Meta-analyses are usually conducted on small amounts of "trusted" data, ideally from randomized, controlled trials. Excluding untrusted (observational) data - such as medical records and related scientific literature - avoids potential confounding and ensures unbiased conclusions. Unfortunately, this exclusion can reduce predictive accuracy to the point of clinical irrelevance, especially when trials are heterogeneous. This paper shows how untrusted data can be safely incorporated into meta-analysis, improving predictions without sacrificing rigor or introducing unproven assumptions. Our approach, called conformal meta-analysis, consists of (1) learning a (potentially flawed) prior distribution from the untrusted data, (2) using the prior and trusted data to derive a simple, fully-conformal prediction interval for the observed trial effect, and (3) analytically extracting an interval for the true (unobserved) effect. In multiple experiments on healthcare datasets, our algorithms deliver tighter, sounder intervals than traditional ones. This paper conceptually realigns meta-analysis as a foundation for evidence-based medicine, embracing heterogeneity and untrusted data for more nuanced, precise predictions.

Every length-$(n+1)$ sequence of orthogonal polynomials is uniquely represented by two length-$(n+1)$ sequences of coefficients $\alpha$ and $\beta$. We make this representation learnable by gradient-based methods. Orthogonal polynomial operations may be automatically differentiated, but this uses $O(n^2)$ memory and is very slow in practice. By exploiting reversibility, we derive a differentiation algorithm which uses $O(n)$ memory and is much faster in practice. Using this algorithm, fixed polynomial transforms (e.g. discrete cosine transforms) can be replaced by learnable layers. These are more expressive, but they retain the computational efficiency and analytic tractability of orthogonal polynomials. As another application, we present an algorithm for approximating the minimal value $f(w^*)$ of a general nonconvex objective $f$, without finding the minimizer $w^*$. It follows a scheme recently proposed by Lasserre (2020), whose core algorithmic problem is to find the sequence of polynomials orthogonal to a given probability distribution. Despite the general intractability of this problem, we observe encouraging initial results on some test cases.

Running nonlinear RNNs for T steps takes Ω(T ) time. Our construction, called LDStack, approximately runs them inO(log T ) parallel time, and obtains arbitrarily low error via repetition. First, we show nonlinear RNNs can be approximated by a stack of multiple-input, multiple-output (MIMO) LDS. This replaces nonlinearity across time with nonlinearity along depth. Next, we show that MIMO LDS can be approximated by an average or a concatenation of single-input, multiple-output (SIMO) LDS. Finally, we present an algorithm for running (and differentiating) SIMO LDS in O(log T ) parallel time. On long sequences, LDStack is much faster than traditional RNNs, yet it achieves similar accuracy in our experiments. Furthermore, LDStack is amenable to linear systems theory. Therefore, it improves not only speed, but also interpretability and mathematical tractability.

Brief, intense exercise can improve health due to its acute effect on the autonomic nervous system, particularly the sympathetic nervous system. Salivary amylase is a marker of sympathetic activity during exercise, but it requires specialized equipment to measure. We investigate the feasibility of estimating the amylase response from heartbeat data recorded by commodity sensors. We collect heartbeat and amylase data for n = 71 sessions of intense exercise performed in a commercial setting. Our machine learning model exploits structure in the heartbeat signal: by identifying and removing the contribution of the parasympathetic nervous system, we obtain a residual with sympathetic information, to which we apply a convolutional neural network. This model has better accuracy than existing measures of exercise response, such as maximum heart rate, even though it doesn't use metadata such as age and gender. This suggests sympathetic activity may be (weakly) discerned from heartbeat data. With a larger dataset, a practical measure of sympathetic response to exercise could potentially be developed. Our quantification of parasympathetic activity is more powerful than existing approaches and may have independent value.

Classical linear/shallow learning is relatively easy to analyze and understand, but the power of deep learning is often desirable. I am developing a hybrid approach in order to obtain learning algorithms that are both trustworthy and accurate. My research has mostly focused on learning from corrupted or inconsistent training data (agnostic learning'). Recently, I, as well as independent researchers, have found these same techniques could help make algorithms more fair.

We use the statistical quantity of margin --- the distance between a decision boundary and a classified point, or the gap between two scores --- to formalize the principle of equal opportunity --- the chance to improve one's outcome, regardless of group status. This leads to a better definition of opportunity which recognizes, for example, that a strongly rejected individual was offered less recourse than a weakly rejected one, despite the shared outcome. It also leads to simpler algorithms, since real-valued margins are easier to analyze and optimize than discrete outcomes. We formalize two ways that a protected group may be guaranteed equal opportunity: (1) (social) mobility: acceptance should be within reach for the group (conversely, the general population shouldn't be cushioned from rejection), and (2) contrast: within the group, good candidates should get substantially higher scores than bad candidates, preventing the so-called 'token' effect. A simple linear classifier seems to offer roughly equal opportunity both experimentally and mathematically.

Upgrading a linear classifier to a nonlinear one leads to statistical or computational compromise. For example, lifting its dimension, as in kernel methods, may lead to overfitting. Nonconvex optimization algorithms, used for (deep) neural networks or decision trees, may not obtain high-quality solutions or even converge. Similarly, boosting can get stuck on increasingly difficult subproblems. We avoid concerns about overfitting and convergence with new nonlinear classifiers — smooth lists of halfspaces — and a new learning algorithm — the sequence of averages. Smooth lists have unbounded depth, yet do not need more data, in the worst case, than single linear classifiers. Theory suggests the algorithm never gets prematurely stuck and monotonically improves the classifier. In experiments inspired by challenging problems in computational learning theory, it performs well when only a moderate amount of data and time are available, and there is non-trivial noise or inconsistency. At a high level, we isolate a kind of deep learning which is easier to reason about but retains some of the advantages of depth.

A learning algorithm is agnostic if it doesn’t presume a perfect model of how input data produce output data. Such algorithms are difficult to design, even for the basic task of classifying data as well as the best linear separator. This has led to a persistent rift between practice and theory: popular algorithms, such as SVMs and logistic regression, are susceptible to noise; provable agnostic algorithms involve brute-force sampling (which uses too much time) or fitting polynomials (which uses too much data.) We recently introduced a new classification algorithm, KG, which is both practical and agnostic. It revisits basic elements of learning: 1. What functions should the algorithm fit? Smooth lists of halfspaces are a novel generalization of halfspaces. They are more flexible than halfspaces, but do not require more data to train in the worst case. 2. How should the algorithm fit such a function to the data? The algorithm involves ‘immutable’ iterations which are fundamentally different than update rules such as gradient descent, multiplicative weights, or perceptrons. KG achieves promising experimental performance for both natural and artificial problems. We seek to deepen our theoretical understanding of the algorithm and expand its practical applications. The main question we shall answer is: when is KG provably fast? It eventually converges to the correct solution for a wide variety of input distributions. However, these intersect with a litany of hardness results, so restricting the input distribution seems necessary. Based on experimental evidence and the mechanics of the algorithm, we believe it is possible the algorithm runs in polynomial time when the inputs are normally distributed. If so, this algorithm would solve a notorious problem in computer science: learning logarithmically-sparse parities with noise. This would resolve a variety of challenges in learning theory, such as learning DNFs (encountered in 1984 by Valiant) and learning log-juntas (the subject of a prize offered in 2003 by Blum). As exciting as this possibility seems, it does not contradict known hardness results, nor does it upset the consensus on related problems in cryptography or complexity theory. We propose to gain more experimental and theoretical evidence for this possibility. In practice, many classification tasks involve multiple classes. When the number of classes is large, we do not believe fast agnostic classification is possible. We posit stronger lower bounds for classification with a growing number of classes which depend on P 6= NP rather than weaker conjectures about refuting random constraint satisfaction problems. We believe the problem remains challenging even when the inputs are normally distributed. This is due to close relations with the learning with errors (LWE) problem, which underpins much of modern cryptography. The difficulty of LWE and its variants depends heavily on its parameters, including the dimension of the inputs, the amount of data, the number of classes, and the amount of noise. For some choices, it is as hard as worst-case lattice problems; for others, it may be solved in polynomial time. We propose a generalization of KG for multiple classes. It may substantially improve upon logistic regression and multiclass SVM for many practical applications. It may also experimentally validate the range of safe parameters for lattice-based cryptography.

A means to take advantage of molecular similarity to lower the computational cost of electronic structure theory is proposed, in which parameters are embedded into a low-cost, low-level (LL) ab initio theory and adjusted to obtain agreement with a higher level (HL) ab initio theory. This approach is explored by training such a model on data for ethane and testing the resulting model on methane, propane and butane. The electronic distribution of the molecules is varied by placing them in strong electrostatic environments consisting of random charges placed on the corners of a cube. The results find that parameters embedded in HF/STO-3G theory can be adjusted to obtain agreement, to within about 2 kcal/mol, with results of HF/6-31G theory. Obtaining this level of agreement requires the use of parameters that are functions of the bond lengths, atomic charges, and bond orders within the molecules. The argument is made that this approach provides a well-controlled means to take advantage of molecular similarity in quantum chemistry.