Towards Stable Machine Learning Model Retraining via Slowly Varying Sequences

Retraining machine learning models remains an important task for real-world machine learning model deployment. Existing methods focus largely on greedy approaches to find the best-performing model without considering the stability of trained model structures across different retraining evolutions. In this study, we develop a mixed integer optimization algorithm that holistically considers the problem of retraining machine learning models across different data batch updates. Our method focuses on retaining consistent analytical insights - which is important to model interpretability, ease of implementation, and fostering trust with users - by using custom-defined distance metrics that can be directly incorporated into the optimization problem. Importantly, our method shows stronger stability than greedily trained models with a small, controllable sacrifice in model performance in a real-world production case study. Finally, important analytical insights, as demonstrated using SHAP feature importance, are shown to be consistent across retraining iterations.

Adaptive Optimization for Prediction with Missing Data

When training predictive models on data with missing entries, the most widely used and versatile approach is a pipeline technique where we first impute missing entries and then compute predictions. In this paper, we view prediction with missing data as a two-stage adaptive optimization problem and propose a new class of models, adaptive linear regression models, where the regression coefficients adapt to the set of observed features. We show that some adaptive linear regression models are equivalent to learning an imputation rule and a downstream linear regression model simultaneously instead of sequentially. We leverage this joint-impute-then-regress interpretation to generalize our framework to non-linear models. In settings where data is strongly not missing at random, our methods achieve a 2-10% improvement in out-of-sample accuracy.

Universal Neurons in GPT2 Language Models

A basic question within the emerging field of mechanistic interpretability is the degree to which neural networks learn the same underlying mechanisms. In other words, are neural mechanisms universal across different models? In this work, we study the universality of individual neurons across GPT2 models trained from different initial random seeds, motivated by the hypothesis that universal neurons are likely to be interpretable. In particular, we compute pairwise correlations of neuron activations over 100 million tokens for every neuron pair across five different seeds and find that 1-5\% of neurons are universal, that is, pairs of neurons which consistently activate on the same inputs. We then study these universal neurons in detail, finding that they usually have clear interpretations and taxonomize them into a small number of neuron families. We conclude by studying patterns in neuron weights to establish several universal functional roles of neurons in simple circuits: deactivating attention heads, changing the entropy of the next token distribution, and predicting the next token to (not) be within a particular set.

Robust Regression over Averaged Uncertainty

We propose a new formulation of robust regression by integrating all realizations of the uncertainty set and taking an averaged approach to obtain the optimal solution for the ordinary least-squared regression problem. We show that this formulation surprisingly recovers ridge regression and establishes the missing link between robust optimization and the mean squared error approaches for existing regression problems. We first prove the equivalence for four uncertainty sets: ellipsoidal, box, diamond, and budget, and provide closed-form formulations of the penalty term as a function of the sample size, feature size, as well as perturbation protection strength. We then show in synthetic datasets with different levels of perturbations, a consistent improvement of the averaged formulation over the existing worst-case formulation in out-of-sample performance. Importantly, as the perturbation level increases, the improvement increases, confirming our method's advantage in high-noise environments. We report similar improvements in the out-of-sample datasets in real-world regression problems obtained from UCI datasets.

Global Optimization: A Machine Learning Approach

Many approaches for addressing Global Optimization problems typically rely on relaxations of nonlinear constraints over specific mathematical primitives. This is restricting in applications with constraints that are black-box, implicit or consist of more general primitives. Trying to address such limitations, Bertsimas and Ozturk (2023) proposed OCTHaGOn as a way of solving black-box global optimization problems by approximating the nonlinear constraints using hyperplane-based Decision-Trees and then using those trees to construct a unified mixed integer optimization (MIO) approximation of the original problem. We provide extensions to this approach, by (i) approximating the original problem using other MIO-representable ML models besides Decision Trees, such as Gradient Boosted Trees, Multi Layer Perceptrons and Suport Vector Machines, (ii) proposing adaptive sampling procedures for more accurate machine learning-based constraint approximations, (iii) utilizing robust optimization to account for the uncertainty of the sample-dependent training of the ML models, and (iv) leveraging a family of relaxations to address the infeasibilities of the final MIO approximation. We then test the enhanced framework in 81 Global Optimization instances. We show improvements in solution feasibility and optimality in the majority of instances. We also compare against BARON, showing improved optimality gaps or solution times in 11 instances.

The R.O.A.D. to precision medicine

We propose a prognostic stratum matching framework that addresses the deficiencies of Randomized trial data subgroup analysis and transforms ObservAtional Data to be used as if they were randomized, thus paving the road for precision medicine. Our approach counters the effects of unobserved confounding in observational data by correcting the estimated probabilities of the outcome under a treatment through a novel two-step process. These probabilities are then used to train Optimal Policy Trees (OPTs), which are decision trees that optimally assign treatments to subgroups of patients based on their characteristics. This facilitates the creation of clinically intuitive treatment recommendations. We applied our framework to observational data of patients with gastrointestinal stromal tumors (GIST) and validated the OPTs in an external cohort using the sensitivity and specificity metrics. We show that these recommendations outperformed those of experts in GIST. We further applied the same framework to randomized clinical trial (RCT) data of patients with extremity sarcomas. Remarkably, despite the initial trial results suggesting that all patients should receive treatment, our framework, after addressing imbalances in patient distribution due to the trial's small sample size, identified through the OPTs a subset of patients with unique characteristics who may not require treatment. Again, we successfully validated our recommendations in an external cohort.

A Machine Learning Approach to Two-Stage Adaptive Robust Optimization

We propose an approach based on machine learning to solve two-stage linear adaptive robust optimization (ARO) problems with binary here-and-now variables and polyhedral uncertainty sets. We encode the optimal here-and-now decisions, the worst-case scenarios associated with the optimal here-and-now decisions, and the optimal wait-and-see decisions into what we denote as the strategy. We solve multiple similar ARO instances in advance using the column and constraint generation algorithm and extract the optimal strategies to generate a training set. We train a machine learning model that predicts high-quality strategies for the here-and-now decisions, the worst-case scenarios associated with the optimal here-and-now decisions, and the wait-and-see decisions. We also introduce an algorithm to reduce the number of different target classes the machine learning algorithm needs to be trained on. We apply the proposed approach to the facility location, the multi-item inventory control and the unit commitment problems. Our approach solves ARO problems drastically faster than the state-of-the-art algorithms with high accuracy.

Optimal Control of Multiclass Fluid Queueing Networks: A Machine Learning Approach

We propose a machine learning approach to the optimal control of multiclass fluid queueing networks (MFQNETs) that provides explicit and insightful control policies. We prove that a threshold type optimal policy exists for MFQNET control problems, where the threshold curves are hyperplanes passing through the origin. We use Optimal Classification Trees with hyperplane splits (OCT-H) to learn an optimal control policy for MFQNETs. We use numerical solutions of MFQNET control problems as a training set and apply OCT-H to learn explicit control policies. We report experimental results with up to 33 servers and 99 classes that demonstrate that the learned policies achieve 100\% accuracy on the test set. While the offline training of OCT-H can take days in large networks, the online application takes milliseconds.

Compressed Sensing: A Discrete Optimization Approach

We study the Compressed Sensing (CS) problem, which is the problem of finding the most sparse vector that satisfies a set of linear measurements up to some numerical tolerance. CS is a central problem in Statistics, Operations Research and Machine Learning which arises in applications such as signal processing, data compression and image reconstruction. We introduce an $\ell_2$ regularized formulation of CS which we reformulate as a mixed integer second order cone program. We derive a second order cone relaxation of this problem and show that under mild conditions on the regularization parameter, the resulting relaxation is equivalent to the well studied basis pursuit denoising problem. We present a semidefinite relaxation that strengthens the second order cone relaxation and develop a custom branch-and-bound algorithm that leverages our second order cone relaxation to solve instances of CS to certifiable optimality. Our numerical results show that our approach produces solutions that are on average $6.22\%$ more sparse than solutions returned by state of the art benchmark methods on synthetic data in minutes. On real world ECG data, for a given $\ell_2$ reconstruction error our approach produces solutions that are on average $9.95\%$ more sparse than benchmark methods, while for a given sparsity level our approach produces solutions that have on average $10.77\%$ lower reconstruction error than benchmark methods in minutes.