delayed target approach can lead to accurate solutions with comparable quality to ones estimated with a large sample size integral. Our implementation is open-source and available at https://github.com/ehsansaleh/btspinn.