John Lafferty

Qi Lin, Zifan Li, John Lafferty et al.

Much of what we remember is not due to intentional selection, but simply a by-product of perceiving. This raises a foundational question about the architecture of the mind: How does perception interface with and influence memory? Here, inspired by a classic proposal relating perceptual processing to memory durability, the level-of-processing theory, we present a sparse coding model for compressing feature embeddings of images, and show that the reconstruction residuals from this model predict how well images are encoded into memory. In an open memorability dataset of scene images, we show that reconstruction error not only explains memory accuracy but also response latencies during retrieval, subsuming, in the latter case, all of the variance explained by powerful vision-only models. We also confirm a prediction of this account with 'model-driven psychophysics'. This work establishes reconstruction error as a novel signal interfacing perception and memory, possibly through adaptive modulation of perceptual processing.

Leon Lufkin, Ashish Puri, Ganlin Song et al.

Local patterns of excitation and inhibition that can generate neural waves are studied as a computational mechanism underlying the organization of neuronal tunings. Sparse coding algorithms based on networks of excitatory and inhibitory neurons are proposed that exhibit topographic maps as the receptive fields are adapted to input stimuli. Motivated by a leaky integrate-and-fire model of neural waves, we propose an activation model that is more typical of artificial neural networks. Computational experiments with the activation model using both natural images and natural language text are presented. In the case of images, familiar "pinwheel" patterns of oriented edge detectors emerge; in the case of text, the resulting topographic maps exhibit a 2-dimensional representation of granular word semantics. Experiments with a synthetic model of somatosensory input are used to investigate how the network dynamics may affect plasticity of neuronal maps under changes to the inputs.

Awni Altabaa, Taylor Webb, Jonathan Cohen et al.

An extension of Transformers is proposed that enables explicit relational reasoning through a novel module called the Abstractor. At the core of the Abstractor is a variant of attention called relational cross-attention. The approach is motivated by an architectural inductive bias for relational learning that disentangles relational information from object-level features. This enables explicit relational reasoning, supporting abstraction and generalization from limited data. The Abstractor is first evaluated on simple discriminative relational tasks and compared to existing relational architectures. Next, the Abstractor is evaluated on purely relational sequence-to-sequence tasks, where dramatic improvements are seen in sample efficiency compared to standard Transformers. Finally, Abstractors are evaluated on a collection of tasks based on mathematical problem solving, where consistent improvements in performance and sample efficiency are observed.

Taylor W. Webb, Steven M. Frankland, Awni Altabaa et al.

A central challenge for cognitive science is to explain how abstract concepts are acquired from limited experience. This has often been framed in terms of a dichotomy between connectionist and symbolic cognitive models. Here, we highlight a recently emerging line of work that suggests a novel reconciliation of these approaches, by exploiting an inductive bias that we term the relational bottleneck. In that approach, neural networks are constrained via their architecture to focus on relations between perceptual inputs, rather than the attributes of individual inputs. We review a family of models that employ this approach to induce abstractions in a data-efficient manner, emphasizing their potential as candidate models for the acquisition of abstract concepts in the human mind and brain.

Awni Altabaa, John Lafferty

An evolving area of research in deep learning is the study of architectures and inductive biases that support the learning of relational feature representations. In this paper, we address the challenge of learning representations of hierarchical relations--that is, higher-order relational patterns among groups of objects. We introduce "relational convolutional networks", a neural architecture equipped with computational mechanisms that capture progressively more complex relational features through the composition of simple modules. A key component of this framework is a novel operation that captures relational patterns in groups of objects by convolving graphlet filters--learnable templates of relational patterns--against subsets of the input. Composing relational convolutions gives rise to a deep architecture that learns representations of higher-order, hierarchical relations. We present the motivation and details of the architecture, together with a set of experiments to demonstrate how relational convolutional networks can provide an effective framework for modeling relational tasks that have hierarchical structure.

Awni Altabaa, John Lafferty

Inner products of neural network feature maps arise in a wide variety of machine learning frameworks as a method of modeling relations between inputs. This work studies the approximation properties of inner products of neural networks. It is shown that the inner product of a multi-layer perceptron with itself is a universal approximator for symmetric positive-definite relation functions. In the case of asymmetric relation functions, it is shown that the inner product of two different multi-layer perceptrons is a universal approximator. In both cases, a bound is obtained on the number of neurons required to achieve a given accuracy of approximation. In the symmetric case, the function class can be identified with kernels of reproducing kernel Hilbert spaces, whereas in the asymmetric case the function class can be identified with kernels of reproducing kernel Banach spaces. Finally, these approximation results are applied to analyzing the attention mechanism underlying Transformers, showing that any retrieval mechanism defined by an abstract preorder can be approximated by attention through its inner product relations. This result uses the Debreu representation theorem in economics to represent preference relations in terms of utility functions.

Awni Altabaa, Siyu Chen, John Lafferty et al.

Systematic, compositional generalization beyond the training distribution remains a core challenge in machine learning -- and a critical bottleneck for the emergent reasoning abilities of modern language models. This work investigates out-of-distribution (OOD) generalization in Transformer networks using a GSM8K-style modular arithmetic on computational graphs task as a testbed. We introduce and explore a set of four architectural mechanisms aimed at enhancing OOD generalization: (i) input-adaptive recurrence; (ii) algorithmic supervision; (iii) anchored latent representations via a discrete bottleneck; and (iv) an explicit error-correction mechanism. Collectively, these mechanisms yield an architectural approach for native and scalable latent space reasoning in Transformer networks with robust algorithmic generalization capabilities. We complement these empirical results with a detailed mechanistic interpretability analysis that reveals how these mechanisms give rise to robust OOD generalization abilities.

Awni Altabaa, Omar Montasser, John Lafferty

Learning complex functions that involve multi-step reasoning poses a significant challenge for standard supervised learning from input-output examples. Chain-of-thought (CoT) supervision, which provides intermediate reasoning steps together with the final output, has emerged as a powerful empirical technique, underpinning much of the recent progress in the reasoning capabilities of large language models. This paper develops a statistical theory of learning under CoT supervision. A key characteristic of the CoT setting, in contrast to standard supervision, is the mismatch between the training objective (CoT risk) and the test objective (end-to-end risk). A central part of our analysis, distinguished from prior work, is explicitly linking those two types of risk to achieve sharper sample complexity bounds. This is achieved via the *CoT information measure* $\mathcal{I}_{\mathcal{D}, h_\star}^{\mathrm{CoT}}(ε; \calH)$, which quantifies the additional discriminative power gained from observing the reasoning process. The main theoretical results demonstrate how CoT supervision can yield significantly faster learning rates compared to standard E2E supervision. Specifically, it is shown that the sample complexity required to achieve a target E2E error $ε$ scales as $d/\mathcal{I}_{\mathcal{D}, h_\star}^{\mathrm{CoT}}(ε; \calH)$, where $d$ is a measure of hypothesis class complexity, which can be much faster than standard $d/ε$ rates. Information-theoretic lower bounds in terms of the CoT information are also obtained. Together, these results suggest that CoT information is a fundamental measure of statistical complexity for learning under chain-of-thought supervision.

Ganlin Song, Ruitu Xu, John Lafferty

Stochastic gradient descent with backpropagation is the workhorse of artificial neural networks. It has long been recognized that backpropagation fails to be a biologically plausible algorithm. Fundamentally, it is a non-local procedure -- updating one neuron's synaptic weights requires knowledge of synaptic weights or receptive fields of downstream neurons. This limits the use of artificial neural networks as a tool for understanding the biological principles of information processing in the brain. Lillicrap et al. (2016) propose a more biologically plausible "feedback alignment" algorithm that uses random and fixed backpropagation weights, and show promising simulations. In this paper we study the mathematical properties of the feedback alignment procedure by analyzing convergence and alignment for two-layer networks under squared error loss. In the overparameterized setting, we prove that the error converges to zero exponentially fast, and also that regularization is necessary in order for the parameters to become aligned with the random backpropagation weights. Simulations are given that are consistent with this analysis and suggest further generalizations. These results contribute to our understanding of how biologically plausible algorithms might carry out weight learning in a manner different from Hebbian learning, with performance that is comparable with the full non-local backpropagation algorithm.

Tuo Zhao, Han Liu, Kathryn Roeder et al.

We describe an R package named huge which provides easy-to-use functions for estimating high dimensional undirected graphs from data. This package implements recent results in the literature, including Friedman et al. (2007), Liu et al. (2009, 2012) and Liu et al. (2010). Compared with the existing graph estimation package glasso, the huge package provides extra features: (1) instead of using Fortan, it is written in C, which makes the code more portable and easier to modify; (2) besides fitting Gaussian graphical models, it also provides functions for fitting high dimensional semiparametric Gaussian copula models; (3) more functions like data-dependent model selection, data generation and graph visualization; (4) a minor convergence problem of the graphical lasso algorithm is corrected; (5) the package allows the user to apply both lossless and lossy screening rules to scale up large-scale problems, making a tradeoff between computational and statistical efficiency.

Chao Gao, John Lafferty

A new type of robust estimation problem is introduced where the goal is to recover a statistical model that has been corrupted after it has been estimated from data. Methods are proposed for "repairing" the model using only the design and not the response values used to fit the model in a supervised learning setting. Theory is developed which reveals that two important ingredients are necessary for model repair---the statistical model must be over-parameterized, and the estimator must incorporate redundancy. In particular, estimators based on stochastic gradient descent are seen to be well suited to model repair, but sparse estimators are not in general repairable. After formulating the problem and establishing a key technical lemma related to robust estimation, a series of results are presented for repair of over-parameterized linear models, random feature models, and artificial neural networks. Simulation studies are presented that corroborate and illustrate the theoretical findings.

Ganlin Song, Zhou Fan, John Lafferty

We investigate a sequential optimization procedure to minimize the empirical risk functional $f_{\hatθ}(x) = \frac{1}{2}\|G_{\hatθ}(x) - y\|^2$ for certain families of deep networks $G_θ(x)$. The approach is to optimize a sequence of objective functions that use network parameters obtained during different stages of the training process. When initialized with random parameters $θ_0$, we show that the objective $f_{θ_0}(x)$ is "nice'' and easy to optimize with gradient descent. As learning is carried out, we obtain a sequence of generative networks $x \mapsto G_{θ_t}(x)$ and associated risk functions $f_{θ_t}(x)$, where $t$ indicates a stage of stochastic gradient descent during training. Since the parameters of the network do not change by very much in each step, the surface evolves slowly and can be incrementally optimized. The algorithm is formalized and analyzed for a family of expansive networks. We call the procedure {\it surfing} since it rides along the peak of the evolving (negative) empirical risk function, starting from a smooth surface at the beginning of learning and ending with a wavy nonconvex surface after learning is complete. Experiments show how surfing can be used to find the global optimum and for compressed sensing even when direct gradient descent on the final learned network fails.

Dana Yang, John Lafferty, David Pollard

Quantile regression is a tool for learning conditional distributions. In this paper we study quantile regression in the setting where a protected attribute is unavailable when fitting the model. This can lead to "unfair'' quantile estimators for which the effective quantiles are very different for the subpopulations defined by the protected attribute. We propose a procedure for adjusting the estimator on a heldout sample where the protected attribute is available. The main result of the paper is an empirical process analysis showing that the adjustment leads to a fair estimator for which the target quantiles are brought into balance, in a statistical sense that we call $\sqrt{n}$-fairness. We illustrate the ideas and adjustment procedure on a dataset of 200,000 live births, where the objective is to characterize the dependence of the birth weights of the babies on demographic attributes of the birth mother; the protected attribute is the mother's race.

Michihiro Yasunaga, John Lafferty

Scientific documents rely on both mathematics and text to communicate ideas. Inspired by the topical correspondence between mathematical equations and word contexts observed in scientific texts, we propose a novel topic model that jointly generates mathematical equations and their surrounding text (TopicEq). Using an extension of the correlated topic model, the context is generated from a mixture of latent topics, and the equation is generated by an RNN that depends on the latent topic activations. To experiment with this model, we create a corpus of 400K equation-context pairs extracted from a range of scientific articles from arXiv, and fit the model using a variational autoencoder approach. Experimental results show that this joint model significantly outperforms existing topic models and equation models for scientific texts. Moreover, we qualitatively show that the model effectively captures the relationship between topics and mathematics, enabling novel applications such as topic-aware equation generation, equation topic inference, and topic-aware alignment of mathematical symbols and words.

Matt Bonakdarpour, Sabyasachi Chatterjee, Rina Foygel Barber et al.

Two methods are proposed for high-dimensional shape-constrained regression and classification. These methods reshape pre-trained prediction rules to satisfy shape constraints like monotonicity and convexity. The first method can be applied to any pre-trained prediction rule, while the second method deals specifically with random forests. In both cases, efficient algorithms are developed for computing the estimators, and experiments are performed to demonstrate their performance on four datasets. We find that reshaping methods enforce shape constraints without compromising predictive accuracy.

Yuancheng Zhu, John Lafferty

This paper studies the problem of nonparametric estimation of a smooth function with data distributed across multiple machines. We assume an independent sample from a white noise model is collected at each machine, and an estimator of the underlying true function needs to be constructed at a central machine. We place limits on the number of bits that each machine can use to transmit information to the central machine. Our results give both asymptotic lower bounds and matching upper bounds on the statistical risk under various settings. We identify three regimes, depending on the relationship among the number of machines, the size of the data available at each machine, and the communication budget. When the communication budget is small, the statistical risk depends solely on this communication bottleneck, regardless of the sample size. In the regime where the communication budget is large, the classic minimax risk in the non-distributed estimation setting is recovered. In an intermediate regime, the statistical risk depends on both the sample size and the communication budget.

Yuancheng Zhu, Sabyasachi Chatterjee, John Duchi et al.

We extend the traditional worst-case, minimax analysis of stochastic convex optimization by introducing a localized form of minimax complexity for individual functions. Our main result gives function-specific lower and upper bounds on the number of stochastic subgradient evaluations needed to optimize either the function or its "hardest local alternative" to a given numerical precision. The bounds are expressed in terms of a localized and computational analogue of the modulus of continuity that is central to statistical minimax analysis. We show how the computational modulus of continuity can be explicitly calculated in concrete cases, and relates to the curvature of the function at the optimum. We also prove a superefficiency result that demonstrates it is a meaningful benchmark, acting as a computational analogue of the Fisher information in statistical estimation. The nature and practical implications of the results are demonstrated in simulations.

Qinqing Zheng, John Lafferty

We address the rectangular matrix completion problem by lifting the unknown matrix to a positive semidefinite matrix in higher dimension, and optimizing a nonconvex objective over the semidefinite factor using a simple gradient descent scheme. With $O( μr^2 κ^2 n \max(μ, \log n))$ random observations of a $n_1 \times n_2$ $μ$-incoherent matrix of rank $r$ and condition number $κ$, where $n = \max(n_1, n_2)$, the algorithm linearly converges to the global optimum with high probability.

Qinqing Zheng, John Lafferty

We propose a simple, scalable, and fast gradient descent algorithm to optimize a nonconvex objective for the rank minimization problem and a closely related family of semidefinite programs. With $O(r^3 κ^2 n \log n)$ random measurements of a positive semidefinite $n \times n$ matrix of rank $r$ and condition number $κ$, our method is guaranteed to converge linearly to the global optimum.

Yuancheng Zhu, John Lafferty

We formulate the notion of minimax estimation under storage or communication constraints, and prove an extension to Pinsker's theorem for nonparametric estimation over Sobolev ellipsoids. Placing limits on the number of bits used to encode any estimator, we give tight lower and upper bounds on the excess risk due to quantization in terms of the number of bits, the signal size, and the noise level. This establishes the Pareto optimal tradeoff between storage and risk under quantization constraints for Sobolev spaces. Our results and proof techniques combine elements of rate distortion theory and minimax analysis. The proposed quantized estimation scheme, which shows achievability of the lower bounds, is adaptive in the usual statistical sense, achieving the optimal quantized minimax rate without knowledge of the smoothness parameter of the Sobolev space. It is also adaptive in a computational sense, as it constructs the code only after observing the data, to dynamically allocate more codewords to blocks where the estimated signal size is large. Simulations are included that illustrate the effect of quantization on statistical risk.

Min Xu, Minhua Chen, John Lafferty

We study the problem of variable selection in convex nonparametric regression. Under the assumption that the true regression function is convex and sparse, we develop a screening procedure to select a subset of variables that contains the relevant variables. Our approach is a two-stage quadratic programming method that estimates a sum of one-dimensional convex functions, followed by one-dimensional concave regression fits on the residuals. In contrast to previous methods for sparse additive models, the optimization is finite dimensional and requires no tuning parameters for smoothness. Under appropriate assumptions, we prove that the procedure is faithful in the population setting, yielding no false negatives. We give a finite sample statistical analysis, and introduce algorithms for efficiently carrying out the required quadratic programs. The approach leads to computational and statistical advantages over fitting a full model, and provides an effective, practical approach to variable screening in convex regression.

Yuancheng Zhu, John Lafferty

A central result in statistical theory is Pinsker's theorem, which characterizes the minimax rate in the normal means model of nonparametric estimation. In this paper, we present an extension to Pinsker's theorem where estimation is carried out under storage or communication constraints. In particular, we place limits on the number of bits used to encode an estimator, and analyze the excess risk in terms of this constraint, the signal size, and the noise level. We give sharp upper and lower bounds for the case of a Euclidean ball, which establishes the Pareto-optimal minimax tradeoff between storage and risk in this setting.

John Lafferty, Larry A. Wasserman

We present an iterative Markov chainMonte Carlo algorithm for computingreference priors and minimax risk forgeneral parametric families. Ourapproach uses MCMC techniques based onthe Blahut-Arimoto algorithm forcomputing channel capacity ininformation theory. We give astatistical analysis of the algorithm,bounding the number of samples requiredfor the stochastic algorithm to closelyapproximate the deterministic algorithmin each iteration. Simulations arepresented for several examples fromexponential families. Although we focuson applications to reference priors andminimax risk, the methods and analysiswe develop are applicable to a muchbroader class of optimization problemsand iterative algorithms.

Rina Foygel, Michael Horrell, Mathias Drton et al.

We propose an approach to multivariate nonparametric regression that generalizes reduced rank regression for linear models. An additive model is estimated for each dimension of a $q$-dimensional response, with a shared $p$-dimensional predictor variable. To control the complexity of the model, we employ a functional form of the Ky-Fan or nuclear norm, resulting in a set of function estimates that have low rank. Backfitting algorithms are derived and justified using a nonparametric form of the nuclear norm subdifferential. Oracle inequalities on excess risk are derived that exhibit the scaling behavior of the procedure in the high dimensional setting. The methods are illustrated on gene expression data.

Pradeep Ravikumar, John Lafferty

Recent research has made significant progress on the problem of bounding log partition functions for exponential family graphical models. Such bounds have associated dual parameters that are often used as heuristic estimates of the marginal probabilities required in inference and learning. However these variational estimates do not give rigorous bounds on marginal probabilities, nor do they give estimates for probabilities of more general events than simple marginals. In this paper we build on this recent work by deriving rigorous upper and lower bounds on event probabilities for graphical models. Our approach is based on the use of generalized Chernoff bounds to express bounds on event probabilities in terms of convex optimization problems; these optimization problems, in turn, require estimates of generalized log partition functions. Simulations indicate that this technique can result in useful, rigorous bounds to complement the heuristic variational estimates, with comparable computational cost.

Han Liu, Fang Han, Ming Yuan et al.

We propose a semiparametric approach, named nonparanormal skeptic, for estimating high dimensional undirected graphical models. In terms of modeling, we consider the nonparanormal family proposed by Liu et al (2009). In terms of estimation, we exploit nonparametric rank-based correlation coefficient estimators including the Spearman's rho and Kendall's tau. In high dimensional settings, we prove that the nonparanormal skeptic achieves the optimal parametric rate of convergence in both graph and parameter estimation. This result suggests that the nonparanormal graphical models are a safe replacement of the Gaussian graphical models, even when the data are Gaussian.

Haijie Gu, John Lafferty

We present algorithms for nonparametric regression in settings where the data are obtained sequentially. While traditional estimators select bandwidths that depend upon the sample size, for sequential data the effective sample size is dynamically changing. We propose a linear time algorithm that adjusts the bandwidth for each new data point, and show that the estimator achieves the optimal minimax rate of convergence. We also propose the use of online expert mixing algorithms to adapt to unknown smoothness of the regression function. We provide simulations that confirm the theoretical results, and demonstrate the effectiveness of the methods.

Min Xu, John Lafferty

We study the problem of multivariate regression where the data are naturally grouped, and a regression matrix is to be estimated for each group. We propose an approach in which a dictionary of low rank parameter matrices is estimated across groups, and a sparse linear combination of the dictionary elements is estimated to form a model within each group. We refer to the method as conditional sparse coding since it is a coding procedure for the response vectors Y conditioned on the covariate vectors X. This approach captures the shared information across the groups while adapting to the structure within each group. It exploits the same intuition behind sparse coding that has been successfully developed in computer vision and computational neuroscience. We propose an algorithm for conditional sparse coding, analyze its theoretical properties in terms of predictive accuracy, and present the results of simulation and brain imaging experiments that compare the new technique to reduced rank regression.

Sivaraman Balakrishnan, Kriti Puniyani, John Lafferty

Canonical Correlation Analysis (CCA) is a classical tool for finding correlations among the components of two random vectors. In recent years, CCA has been widely applied to the analysis of genomic data, where it is common for researchers to perform multiple assays on a single set of patient samples. Recent work has proposed sparse variants of CCA to address the high dimensionality of such data. However, classical and sparse CCA are based on linear models, and are thus limited in their ability to find general correlations. In this paper, we present two approaches to high-dimensional nonparametric CCA, building on recent developments in high-dimensional nonparametric regression. We present estimation procedures for both approaches, and analyze their theoretical properties in the high-dimensional setting. We demonstrate the effectiveness of these procedures in discovering nonlinear correlations via extensive simulations, as well as through experiments with genomic data.

Han Liu, Fang Han, Ming Yuan et al.

In this paper, we propose a semiparametric approach, named nonparanormal skeptic, for efficiently and robustly estimating high dimensional undirected graphical models. To achieve modeling flexibility, we consider Gaussian Copula graphical models (or the nonparanormal) as proposed by Liu et al. (2009). To achieve estimation robustness, we exploit nonparametric rank-based correlation coefficient estimators, including Spearman's rho and Kendall's tau. In high dimensional settings, we prove that the nonparanormal skeptic achieves the optimal parametric rate of convergence in both graph and parameter estimation. This celebrating result suggests that the Gaussian copula graphical models can be used as a safe replacement of the popular Gaussian graphical models, even when the data are truly Gaussian. Besides theoretical analysis, we also conduct thorough numerical simulations to compare different estimators for their graph recovery performance under both ideal and noisy settings. The proposed methods are then applied on a large-scale genomic dataset to illustrate their empirical usefulness. The R language software package huge implementing the proposed methods is available on the Comprehensive R Archive Network: http://cran. r-project.org/.

John Lafferty, Han Liu, Larry Wasserman

We present some nonparametric methods for graphical modeling. In the discrete case, where the data are binary or drawn from a finite alphabet, Markov random fields are already essentially nonparametric, since the cliques can take only a finite number of values. Continuous data are different. The Gaussian graphical model is the standard parametric model for continuous data, but it makes distributional assumptions that are often unrealistic. We discuss two approaches to building more flexible graphical models. One allows arbitrary graphs and a nonparametric extension of the Gaussian; the other uses kernel density estimation and restricts the graphs to trees and forests. Examples of both methods are presented. We also discuss possible future research directions for nonparametric graphical modeling.