Christopher Tosh

Christopher Tosh, Mauricio Tec, Wesley Tansey

A fundamental task in science is to design experiments that yield valuable insights about the system under study. Mathematically, these insights can be represented as a utility or risk function that shapes the value of conducting each experiment. We present PDBAL, a targeted active learning method that adaptively designs experiments to maximize scientific utility. PDBAL takes a user-specified risk function and combines it with a probabilistic model of the experimental outcomes to choose designs that rapidly converge on a high-utility model. We prove theoretical bounds on the label complexity of PDBAL and provide fast closed-form solutions for designing experiments with common exponential family likelihoods. In simulation studies, PDBAL consistently outperforms standard untargeted approaches that focus on maximizing expected information gain over the design space. Finally, we demonstrate the scientific potential of PDBAL through a study on a large cancer drug screen dataset where PDBAL quickly recovers the most efficacious drugs with a small fraction of the total number of experiments.

Kaushik Sinha, Christopher Tosh

Expand-and-sparsify representations are a class of theoretical models that capture sparse representation phenomena observed in the sensory systems of many animals. At a high level, these representations map an input $x \in \mathbb{R}^d$ to a much higher dimension $m \gg d$ via random linear projections before zeroing out all but the $k \ll m$ largest entries. The result is a $k$-sparse vector in $\{0,1\}^m$. We study the suitability of this representation for two fundamental statistical problems: density estimation and mode estimation. For density estimation, we show that a simple linear function of the expand-and-sparsify representation produces an estimator with minimax-optimal $\ell_{\infty}$ convergence rates. In mode estimation, we provide simple algorithms on top of our density estimator that recover single or multiple modes at optimal rates up to logarithmic factors under mild conditions.

Christopher Tosh, Boyuan Zhang, Wesley Tansey

Scientists often need to analyze the samples in a study that responded to treatment in order to refine their hypotheses and find potential causal drivers of response. Natural variation in outcomes makes teasing apart responders from non-responders a statistical inference problem. To handle latent responses, we introduce the causal two-groups (C2G) model, a causal extension of the classical two-groups model. The C2G model posits that treated samples may or may not experience an effect, according to some prior probability. We propose two empirical Bayes procedures for the causal two-groups model, one under semi-parametric conditions and another under fully nonparametric conditions. The semi-parametric model assumes additive treatment effects and is identifiable from observed data. The nonparametric model is unidentifiable, but we show it can still be used to test for response in each treated sample. We show empirically and theoretically that both methods for selecting responders control the false discovery rate at the target level with near-optimal power. We also propose two novel estimands of interest and provide a strategy for deriving estimand intervals in the unidentifiable nonparametric model. On a cancer immunotherapy dataset, the nonparametric C2G model recovers clinically-validated predictive biomarkers of both positive and negative outcomes. Code is available at https://github.com/tansey-lab/causal2groups.

Wesley Tansey, Christopher Tosh, David M. Blei

Exploratory cancer drug studies test multiple tumor cell lines against multiple candidate drugs. The goal in each paired (cell line, drug) experiment is to map out the dose-response curve of the cell line as the dose level of the drug increases. We propose Bayesian Tensor Filtering (BTF), a hierarchical Bayesian model for dose-response modeling in multi-sample, multi-treatment cancer drug studies. BTF uses low-dimensional embeddings to share statistical strength between similar drugs and similar cell lines. Structured shrinkage priors in BTF encourage smoothness in the dose-response curves while remaining adaptive to sharp jumps when the data call for it. We focus on a pair of cancer drug studies exhibiting a particular pathology in their experimental design, leading us to a non-conjugate monotone mixture-of-Gammas likelihood. To perform posterior inference, we develop a variant of the elliptical slice sampling algorithm for sampling from linearly-constrained multivariate normal priors with non-conjugate likelihoods. In benchmarks, BTF outperforms state-of-the-art methods for covariance regression and dynamic Poisson matrix factorization. On the two cancer drug studies, BTF outperforms the current standard approach in biology and reveals potential new biomarkers of drug sensitivity in cancer. Code is available at https://github.com/tansey/functionalmf.

Antoine de Mathelin, Christopher Tosh, Wesley Tansey

Personalizing combination therapies in oncology requires navigating an immense space of possible drug and dose combinations, a task that remains largely infeasible through exhaustive experimentation. Recent developments in patient-derived models have enabled high-throughput ex vivo screening, but the number of feasible experiments is limited. Further, a tight therapeutic window makes gathering molecular profiling information (e.g. RNA-seq) impractical as a means of guiding drug response prediction. This leads to a challenging cold-start problem: how do we select the most informative combinations to test early, when no prior information about the patient is available? We propose a strategy that leverages a pretrained deep learning model built on historical drug response data. The model provides both embeddings for drug combinations and dose-level importance scores, enabling a principled selection of initial experiments. We combine clustering of drug embeddings to ensure functional diversity with a dose-weighting mechanism that prioritizes doses based on their historical informativeness. Retrospective simulations on large-scale drug combination datasets show that our method substantially improves initial screening efficiency compared to baselines, offering a viable path for more effective early-phase decision-making in personalized combination drug screens.

Christopher Tosh, Daniel Hsu

Multi-group agnostic learning is a formal learning criterion that is concerned with the conditional risks of predictors within subgroups of a population. The criterion addresses recent practical concerns such as subgroup fairness and hidden stratification. This paper studies the structure of solutions to the multi-group learning problem, and provides simple and near-optimal algorithms for the learning problem.

Max Simchowitz, Christopher Tosh, Akshay Krishnamurthy et al.

Thompson sampling and other Bayesian sequential decision-making algorithms are among the most popular approaches to tackle explore/exploit trade-offs in (contextual) bandits. The choice of prior in these algorithms offers flexibility to encode domain knowledge but can also lead to poor performance when misspecified. In this paper, we demonstrate that performance degrades gracefully with misspecification. We prove that the expected reward accrued by Thompson sampling (TS) with a misspecified prior differs by at most $\tilde{\mathcal{O}}(H^2 ε)$ from TS with a well specified prior, where $ε$ is the total-variation distance between priors and $H$ is the learning horizon. Our bound does not require the prior to have any parametric form. For priors with bounded support, our bound is independent of the cardinality or structure of the action space, and we show that it is tight up to universal constants in the worst case. Building on our sensitivity analysis, we establish generic PAC guarantees for algorithms in the recently studied Bayesian meta-learning setting and derive corollaries for various families of priors. Our results generalize along two axes: (1) they apply to a broader family of Bayesian decision-making algorithms, including a Monte-Carlo implementation of the knowledge gradient algorithm (KG), and (2) they apply to Bayesian POMDPs, the most general Bayesian decision-making setting, encompassing contextual bandits as a special case. Through numerical simulations, we illustrate how prior misspecification and the deployment of one-step look-ahead (as in KG) can impact the convergence of meta-learning in multi-armed and contextual bandits with structured and correlated priors.

Christopher Tosh, Akshay Krishnamurthy, Daniel Hsu

Self-supervised learning is an empirically successful approach to unsupervised learning based on creating artificial supervised learning problems. A popular self-supervised approach to representation learning is contrastive learning, which leverages naturally occurring pairs of similar and dissimilar data points, or multiple views of the same data. This work provides a theoretical analysis of contrastive learning in the multi-view setting, where two views of each datum are available. The main result is that linear functions of the learned representations are nearly optimal on downstream prediction tasks whenever the two views provide redundant information about the label.

Sanjoy Dasgupta, Christopher Tosh

A simple sparse coding mechanism appears in the sensory systems of several organisms: to a coarse approximation, an input $x \in \R^d$ is mapped to much higher dimension $m \gg d$ by a random linear transformation, and is then sparsified by a winner-take-all process in which only the positions of the top $k$ values are retained, yielding a $k$-sparse vector $z \in \{0,1\}^m$. We study the benefits of this representation for subsequent learning. We first show a universal approximation property, that arbitrary continuous functions of $x$ are well approximated by linear functions of $z$, provided $m$ is large enough. This can be interpreted as saying that $z$ unpacks the information in $x$ and makes it more readily accessible. The linear functions can be specified explicitly and are easy to learn, and we give bounds on how large $m$ needs to be as a function of the input dimension $d$ and the smoothness of the target function. Next, we consider whether the representation is adaptive to manifold structure in the input space. This is highly dependent on the specific method of sparsification: we show that adaptivity is not obtained under the winner-take-all mechanism, but does hold under a slight variant. Finally we consider mappings to the representation space that are random but are attuned to the data distribution, and we give favorable approximation bounds in this setting.

Christopher Tosh, Akshay Krishnamurthy, Daniel Hsu

Contrastive learning is an approach to representation learning that utilizes naturally occurring similar and dissimilar pairs of data points to find useful embeddings of data. In the context of document classification under topic modeling assumptions, we prove that contrastive learning is capable of recovering a representation of documents that reveals their underlying topic posterior information to linear models. We apply this procedure in a semi-supervised setup and demonstrate empirically that linear classifiers with these representations perform well in document classification tasks with very few training examples.

Sanjoy Dasgupta, Stefanos Poulis, Christopher Tosh

The formalism of anchor words has enabled the development of fast topic modeling algorithms with provable guarantees. In this paper, we introduce a protocol that allows users to interact with anchor words to build customized and interpretable topic models. Experimental evidence validating the usefulness of our approach is also presented.

Christopher Tosh, Daniel Hsu

We introduce interactive structure discovery, a generic framework that encompasses many interactive learning settings, including active learning, top-k item identification, interactive drug discovery, and others. We adapt a recently developed active learning algorithm of Tosh and Dasgupta (2017) for interactive structure discovery, and show that the new algorithm can be made noise-tolerant and enjoys favorable query complexity bounds.

Christopher Tosh, Sanjoy Dasgupta

In this work, we describe a framework that unifies many different interactive learning tasks. We present a generalization of the {\it query-by-committee} active learning algorithm for this setting, and we study its consistency and rate of convergence, both theoretically and empirically, with and without noise.

Christopher Tosh, Sanjoy Dasgupta

To date, the tightest upper and lower-bounds for the active learning of general concept classes have been in terms of a parameter of the learning problem called the splitting index. We provide, for the first time, an efficient algorithm that is able to realize this upper bound, and we empirically demonstrate its good performance.