Caroline Uhler

Wengong Jin, Siranush Sarkizova, Xun Chen et al.

Protein-ligand binding prediction is a fundamental problem in AI-driven drug discovery. Prior work focused on supervised learning methods using a large set of binding affinity data for small molecules, but it is hard to apply the same strategy to other drug classes like antibodies as labelled data is limited. In this paper, we explore unsupervised approaches and reformulate binding energy prediction as a generative modeling task. Specifically, we train an energy-based model on a set of unlabelled protein-ligand complexes using SE(3) denoising score matching and interpret its log-likelihood as binding affinity. Our key contribution is a new equivariant rotation prediction network called Neural Euler's Rotation Equations (NERE) for SE(3) score matching. It predicts a rotation by modeling the force and torque between protein and ligand atoms, where the force is defined as the gradient of an energy function with respect to atom coordinates. We evaluate NERE on protein-ligand and antibody-antigen binding affinity prediction benchmarks. Our model outperforms all unsupervised baselines (physics-based and statistical potentials) and matches supervised learning methods in the antibody case.

Jiaqi Zhang, Chandler Squires, Kristjan Greenewald et al.

Causal disentanglement aims to uncover a representation of data using latent variables that are interrelated through a causal model. Such a representation is identifiable if the latent model that explains the data is unique. In this paper, we focus on the scenario where unpaired observational and interventional data are available, with each intervention changing the mechanism of a latent variable. When the causal variables are fully observed, statistically consistent algorithms have been developed to identify the causal model under faithfulness assumptions. We here show that identifiability can still be achieved with unobserved causal variables, given a generalized notion of faithfulness. Our results guarantee that we can recover the latent causal model up to an equivalence class and predict the effect of unseen combinations of interventions, in the limit of infinite data. We implement our causal disentanglement framework by developing an autoencoding variational Bayes algorithm and apply it to the problem of predicting combinatorial perturbation effects in genomics.

Chandler Squires, Anna Seigal, Salil Bhate et al.

Causal disentanglement seeks a representation of data involving latent variables that relate to one another via a causal model. A representation is identifiable if both the latent model and the transformation from latent to observed variables are unique. In this paper, we study observed variables that are a linear transformation of a linear latent causal model. Data from interventions are necessary for identifiability: if one latent variable is missing an intervention, we show that there exist distinct models that cannot be distinguished. Conversely, we show that a single intervention on each latent variable is sufficient for identifiability. Our proof uses a generalization of the RQ decomposition of a matrix that replaces the usual orthogonal and upper triangular conditions with analogues depending on a partial order on the rows of the matrix, with partial order determined by a latent causal model. We corroborate our theoretical results with a method for causal disentanglement that accurately recovers a latent causal model.

Chandler Squires, Caroline Uhler

In this review, we discuss approaches for learning causal structure from data, also called causal discovery. In particular, we focus on approaches for learning directed acyclic graphs (DAGs) and various generalizations which allow for some variables to be unobserved in the available data. We devote special attention to two fundamental combinatorial aspects of causal structure learning. First, we discuss the structure of the search space over causal graphs. Second, we discuss the structure of equivalence classes over causal graphs, i.e., sets of graphs which represent what can be learned from observational data alone, and how these equivalence classes can be refined by adding interventional data.

Jiaqi Zhang, Louis Cammarata, Chandler Squires et al.

Sequential experimental design to discover interventions that achieve a desired outcome is a key problem in various domains including science, engineering and public policy. When the space of possible interventions is large, making an exhaustive search infeasible, experimental design strategies are needed. In this context, encoding the causal relationships between the variables, and thus the effect of interventions on the system, is critical for identifying desirable interventions more efficiently. Here, we develop a causal active learning strategy to identify interventions that are optimal, as measured by the discrepancy between the post-interventional mean of the distribution and a desired target mean. The approach employs a Bayesian update for the causal model and prioritizes interventions using a carefully designed, causally informed acquisition function. This acquisition function is evaluated in closed form, allowing for fast optimization. The resulting algorithms are theoretically grounded with information-theoretic bounds and provable consistency results for linear causal models with known causal graph. We apply our approach to both synthetic data and single-cell transcriptomic data from Perturb-CITE-seq experiments to identify optimal perturbations that induce a specific cell state transition. The causally informed acquisition function generally outperforms existing criteria allowing for optimal intervention design with fewer but carefully selected samples.

Adityanarayanan Radhakrishnan, Mikhail Belkin, Caroline Uhler

While neural networks are used for classification tasks across domains, a long-standing open problem in machine learning is determining whether neural networks trained using standard procedures are optimal for classification, i.e., whether such models minimize the probability of misclassification for arbitrary data distributions. In this work, we identify and construct an explicit set of neural network classifiers that achieve optimality. Since effective neural networks in practice are typically both wide and deep, we analyze infinitely wide networks that are also infinitely deep. In particular, using the recent connection between infinitely wide neural networks and Neural Tangent Kernels, we provide explicit activation functions that can be used to construct networks that achieve optimality. Interestingly, these activation functions are simple and easy to implement, yet differ from commonly used activations such as ReLU or sigmoid. More generally, we create a taxonomy of infinitely wide and deep networks and show that these models implement one of three well-known classifiers depending on the activation function used: (1) 1-nearest neighbor (model predictions are given by the label of the nearest training example); (2) majority vote (model predictions are given by the label of the class with greatest representation in the training set); or (3) singular kernel classifiers (a set of classifiers containing those that achieve optimality). Our results highlight the benefit of using deep networks for classification tasks, in contrast to regression tasks, where excessive depth is harmful.

Nils Sturma, Chandler Squires, Mathias Drton et al.

The goal of causal representation learning is to find a representation of data that consists of causally related latent variables. We consider a setup where one has access to data from multiple domains that potentially share a causal representation. Crucially, observations in different domains are assumed to be unpaired, that is, we only observe the marginal distribution in each domain but not their joint distribution. In this paper, we give sufficient conditions for identifiability of the joint distribution and the shared causal graph in a linear setup. Identifiability holds if we can uniquely recover the joint distribution and the shared causal representation from the marginal distributions in each domain. We transform our identifiability results into a practical method to recover the shared latent causal graph.

Adityanarayanan Radhakrishnan, Max Ruiz Luyten, Neha Prasad et al.

Transfer learning refers to the process of adapting a model trained on a source task to a target task. While kernel methods are conceptually and computationally simple machine learning models that are competitive on a variety of tasks, it has been unclear how to perform transfer learning for kernel methods. In this work, we propose a transfer learning framework for kernel methods by projecting and translating the source model to the target task. We demonstrate the effectiveness of our framework in applications to image classification and virtual drug screening. In particular, we show that transferring modern kernels trained on large-scale image datasets can result in substantial performance increase as compared to using the same kernel trained directly on the target task. In addition, we show that transfer-learned kernels allow a more accurate prediction of the effect of drugs on cancer cell lines. For both applications, we identify simple scaling laws that characterize the performance of transfer-learned kernels as a function of the number of target examples. We explain this phenomenon in a simplified linear setting, where we are able to derive the exact scaling laws. By providing a simple and effective transfer learning framework for kernel methods, our work enables kernel methods trained on large datasets to be easily adapted to a variety of downstream target tasks.

Renzo G. Soatto, Anders Hoel, Greycen Ren et al.

Diffusion models have excelled at generative tasks for both continuous and token-based domains, but their application to discrete ordinal data remains underdeveloped. We present CountsDiff, a diffusion framework designed to natively model distributions on the natural numbers. CountsDiff extends the Blackout diffusion framework by simplifying its formulation through a direct parameterization in terms of a survival probability schedule and an explicit loss weighting. This introduces flexibility through design parameters with direct analogues in existing diffusion modeling frameworks. Beyond this reparameterization, CountsDiff introduces features from modern diffusion models, previously absent in counts-based domains, including continuous-time training, classifier-free guidance, and churn/remasking reverse dynamics that allow non-monotone reverse trajectories. We propose an initial instantiation of CountsDiff and validate it on natural image datasets (CIFAR-10, CelebA), exploring the effects of varying the introduced design parameters in a complex, well-studied, and interpretable data domain. We then highlight biological count assays as a natural use case, evaluating CountsDiff on single-cell RNA-seq imputation in a fetal cell and heart cell atlas. Remarkably, we find that even this simple instantiation matches or surpasses the performance of a state-of-the-art discrete generative model and leading RNA-seq imputation methods, while leaving substantial headroom for further gains through optimized design choices in future work.

Kirankumar Shiragur, Jiaqi Zhang, Caroline Uhler

Learning causal structures from interventional data is a fundamental problem with broad applications across various fields. While many previous works have focused on recovering the entire causal graph, in practice, there are scenarios where learning only part of the causal graph suffices. This is called $targeted$ causal discovery. In our work, we focus on two such well-motivated problems: subset search and causal matching. We aim to minimize the number of interventions in both cases. Towards this, we introduce the $Meek~separator$, which is a subset of vertices that, when intervened, decomposes the remaining unoriented edges into smaller connected components. We then present an efficient algorithm to find Meek separators that are of small sizes. Such a procedure is helpful in designing various divide-and-conquer-based approaches. In particular, we propose two randomized algorithms that achieve logarithmic approximation for subset search and causal matching, respectively. Our results provide the first known average-case provable guarantees for both problems. We believe that this opens up possibilities to design near-optimal methods for many other targeted causal structure learning problems arising from various applications.

Chenyu Wang, Sharut Gupta, Xinyi Zhang et al.

Multimodal representation learning seeks to relate and decompose information inherent in multiple modalities. By disentangling modality-specific information from information that is shared across modalities, we can improve interpretability and robustness and enable downstream tasks such as the generation of counterfactual outcomes. Separating the two types of information is challenging since they are often deeply entangled in many real-world applications. We propose Disentangled Self-Supervised Learning (DisentangledSSL), a novel self-supervised approach for learning disentangled representations. We present a comprehensive analysis of the optimality of each disentangled representation, particularly focusing on the scenario not covered in prior work where the so-called Minimum Necessary Information (MNI) point is not attainable. We demonstrate that DisentangledSSL successfully learns shared and modality-specific features on multiple synthetic and real-world datasets and consistently outperforms baselines on various downstream tasks, including prediction tasks for vision-language data, as well as molecule-phenotype retrieval tasks for biological data. The code is available at https://github.com/uhlerlab/DisentangledSSL.

Marc Franquesa Monés, Jiaqi Zhang, Caroline Uhler

Learning causal relations from observational data is a fundamental problem with wide-ranging applications across many fields. Constraint-based methods infer the underlying causal structure by performing conditional independence tests. However, existing algorithms such as the prominent PC algorithm need to perform a large number of independence tests, which in the worst case is exponential in the maximum degree of the causal graph. Despite extensive research, it remains unclear if there exist algorithms with better complexity without additional assumptions. Here, we establish an algorithm that achieves a better complexity of $p^{\mathcal{O}(s)}$ tests, where $p$ is the number of nodes in the graph and $s$ denotes the maximum undirected clique size of the underlying essential graph. Complementing this result, we prove that any constraint-based algorithm must perform at least $2^{Ω(s)}$ conditional independence tests, establishing that our proposed algorithm achieves exponent-optimality up to a logarithmic factor in terms of the number of conditional independence tests needed. Finally, we validate our theoretical findings through simulations, on semi-synthetic gene-expression data, and real-world data, demonstrating the efficiency of our algorithm compared to existing methods in terms of number of conditional independence tests needed.

Caroline Uhler, Jiaqi Zhang

Massive data collection holds the promise of a better understanding of complex phenomena and, ultimately, better decisions. Representation learning has become a key driver of deep learning applications, as it allows learning latent spaces that capture important properties of the data without requiring any supervised annotations. Although representation learning has been hugely successful in predictive tasks, it can fail miserably in causal tasks including predicting the effect of a perturbation/intervention. This calls for a marriage between representation learning and causal inference. An exciting opportunity in this regard stems from the growing availability of multi-modal data (observational and perturbational, imaging-based and sequencing-based, at the single-cell level, tissue-level, and organism-level). We outline a statistical and computational framework for causal structure and representation learning motivated by fundamental biomedical questions: how to effectively use observational and perturbational data to perform causal discovery on observed causal variables; how to use multi-modal views of the system to learn causal variables; and how to design optimal perturbations.

Thomas Gaudelet, Alice Del Vecchio, Eli M Carrami et al.

Interventions play a pivotal role in the study of complex biological systems. In drug discovery, genetic interventions (such as CRISPR base editing) have become central to both identifying potential therapeutic targets and understanding a drug's mechanism of action. With the advancement of CRISPR and the proliferation of genome-scale analyses such as transcriptomics, a new challenge is to navigate the vast combinatorial space of concurrent genetic interventions. Addressing this, our work concentrates on estimating the effects of pairwise genetic combinations on the cellular transcriptome. We introduce two novel contributions: Salt, a biologically-inspired baseline that posits the mostly additive nature of combination effects, and Peper, a deep learning model that extends Salt's additive assumption to achieve unprecedented accuracy. Our comprehensive comparison against existing state-of-the-art methods, grounded in diverse metrics, and our out-of-distribution analysis highlight the limitations of current models in realistic settings. This analysis underscores the necessity for improved modelling techniques and data acquisition strategies, paving the way for more effective exploration of genetic intervention effects.

Ryan Welch, Jiaqi Zhang, Caroline Uhler

Causal disentanglement aims to learn about latent causal factors behind data, holding the promise to augment existing representation learning methods in terms of interpretability and extrapolation. Recent advances establish identifiability results assuming that interventions on (single) latent factors are available; however, it remains debatable whether such assumptions are reasonable due to the inherent nature of intervening on latent variables. Accordingly, we reconsider the fundamentals and ask what can be learned using just observational data. We provide a precise characterization of latent factors that can be identified in nonlinear causal models with additive Gaussian noise and linear mixing, without any interventions or graphical restrictions. In particular, we show that the causal variables can be identified up to a layer-wise transformation and that further disentanglement is not possible. We transform these theoretical results into a practical algorithm consisting of solving a quadratic program over the score estimation of the observed data. We provide simulation results to support our theoretical guarantees and demonstrate that our algorithm can derive meaningful causal representations from purely observational data.

Divya Nori, Anisha Parsan, Caroline Uhler et al.

Protein binder design has been transformed by hallucination-based methods that optimize structure prediction confidence metrics, such as the interface predicted TM-score (ipTM), via backpropagation. However, these metrics do not reflect the statistical likelihood of a binder-target complex under the learned distribution and yield sparse gradients for optimization. In this work, we propose a method to extract such likelihoods from structure predictors by reinterpreting their confidence outputs as an energy-based model (EBM). By leveraging the Joint Energy-based Modeling (JEM) framework, we introduce pTMEnergy, a statistical energy function derived from predicted inter-residue error distributions. We incorporate pTMEnergy into BindEnergyCraft (BECraft), a design pipeline that maintains the same optimization framework as BindCraft but replaces ipTM with our energy-based objective. BECraft outperforms BindCraft, RFDiffusion, and ESM3 across multiple challenging targets, achieving higher in silico binder success rates while reducing structural clashes. Furthermore, pTMEnergy establishes a new state-of-the-art in structure-based virtual screening tasks for miniprotein and RNA aptamer binders.

Luka Kovačević, Thomas Gaudelet, James Opzoomer et al.

Despite substantial efforts, deep learning has not yet delivered a transformative impact on elucidating regulatory biology, particularly in the realm of predicting gene expression profiles. Here, we argue that genuine "foundation models" of regulatory biology will remain out of reach unless guided by frameworks that integrate mechanistic insight with principled experimental design. We present one such ground-up, semi-mechanistic framework that unifies perturbation-based experimental designs across both in vitro and in vivo CRISPR screens, accounting for differentiating and non-differentiating cellular systems. By revealing previously unrecognised assumptions in published machine learning methods, our approach clarifies links with popular techniques such as variational autoencoders and structural causal models. In practice, this framework suggests a modified loss function that we demonstrate can improve predictive performance, and further suggests an error analysis that informs batching strategies. Ultimately, since cellular regulation emerges from innumerable interactions amongst largely uncharted molecular components, we contend that systems-level understanding cannot be achieved through structural biology alone. Instead, we argue that real progress will require a first-principles perspective on how experiments capture biological phenomena, how data are generated, and how these processes can be reflected in more faithful modelling architectures.

Alvaro Ribot, Chandler Squires, Caroline Uhler

We consider the task of causal imputation, where we aim to predict the outcomes of some set of actions across a wide range of possible contexts. As a running example, we consider predicting how different drugs affect cells from different cell types. We study the index-only setting, where the actions and contexts are categorical variables with a finite number of possible values. Even in this simple setting, a practical challenge arises, since often only a small subset of possible action-context pairs have been studied. Thus, models must extrapolate to novel action-context pairs, which can be framed as a form of matrix completion with rows indexed by actions, columns indexed by contexts, and matrix entries corresponding to outcomes. We introduce a novel SCM-based model class, where the outcome is expressed as a counterfactual, actions are expressed as interventions on an instrumental variable, and contexts are defined based on the initial state of the system. We show that, under a linearity assumption, this setup induces a latent factor model over the matrix of outcomes, with an additional fixed effect term. To perform causal prediction based on this model class, we introduce simple extension to the Synthetic Interventions estimator (Agarwal et al., 2020). We evaluate several matrix completion approaches on the PRISM drug repurposing dataset, showing that our method outperforms all other considered matrix completion approaches.

Davin Choo, Kirankumar Shiragur, Caroline Uhler

Causal graph discovery is a significant problem with applications across various disciplines. However, with observational data alone, the underlying causal graph can only be recovered up to its Markov equivalence class, and further assumptions or interventions are necessary to narrow down the true graph. This work addresses the causal discovery problem under the setting of stochastic interventions with the natural goal of minimizing the number of interventions performed. We propose the following stochastic intervention model which subsumes existing adaptive noiseless interventions in the literature while capturing scenarios such as fat-hand interventions and CRISPR gene knockouts: any intervention attempt results in an actual intervention on a random subset of vertices, drawn from a distribution dependent on attempted action. Under this model, we study the two fundamental problems in causal discovery of verification and search and provide approximation algorithms with polylogarithmic competitive ratios and provide some preliminary experimental results.

Bijan Mazaheri, Jiaqi Zhang, Caroline Uhler

Constraint-based causal discovery algorithms utilize many statistical tests for conditional independence to uncover networks of causal dependencies. These approaches to causal discovery rely on an assumed correspondence between the graphical properties of a causal structure and the conditional independence properties of observed variables, known as the causal Markov condition and faithfulness. Finite data yields an empirical distribution that is "close" to the actual distribution. Across these many possible empirical distributions, the correspondence to the graphical properties can break down for different conditional independencies, and multiple violations can occur at the same time. We study this "meta-dependence" between conditional independence properties using the following geometric intuition: each conditional independence property constrains the space of possible joint distributions to a manifold. The "meta-dependence" between conditional independences is informed by the position of these manifolds relative to the true probability distribution. We provide a simple-to-compute measure of this meta-dependence using information projections and consolidate our findings empirically using both synthetic and real-world data.

Divya Shyamal, Jiaqi Zhang, Caroline Uhler

A combinatorial intervention, consisting of multiple treatments applied to a single unit with potentially interactive effects, has substantial applications in fields such as biomedicine, engineering, and beyond. Given $p$ possible treatments, conducting all possible $2^p$ combinatorial interventions can be laborious and quickly becomes infeasible as $p$ increases. Here we introduce probabilistic factorial experimental design, formalized from how scientists perform lab experiments. In this framework, the experimenter selects a dosage for each possible treatment and applies it to a group of units. Each unit independently receives a random combination of treatments, sampled from a product Bernoulli distribution determined by the dosages. Additionally, the experimenter can carry out such experiments over multiple rounds, adapting the design in an active manner. We address the optimal experimental design problem within an intervention model that imposes bounded-degree interactions between treatments. In the passive setting, we provide a closed-form solution for the near-optimal design. Our results prove that a dosage of $\tfrac{1}{2}$ for each treatment is optimal up to a factor of $1+O(\tfrac{\ln(n)}{n})$ for estimating any $k$-way interaction model, regardless of $k$, and imply that $O\big(kp^{3k}\ln(p)\big)$ observations are required to accurately estimate this model. For the multi-round setting, we provide a near-optimal acquisition function that can be numerically optimized. We also explore several extensions of the design problem and finally validate our findings through simulations.

Abhinav Kumar, Kirankumar Shiragur, Caroline Uhler

The ability to conduct interventions plays a pivotal role in learning causal relationships among variables, thus facilitating applications across diverse scientific disciplines such as genomics, economics, and machine learning. However, in many instances within these applications, the process of generating interventional data is subject to noise: rather than data being sampled directly from the intended interventional distribution, interventions often yield data sampled from a blend of both intended and unintended interventional distributions. We consider the fundamental challenge of disentangling mixed interventional and observational data within linear Structural Equation Models (SEMs) with Gaussian additive noise without the knowledge of the true causal graph. We demonstrate that conducting interventions, whether do or soft, yields distributions with sufficient diversity and properties conducive to efficiently recovering each component within the mixture. Furthermore, we establish that the sample complexity required to disentangle mixed data inversely correlates with the extent of change induced by an intervention in the equations governing the affected variable values. As a result, the causal graph can be identified up to its interventional Markov Equivalence Class, similar to scenarios where no noise influences the generation of interventional data. We further support our theoretical findings by conducting simulations wherein we perform causal discovery from such mixed data.

Kirankumar Shiragur, Jiaqi Zhang, Caroline Uhler

Many questions in science center around the fundamental problem of understanding causal relationships. However, most constraint-based causal discovery algorithms, including the well-celebrated PC algorithm, often incur an exponential number of conditional independence (CI) tests, posing limitations in various applications. Addressing this, our work focuses on characterizing what can be learned about the underlying causal graph with a reduced number of CI tests. We show that it is possible to a learn a coarser representation of the hidden causal graph with a polynomial number of tests. This coarser representation, named Causal Consistent Partition Graph (CCPG), comprises of a partition of the vertices and a directed graph defined over its components. CCPG satisfies consistency of orientations and additional constraints which favor finer partitions. Furthermore, it reduces to the underlying causal graph when the causal graph is identifiable. As a consequence, our results offer the first efficient algorithm for recovering the true causal graph with a polynomial number of tests, in special cases where the causal graph is fully identifiable through observational data and potentially additional interventions.

Jiaqi Zhang, Kirankumar Shiragur, Caroline Uhler

Understanding causal relationships between variables is a fundamental problem with broad impact in numerous scientific fields. While extensive research has been dedicated to learning causal graphs from data, its complementary concept of testing causal relationships has remained largely unexplored. While learning involves the task of recovering the Markov equivalence class (MEC) of the underlying causal graph from observational data, the testing counterpart addresses the following critical question: Given a specific MEC and observational data from some causal graph, can we determine if the data-generating causal graph belongs to the given MEC? We explore constraint-based testing methods by establishing bounds on the required number of conditional independence tests. Our bounds are in terms of the size of the maximum undirected clique ($s$) of the given MEC. In the worst case, we show a lower bound of $\exp(Ω(s))$ independence tests. We then give an algorithm that resolves the task with $\exp(O(s))$ tests, matching our lower bound. Compared to the learning problem, where algorithms often use a number of independence tests that is exponential in the maximum in-degree, this shows that testing is relatively easier. In particular, it requires exponentially less independence tests in graphs featuring high in-degrees and small clique sizes. Additionally, using the DAG associahedron, we provide a geometric interpretation of testing versus learning and discuss how our testing result can aid learning.

Adityanarayanan Radhakrishnan, Mikhail Belkin, Caroline Uhler

Establishing a fast rate of convergence for optimization methods is crucial to their applicability in practice. With the increasing popularity of deep learning over the past decade, stochastic gradient descent and its adaptive variants (e.g. Adagrad, Adam, etc.) have become prominent methods of choice for machine learning practitioners. While a large number of works have demonstrated that these first order optimization methods can achieve sub-linear or linear convergence, we establish local quadratic convergence for stochastic gradient descent with adaptive step size for problems such as matrix inversion.

Adityanarayanan Radhakrishnan, George Stefanakis, Mikhail Belkin et al.

Matrix completion problems arise in many applications including recommendation systems, computer vision, and genomics. Increasingly larger neural networks have been successful in many of these applications, but at considerable computational costs. Remarkably, taking the width of a neural network to infinity allows for improved computational performance. In this work, we develop an infinite width neural network framework for matrix completion that is simple, fast, and flexible. Simplicity and speed come from the connection between the infinite width limit of neural networks and kernels known as neural tangent kernels (NTK). In particular, we derive the NTK for fully connected and convolutional neural networks for matrix completion. The flexibility stems from a feature prior, which allows encoding relationships between coordinates of the target matrix, akin to semi-supervised learning. The effectiveness of our framework is demonstrated through competitive results for virtual drug screening and image inpainting/reconstruction. We also provide an implementation in Python to make our framework accessible on standard hardware to a broad audience.

Jiaqi Zhang, Chandler Squires, Caroline Uhler

Transforming a causal system from a given initial state to a desired target state is an important task permeating multiple fields including control theory, biology, and materials science. In causal models, such transformations can be achieved by performing a set of interventions. In this paper, we consider the problem of identifying a shift intervention that matches the desired mean of a system through active learning. We define the Markov equivalence class that is identifiable from shift interventions and propose two active learning strategies that are guaranteed to exactly match a desired mean. We then derive a worst-case lower bound for the number of interventions required and show that these strategies are optimal for certain classes of graphs. In particular, we show that our strategies may require exponentially fewer interventions than the previously considered approaches, which optimize for structure learning in the underlying causal graph. In line with our theoretical results, we also demonstrate experimentally that our proposed active learning strategies require fewer interventions compared to several baselines.

Saachi Jain, Adityanarayanan Radhakrishnan, Caroline Uhler

Aligned latent spaces, where meaningful semantic shifts in the input space correspond to a translation in the embedding space, play an important role in the success of downstream tasks such as unsupervised clustering and data imputation. In this work, we prove that linear and nonlinear autoencoders produce aligned latent spaces by stretching along the left singular vectors of the data. We fully characterize the amount of stretching in linear autoencoders and provide an initialization scheme to arbitrarily stretch along the top directions using these networks. We also quantify the amount of stretching in nonlinear autoencoders in a simplified setting. We use our theoretical results to align drug signatures across cell types in gene expression space and semantic shifts in word embedding spaces.

Scott Sussex, Andreas Krause, Caroline Uhler

Causal structure learning is a key problem in many domains. Causal structures can be learnt by performing experiments on the system of interest. We address the largely unexplored problem of designing a batch of experiments that each simultaneously intervene on multiple variables. While potentially more informative than the commonly considered single-variable interventions, selecting such interventions is algorithmically much more challenging, due to the doubly-exponential combinatorial search space over sets of composite interventions. In this paper, we develop efficient algorithms for optimizing different objective functions quantifying the informativeness of a budget-constrained batch of experiments. By establishing novel submodularity properties of these objectives, we provide approximation guarantees for our algorithms. Our algorithms empirically perform superior to both random interventions and algorithms that only select single-variable interventions.

Chandler Squires, Joshua Amaniampong, Caroline Uhler

The problem of learning a directed acyclic graph (DAG) up to Markov equivalence is equivalent to the problem of finding a permutation of the variables that induces the sparsest graph. Without additional assumptions, this task is known to be NP-hard. Building on the minimum degree algorithm for sparse Cholesky decomposition, but utilizing DAG-specific problem structure, we introduce an efficient algorithm for finding such sparse permutations. We show that on jointly Gaussian distributions, our method with depth $w$ runs in $O(p^{w+3})$ time. We compare our method with $w = 1$ to algorithms for finding sparse elimination orderings of undirected graphs, and show that taking advantage of DAG-specific problem structure leads to a significant improvement in the discovered permutation. We also compare our algorithm to provably consistent causal structure learning algorithms, such as the PC algorithm, GES, and GSP, and show that our method achieves comparable performance with a shorter runtime. Thus, our method can be used on its own for causal structure discovery. Finally, we show that there exist dense graphs on which our method achieves almost perfect performance, so that unlike most existing causal structure learning algorithms, the situations in which our algorithm achieves both good performance and good runtime are not limited to sparse graphs.

Eshaan Nichani, Adityanarayanan Radhakrishnan, Caroline Uhler

Recent works have demonstrated that increasing model capacity through width in over-parameterized neural networks leads to a decrease in test risk. For neural networks, however, model capacity can also be increased through depth, yet understanding the impact of increasing depth on test risk remains an open question. In this work, we demonstrate that the test risk of over-parameterized convolutional networks is a U-shaped curve (i.e. monotonically decreasing, then increasing) with increasing depth. We first provide empirical evidence for this phenomenon via image classification experiments using both ResNets and the convolutional neural tangent kernel (CNTK). We then present a novel linear regression framework for characterizing the impact of depth on test risk, and show that increasing depth leads to a U-shaped test risk for the linear CNTK. In particular, we prove that the linear CNTK corresponds to a depth-dependent linear transformation on the original space and characterize properties of this transformation. We then analyze over-parameterized linear regression under arbitrary linear transformations and, in simplified settings, provably identify the depths which minimize each of the bias and variance terms of the test risk.

Madeline Navarro, Yuhao Wang, Antonio G. Marques et al.

Inferring graph structure from observations on the nodes is an important and popular network science task. Departing from the more common inference of a single graph and motivated by social and biological networks, we study the problem of jointly inferring multiple graphs from the observation of signals at their nodes (graph signals), which are assumed to be stationary in the sought graphs. From a mathematical point of view, graph stationarity implies that the mapping between the covariance of the signals and the sparse matrix representing the underlying graph is given by a matrix polynomial. A prominent example is that of Markov random fields, where the inverse of the covariance yields the sparse matrix of interest. From a modeling perspective, stationary graph signals can be used to model linear network processes evolving on a set of (not necessarily known) networks. Leveraging that matrix polynomials commute, a convex optimization method along with sufficient conditions that guarantee the recovery of the true graphs are provided when perfect covariance information is available. Particularly important from an empirical viewpoint, we provide high-probability bounds on the recovery error as a function of the number of signals observed and other key problem parameters. Numerical experiments using synthetic and real-world data demonstrate the effectiveness of the proposed method with perfect covariance information as well as its robustness in the noisy regime.

Adityanarayanan Radhakrishnan, Mikhail Belkin, Caroline Uhler

The Polyak-Lojasiewicz (PL) inequality is a sufficient condition for establishing linear convergence of gradient descent, even in non-convex settings. While several recent works use a PL-based analysis to establish linear convergence of stochastic gradient descent methods, the question remains as to whether a similar analysis can be conducted for more general optimization methods. In this work, we present a PL-based analysis for linear convergence of generalized mirror descent (GMD), a generalization of mirror descent with a possibly time-dependent mirror. GMD subsumes popular first order optimization methods including gradient descent, mirror descent, and preconditioned gradient descent methods such as Adagrad. Since the standard PL analysis cannot be extended naturally from GMD to stochastic GMD, we present a Taylor-series based analysis to establish sufficient conditions for linear convergence of stochastic GMD. As a corollary, our result establishes sufficient conditions and provides learning rates for linear convergence of stochastic mirror descent and Adagrad. Lastly, for functions that are locally PL*, our analysis implies existence of an interpolating solution and convergence of GMD to this solution.

Neha Prasad, Karren Yang, Caroline Uhler

In this paper, we present Super-OT, a novel approach to computational lineage tracing that combines a supervised learning framework with optimal transport based on Generative Adversarial Networks (GANs). Unlike previous approaches to lineage tracing, Super-OT has the flexibility to integrate paired data. We benchmark Super-OT based on single-cell RNA-seq data against Waddington-OT, a popular approach for lineage tracing that also employs optimal transport. We show that Super-OT achieves gains over Waddington-OT in predicting the class outcome of cells during differentiation, since it allows the integration of additional information during training.

Pantelis R. Vlachas, Georgios Arampatzis, Caroline Uhler et al.

Predictive simulations of complex systems are essential for applications ranging from weather forecasting to drug design. The veracity of these predictions hinges on their capacity to capture the effective system dynamics. Massively parallel simulations predict the system dynamics by resolving all spatiotemporal scales, often at a cost that prevents experimentation while their findings may not allow for generalisation. On the other hand reduced order models are fast but limited by the frequently adopted linearization of the system dynamics and/or the utilization of heuristic closures. Here we present a novel systematic framework that bridges large scale simulations and reduced order models to Learn the Effective Dynamics (LED) of diverse complex systems. The framework forms algorithmic alloys between non-linear machine learning algorithms and the Equation-Free approach for modeling complex systems. LED deploys autoencoders to formulate a mapping between fine and coarse-grained representations and evolves the latent space dynamics using recurrent neural networks. The algorithm is validated on benchmark problems and we find that it outperforms state of the art reduced order models in terms of predictability and large scale simulations in terms of cost. LED is applicable to systems ranging from chemistry to fluid mechanics and reduces the computational effort by up to two orders of magnitude while maintaining the prediction accuracy of the full system dynamics. We argue that LED provides a novel potent modality for the accurate prediction of complex systems.

Karren Yang, Samuel Goldman, Wengong Jin et al.

In this paper, we aim to synthesize cell microscopy images under different molecular interventions, motivated by practical applications to drug development. Building on the recent success of graph neural networks for learning molecular embeddings and flow-based models for image generation, we propose Mol2Image: a flow-based generative model for molecule to cell image synthesis. To generate cell features at different resolutions and scale to high-resolution images, we develop a novel multi-scale flow architecture based on a Haar wavelet image pyramid. To maximize the mutual information between the generated images and the molecular interventions, we devise a training strategy based on contrastive learning. To evaluate our model, we propose a new set of metrics for biological image generation that are robust, interpretable, and relevant to practitioners. We show quantitatively that our method learns a meaningful embedding of the molecular intervention, which is translated into an image representation reflecting the biological effects of the intervention.

Adityanarayanan Radhakrishnan, Eshaan Nichani, Daniel Bernstein et al.

We study the properties of alignment, a form of implicit regularization, in linear neural networks under gradient descent. We define alignment for fully connected networks with multidimensional outputs and show that it is a natural extension of alignment in networks with 1-dimensional outputs as defined by Ji and Telgarsky, 2018. While in fully connected networks, there always exists a global minimum corresponding to an aligned solution, we analyze alignment as it relates to the training process. Namely, we characterize when alignment is an invariant of training under gradient descent by providing necessary and sufficient conditions for this invariant to hold. In such settings, the dynamics of gradient descent simplify, thereby allowing us to provide an explicit learning rate under which the network converges linearly to a global minimum. We then analyze networks with layer constraints such as convolutional networks. In this setting, we prove that gradient descent is equivalent to projected gradient descent, and that alignment is impossible with sufficiently large datasets.

Basil Saeed, Snigdha Panigrahi, Caroline Uhler

We consider distributions arising from a mixture of causal models, where each model is represented by a directed acyclic graph (DAG). We provide a graphical representation of such mixture distributions and prove that this representation encodes the conditional independence relations of the mixture distribution. We then consider the problem of structure learning based on samples from such distributions. Since the mixing variable is latent, we consider causal structure discovery algorithms such as FCI that can deal with latent variables. We show that such algorithms recover a "union" of the component DAGs and can identify variables whose conditional distribution across the component DAGs vary. We demonstrate our results on synthetic and real data showing that the inferred graph identifies nodes that vary between the different mixture components. As an immediate application, we demonstrate how retrieval of this causal information can be used to cluster samples according to each mixture component.

Daniel Irving Bernstein, Basil Saeed, Chandler Squires et al.

We consider the task of learning a causal graph in the presence of latent confounders given i.i.d.~samples from the model. While current algorithms for causal structure discovery in the presence of latent confounders are constraint-based, we here propose a score-based approach. We prove that under assumptions weaker than faithfulness, any sparsest independence map (IMAP) of the distribution belongs to the Markov equivalence class of the true model. This motivates the \emph{Sparsest Poset} formulation - that posets can be mapped to minimal IMAPs of the true model such that the sparsest of these IMAPs is Markov equivalent to the true model. Motivated by this result, we propose a greedy algorithm over the space of posets for causal structure discovery in the presence of latent confounders and compare its performance to the current state-of-the-art algorithms FCI and FCI+ on synthetic data.

Adityanarayanan Radhakrishnan, Mikhail Belkin, Caroline Uhler

Identifying computational mechanisms for memorization and retrieval of data is a long-standing problem at the intersection of machine learning and neuroscience. Our main finding is that standard overparameterized deep neural networks trained using standard optimization methods implement such a mechanism for real-valued data. Empirically, we show that: (1) overparameterized autoencoders store training samples as attractors, and thus, iterating the learned map leads to sample recovery; (2) the same mechanism allows for encoding sequences of examples, and serves as an even more efficient mechanism for memory than autoencoding. Theoretically, we prove that when trained on a single example, autoencoders store the example as an attractor. Lastly, by treating a sequence encoder as a composition of maps, we prove that sequence encoding provides a more efficient mechanism for memory than autoencoding.

Dmitriy Katz, Karthikeyan Shanmugam, Chandler Squires et al.

Directed acyclic graph (DAG) models are popular for capturing causal relationships. From observational and interventional data, a DAG model can only be determined up to its \emph{interventional Markov equivalence class} (I-MEC). We investigate the size of MECs for random DAG models generated by uniformly sampling and ordering an Erdős-Rényi graph. For constant density, we show that the expected $\log$ observational MEC size asymptotically (in the number of vertices) approaches a constant. We characterize I-MEC size in a similar fashion in the above settings with high precision. We show that the asymptotic expected number of interventions required to fully identify a DAG is a constant. These results are obtained by exploiting Meek rules and coupling arguments to provide sharp upper and lower bounds on the asymptotic quantities, which are then calculated numerically up to high precision. Our results have important consequences for experimental design of interventions and the development of algorithms for causal inference.

Karren D. Yang, Caroline Uhler

Multi-domain translation seeks to learn a probabilistic coupling between marginal distributions that reflects the correspondence between different domains. We assume that data from different domains are generated from a shared latent representation based on a structural equation model. Under this assumption, we show that the problem of computing a probabilistic coupling between marginals is equivalent to learning multiple uncoupled autoencoders that embed to a given shared latent distribution. In addition, we propose a new framework and algorithm for multi-domain translation based on learning the shared latent distribution and training autoencoders under distributional constraints. A key practical advantage of our framework is that new autoencoders (i.e., new domains) can be added sequentially to the model without retraining on the other domains, which we demonstrate experimentally on image as well as genomics datasets.

Karren D. Yang, Caroline Uhler

Generative adversarial networks (GANs) are an expressive class of neural generative models with tremendous success in modeling high-dimensional continuous measures. In this paper, we present a scalable method for unbalanced optimal transport (OT) based on the generative-adversarial framework. We formulate unbalanced OT as a problem of simultaneously learning a transport map and a scaling factor that push a source measure to a target measure in a cost-optimal manner. In addition, we propose an algorithm for solving this problem based on stochastic alternating gradient updates, similar in practice to GANs. We also provide theoretical justification for this formulation, showing that it is closely related to an existing static formulation by Liero et al. (2018), and perform numerical experiments demonstrating how this methodology can be applied to population modeling.

Adityanarayanan Radhakrishnan, Karren Yang, Mikhail Belkin et al.

The ability of deep neural networks to generalize well in the overparameterized regime has become a subject of significant research interest. We show that overparameterized autoencoders exhibit memorization, a form of inductive bias that constrains the functions learned through the optimization process to concentrate around the training examples, although the network could in principle represent a much larger function class. In particular, we prove that single-layer fully-connected autoencoders project data onto the (nonlinear) span of the training examples. In addition, we show that deep fully-connected autoencoders learn a map that is locally contractive at the training examples, and hence iterating the autoencoder results in convergence to the training examples. Finally, we prove that depth is necessary and provide empirical evidence that it is also sufficient for memorization in convolutional autoencoders. Understanding this inductive bias may shed light on the generalization properties of overparametrized deep neural networks that are currently unexplained by classical statistical theory.

Raj Agrawal, Tamara Broderick, Caroline Uhler

Learning a Bayesian network (BN) from data can be useful for decision-making or discovering causal relationships. However, traditional methods often fail in modern applications, which exhibit a larger number of observed variables than data points. The resulting uncertainty about the underlying network as well as the desire to incorporate prior information recommend a Bayesian approach to learning the BN, but the highly combinatorial structure of BNs poses a striking challenge for inference. The current state-of-the-art methods such as order MCMC are faster than previous methods but prevent the use of many natural structural priors and still have running time exponential in the maximum indegree of the true directed acyclic graph (DAG) of the BN. We here propose an alternative posterior approximation based on the observation that, if we incorporate empirical conditional independence tests, we can focus on a high-probability DAG associated with each order of the vertices. We show that our method allows the desired flexibility in prior specification, removes timing dependence on the maximum indegree and yields provably good posterior approximations; in addition, we show that it achieves superior accuracy, scalability, and sampler mixing on several datasets.

Adityanarayanan Radhakrishnan, Charles Durham, Ali Soylemezoglu et al.

Understanding how a complex machine learning model makes a classification decision is essential for its acceptance in sensitive areas such as health care. Towards this end, we present PatchNet, a method that provides the features indicative of each class in an image using a tradeoff between restricting global image context and classification error. We mathematically analyze this tradeoff, demonstrate Patchnet's ability to construct sharp visual heatmap representations of the learned features, and quantitatively compare these features with features selected by domain experts by applying PatchNet to the classification of benign/malignant skin lesions from the ISBI-ISIC 2017 melanoma classification challenge.

Fei Yu, Michal Rybar, Caroline Uhler et al.

Following the publication of an attack on genome-wide association studies (GWAS) data proposed by Homer et al., considerable attention has been given to developing methods for releasing GWAS data in a privacy-preserving way. Here, we develop an end-to-end differentially private method for solving regression problems with convex penalty functions and selecting the penalty parameters by cross-validation. In particular, we focus on penalized logistic regression with elastic-net regularization, a method widely used to in GWAS analyses to identify disease-causing genes. We show how a differentially private procedure for penalized logistic regression with elastic-net regularization can be applied to the analysis of GWAS data and evaluate our method's performance.

Garvesh Raskutti, Caroline Uhler

We consider the problem of learning a Bayesian network or directed acyclic graph (DAG) model from observational data. A number of constraint-based, score-based and hybrid algorithms have been developed for this purpose. For constraint-based methods, statistical consistency guarantees typically rely on the faithfulness assumption, which has been show to be restrictive especially for graphs with cycles in the skeleton. However, there is only limited work on consistency guarantees for score-based and hybrid algorithms and it has been unclear whether consistency guarantees can be proven under weaker conditions than the faithfulness assumption. In this paper, we propose the sparsest permutation (SP) algorithm. This algorithm is based on finding the causal ordering of the variables that yields the sparsest DAG. We prove that this new score-based method is consistent under strictly weaker conditions than the faithfulness assumption. We also demonstrate through simulations on small DAGs that the SP algorithm compares favorably to the constraint-based PC and SGS algorithms as well as the score-based Greedy Equivalence Search and hybrid Max-Min Hill-Climbing method. In the Gaussian setting, we prove that our algorithm boils down to finding the permutation of the variables with sparsest Cholesky decomposition for the inverse covariance matrix. Using this connection, we show that in the oracle setting, where the true covariance matrix is known, the SP algorithm is in fact equivalent to $\ell_0$-penalized maximum likelihood estimation.

Caroline Uhler, Aleksandra B. Slavkovic, Stephen E. Fienberg

Traditional statistical methods for confidentiality protection of statistical databases do not scale well to deal with GWAS (genome-wide association studies) databases especially in terms of guarantees regarding protection from linkage to external information. The more recent concept of differential privacy, introduced by the cryptographic community, is an approach which provides a rigorous definition of privacy with meaningful privacy guarantees in the presence of arbitrary external information, although the guarantees come at a serious price in terms of data utility. Building on such notions, we propose new methods to release aggregate GWAS data without compromising an individual's privacy. We present methods for releasing differentially private minor allele frequencies, chi-square statistics and p-values. We compare these approaches on simulated data and on a GWAS study of canine hair length involving 685 dogs. We also propose a privacy-preserving method for finding genome-wide associations based on a differentially-private approach to penalized logistic regression.