SmallToLarge (S2L): Scalable Data Selection for Fine-tuning Large Language Models by Summarizing Training Trajectories of Small Models

Despite the effectiveness of data selection for large language models (LLMs) during pretraining and instruction fine-tuning phases, improving data efficiency in supervised fine-tuning (SFT) for specialized domains poses significant challenges due to the complexity of fine-tuning data. To bridge this gap, we introduce an effective and scalable data selection method for SFT, SmallToLarge (S2L), which leverages training trajectories from small models to guide the data selection for larger models. We demonstrate through extensive experiments that S2L significantly improves data efficiency in SFT for mathematical problem-solving, reducing the training data to just 11% of the original MathInstruct dataset (Yue et al., 2023) to match full dataset performance while outperforming state-of-the-art data selection algorithms by an average of 4.7% across 6 in- and out-domain evaluation datasets. Remarkably, selecting only 50K data for SFT, S2L achieves a 32.7% accuracy on the most challenging MATH (Hendrycks et al., 2021) benchmark, improving Phi-2 (Li et al., 2023b) by 16.6%. In clinical text summarization on the MIMIC-III dataset (Johnson et al., 2016), S2L again outperforms training on the full dataset using only 50% of the data. Notably, S2L can perform data selection using a reference model 40x smaller than the target model, proportionally reducing the cost of data selection.

A universal approximation theorem for nonlinear resistive networks

Resistor networks have recently had a surge of interest as substrates for energy-efficient self-learning machines. This work studies the computational capabilities of these resistor networks. We show that electrical networks composed of voltage sources, linear resistors, diodes and voltage-controlled voltage sources (VCVS) can implement any continuous functions. To prove it, we assume that the circuit elements are ideal and that the conductances of variable resistors and the amplification factors of the VCVS's can take arbitrary values -- arbitrarily small or arbitrarily large. The constructive nature of our proof could also inform the design of such self-learning electrical networks.

An operator preconditioning perspective on training in physics-informed machine learning

In this paper, we investigate the behavior of gradient descent algorithms in physics-informed machine learning methods like PINNs, which minimize residuals connected to partial differential equations (PDEs). Our key result is that the difficulty in training these models is closely related to the conditioning of a specific differential operator. This operator, in turn, is associated to the Hermitian square of the differential operator of the underlying PDE. If this operator is ill-conditioned, it results in slow or infeasible training. Therefore, preconditioning this operator is crucial. We employ both rigorous mathematical analysis and empirical evaluations to investigate various strategies, explaining how they better condition this critical operator, and consequently improve training.

How does over-squashing affect the power of GNNs?

Graph Neural Networks (GNNs) are the state-of-the-art model for machine learning on graph-structured data. The most popular class of GNNs operate by exchanging information between adjacent nodes, and are known as Message Passing Neural Networks (MPNNs). Given their widespread use, understanding the expressive power of MPNNs is a key question. However, existing results typically consider settings with uninformative node features. In this paper, we provide a rigorous analysis to determine which function classes of node features can be learned by an MPNN of a given capacity. We do so by measuring the level of pairwise interactions between nodes that MPNNs allow for. This measure provides a novel quantitative characterization of the so-called over-squashing effect, which is observed to occur when a large volume of messages is aggregated into fixed-size vectors. Using our measure, we prove that, to guarantee sufficient communication between pairs of nodes, the capacity of the MPNN must be large enough, depending on properties of the input graph structure, such as commute times. For many relevant scenarios, our analysis results in impossibility statements in practice, showing that over-squashing hinders the expressive power of MPNNs. We validate our theoretical findings through extensive controlled experiments and ablation studies.

Are Neural Operators Really Neural Operators? Frame Theory Meets Operator Learning

Recently, there has been significant interest in operator learning, i.e. learning mappings between infinite-dimensional function spaces. This has been particularly relevant in the context of learning partial differential equations from data. However, it has been observed that proposed models may not behave as operators when implemented on a computer, questioning the very essence of what operator learning should be. We contend that in addition to defining the operator at the continuous level, some form of continuous-discrete equivalence is necessary for an architecture to genuinely learn the underlying operator, rather than just discretizations of it. To this end, we propose to employ frames, a concept in applied harmonic analysis and signal processing that gives rise to exact and stable discrete representations of continuous signals. Extending these concepts to operators, we introduce a unifying mathematical framework of Representation equivalent Neural Operator (ReNO) to ensure operations at the continuous and discrete level are equivalent. Lack of this equivalence is quantified in terms of aliasing errors. We analyze various existing operator learning architectures to determine whether they fall within this framework, and highlight implications when they fail to do so.

Neural Oscillators are Universal

Coupled oscillators are being increasingly used as the basis of machine learning (ML) architectures, for instance in sequence modeling, graph representation learning and in physical neural networks that are used in analog ML devices. We introduce an abstract class of neural oscillators that encompasses these architectures and prove that neural oscillators are universal, i.e, they can approximate any continuous and casual operator mapping between time-varying functions, to desired accuracy. This universality result provides theoretical justification for the use of oscillator based ML systems. The proof builds on a fundamental result of independent interest, which shows that a combination of forced harmonic oscillators with a nonlinear read-out suffices to approximate the underlying operators.

A Survey on Oversmoothing in Graph Neural Networks

Node features of graph neural networks (GNNs) tend to become more similar with the increase of the network depth. This effect is known as over-smoothing, which we axiomatically define as the exponential convergence of suitable similarity measures on the node features. Our definition unifies previous approaches and gives rise to new quantitative measures of over-smoothing. Moreover, we empirically demonstrate this behavior for several over-smoothing measures on different graphs (small-, medium-, and large-scale). We also review several approaches for mitigating over-smoothing and empirically test their effectiveness on real-world graph datasets. Through illustrative examples, we demonstrate that mitigating over-smoothing is a necessary but not sufficient condition for building deep GNNs that are expressive on a wide range of graph learning tasks. Finally, we extend our definition of over-smoothing to the rapidly emerging field of continuous-time GNNs.

Multi-Scale Message Passing Neural PDE Solvers

We propose a novel multi-scale message passing neural network algorithm for learning the solutions of time-dependent PDEs. Our algorithm possesses both temporal and spatial multi-scale resolution features by incorporating multi-scale sequence models and graph gating modules in the encoder and processor, respectively. Benchmark numerical experiments are presented to demonstrate that the proposed algorithm outperforms baselines, particularly on a PDE with a range of spatial and temporal scales.

Convolutional Neural Operators

Although very successfully used in machine learning, convolution based neural network architectures -- believed to be inconsistent in function space -- have been largely ignored in the context of learning solution operators of PDEs. Here, we adapt convolutional neural networks to demonstrate that they are indeed able to process functions as inputs and outputs. The resulting architecture, termed as convolutional neural operators (CNOs), is shown to significantly outperform competing models on benchmark experiments, paving the way for the design of an alternative robust and accurate framework for learning operators.

Neural Inverse Operators for Solving PDE Inverse Problems

A large class of inverse problems for PDEs are only well-defined as mappings from operators to functions. Existing operator learning frameworks map functions to functions and need to be modified to learn inverse maps from data. We propose a novel architecture termed Neural Inverse Operators (NIOs) to solve these PDE inverse problems. Motivated by the underlying mathematical structure, NIO is based on a suitable composition of DeepONets and FNOs to approximate mappings from operators to functions. A variety of experiments are presented to demonstrate that NIOs significantly outperform baselines and solve PDE inverse problems robustly, accurately and are several orders of magnitude faster than existing direct and PDE-constrained optimization methods.