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The impressive performance of Large Language Models (LLMs) across various natural language processing tasks comes at the cost of vast computational resources and storage requirements. One-shot pruning techniques offer a way to alleviate these burdens by removing redundant weights without the need for retraining. Yet, the massive scale of LLMs often forces current pruning approaches to rely on heuristics instead of optimization-based techniques, potentially resulting in suboptimal compression. In this paper, we introduce ALPS, an optimization-based framework that tackles the pruning problem using the operator splitting technique and a preconditioned conjugate gradient-based post-processing step. Our approach incorporates novel techniques to accelerate and theoretically guarantee convergence while leveraging vectorization and GPU parallelism for efficiency. ALPS substantially outperforms state-of-the-art methods in terms of the pruning objective and perplexity reduction, particularly for highly sparse models. On the OPT-30B model with 70% sparsity, ALPS achieves a 13% reduction in test perplexity on the WikiText dataset and a 19% improvement in zero-shot benchmark performance compared to existing methods.

The increasing computational demands of modern neural networks present deployment challenges on resource-constrained devices. Network pruning offers a solution to reduce model size and computational cost while maintaining performance. However, most current pruning methods focus primarily on improving sparsity by reducing the number of nonzero parameters, often neglecting other deployment costs such as inference time, which are closely related to the number of floating-point operations (FLOPs). In this paper, we propose FALCON, a novel combinatorial-optimization-based framework for network pruning that jointly takes into account model accuracy (fidelity), FLOPs, and sparsity constraints. A main building block of our approach is an integer linear program (ILP) that simultaneously handles FLOP and sparsity constraints. We present a novel algorithm to approximately solve the ILP. We propose a novel first-order method for our optimization framework which makes use of our ILP solver. Using problem structure (e.g., the low-rank structure of approx. Hessian), we can address instances with millions of parameters. Our experiments demonstrate that FALCON achieves superior accuracy compared to other pruning approaches within a fixed FLOP budget. For instance, for ResNet50 with 20% of the total FLOPs retained, our approach improves the accuracy by 48% relative to state-of-the-art. Furthermore, in gradual pruning settings with re-training between pruning steps, our framework outperforms existing pruning methods, emphasizing the significance of incorporating both FLOP and sparsity constraints for effective network pruning.

Structured pruning is a promising approach for reducing the inference costs of large vision and language models. By removing carefully chosen structures, e.g., neurons or attention heads, the improvements from this approach can be realized on standard deep learning hardware. In this work, we focus on structured pruning in the one-shot (post-training) setting, which does not require model retraining after pruning. We propose a novel combinatorial optimization framework for this problem, based on a layer-wise reconstruction objective and a careful reformulation that allows for scalable optimization. Moreover, we design a new local combinatorial optimization algorithm, which exploits low-rank updates for efficient local search. Our framework is time and memory-efficient and considerably improves upon state-of-the-art one-shot methods on vision models (e.g., ResNet50, MobileNet) and language models (e.g., OPT-1.3B -- OPT-30B). For language models, e.g., OPT-2.7B, OSSCAR can lead to $125\times$ lower test perplexity on WikiText with $2\times$ inference time speedup in comparison to the state-of-the-art ZipLM approach. Our framework is also $6\times$ -- $8\times$ faster. Notably, our work considers models with tens of billions of parameters, which is up to $100\times$ larger than what has been previously considered in the structured pruning literature.

We study the often overlooked phenomenon, first noted in \cite{breiman2001random}, that random forests appear to reduce bias compared to bagging. Motivated by an interesting paper by \cite{mentch2020randomization}, where the authors argue that random forests reduce effective degrees of freedom and only outperform bagging ensembles in low signal-to-noise ratio (SNR) settings, we explore how random forests can uncover patterns in the data missed by bagging. We empirically demonstrate that in the presence of such patterns, random forests reduce bias along with variance and increasingly outperform bagging ensembles when SNR is high. Our observations offer insights into the real-world success of random forests across a range of SNRs and enhance our understanding of the difference between random forests and bagging ensembles with respect to the randomization injected into each split. Our investigations also yield practical insights into the importance of tuning $mtry$ in random forests.

We present FAST, an optimization framework for fast additive segmentation. FAST segments piecewise constant shape functions for each feature in a dataset to produce transparent additive models. The framework leverages a novel optimization procedure to fit these models $\sim$2 orders of magnitude faster than existing state-of-the-art methods, such as explainable boosting machines \citep{nori2019interpretml}. We also develop new feature selection algorithms in the FAST framework to fit parsimonious models that perform well. Through experiments and case studies, we show that FAST improves the computational efficiency and interpretability of additive models.

We propose a new optimization-based approach for feature selection in tree ensembles, an important problem in statistics and machine learning. Popular tree ensemble toolkits e.g., Gradient Boosted Trees and Random Forests support feature selection post-training based on feature importance scores, while very popular, they are known to have drawbacks. We propose Skinny Trees: an end-to-end toolkit for feature selection in tree ensembles where we train a tree ensemble while controlling the number of selected features. Our optimization-based approach learns an ensemble of differentiable trees, and simultaneously performs feature selection using a grouped $\ell_0$-regularizer. We use first-order methods for optimization and present convergence guarantees for our approach. We use a dense-to-sparse regularization scheduling scheme that can lead to more expressive and sparser tree ensembles. On 15 synthetic and real-world datasets, Skinny Trees can achieve $1.5\!\times\! -~620~\!\times\!$ feature compression rates, leading up to $10\times$ faster inference over dense trees, without any loss in performance. Skinny Trees lead to superior feature selection than many existing toolkits e.g., in terms of AUC performance for 25\% feature budget, Skinny Trees outperforms LightGBM by $10.2\%$ (up to $37.7\%$), and Random Forests by $3\%$ (up to $12.5\%$)., Comment: Accepted in AISTATS 2024

The decision tree is a flexible machine learning model that finds its success in numerous applications. It is usually fitted in a recursively greedy manner using CART. In this paper, we investigate the convergence rate of CART under a regression setting. First, we establish an upper bound on the prediction error of CART under a sufficient impurity decrease (SID) condition \cite{chi2022asymptotic} -- our result improves upon the known result by \cite{chi2022asymptotic} under a similar assumption. Furthermore, we provide examples that demonstrate the error bound cannot be further improved by more than a constant or a logarithmic factor. Second, we introduce a set of easily verifiable sufficient conditions for the SID condition. Specifically, we demonstrate that the SID condition can be satisfied in the case of an additive model, provided that the component functions adhere to a ``locally reverse Poincar{\'e} inequality". We discuss several well-known function classes in non-parametric estimation to illustrate the practical utility of this concept.

With the rising popularity of Large Language Models (LLMs), there has been an increasing interest in compression techniques that enable their efficient deployment. This study focuses on the Post-Training Quantization (PTQ) of LLMs. Drawing from recent advances, our work introduces QuantEase, a layer-wise quantization framework where individual layers undergo separate quantization. The problem is framed as a discrete-structured non-convex optimization, prompting the development of algorithms rooted in Coordinate Descent (CD) techniques. These CD-based methods provide high-quality solutions to the complex non-convex layer-wise quantization problems. Notably, our CD-based approach features straightforward updates, relying solely on matrix and vector operations, circumventing the need for matrix inversion or decomposition. We also explore an outlier-aware variant of our approach, allowing for retaining significant weights (outliers) with complete precision. Our proposal attains state-of-the-art performance in terms of perplexity and zero-shot accuracy in empirical evaluations across various LLMs and datasets, with relative improvements up to 15% over methods such as GPTQ. Leveraging careful linear algebra optimizations, QuantEase can quantize models like Falcon-180B on a single NVIDIA A100 GPU in $\sim$3 hours. Particularly noteworthy is our outlier-aware algorithm's capability to achieve near or sub-3-bit quantization of LLMs with an acceptable drop in accuracy, obviating the need for non-uniform quantization or grouping techniques, improving upon methods such as SpQR by up to two times in terms of perplexity.

We consider the problem of learning a sparse graph underlying an undirected Gaussian graphical model, a key problem in statistical machine learning. Given $n$ samples from a multivariate Gaussian distribution with $p$ variables, the goal is to estimate the $p \times p$ inverse covariance matrix (aka precision matrix), assuming it is sparse (i.e., has a few nonzero entries). We propose GraphL0BnB, a new estimator based on an $\ell_0$-penalized version of the pseudolikelihood function, while most earlier approaches are based on the $\ell_1$-relaxation. Our estimator can be formulated as a convex mixed integer program (MIP) which can be difficult to compute at scale using off-the-shelf commercial solvers. To solve the MIP, we propose a custom nonlinear branch-and-bound (BnB) framework that solves node relaxations with tailored first-order methods. As a by-product of our BnB framework, we propose large-scale solvers for obtaining good primal solutions that are of independent interest. We derive novel statistical guarantees (estimation and variable selection) for our estimator and discuss how our approach improves upon existing estimators. Our numerical experiments on real/synthetic datasets suggest that our method can solve, to near-optimality, problem instances with $p = 10^4$ -- corresponding to a symmetric matrix of size $p \times p$ with $p^2/2$ binary variables. We demonstrate the usefulness of GraphL0BnB versus various state-of-the-art approaches on a range of datasets.

We present FIRE, Fast Interpretable Rule Extraction, an optimization-based framework to extract a small but useful collection of decision rules from tree ensembles. FIRE selects sparse representative subsets of rules from tree ensembles, that are easy for a practitioner to examine. To further enhance the interpretability of the extracted model, FIRE encourages fusing rules during selection, so that many of the selected decision rules share common antecedents. The optimization framework utilizes a fusion regularization penalty to accomplish this, along with a non-convex sparsity-inducing penalty to aggressively select rules. Optimization problems in FIRE pose a challenge to off-the-shelf solvers due to problem scale and the non-convexity of the penalties. To address this, making use of problem-structure, we develop a specialized solver based on block coordinate descent principles; our solver performs up to 40x faster than existing solvers. We show in our experiments that FIRE outperforms state-of-the-art rule ensemble algorithms at building sparse rule sets, and can deliver more interpretable models compared to existing methods.

The sparse Mixture-of-Experts (Sparse-MoE) framework efficiently scales up model capacity in various domains, such as natural language processing and vision. Sparse-MoEs select a subset of the "experts" (thus, only a portion of the overall network) for each input sample using a sparse, trainable gate. Existing sparse gates are prone to convergence and performance issues when training with first-order optimization methods. In this paper, we introduce two improvements to current MoE approaches. First, we propose a new sparse gate: COMET, which relies on a novel tree-based mechanism. COMET is differentiable, can exploit sparsity to speed up computation, and outperforms state-of-the-art gates. Second, due to the challenging combinatorial nature of sparse expert selection, first-order methods are typically prone to low-quality solutions. To deal with this challenge, we propose a novel, permutation-based local search method that can complement first-order methods in training any sparse gate, e.g., Hash routing, Top-k, DSelect-k, and COMET. We show that local search can help networks escape bad initializations or solutions. We performed large-scale experiments on various domains, including recommender systems, vision, and natural language processing. On standard vision and recommender systems benchmarks, COMET+ (COMET with local search) achieves up to 13% improvement in ROC AUC over popular gates, e.g., Hash routing and Top-k, and up to 9% over prior differentiable gates e.g., DSelect-k. When Top-k and Hash gates are combined with local search, we see up to $100\times$ reduction in the budget needed for hyperparameter tuning. Moreover, for language modeling, our approach improves over the state-of-the-art MoEBERT model for distilling BERT on 5/7 GLUE benchmarks as well as SQuAD dataset., Comment: Accepted in KDD 2023

Most of the existing works on provable guarantees for low-rank matrix completion algorithms rely on some unrealistic assumptions such that matrix entries are sampled randomly or the sampling pattern has a specific structure. In this work, we establish theoretical guarantee for the exact and approximate low-rank matrix completion problems which can be applied to any deterministic sampling schemes. For this, we introduce a graph having observed entries as its edge set, and investigate its graph properties involving the performance of the standard constrained nuclear norm minimization algorithm. We theoretically and experimentally show that the algorithm can be successful as the observation graph is well-connected and has similar node degrees. Our result can be viewed as an extension of the works by Bhojanapalli and Jain [2014] and Burnwal and Vidyasagar [2020], in which the node degrees of the observation graph were assumed to be the same. In particular, our theory significantly improves their results when the underlying matrix is symmetric.

The sheer size of modern neural networks makes model serving a serious computational challenge. A popular class of compression techniques overcomes this challenge by pruning or sparsifying the weights of pretrained networks. While useful, these techniques often face serious tradeoffs between computational requirements and compression quality. In this work, we propose a novel optimization-based pruning framework that considers the combined effect of pruning (and updating) multiple weights subject to a sparsity constraint. Our approach, CHITA, extends the classical Optimal Brain Surgeon framework and results in significant improvements in speed, memory, and performance over existing optimization-based approaches for network pruning. CHITA's main workhorse performs combinatorial optimization updates on a memory-friendly representation of local quadratic approximation(s) of the loss function. On a standard benchmark of pretrained models and datasets, CHITA leads to significantly better sparsity-accuracy tradeoffs than competing methods. For example, for MLPNet with only 2% of the weights retained, our approach improves the accuracy by 63% relative to the state of the art. Furthermore, when used in conjunction with fine-tuning SGD steps, our method achieves significant accuracy gains over the state-of-the-art approaches.

Sharpness-Aware Minimization (SAM) is a recent optimization framework aiming to improve the deep neural network generalization, through obtaining flatter (i.e. less sharp) solutions. As SAM has been numerically successful, recent papers have studied the theoretical aspects of the framework and have shown SAM solutions are indeed flat. However, there has been limited theoretical exploration regarding statistical properties of SAM. In this work, we directly study the statistical performance of SAM, and present a new theoretical explanation of why SAM generalizes well. To this end, we study two statistical problems, neural networks with a hidden layer and kernel regression, and prove under certain conditions, SAM has smaller prediction error over Gradient Descent (GD). Our results concern both convex and non-convex settings, and show that SAM is particularly well-suited for non-convex problems. Additionally, we prove that in our setup, SAM solutions are less sharp as well, showing our results are in agreement with the previous work. Our theoretical findings are validated using numerical experiments on numerous scenarios, including deep neural networks.

Modern deep learning models are over-parameterized, where different optima can result in widely varying generalization performance. The Sharpness-Aware Minimization (SAM) technique modifies the fundamental loss function that steers gradient descent methods toward flatter minima, which are believed to exhibit enhanced generalization prowess. Our study delves into a specific variant of SAM known as micro-batch SAM (mSAM). This variation involves aggregating updates derived from adversarial perturbations across multiple shards (micro-batches) of a mini-batch during training. We extend a recently developed and well-studied general framework for flatness analysis to theoretically show that SAM achieves flatter minima than SGD, and mSAM achieves even flatter minima than SAM. We provide a thorough empirical evaluation of various image classification and natural language processing tasks to substantiate this theoretical advancement. We also show that contrary to previous work, mSAM can be implemented in a flexible and parallelizable manner without significantly increasing computational costs. Our implementation of mSAM yields superior generalization performance across a wide range of tasks compared to SAM, further supporting our theoretical framework., Comment: arXiv admin note: substantial text overlap with arXiv:2212.04343

Modern deep learning models are over-parameterized, where the optimization setup strongly affects the generalization performance. A key element of reliable optimization for these systems is the modification of the loss function. Sharpness-Aware Minimization (SAM) modifies the underlying loss function to guide descent methods towards flatter minima, which arguably have better generalization abilities. In this paper, we focus on a variant of SAM known as mSAM, which, during training, averages the updates generated by adversarial perturbations across several disjoint shards of a mini-batch. Recent work suggests that mSAM can outperform SAM in terms of test accuracy. However, a comprehensive empirical study of mSAM is missing from the literature -- previous results have mostly been limited to specific architectures and datasets. To that end, this paper presents a thorough empirical evaluation of mSAM on various tasks and datasets. We provide a flexible implementation of mSAM and compare the generalization performance of mSAM to the performance of SAM and vanilla training on different image classification and natural language processing tasks. We also conduct careful experiments to understand the computational cost of training with mSAM, its sensitivity to hyperparameters and its correlation with the flatness of the loss landscape. Our analysis reveals that mSAM yields superior generalization performance and flatter minima, compared to SAM, across a wide range of tasks without significantly increasing computational costs.

The impossibility theorem of fairness is a foundational result in the algorithmic fairness literature. It states that outside of special cases, one cannot exactly and simultaneously satisfy all three common and intuitive definitions of fairness - demographic parity, equalized odds, and predictive rate parity. This result has driven most works to focus on solutions for one or two of the metrics. Rather than follow suit, in this paper we present a framework that pushes the limits of the impossibility theorem in order to satisfy all three metrics to the best extent possible. We develop an integer-programming based approach that can yield a certifiably optimal post-processing method for simultaneously satisfying multiple fairness criteria under small violations. We show experiments demonstrating that our post-processor can improve fairness across the different definitions simultaneously with minimal model performance reduction. We also discuss applications of our framework for model selection and fairness explainability, thereby attempting to answer the question: who's the fairest of them all?

Decision trees are one of the most useful and popular methods in the machine learning toolbox. In this paper, we consider the problem of learning optimal decision trees, a combinatorial optimization problem that is challenging to solve at scale. A common approach in the literature is to use greedy heuristics, which may not be optimal. Recently there has been significant interest in learning optimal decision trees using various approaches (e.g., based on integer programming, dynamic programming) -- to achieve computational scalability, most of these approaches focus on classification tasks with binary features. In this paper, we present a new discrete optimization method based on branch-and-bound (BnB) to obtain optimal decision trees. Different from existing customized approaches, we consider both regression and classification tasks with continuous features. The basic idea underlying our approach is to split the search space based on the quantiles of the feature distribution -- leading to upper and lower bounds for the underlying optimization problem along the BnB iterations. Our proposed algorithm Quant-BnB shows significant speedups compared to existing approaches for shallow optimal trees on various real datasets.

Tree ensembles are powerful models that achieve excellent predictive performances, but can grow to unwieldy sizes. These ensembles are often post-processed (pruned) to reduce memory footprint and improve interpretability. We present ForestPrune, a novel optimization framework to post-process tree ensembles by pruning depth layers from individual trees. Since the number of nodes in a decision tree increases exponentially with tree depth, pruning deep trees drastically compactifies ensembles. We develop a specialized optimization algorithm to efficiently obtain high-quality solutions to problems under ForestPrune. Our algorithm typically reaches good solutions in seconds for medium-size datasets and ensembles, with 10000s of rows and 100s of trees, resulting in significant speedups over existing approaches. Our experiments demonstrate that ForestPrune produces parsimonious models that outperform models extracted by existing post-processing algorithms.

Decision tree ensembles are widely used and competitive learning models. Despite their success, popular toolkits for learning tree ensembles have limited modeling capabilities. For instance, these toolkits support a limited number of loss functions and are restricted to single task learning. We propose a flexible framework for learning tree ensembles, which goes beyond existing toolkits to support arbitrary loss functions, missing responses, and multi-task learning. Our framework builds on differentiable (a.k.a. soft) tree ensembles, which can be trained using first-order methods. However, unlike classical trees, differentiable trees are difficult to scale. We therefore propose a novel tensor-based formulation of differentiable trees that allows for efficient vectorization on GPUs. We perform experiments on a collection of 28 real open-source and proprietary datasets, which demonstrate that our framework can lead to 100x more compact and 23% more expressive tree ensembles than those by popular toolkits., Comment: Accepted at SIGKDD'2022

We present L0Learn: an open-source package for sparse linear regression and classification using $\ell_0$ regularization. L0Learn implements scalable, approximate algorithms, based on coordinate descent and local combinatorial optimization. The package is built using C++ and has user-friendly R and Python interfaces. L0Learn can address problems with millions of features, achieving competitive run times and statistical performance with state-of-the-art sparse learning packages. L0Learn is available on both CRAN and GitHub (https://cran.r-project.org/package=L0Learn and https://github.com/hazimehh/L0Learn)., Comment: Accepted to JMLR (MLOSS)

Fine-tuning of large pre-trained image and language models on small customized datasets has become increasingly popular for improved prediction and efficient use of limited resources. Fine-tuning requires identification of best models to transfer-learn from and quantifying transferability prevents expensive re-training on all of the candidate models/tasks pairs. In this paper, we show that the statistical problems with covariance estimation drive the poor performance of H-score -- a common baseline for newer metrics -- and propose shrinkage-based estimator. This results in up to 80% absolute gain in H-score correlation performance, making it competitive with the state-of-the-art LogME measure. Our shrinkage-based H-score is $3\times$-10$\times$ faster to compute compared to LogME. Additionally, we look into a less common setting of target (as opposed to source) task selection. We demonstrate previously overlooked problems in such settings with different number of labels, class-imbalance ratios etc. for some recent metrics e.g., NCE, LEEP that resulted in them being misrepresented as leading measures. We propose a correction and recommend measuring correlation performance against relative accuracy in such settings. We support our findings with ~164,000 (fine-tuning trials) experiments on both vision models and graph neural networks., Comment: Accepted in ECMLPKDD 2022

It is increasingly common to encounter prediction tasks in the biomedical sciences for which multiple datasets are available for model training. Common approaches such as pooling datasets and applying standard statistical learning methods can result in poor out-of-study prediction performance when datasets are heterogeneous. Theoretical and applied work has shown $\textit{multi-study ensembling}$ to be a viable alternative that leverages the variability across datasets in a manner that promotes model generalizability. Multi-study ensembling uses a two-stage $\textit{stacking}$ strategy which fits study-specific models and estimates ensemble weights separately. This approach ignores, however, the ensemble properties at the model-fitting stage, potentially resulting in a loss of efficiency. We therefore propose $\textit{optimal ensemble construction}$, an $\textit{all-in-one}$ approach to multi-study stacking whereby we jointly estimate ensemble weights as well as parameters associated with each study-specific model. We prove that limiting cases of our approach yield existing methods such as multi-study stacking and pooling datasets before model fitting. We propose an efficient block coordinate descent algorithm to optimize the proposed loss function. We compare our approach to standard methods by applying it to a multi-country COVID-19 dataset for baseline mortality prediction. We show that when little data is available for a country before the onset of the pandemic, leveraging data from other countries can substantially improve prediction accuracy. Importantly, our approach outperforms multi-study stacking and other standard methods in this application. We further characterize the method's performance in simulations. Our method remains competitive with or outperforms multi-study stacking and other earlier methods across a range of between-study heterogeneity levels., Comment: Manuscript: 27 pages, 6 figures, 4 tables; Supplement: 18 pages, 11 figures, 10 tables

In this paper we consider the problem of predicting survey response rates using a family of flexible and interpretable nonparametric models. The study is motivated by the US Census Bureau's well-known ROAM application which uses a linear regression model trained on the US Census Planning Database data to identify hard-to-survey areas. A crowdsourcing competition (Erdman and Bates, 2016) organized around ten years ago revealed that machine learning methods based on ensembles of regression trees led to the best performance in predicting survey response rates; however, the corresponding models could not be adopted for the intended application due to their black-box nature. We consider nonparametric additive models with small number of main and pairwise interaction effects using $\ell_0$-based penalization. From a methodological viewpoint, we study both computational and statistical aspects of our estimator; and discuss variants that incorporate strong hierarchical interactions. Our algorithms (opensourced on github) extend the computational frontiers of existing algorithms for sparse additive models, to be able to handle datasets relevant for the application we consider. We discuss and interpret findings from our model on the US Census Planning Database. In addition to being useful from an interpretability standpoint, our models lead to predictions that appear to be better than popular black-box machine learning methods based on gradient boosting and feedforward neural networks - suggesting that it is possible to have models that have the best of both worlds: good model accuracy and interpretability., Comment: 40 pages, 7 figures

The Mixture-of-Experts (MoE) architecture is showing promising results in improving parameter sharing in multi-task learning (MTL) and in scaling high-capacity neural networks. State-of-the-art MoE models use a trainable sparse gate to select a subset of the experts for each input example. While conceptually appealing, existing sparse gates, such as Top-k, are not smooth. The lack of smoothness can lead to convergence and statistical performance issues when training with gradient-based methods. In this paper, we develop DSelect-k: a continuously differentiable and sparse gate for MoE, based on a novel binary encoding formulation. The gate can be trained using first-order methods, such as stochastic gradient descent, and offers explicit control over the number of experts to select. We demonstrate the effectiveness of DSelect-k on both synthetic and real MTL datasets with up to $128$ tasks. Our experiments indicate that DSelect-k can achieve statistically significant improvements in prediction and expert selection over popular MoE gates. Notably, on a real-world, large-scale recommender system, DSelect-k achieves over $22\%$ improvement in predictive performance compared to Top-k. We provide an open-source implementation of DSelect-k., Comment: Appeared in NeurIPS 2021

We present a new algorithmic framework for grouped variable selection that is based on discrete mathematical optimization. While there exist several appealing approaches based on convex relaxations and nonconvex heuristics, we focus on optimal solutions for the $\ell_0$-regularized formulation, a problem that is relatively unexplored due to computational challenges. Our methodology covers both high-dimensional linear regression and nonparametric sparse additive modeling with smooth components. Our algorithmic framework consists of approximate and exact algorithms. The approximate algorithms are based on coordinate descent and local search, with runtimes comparable to popular sparse learning algorithms. Our exact algorithm is based on a standalone branch-and-bound (BnB) framework, which can solve the associated mixed integer programming (MIP) problem to certified optimality. By exploiting the problem structure, our custom BnB algorithm can solve to optimality problem instances with $5 \times 10^6$ features and $10^3$ observations in minutes to hours -- over $1000$ times larger than what is currently possible using state-of-the-art commercial MIP solvers. We also explore statistical properties of the $\ell_0$-based estimators. We demonstrate, theoretically and empirically, that our proposed estimators have an edge over popular group-sparse estimators in terms of statistical performance in various regimes. We provide an open-source implementation of our proposed framework.

We consider the problem of sparse nonnegative matrix factorization (NMF) using archetypal regularization. The goal is to represent a collection of data points as nonnegative linear combinations of a few nonnegative sparse factors with appealing geometric properties, arising from the use of archetypal regularization. We generalize the notion of robustness studied in Javadi and Montanari (2019) (without sparsity) to the notions of (a) strong robustness that implies each estimated archetype is close to the underlying archetypes and (b) weak robustness that implies there exists at least one recovered archetype that is close to the underlying archetypes. Our theoretical results on robustness guarantees hold under minimal assumptions on the underlying data, and applies to settings where the underlying archetypes need not be sparse. We present theoretical results and illustrative examples to strengthen the insights underlying the notions of robustness. We propose new algorithms for our optimization problem; and present numerical experiments on synthetic and real data sets that shed further insights into our proposed framework and theoretical developments.

We consider the least squares regression problem, penalized with a combination of the $\ell_{0}$ and squared $\ell_{2}$ penalty functions (a.k.a. $\ell_0 \ell_2$ regularization). Recent work shows that the resulting estimators are of key importance in many high-dimensional statistical settings. However, exact computation of these estimators remains a major challenge. Indeed, modern exact methods, based on mixed integer programming (MIP), face difficulties when the number of features $p \sim 10^4$. In this work, we present a new exact MIP framework for $\ell_0\ell_2$-regularized regression that can scale to $p \sim 10^7$, achieving speedups of at least $5000$x, compared to state-of-the-art exact methods. Unlike recent work, which relies on modern commercial MIP solvers, we design a specialized nonlinear branch-and-bound (BnB) framework, by critically exploiting the problem structure. A key distinguishing component in our framework lies in efficiently solving the node relaxations using a specialized first-order method, based on coordinate descent (CD). Our CD-based method effectively leverages information across the BnB nodes, through using warm starts, active sets, and gradient screening. In addition, we design a novel method for obtaining dual bounds from primal CD solutions, which certifiably works in high dimensions. Experiments on synthetic and real high-dimensional datasets demonstrate that our framework is not only significantly faster than the state of the art, but can also deliver certifiably optimal solutions to statistically challenging instances that cannot be handled with existing methods. We open source the implementation through our toolkit L0BnB.

Neural networks and tree ensembles are state-of-the-art learners, each with its unique statistical and computational advantages. We aim to combine these advantages by introducing a new layer for neural networks, composed of an ensemble of differentiable decision trees (a.k.a. soft trees). While differentiable trees demonstrate promising results in the literature, they are typically slow in training and inference as they do not support conditional computation. We mitigate this issue by introducing a new sparse activation function for sample routing, and implement true conditional computation by developing specialized forward and backward propagation algorithms that exploit sparsity. Our efficient algorithms pave the way for jointly training over deep and wide tree ensembles using first-order methods (e.g., SGD). Experiments on 23 classification datasets indicate over 10x speed-ups compared to the differentiable trees used in the literature and over 20x reduction in the number of parameters compared to gradient boosted trees, while maintaining competitive performance. Moreover, experiments on CIFAR, MNIST, and Fashion MNIST indicate that replacing dense layers in CNNs with our tree layer reduces the test loss by 7-53% and the number of parameters by 8x. We provide an open-source TensorFlow implementation with a Keras API., Comment: ICML 2020

We consider a discrete optimization formulation for learning sparse classifiers, where the outcome depends upon a linear combination of a small subset of features. Recent work has shown that mixed integer programming (MIP) can be used to solve (to optimality) $\ell_0$-regularized regression problems at scales much larger than what was conventionally considered possible. Despite their usefulness, MIP-based global optimization approaches are significantly slower compared to the relatively mature algorithms for $\ell_1$-regularization and heuristics for nonconvex regularized problems. We aim to bridge this gap in computation times by developing new MIP-based algorithms for $\ell_0$-regularized classification. We propose two classes of scalable algorithms: an exact algorithm that can handle $p\approx 50,000$ features in a few minutes, and approximate algorithms that can address instances with $p\approx 10^6$ in times comparable to the fast $\ell_1$-based algorithms. Our exact algorithm is based on the novel idea of \textsl{integrality generation}, which solves the original problem (with $p$ binary variables) via a sequence of mixed integer programs that involve a small number of binary variables. Our approximate algorithms are based on coordinate descent and local combinatorial search. In addition, we present new estimation error bounds for a class of $\ell_0$-regularized estimators. Experiments on real and synthetic data demonstrate that our approach leads to models with considerably improved statistical performance (especially, variable selection) when compared to competing methods., Comment: To appear in JMLR

In many learning settings, it is beneficial to augment the main features with pairwise interactions. Such interaction models can be often enhanced by performing variable selection under the so-called strong hierarchy constraint: an interaction is non-zero only if its associated main features are non-zero. Existing convex optimization based algorithms face difficulties in handling problems where the number of main features $p \sim 10^3$ (with total number of features $\sim p^2$). In this paper, we study a convex relaxation which enforces strong hierarchy and develop a highly scalable algorithm based on proximal gradient descent. We introduce novel screening rules that allow for solving the complicated proximal problem in parallel. In addition, we introduce a specialized active-set strategy with gradient screening for avoiding costly gradient computations. The framework can handle problems having dense design matrices, with $p = 50,000$ ($\sim 10^9$ interactions)---instances that are much larger than current state of the art. Experiments on real and synthetic data suggest that our toolkit hierScale outperforms the state of the art in terms of prediction and variable selection and can achieve over a 4900x speed-up., Comment: AISTATS 2020

The linear Support Vector Machine (SVM) is a classic classification technique in machine learning. Motivated by applications in modern high dimensional statistics, we consider penalized SVM problems involving the minimization of a hinge-loss function with a convex sparsity-inducing regularizer such as: the L1-norm on the coefficients, its grouped generalization and the sorted L1-penalty (aka Slope). Each problem can be expressed as a Linear Program (LP) and is computationally challenging when the number of features and/or samples is large -- the current state of algorithms for these problems is rather nascent when compared to the usual L2-regularized linear SVM. To this end, we propose new computational algorithms for these LPs by bringing together techniques from (a) classical column (and constraint) generation methods and (b) first order methods for non-smooth convex optimization -- techniques that are rarely used together for solving large scale LPs. These components have their respective strengths; and while they are found to be useful as separate entities, they have not been used together in the context of solving large scale LPs such as the ones studied herein. Our approach complements the strengths of (a) and (b) -- leading to a scheme that seems to significantly outperform commercial solvers as well as specialized implementations for these problems. We present numerical results on a series of real and synthetic datasets demonstrating the surprising effectiveness of classic column/constraint generation methods in the context of challenging LP-based machine learning tasks.

Gradient Boosting Machine (GBM) introduced by Friedman is a powerful supervised learning algorithm that is very widely used in practice---it routinely features as a leading algorithm in machine learning competitions such as Kaggle and the KDDCup. In spite of the usefulness of GBM in practice, our current theoretical understanding of this method is rather limited. In this work, we propose Randomized Gradient Boosting Machine (RGBM) which leads to substantial computational gains compared to GBM, by using a randomization scheme to reduce search in the space of weak-learners. We derive novel computational guarantees for RGBM. We also provide a principled guideline towards better step-size selection in RGBM that does not require a line search. Our proposed framework is inspired by a special variant of coordinate descent that combines the benefits of randomized coordinate descent and greedy coordinate descent; and may be of independent interest as an optimization algorithm. As a special case, our results for RGBM lead to superior computational guarantees for GBM. Our computational guarantees depend upon a curious geometric quantity that we call Minimal Cosine Angle, which relates to the density of weak-learners in the prediction space. On a series of numerical experiments on real datasets, we demonstrate the effectiveness of RGBM over GBM in terms of obtaining a model with good training and/or testing data fidelity with a fraction of the computational cost.

Logistic regression is one of the most popular methods in binary classification, wherein estimation of model parameters is carried out by solving the maximum likelihood (ML) optimization problem, and the ML estimator is defined to be the optimal solution of this problem. It is well known that the ML estimator exists when the data is non-separable, but fails to exist when the data is separable. First-order methods are the algorithms of choice for solving large-scale instances of the logistic regression problem. In this paper, we introduce a pair of condition numbers that measure the degree of non-separability or separability of a given dataset in the setting of binary classification, and we study how these condition numbers relate to and inform the properties and the convergence guarantees of first-order methods. When the training data is non-separable, we show that the degree of non-separability naturally enters the analysis and informs the properties and convergence guarantees of two standard first-order methods: steepest descent (for any given norm) and stochastic gradient descent. Expanding on the work of Bach, we also show how the degree of non-separability enters into the analysis of linear convergence of steepest descent (without needing strong convexity), as well as the adaptive convergence of stochastic gradient descent. When the training data is separable, first-order methods rather curiously have good empirical success, which is not well understood in theory. In the case of separable data, we demonstrate how the degree of separability enters into the analysis of $\ell_2$ steepest descent and stochastic gradient descent for delivering approximate-maximum-margin solutions with associated computational guarantees as well. This suggests that first-order methods can lead to statistically meaningful solutions in the separable case, even though the ML solution does not exist., Comment: 38 pages

An important metric of users' satisfaction and engagement within on-line streaming services is the user session length, i.e. the amount of time they spend on a service continuously without interruption. Being able to predict this value directly benefits the recommendation and ad pacing contexts in music and video streaming services. Recent research has shown that predicting the exact amount of time spent is highly nontrivial due to many external factors for which a user can end a session, and the lack of predictive covariates. Most of the other related literature on duration based user engagement has focused on dwell time for websites, for search and display ads, mainly for post-click satisfaction prediction or ad ranking. In this work we present a novel framework inspired by hierarchical Bayesian modeling to predict, at the moment of login, the amount of time a user will spend in the streaming service. The time spent by a user on a platform depends upon user-specific latent variables which are learned via hierarchical shrinkage. Our framework enjoys theoretical guarantees and naturally incorporates flexible parametric/nonparametric models on the covariates, including models robust to outliers. Our proposal is found to outperform state-of- the-art estimators in terms of efficiency and predictive performance on real world public and private datasets., Comment: 20 pages