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Understanding how language model performance varies with scale is critical to benchmark and algorithm development. Scaling laws are one approach to building this understanding, but the requirement of training models across many different scales has limited their use. We propose an alternative, observational approach that bypasses model training and instead builds scaling laws from ~80 publically available models. Building a single scaling law from multiple model families is challenging due to large variations in their training compute efficiencies and capabilities. However, we show that these variations are consistent with a simple, generalized scaling law where language model performance is a function of a low-dimensional capability space, and model families only vary in their efficiency in converting training compute to capabilities. Using this approach, we show the surprising predictability of complex scaling phenomena: we show that several emergent phenomena follow a smooth, sigmoidal behavior and are predictable from small models; we show that the agent performance of models such as GPT-4 can be precisely predicted from simpler non-agentic benchmarks; and we show how to predict the impact of post-training interventions like Chain-of-Thought and Self-Consistency as language model capabilities continue to improve.

We consider the problem of selecting an optimal subset of information sources for a hypothesis testing/classification task where the goal is to identify the true state of the world from a finite set of hypotheses, based on finite observation samples from the sources. In order to characterize the learning performance, we propose a misclassification penalty framework, which enables non-uniform treatment of different misclassification errors. In a centralized Bayesian learning setting, we study two variants of the subset selection problem: (i) selecting a minimum cost information set to ensure that the maximum penalty of misclassifying the true hypothesis remains bounded and (ii) selecting an optimal information set under a limited budget to minimize the maximum penalty of misclassifying the true hypothesis. Under mild assumptions, we prove that the objective (or constraints) of these combinatorial optimization problems are weak (or approximate) submodular, and establish high-probability performance guarantees for greedy algorithms. Further, we propose an alternate metric for information set selection which is based on the total penalty of misclassification. We prove that this metric is submodular and establish near-optimal guarantees for the greedy algorithms for both the information set selection problems. Finally, we present numerical simulations to validate our theoretical results over several randomly generated instances., Comment: 23 pages, 4 figures

In this paper, we focus on the problem of shrinking-horizon Model Predictive Control (MPC) in uncertain dynamic environments. We consider controlling a deterministic autonomous system that interacts with uncontrollable stochastic agents during its mission. Employing tools from conformal prediction, existing works derive high-confidence prediction regions for the unknown agent trajectories, and integrate these regions in the design of suitable safety constraints for MPC. Despite guaranteeing probabilistic safety of the closed-loop trajectories, these constraints do not ensure feasibility of the respective MPC schemes for the entire duration of the mission. We propose a shrinking-horizon MPC that guarantees recursive feasibility via a gradual relaxation of the safety constraints as new prediction regions become available online. This relaxation enforces the safety constraints to hold over the least restrictive prediction region from the set of all available prediction regions. In a comparative case study with the state of the art, we empirically show that our approach results in tighter prediction regions and verify recursive feasibility of our MPC scheme.

One can improve predictability in the unknown domain by combining forecasts of imperfect complex computational models using a Bayesian statistical machine learning framework. In many cases, however, the models used in the mixing process are similar. In addition to contaminating the model space, the existence of such similar, or even redundant, models during the multimodeling process can result in misinterpretation of results and deterioration of predictive performance. In this work we describe a method based on the Principal Component Analysis that eliminates model redundancy. We show that by adding model orthogonalization to the proposed Bayesian Model Combination framework, one can arrive at better prediction accuracy and reach excellent uncertainty quantification performance., Comment: 12 pages, 4 figures

We consider stochastic optimization problems involving an expected value of a nonlinear function of a base random vector and a conditional expectation of another function depending on the base random vector, a dependent random vector, and the decision variables. We call such problems conditional stochastic optimization problems. They arise in many applications, such as uplift modeling, reinforcement learning, and contextual optimization. We propose a specialized single time-scale stochastic method for nonconvex constrained conditional stochastic optimization problems with a Lipschitz smooth outer function and a generalized differentiable inner function. In the method, we approximate the inner conditional expectation with a rich parametric model whose mean squared error satisfies a stochastic version of a {\L}ojasiewicz condition. The model is used by an inner learning algorithm. The main feature of our approach is that unbiased stochastic estimates of the directions used by the method can be generated with one observation from the joint distribution per iteration, which makes it applicable to real-time learning. The directions, however, are not gradients or subgradients of any overall objective function. We prove the convergence of the method with probability one, using the method of differential inclusions and a specially designed Lyapunov function, involving a stochastic generalization of the Bregman distance. Finally, a numerical illustration demonstrates the viability of our approach.

A more general formulation of the linear bandit problem is considered to allow for dependencies over time. Specifically, it is assumed that there exists an unknown $\mathbb{R}^d$-valued stationary $\varphi$-mixing sequence of parameters $(\theta_t,~t \in \mathbb{N})$ which gives rise to pay-offs. This instance of the problem can be viewed as a generalization of both the classical linear bandits with iid noise, and the finite-armed restless bandits. In light of the well-known computational hardness of optimal policies for restless bandits, an approximation is proposed whose error is shown to be controlled by the $\varphi$-dependence between consecutive $\theta_t$. An optimistic algorithm, called LinMix-UCB, is proposed for the case where $\theta_t$ has an exponential mixing rate. The proposed algorithm is shown to incur a sub-linear regret of $\mathcal{O}\left(\sqrt{d n\mathrm{polylog}(n) }\right)$ with respect to an oracle that always plays a multiple of $\mathbb{E}\theta_t$. The main challenge in this setting is to ensure that the exploration-exploitation strategy is robust against long-range dependencies. The proposed method relies on Berbee's coupling lemma to carefully select near-independent samples and construct confidence ellipsoids around empirical estimates of $\mathbb{E}\theta_t$.

In this paper we propose an $\ell_1$-regularized GLS estimator for high-dimensional regressions with potentially autocorrelated errors. We establish non-asymptotic oracle inequalities for estimation accuracy in a framework that allows for highly persistent autoregressive errors. In practice, the Whitening matrix required to implement the GLS is unkown, we present a feasible estimator for this matrix, derive consistency results and ultimately show how our proposed feasible GLS can recover closely the optimal performance (as if the errors were a white noise) of the LASSO. A simulation study verifies the performance of the proposed method, demonstrating that the penalized (feasible) GLS-LASSO estimator performs on par with the LASSO in the case of white noise errors, whilst outperforming it in terms of sign-recovery and estimation error when the errors exhibit significant correlation., Comment: 13 pages, 6 figures

We consider a distributed learning problem, where agents minimize a global objective function by exchanging information over a network. Our approach has two distinct features: (i) It substantially reduces communication by triggering communication only when necessary, and (ii) it is agnostic to the data-distribution among the different agents. We can therefore guarantee convergence even if the local data-distributions of the agents are arbitrarily distinct. We analyze the convergence rate of the algorithm and derive accelerated convergence rates in a convex setting. We also characterize the effect of communication drops and demonstrate that our algorithm is robust to communication failures. The article concludes by presenting numerical results from a distributed LASSO problem, and distributed learning tasks on MNIST and CIFAR-10 datasets. The experiments underline communication savings of 50% or more due to the event-based communication strategy, show resilience towards heterogeneous data-distributions, and highlight that our approach outperforms common baselines such as FedAvg, FedProx, and FedADMM., Comment: 29 pages, 12 figures

Society's capacity for algorithmic problem-solving has never been greater. Artificial Intelligence is now applied across more domains than ever, a consequence of powerful abstractions, abundant data, and accessible software. As capabilities have expanded, so have risks, with models often deployed without fully understanding their potential impacts. Interpretable and interactive machine learning aims to make complex models more transparent and controllable, enhancing user agency. This review synthesizes key principles from the growing literature in this field. We first introduce precise vocabulary for discussing interpretability, like the distinction between glass box and explainable algorithms. We then explore connections to classical statistical and design principles, like parsimony and the gulfs of interaction. Basic explainability techniques -- including learned embeddings, integrated gradients, and concept bottlenecks -- are illustrated with a simple case study. We also review criteria for objectively evaluating interpretability approaches. Throughout, we underscore the importance of considering audience goals when designing interactive algorithmic systems. Finally, we outline open challenges and discuss the potential role of data science in addressing them. Code to reproduce all examples can be found at https://go.wisc.edu/3k1ewe.

We consider the problem of estimating an unknown function f* and its partial derivatives from a noisy data set of n observations, where we make no assumptions about f* except that it is smooth in the sense that it has square integrable partial derivatives of order m. A natural candidate for the estimator of f* in such a case is the best fit to the data set that satisfies a certain smoothness condition. This estimator can be seen as a least squares estimator subject to an upper bound on some measure of smoothness. Another useful estimator is the one that minimizes the degree of smoothness subject to an upper bound on the average of squared errors. We prove that these two estimators are computable as solutions to quadratic programs, establish the consistency of these estimators and their partial derivatives, and study the convergence rate as n increases to infinity. The effectiveness of the estimators is illustrated numerically in a setting where the value of a stock option and its second derivative are estimated as functions of the underlying stock price., Comment: 27 pages. Mathematics of Operations Research 2024

We introduce LatentTimePFN (LaT-PFN), a foundational Time Series model with a strong embedding space that enables zero-shot forecasting. To achieve this, we perform in-context learning in latent space utilizing a novel integration of the Prior-data Fitted Networks (PFN) and Joint Embedding Predictive Architecture (JEPA) frameworks. We leverage the JEPA framework to create a prediction-optimized latent representation of the underlying stochastic process that generates time series and combines it with contextual learning, using a PFN. Furthermore, we improve on preceding works by utilizing related time series as a context and introducing an abstract time axis. This drastically reduces training time and increases the versatility of the model by allowing any time granularity and forecast horizon. We show that this results in superior zero-shot predictions compared to established baselines. We also demonstrate our latent space produces informative embeddings of both individual time steps and fixed-length summaries of entire series. Finally, we observe the emergence of multi-step patch embeddings without explicit training, suggesting the model actively learns discrete tokens that encode local structures in the data, analogous to vision transformers., Comment: 9 pages plus references and appendix, 2 tables, 11 figures

We revisit the classical problem of multiclass classification with bandit feedback (Kakade, Shalev-Shwartz and Tewari, 2008), where each input classifies to one of $K$ possible labels and feedback is restricted to whether the predicted label is correct or not. Our primary inquiry is with regard to the dependency on the number of labels $K$, and whether $T$-step regret bounds in this setting can be improved beyond the $\smash{\sqrt{KT}}$ dependence exhibited by existing algorithms. Our main contribution is in showing that the minimax regret of bandit multiclass is in fact more nuanced, and is of the form $\smash{\widetilde{\Theta}\left(\min \left\{|\mathcal{H}| + \sqrt{T}, \sqrt{KT \log |{\mathcal{H}|}} \right\} \right) }$, where $\mathcal{H}$ is the underlying (finite) hypothesis class. In particular, we present a new bandit classification algorithm that guarantees regret $\smash{\widetilde{O}(|\mathcal{H}|+\sqrt{T})}$, improving over classical algorithms for moderately-sized hypothesis classes, and give a matching lower bound establishing tightness of the upper bounds (up to log-factors) in all parameter regimes.

With the proliferation of screening tools for chemical testing, it is now possible to create vast databases of chemicals easily. However, rigorous statistical methodologies employed to analyse these databases are in their infancy, and further development to facilitate chemical discovery is imperative. In this paper, we present conditional Gaussian process models to predict ordinal outcomes from chemical experiments, where the inputs are chemical compounds. We implement the Tanimoto distance, a metric on the chemical space, within the covariance of the Gaussian processes to capture correlated effects in the chemical space. A novel aspect of our model is that the kernel contains a scaling parameter, a feature not previously examined in the literature, that controls the strength of the correlation between elements of the chemical space. Using molecular fingerprints, a numerical representation of a compound's location within the chemical space, we show that accounting for correlation amongst chemical compounds improves predictive performance over the uncorrelated model, where effects are assumed to be independent. Moreover, we present a genetic algorithm for the facilitation of chemical discovery and identification of important features to the compound's efficacy. A simulation study is conducted to demonstrate the suitability of the proposed methods. Our proposed methods are demonstrated on a hazard classification problem of organic solvents.

We leverage the framework of hyperplane arrangements to analyze potential regions of (stable) fixed points. We provide an upper bound on the number of fixed points for multi-layer neural networks equipped with piecewise linear (PWL) activation functions with arbitrary many linear pieces. The theoretical optimality of the exponential growth in the number of layers of the latter bound is shown. Specifically, we also derive a sharper upper bound on the number of stable fixed points for one-hidden-layer networks with hard tanh activation.

Exploring and detecting community structures hold significant importance in genetics, social sciences, neuroscience, and finance. Especially in graphical models, community detection can encourage the exploration of sets of variables with group-like properties. In this paper, within the framework of Gaussian graphical models, we introduce a novel decomposition of the underlying graphical structure into a sparse part and low-rank diagonal blocks (non-overlapped communities). We illustrate the significance of this decomposition through two modeling perspectives and propose a three-stage estimation procedure with a fast and efficient algorithm for the identification of the sparse structure and communities. Also on the theoretical front, we establish conditions for local identifiability and extend the traditional irrepresentability condition to an adaptive form by constructing an effective norm, which ensures the consistency of model selection for the adaptive $\ell_1$ penalized estimator in the second stage. Moreover, we also provide the clustering error bound for the K-means procedure in the third stage. Extensive numerical experiments are conducted to demonstrate the superiority of the proposed method over existing approaches in estimating graph structures. Furthermore, we apply our method to the stock return data, revealing its capability to accurately identify non-overlapped community structures., Comment: 61 pages, 11 figures, 4 tables

In this paper, we investigate the contextual multinomial logit (MNL) bandit problem in which a learning agent sequentially selects an assortment based on contextual information, and user feedback follows an MNL choice model. There has been a significant discrepancy between lower and upper regret bounds, particularly regarding the feature dimension $d$ and the maximum assortment size $K$. Additionally, the variation in reward structures between these bounds complicates the quest for optimality. Under uniform rewards, where all items have the same expected reward, we establish a regret lower bound of $\Omega(d\sqrt{\smash[b]{T/K}})$ and propose a constant-time algorithm, OFU-MNL+, that achieves a matching upper bound of $\tilde{\mathcal{O}}(d\sqrt{\smash[b]{T/K}})$. Under non-uniform rewards, we prove a lower bound of $\Omega(d\sqrt{T})$ and an upper bound of $\tilde{\mathcal{O}}(d\sqrt{T})$, also achievable by OFU-MNL+. Our empirical studies support these theoretical findings. To the best of our knowledge, this is the first work in the MNL contextual bandit literature to prove minimax optimality -- for either uniform or non-uniform reward setting -- and to propose a computationally efficient algorithm that achieves this optimality up to logarithmic factors., Comment: Preprint. Under review

Despite their cost, randomized controlled trials (RCTs) are widely regarded as gold-standard evidence in disciplines ranging from social science to medicine. In recent decades, researchers have increasingly sought to reduce the resource burden of repeated RCTs with factorial designs that simultaneously test multiple hypotheses, e.g. experiments that evaluate the effects of many medications or products simultaneously. Here I show that when multiple interventions are randomized in experiments, the effect any single intervention would have outside the experimental setting is not identified absent heroic assumptions, even if otherwise perfectly realistic conditions are achieved. This happens because single-treatment effects involve a counterfactual world with a single focal intervention, allowing other variables to take their natural values (which may be confounded or modified by the focal intervention). In contrast, observational studies and factorial experiments provide information about potential-outcome distributions with zero and multiple interventions, respectively. In this paper, I formalize sufficient conditions for the identifiability of those isolated quantities. I show that researchers who rely on this type of design have to justify either linearity of functional forms or -- in the nonparametric case -- specify with Directed Acyclic Graphs how variables are related in the real world. Finally, I develop nonparametric sharp bounds -- i.e., maximally informative best-/worst-case estimates consistent with limited RCT data -- that show when extrapolations about effect signs are empirically justified. These new results are illustrated with simulated data.

The development of open benchmarking platforms could greatly accelerate the adoption of AI agents in retail. This paper presents comprehensive simulations of customer shopping behaviors for the purpose of benchmarking reinforcement learning (RL) agents that optimize coupon targeting. The difficulty of this learning problem is largely driven by the sparsity of customer purchase events. We trained agents using offline batch data comprising summarized customer purchase histories to help mitigate this effect. Our experiments revealed that contextual bandit and deep RL methods that are less prone to over-fitting the sparse reward distributions significantly outperform static policies. This study offers a practical framework for simulating AI agents that optimize the entire retail customer journey. It aims to inspire the further development of simulation tools for retail AI systems.

In the study of stochastic dynamics, the committor function describes the probability that a process starting from an initial configuration $x$ will reach set $A$ before set $B$. This paper introduces a fast and interpretable method for approximating the committor, called the "fast committor machine" (FCM). The FCM is based on simulated trajectory data, and it uses this data to train a kernel model. The FCM identifies low-dimensional subspaces that optimally describe the $A$ to $B$ transitions, and the subspaces are emphasized in the kernel model. The FCM uses randomized numerical linear algebra to train the model with runtime that scales linearly in the number of data points. This paper applies the FCM to example systems including the alanine dipeptide miniprotein: in these experiments, the FCM is generally more accurate and trains more quickly than a neural network with a similar number of parameters., Comment: 10 pages, 7 figures

In online learning, the data is provided in a sequential order, and the goal of the learner is to make online decisions to minimize overall regrets. This note is concerned with continuous-time models and algorithms for several online learning problems: online linear optimization, adversarial bandit, and adversarial linear bandit. For each problem, we extend the discrete-time algorithm to the continuous-time setting and provide a concise proof of the optimal regret bound.

In complex environments with large discrete action spaces, effective decision-making is critical in reinforcement learning (RL). Despite the widespread use of value-based RL approaches like Q-learning, they come with a computational burden, necessitating the maximization of a value function over all actions in each iteration. This burden becomes particularly challenging when addressing large-scale problems and using deep neural networks as function approximators. In this paper, we present stochastic value-based RL approaches which, in each iteration, as opposed to optimizing over the entire set of $n$ actions, only consider a variable stochastic set of a sublinear number of actions, possibly as small as $\mathcal{O}(\log(n))$. The presented stochastic value-based RL methods include, among others, Stochastic Q-learning, StochDQN, and StochDDQN, all of which integrate this stochastic approach for both value-function updates and action selection. The theoretical convergence of Stochastic Q-learning is established, while an analysis of stochastic maximization is provided. Moreover, through empirical validation, we illustrate that the various proposed approaches outperform the baseline methods across diverse environments, including different control problems, achieving near-optimal average returns in significantly reduced time.

As machine learning models are increasingly deployed in dynamic environments, it becomes paramount to assess and quantify uncertainties associated with distribution shifts. A distribution shift occurs when the underlying data-generating process changes, leading to a deviation in the model's performance. The prediction interval, which captures the range of likely outcomes for a given prediction, serves as a crucial tool for characterizing uncertainties induced by their underlying distribution. In this paper, we propose methodologies for aggregating prediction intervals to obtain one with minimal width and adequate coverage on the target domain under unsupervised domain shift, under which we have labeled samples from a related source domain and unlabeled covariates from the target domain. Our analysis encompasses scenarios where the source and the target domain are related via i) a bounded density ratio, and ii) a measure-preserving transformation. Our proposed methodologies are computationally efficient and easy to implement. Beyond illustrating the performance of our method through a real-world dataset, we also delve into the theoretical details. This includes establishing rigorous theoretical guarantees, coupled with finite sample bounds, regarding the coverage and width of our prediction intervals. Our approach excels in practical applications and is underpinned by a solid theoretical framework, ensuring its reliability and effectiveness across diverse contexts.

Before deploying outputs from foundation models in high-stakes tasks, it is imperative to ensure that they align with human values. For instance, in radiology report generation, reports generated by a vision-language model must align with human evaluations before their use in medical decision-making. This paper presents Conformal Alignment, a general framework for identifying units whose outputs meet a user-specified alignment criterion. It is guaranteed that on average, a prescribed fraction of selected units indeed meet the alignment criterion, regardless of the foundation model or the data distribution. Given any pre-trained model and new units with model-generated outputs, Conformal Alignment leverages a set of reference data with ground-truth alignment status to train an alignment predictor. It then selects new units whose predicted alignment scores surpass a data-dependent threshold, certifying their corresponding outputs as trustworthy. Through applications to question answering and radiology report generation, we demonstrate that our method is able to accurately identify units with trustworthy outputs via lightweight training over a moderate amount of reference data. En route, we investigate the informativeness of various features in alignment prediction and combine them with standard models to construct the alignment predictor.

In nonsmooth, nonconvex stochastic optimization, understanding the uniform convergence of subdifferential mappings is crucial for analyzing stationary points of sample average approximations of risk as they approach the population risk. Yet, characterizing this convergence remains a fundamental challenge. This work introduces a novel perspective by connecting the uniform convergence of subdifferential mappings to that of subgradient mappings as empirical risk converges to the population risk. We prove that, for stochastic weakly-convex objectives, and within any open set, a uniform bound on the convergence of subgradients -- chosen arbitrarily from the corresponding subdifferential sets -- translates to a uniform bound on the convergence of the subdifferential sets itself, measured by the Hausdorff metric. Using this technique, we derive uniform convergence rates for subdifferential sets of stochastic convex-composite objectives. Our results do not rely on key distributional assumptions in the literature, which require the population and finite sample subdifferentials to be continuous in the Hausdorff metric, yet still provide tight convergence rates. These guarantees lead to new insights into the nonsmooth landscapes of such objectives within finite samples.

The problem of an optimal mapping between Hilbert spaces $IN$ of $\left|\psi\right\rangle$ and $OUT$ of $\left|\phi\right\rangle$ based on a set of wavefunction measurements (within a phase) $\psi_l \to \phi_l$, $l=1\dots M$, is formulated as an optimization problem maximizing the total fidelity $\sum_{l=1}^{M} \omega^{(l)} \left|\langle\phi_l|\mathcal{U}|\psi_l\rangle\right|^2$ subject to probability preservation constraints on $\mathcal{U}$ (partial unitarity). Constructed operator $\mathcal{U}$ can be considered as a $IN$ to $OUT$ quantum channel; it is a partially unitary rectangular matrix of the dimension $\dim(OUT) \times \dim(IN)$ transforming operators as $A^{OUT}=\mathcal{U} A^{IN} \mathcal{U}^{\dagger}$. An iteration algorithm finding the global maximum of this optimization problem is developed and it's application to a number of problems is demonstrated. A software product implementing the algorithm is available from the authors., Comment: A working algorithm implementing Partially Unitary Learning arXiv:2212.14810 is developed and generalized

We consider a large class of shallow neural networks with randomly initialized parameters and rectified linear unit activation functions. We prove that these random neural networks are well-defined non-Gaussian processes. As a by-product, we demonstrate that these networks are solutions to stochastic differential equations driven by impulsive white noise (combinations of random Dirac measures). These processes are parameterized by the law of the weights and biases as well as the density of activation thresholds in each bounded region of the input domain. We prove that these processes are isotropic and wide-sense self-similar with Hurst exponent $3/2$. We also derive a remarkably simple closed-form expression for their autocovariance function. Our results are fundamentally different from prior work in that we consider a non-asymptotic viewpoint: The number of neurons in each bounded region of the input domain (i.e., the width) is itself a random variable with a Poisson law with mean proportional to the density parameter. Finally, we show that, under suitable hypotheses, as the expected width tends to infinity, these processes can converge in law not only to Gaussian processes, but also to non-Gaussian processes depending on the law of the weights. Our asymptotic results provide a new take on several classical results (wide networks converge to Gaussian processes) as well as some new ones (wide networks can converge to non-Gaussian processes).

Robust optimisation is a well-established framework for optimising functions in the presence of uncertainty. The inherent goal of this problem is to identify a collection of inputs whose outputs are both desirable for the decision maker, whilst also being robust to the underlying uncertainties in the problem. In this work, we study the multi-objective extension of this problem from a computational standpoint. We identify that the majority of all robust multi-objective algorithms rely on two key operations: robustification and scalarisation. Robustification refers to the strategy that is used to marginalise over the uncertainty in the problem. Whilst scalarisation refers to the procedure that is used to encode the relative importance of each objective. As these operations are not necessarily commutative, the order that they are performed in has an impact on the resulting solutions that are identified and the final decisions that are made. This work aims to give an exposition on the philosophical differences between these two operations and highlight when one should opt for one ordering over the other. As part of our analysis, we showcase how many existing risk concepts can be easily integrated into the specification and solution of a robust multi-objective optimisation problem. Besides this, we also demonstrate how one can principally define the notion of a robust Pareto front and a robust performance metric based on our robustify and scalarise methodology. To illustrate the efficacy of these new ideas, we present two insightful numerical case studies which are based on real-world data sets., Comment: The code is available at: https://github.com/benmltu/scalarize

We study the problem of online unweighted bipartite matching with $n$ offline vertices and $n$ online vertices where one wishes to be competitive against the optimal offline algorithm. While the classic RANKING algorithm of Karp et al. [1990] provably attains competitive ratio of $1-1/e > 1/2$, we show that no learning-augmented method can be both 1-consistent and strictly better than $1/2$-robust under the adversarial arrival model. Meanwhile, under the random arrival model, we show how one can utilize methods from distribution testing to design an algorithm that takes in external advice about the online vertices and provably achieves competitive ratio interpolating between any ratio attainable by advice-free methods and the optimal ratio of 1, depending on the advice quality., Comment: Accepted into ICML 2024

Classical results in asymptotic statistics show that the Fisher information matrix controls the difficulty of estimating a statistical model from observed data. In this work, we introduce a companion measure of robustness of an estimation problem: the radius of statistical efficiency (RSE) is the size of the smallest perturbation to the problem data that renders the Fisher information matrix singular. We compute RSE up to numerical constants for a variety of test bed problems, including principal component analysis, generalized linear models, phase retrieval, bilinear sensing, and matrix completion. In all cases, the RSE quantifies the compatibility between the covariance of the population data and the latent model parameter. Interestingly, we observe a precise reciprocal relationship between RSE and the intrinsic complexity/sensitivity of the problem instance, paralleling the classical Eckart-Young theorem in numerical analysis.

Real 3-manifold triangulations can be uniquely represented by isomorphism signatures. Databases of these isomorphism signatures are generated for a variety of 3-manifolds and knot complements, using SnapPy and Regina, then these language-like inputs are used to train various machine learning architectures to differentiate the manifolds, as well as their Dehn surgeries, via their triangulations. Gradient saliency analysis then extracts key parts of this language-like encoding scheme from the trained models. The isomorphism signature databases are taken from the 3-manifolds' Pachner graphs, which are also generated in bulk for some selected manifolds of focus and for the subset of the SnapPy orientable cusped census with $<8$ initial tetrahedra. These Pachner graphs are further analysed through the lens of network science to identify new structure in the triangulation representation; in particular for the hyperbolic case, a relation between the length of the shortest geodesic (systole) and the size of the Pachner graph's ball is observed., Comment: 35 pages; 23 figures; 7 tables

The stochastic multi-armed bandit problem studies decision-making under uncertainty. In the problem, the learner interacts with an environment by choosing an action at each round, where a round is an instance of an interaction. In response, the environment reveals a reward, which is sampled from a stochastic process, to the learner. The goal of the learner is to maximize cumulative reward. A specific variation of the stochastic multi-armed bandit problem is the restless bandit, where the reward for each action is sampled from a Markov chain. The restless bandit with a discrete state-space is a well-studied problem, but to the best of our knowledge, not many results exist for the continuous state-space version which has many applications such as hyperparameter optimization. In this work, we tackle the restless bandit with continuous state-space by assuming the rewards are the inner product of an action vector and a state vector generated by a linear Gaussian dynamical system. To predict the reward for each action, we propose a method that takes a linear combination of previously observed rewards for predicting each action's next reward. We show that, regardless of the sequence of previous actions chosen, the reward sampled for any previously chosen action can be used for predicting another action's future reward, i.e. the reward sampled for action 1 at round $t-1$ can be used for predicting the reward for action $2$ at round $t$. This is accomplished by designing a modified Kalman filter with a matrix representation that can be learned for reward prediction. Numerical evaluations are carried out on a set of linear Gaussian dynamical systems.

It is well-known that randomly initialized, push-forward, fully-connected neural networks weakly converge to isotropic Gaussian processes, in the limit where the width of all layers goes to infinity. In this paper, we propose to use the angular power spectrum of the limiting field to characterize the complexity of the network architecture. In particular, we define sequences of random variables associated with the angular power spectrum, and provide a full characterization of the network complexity in terms of the asymptotic distribution of these sequences as the depth diverges. On this basis, we classify neural networks as low-disorder, sparse, or high-disorder; we show how this classification highlights a number of distinct features for standard activation functions, and in particular, sparsity properties of ReLU networks. Our theoretical results are also validated by numerical simulations.

Gradient boosting is a sequential ensemble method that fits a new base learner to the gradient of the remaining loss at each step. We propose a novel family of gradient boosting, Wasserstein gradient boosting, which fits a new base learner to an exactly or approximately available Wasserstein gradient of a loss functional on the space of probability distributions. Wasserstein gradient boosting returns a set of particles that approximates a target probability distribution assigned at each input. In probabilistic prediction, a parametric probability distribution is often specified on the space of output variables, and a point estimate of the output-distribution parameter is produced for each input by a model. Our main application of Wasserstein gradient boosting is a novel distributional estimate of the output-distribution parameter, which approximates the posterior distribution over the output-distribution parameter determined pointwise at each data point. We empirically demonstrate the superior performance of the probabilistic prediction by Wasserstein gradient boosting in comparison with various existing methods.

Many algorithms have been recently proposed for causal machine learning. Yet, there is little to no theory on their quality, especially considering finite samples. In this work, we propose a theory based on generalization bounds that provides such guarantees. By introducing a novel change-of-measure inequality, we are able to tightly bound the model loss in terms of the deviation of the treatment propensities over the population, which we show can be empirically limited. Our theory is fully rigorous and holds even in the face of hidden confounding and violations of positivity. We demonstrate our bounds on semi-synthetic and real data, showcasing their remarkable tightness and practical utility.

Causal estimation (e.g. of the average treatment effect) requires estimating complex nuisance parameters (e.g. outcome models). To adjust for errors in nuisance parameter estimation, we present a novel correction method that solves for the best plug-in estimator under the constraint that the first-order error of the estimator with respect to the nuisance parameter estimate is zero. Our constrained learning framework provides a unifying perspective to prominent first-order correction approaches including debiasing (a.k.a. augmented inverse probability weighting) and targeting (a.k.a. targeted maximum likelihood estimation). Our semiparametric inference approach, which we call the "C-Learner", can be implemented with modern machine learning methods such as neural networks and tree ensembles, and enjoys standard guarantees like semiparametric efficiency and double robustness. Empirically, we demonstrate our approach on several datasets, including those with text features that require fine-tuning language models. We observe the C-Learner matches or outperforms other asymptotically optimal estimators, with better performance in settings with less estimated overlap.

The saturation effect refers to the phenomenon that the kernel ridge regression (KRR) fails to achieve the information theoretical lower bound when the smoothness of the underground truth function exceeds certain level. The saturation effect has been widely observed in practices and a saturation lower bound of KRR has been conjectured for decades. In this paper, we provide a proof of this long-standing conjecture., Comment: ICLR 2023; Minor errors are corrected in this version

Fairness in decision-making processes is often quantified using probabilistic metrics. However, these metrics may not fully capture the real-world consequences of unfairness. In this article, we adopt a utility-based approach to more accurately measure the real-world impacts of decision-making process. In particular, we show that if the concept of $\varepsilon$-fairness is employed, it can possibly lead to outcomes that are maximally unfair in the real-world context. Additionally, we address the common issue of unavailable data on false negatives by proposing a reduced setting that still captures essential fairness considerations. We illustrate our findings with two real-world examples: college admissions and credit risk assessment. Our analysis reveals that while traditional probability-based evaluations might suggest fairness, a utility-based approach uncovers the necessary actions to truly achieve equality. For instance, in the college admission case, we find that enhancing completion rates is crucial for ensuring fairness. Summarizing, this paper highlights the importance of considering the real-world context when evaluating fairness.

Recent years have experienced increasing utilization of complex machine learning models across multiple sources of data to inform more generalizable decision-making. However, distribution shifts across data sources and privacy concerns related to sharing individual-level data, coupled with a lack of uncertainty quantification from machine learning predictions, make it challenging to achieve valid inferences in multi-source environments. In this paper, we consider the problem of obtaining distribution-free prediction intervals for a target population, leveraging multiple potentially biased data sources. We derive the efficient influence functions for the quantiles of unobserved outcomes in the target and source populations, and show that one can incorporate machine learning prediction algorithms in the estimation of nuisance functions while still achieving parametric rates of convergence to nominal coverage probabilities. Moreover, when conditional outcome invariance is violated, we propose a data-adaptive strategy to upweight informative data sources for efficiency gain and downweight non-informative data sources for bias reduction. We highlight the robustness and efficiency of our proposals for a variety of conformal scores and data-generating mechanisms via extensive synthetic experiments. Hospital length of stay prediction intervals for pediatric patients undergoing a high-risk cardiac surgical procedure between 2016-2022 in the U.S. illustrate the utility of our methodology., Comment: Accepted to ICML 2024, 39 pages, 13 figures

Stochastic Gradient (SG) Markov Chain Monte Carlo algorithms (MCMC) are popular algorithms for Bayesian sampling in the presence of large datasets. However, they come with little theoretical guarantees and assessing their empirical performances is non-trivial. In such context, it is crucial to develop algorithms that are robust to the choice of hyperparameters and to gradients heterogeneity since, in practice, both the choice of step-size and behaviour of target gradients induce hard-to-control biases in the invariant distribution. In this work we introduce the stochastic gradient Barker dynamics (SGBD) algorithm, extending the recently developed Barker MCMC scheme, a robust alternative to Langevin-based sampling algorithms, to the stochastic gradient framework. We characterize the impact of stochastic gradients on the Barker transition mechanism and develop a bias-corrected version that, under suitable assumptions, eliminates the error due to the gradient noise in the proposal. We illustrate the performance on a number of high-dimensional examples, showing that SGBD is more robust to hyperparameter tuning and to irregular behavior of the target gradients compared to the popular stochastic gradient Langevin dynamics algorithm.

We build a valid p-value based on a concentration inequality for bounded random variables introduced by Pelekis, Ramon and Wang. The motivation behind this work is the calibration of predictive algorithms in a distribution-free setting. The super-uniform p-value is tighter than Hoeffding and Bentkus alternatives in certain regions. Even though we are motivated by a calibration setting in a machine learning context, the ideas presented in this work are also relevant in classical statistical inference. Furthermore, we compare the power of a collection of valid p- values for bounded losses, which are presented in previous literature.

We investigate modifications to Bayesian Optimization for a resource-constrained setting of sequential experimental design where changes to certain design variables of the search space incur a switching cost. This models the scenario where there is a trade-off between evaluating more while maintaining the same setup, or switching and restricting the number of possible evaluations due to the incurred cost. We adapt two process-constrained batch algorithms to this sequential problem formulation, and propose two new methods: one cost-aware and one cost-ignorant. We validate and compare the algorithms using a set of 7 scalable test functions in different dimensionalities and switching-cost settings for 30 total configurations. Our proposed cost-aware hyperparameter-free algorithm yields comparable results to tuned process-constrained algorithms in all settings we considered, suggesting some degree of robustness to varying landscape features and cost trade-offs. This method starts to outperform the other algorithms with increasing switching-cost. Our work broadens out from other recent Bayesian Optimization studies in resource-constrained settings that consider a batch setting only. While the contributions of this work are relevant to the general class of resource-constrained problems, they are particularly relevant to problems where adaptability to varying resource availability is of high importance

Kalman filtering and smoothing are the foundational mechanisms for efficient inference in Gauss-Markov models. However, their time and memory complexities scale prohibitively with the size of the state space. This is particularly problematic in spatiotemporal regression problems, where the state dimension scales with the number of spatial observations. Existing approximate frameworks leverage low-rank approximations of the covariance matrix. Since they do not model the error introduced by the computational approximation, their predictive uncertainty estimates can be overly optimistic. In this work, we propose a probabilistic numerical method for inference in high-dimensional Gauss-Markov models which mitigates these scaling issues. Our matrix-free iterative algorithm leverages GPU acceleration and crucially enables a tunable trade-off between computational cost and predictive uncertainty. Finally, we demonstrate the scalability of our method on a large-scale climate dataset.

The Koopman operator has entered and transformed many research areas over the last years. Although the underlying concept$\unicode{x2013}$representing highly nonlinear dynamical systems by infinite-dimensional linear operators$\unicode{x2013}$has been known for a long time, the availability of large data sets and efficient machine learning algorithms for estimating the Koopman operator from data make this framework extremely powerful and popular. Koopman operator theory allows us to gain insights into the characteristic global properties of a system without requiring detailed mathematical models. We will show how these methods can also be used to analyze complex networks and highlight relationships between Koopman operators and graph Laplacians.

Many Machine Learning (ML) models are referred to as black box models, providing no real insights into why a prediction is made. Feature importance and explainability are important for increasing transparency and trust in ML models, particularly in settings such as healthcare and finance. With quantum computing's unique capabilities, such as leveraging quantum mechanical phenomena like superposition, which can be combined with ML techniques to create the field of Quantum Machine Learning (QML), and such techniques may be applied to QML models. This article explores feature importance and explainability insights in QML compared to Classical ML models. Utilizing the widely recognized Iris dataset, classical ML algorithms such as SVM and Random Forests, are compared against hybrid quantum counterparts, implemented via IBM's Qiskit platform: the Variational Quantum Classifier (VQC) and Quantum Support Vector Classifier (QSVC). This article aims to provide a comparison of the insights generated in ML by employing permutation and leave one out feature importance methods, alongside ALE (Accumulated Local Effects) and SHAP (SHapley Additive exPlanations) explainers., Comment: Amended final year project. 23 pages

A recent study by De et al. (2022) has reported that large-scale representation learning through pre-training on a public dataset significantly enhances differentially private (DP) learning in downstream tasks, despite the high dimensionality of the feature space. To theoretically explain this phenomenon, we consider the setting of a layer-peeled model in representation learning, which results in interesting phenomena related to learned features in deep learning and transfer learning, known as Neural Collapse (NC). Within the framework of NC, we establish an error bound indicating that the misclassification error is independent of dimension when the distance between actual features and the ideal ones is smaller than a threshold. Additionally, the quality of the features in the last layer is empirically evaluated under different pre-trained models within the framework of NC, showing that a more powerful transformer leads to a better feature representation. Furthermore, we reveal that DP fine-tuning is less robust compared to fine-tuning without DP, particularly in the presence of perturbations. These observations are supported by both theoretical analyses and experimental evaluation. Moreover, to enhance the robustness of DP fine-tuning, we suggest several strategies, such as feature normalization or employing dimension reduction methods like Principal Component Analysis (PCA). Empirically, we demonstrate a significant improvement in testing accuracy by conducting PCA on the last-layer features., Comment: To appear in ICML 2024

In safety-critical applications such as medical imaging and autonomous driving, where decisions have profound implications for patient health and road safety, it is imperative to maintain both high adversarial robustness to protect against potential adversarial attacks and reliable uncertainty quantification in decision-making. With extensive research focused on enhancing adversarial robustness through various forms of adversarial training (AT), a notable knowledge gap remains concerning the uncertainty inherent in adversarially trained models. To address this gap, this study investigates the uncertainty of deep learning models by examining the performance of conformal prediction (CP) in the context of standard adversarial attacks within the adversarial defense community. It is first unveiled that existing CP methods do not produce informative prediction sets under the commonly used $l_{\infty}$-norm bounded attack if the model is not adversarially trained, which underpins the importance of adversarial training for CP. Our paper next demonstrates that the prediction set size (PSS) of CP using adversarially trained models with AT variants is often worse than using standard AT, inspiring us to research into CP-efficient AT for improved PSS. We propose to optimize a Beta-weighting loss with an entropy minimization regularizer during AT to improve CP-efficiency, where the Beta-weighting loss is shown to be an upper bound of PSS at the population level by our theoretical analysis. Moreover, our empirical study on four image classification datasets across three popular AT baselines validates the effectiveness of the proposed Uncertainty-Reducing AT (AT-UR)., Comment: ICML2024

Driven by steady progress in generative modeling, simulation-based inference (SBI) has enabled inference over stochastic simulators. However, recent work has demonstrated that model misspecification can harm SBI's reliability. This work introduces robust posterior estimation (ROPE), a framework that overcomes model misspecification with a small real-world calibration set of ground truth parameter measurements. We formalize the misspecification gap as the solution of an optimal transport problem between learned representations of real-world and simulated observations. Assuming the prior distribution over the parameters of interest is known and well-specified, our method offers a controllable balance between calibrated uncertainty and informative inference under all possible misspecifications of the simulator. Our empirical results on four synthetic tasks and two real-world problems demonstrate that ROPE outperforms baselines and consistently returns informative and calibrated credible intervals.

Causal discovery based on observational data is important for deciphering the causal mechanism behind complex systems. However, the effectiveness of existing causal discovery methods is limited due to inferior prior knowledge, domain inconsistencies, and the challenges of high-dimensional datasets with small sample sizes. To address this gap, we propose a novel weakly-supervised fuzzy knowledge and data co-driven causal discovery method named KEEL. KEEL adopts a fuzzy causal knowledge schema to encapsulate diverse types of fuzzy knowledge, and forms corresponding weakened constraints. This schema not only lessens the dependency on expertise but also allows various types of limited and error-prone fuzzy knowledge to guide causal discovery. It can enhance the generalization and robustness of causal discovery, especially in high-dimensional and small-sample scenarios. In addition, we integrate the extended linear causal model (ELCM) into KEEL for dealing with the multi-distribution and incomplete data. Extensive experiments with different datasets demonstrate the superiority of KEEL over several state-of-the-art methods in accuracy, robustness and computational efficiency. For causal discovery in real protein signal transduction processes, KEEL outperforms the benchmark method with limited data. In summary, KEEL is effective to tackle the causal discovery tasks with higher accuracy while alleviating the requirement for extensive domain expertise.

We introduce an algorithm that simplifies the construction of efficient estimators, making them accessible to a broader audience. 'Dimple' takes as input computer code representing a parameter of interest and outputs an efficient estimator. Unlike standard approaches, it does not require users to derive a functional derivative known as the efficient influence function. Dimple avoids this task by applying automatic differentiation to the statistical functional of interest. Doing so requires expressing this functional as a composition of primitives satisfying a novel differentiability condition. Dimple also uses this composition to determine the nuisances it must estimate. In software, primitives can be implemented independently of one another and reused across different estimation problems. We provide a proof-of-concept Python implementation and showcase through examples how it allows users to go from parameter specification to efficient estimation with just a few lines of code.

A common issue in learning decision-making policies in data-rich settings is spurious correlations in the offline dataset, which can be caused by hidden confounders. Instrumental variable (IV) regression, which utilises a key unconfounded variable known as the instrument, is a standard technique for learning causal relationships between confounded action, outcome, and context variables. Most recent IV regression algorithms use a two-stage approach, where a deep neural network (DNN) estimator learnt in the first stage is directly plugged into the second stage, in which another DNN is used to estimate the causal effect. Naively plugging the estimator can cause heavy bias in the second stage, especially when regularisation bias is present in the first stage estimator. We propose DML-IV, a non-linear IV regression method that reduces the bias in two-stage IV regressions and effectively learns high-performing policies. We derive a novel learning objective to reduce bias and design the DML-IV algorithm following the double/debiased machine learning (DML) framework. The learnt DML-IV estimator has strong convergence rate and $O(N^{-1/2})$ suboptimality guarantees that match those when the dataset is unconfounded. DML-IV outperforms state-of-the-art IV regression methods on IV regression benchmarks and learns high-performing policies in the presence of instruments., Comment: Accepted at ICML 2024