Jeff Bilmes

Adhyyan Narang, Omid Sadeghi, Lillian J Ratliff et al. · uw

In the context of online interactive machine learning with combinatorial objectives, we extend purely submodular prior work to more general non-submodular objectives. This includes: (1) those that are additively decomposable into a sum of two terms (a monotone submodular and monotone supermodular term, known as a BP decomposition); and (2) those that are only weakly submodular. In both cases, this allows representing not only competitive (submodular) but also complementary (supermodular) relationships between objects, enhancing this setting to a broader range of applications (e.g., movie recommendations, medical treatments, etc.) where this is beneficial. In the two-term case, moreover, we study not only the more typical monolithic feedback approach but also a novel framework where feedback is available separately for each term. With real-world practicality and scalability in mind, we integrate Nystrom sketching techniques to significantly reduce the computational cost, including for the purely submodular case. In the Gaussian process contextual bandits setting, we show sub-linear theoretical regret bounds in all cases. We also empirically show good applicability to recommendation systems and data subset selection.

Jeff A. Bilmes, Gantavya Bhatt, Arnav M. Das · uw

Neural scaling laws appraise data through dataset size, while the Vendi Score uses quantum entropy to measure dataset value. We show both that common neural-scaling-law objectives and the Vendi Score are submodular. We further show that the Vendi Score is a special case of a broader class of submodular objectives that we call matrix spectral functions. This also includes determinantal (DPP) objectives, as well as many others. We also introduce weakly matrix monotone functions and show how they lead to weakly submodular matrix spectral functions, yielding a broad family of practical objectives for data appraisal. We develop secular-equation-based updates that avoid repeated eigendecompositions during greedy optimization, reducing marginal-gain evaluation for $m$-dimensional embeddings by an $O(m)$ factor relative to oracle queries. This yields an average empirical speedup of about 35,000x, making direct optimization of the Vendi Score feasible on ImageNet-1K-scale datasets. Thus enabled, we compare how well several objectives predict the value of training subsets for held-out test performance under fixed-size, class-balanced, and fixed training-budget regimes, including the Vendi Score, DPPs, facility location, and three new matrix spectral variants. Across multiple datasets, facility location performs the best. Direct optimization also reveals that, while the Vendi Score is predictive over moderate score ranges, pushing the objective to higher values can make it a poor downstream performance proxy. We also find that uniformly at random fixed-size subsets, both unconstrained and class-balanced, are remarkably concentrated in both appraisal scores and held-out performance. Finally, we show that size, class balance, and training budget do not alone determine data value: even when controlling for these factors, performance ranges smoothly from good to bad.

Sahil Verma, Keegan Hines, Jeff Bilmes et al.

The emerging capabilities of large language models (LLMs) have sparked concerns about their immediate potential for harmful misuse. The core approach to mitigate these concerns is the detection of harmful queries to the model. Current detection approaches are fallible, and are particularly susceptible to attacks that exploit mismatched generalization of model capabilities (e.g., prompts in low-resource languages or prompts provided in non-text modalities such as image and audio). To tackle this challenge, we propose OMNIGUARD, an approach for detecting harmful prompts across languages and modalities. Our approach (i) identifies internal representations of an LLM/MLLM that are aligned across languages or modalities and then (ii) uses them to build a language-agnostic or modality-agnostic classifier for detecting harmful prompts. OMNIGUARD improves harmful prompt classification accuracy by 11.57\% over the strongest baseline in a multilingual setting, by 20.44\% for image-based prompts, and sets a new SOTA for audio-based prompts. By repurposing embeddings computed during generation, OMNIGUARD is also very efficient ($\approx 120 \times$ faster than the next fastest baseline). Code and data are available at: https://github.com/vsahil/OmniGuard.

Sahil Verma, Royi Rassin, Arnav Das et al. · uw

Text-to-image models are trained using large datasets collected by scraping image-text pairs from the internet. These datasets often include private, copyrighted, and licensed material. Training models on such datasets enables them to generate images with such content, which might violate copyright laws and individual privacy. This phenomenon is termed imitation -- generation of images with content that has recognizable similarity to its training images. In this work we study the relationship between a concept's frequency in the training dataset and the ability of a model to imitate it. We seek to determine the point at which a model was trained on enough instances to imitate a concept -- the imitation threshold. We posit this question as a new problem: Finding the Imitation Threshold (FIT) and propose an efficient approach that estimates the imitation threshold without incurring the colossal cost of training multiple models from scratch. We experiment with two domains -- human faces and art styles -- for which we create four datasets, and evaluate three text-to-image models which were trained on two pretraining datasets. Our results reveal that the imitation threshold of these models is in the range of 200-600 images, depending on the domain and the model. The imitation threshold can provide an empirical basis for copyright violation claims and acts as a guiding principle for text-to-image model developers that aim to comply with copyright and privacy laws. We release the code and data at \url{https://github.com/vsahil/MIMETIC-2.git} and the project's website is hosted at \url{https://how-many-van-goghs-does-it-take.github.io}.

Sunil Thulasidasan, Tanmoy Bhattacharya, Jeff Bilmes et al.

We introduce a novel method to combat label noise when training deep neural networks for classification. We propose a loss function that permits abstention during training thereby allowing the DNN to abstain on confusing samples while continuing to learn and improve classification performance on the non-abstained samples. We show how such a deep abstaining classifier (DAC) can be used for robust learning in the presence of different types of label noise. In the case of structured or systematic label noise -- where noisy training labels or confusing examples are correlated with underlying features of the data-- training with abstention enables representation learning for features that are associated with unreliable labels. In the case of unstructured (arbitrary) label noise, abstention during training enables the DAC to be used as an effective data cleaner by identifying samples that are likely to have label noise. We provide analytical results on the loss function behavior that enable dynamic adaption of abstention rates based on learning progress during training. We demonstrate the utility of the deep abstaining classifier for various image classification tasks under different types of label noise; in the case of arbitrary label noise, we show significant improvements over previously published results on multiple image benchmarks. Source code is available at https://github.com/thulas/dac-label-noise

Arnav M. Das, Gantavya Bhatt, Lilly Kumari et al. · uw

Retrieval augmentation, the practice of retrieving additional data from large auxiliary pools, has emerged as an effective technique for enhancing model performance in the low-data regime. Prior approaches have employed only nearest-neighbor based strategies for data selection, which retrieve auxiliary samples with high similarity to instances in the target task. However, these approaches are prone to selecting highly redundant samples, since they fail to incorporate any notion of diversity. In our work, we first demonstrate that data selection strategies used in prior retrieval-augmented few-shot adaptation settings can be generalized using a class of functions known as Combinatorial Mutual Information (CMI) measures. We then propose COBRA (COmBinatorial Retrieval Augmentation), which employs an alternative CMI measure that considers both diversity and similarity to a target dataset. COBRA consistently outperforms previous retrieval approaches across image classification tasks and few-shot learning techniques when used to retrieve samples from LAION-2B. COBRA introduces negligible computational overhead to the cost of retrieval while providing significant gains in downstream model performance.

Gantavya Bhatt, Arnav Das, Jeff Bilmes · uw

Submodular functions, crucial for various applications, often lack practical learning methods for their acquisition. Seemingly unrelated, learning a scaling from oracles offering graded pairwise preferences (GPC) is underexplored, despite a rich history in psychometrics. In this paper, we introduce deep submodular peripteral networks (DSPNs), a novel parametric family of submodular functions, and methods for their training using a GPC-based strategy to connect and then tackle both of the above challenges. We introduce newly devised GPC-style ``peripteral'' loss which leverages numerically graded relationships between pairs of objects (sets in our case). Unlike traditional contrastive learning, or RHLF preference ranking, our method utilizes graded comparisons, extracting more nuanced information than just binary-outcome comparisons, and contrasts sets of any size (not just two). We also define a novel suite of automatic sampling strategies for training, including active-learning inspired submodular feedback. We demonstrate DSPNs' efficacy in learning submodularity from a costly target submodular function and demonstrate its superiority both for experimental design and online streaming applications.

Arnav Das, Gantavya Bhatt, Megh Bhalerao et al.

A major problem with Active Learning (AL) is high training costs since models are typically retrained from scratch after every query round. We start by demonstrating that standard AL on neural networks with warm starting fails, both to accelerate training and to avoid catastrophic forgetting when using fine-tuning over AL query rounds. We then develop a new class of techniques, circumventing this problem, by biasing further training towards previously labeled sets. We accomplish this by employing existing, and developing novel, replay-based Continual Learning (CL) algorithms that are effective at quickly learning the new without forgetting the old, especially when data comes from an evolving distribution. We call this paradigm Continual Active Learning (CAL). We show CAL achieves significant speedups using a plethora of replay schemes that use model distillation and that select diverse, uncertain points from the history. We conduct experiments across many data domains, including natural language, vision, medical imaging, and computational biology, each with different neural architectures and dataset sizes. CAL consistently provides a 3x reduction in training time, while retaining performance.

Jeff Bilmes

In this manuscript, we offer a gentle review of submodularity and supermodularity and their properties. We offer a plethora of submodular definitions; a full description of a number of example submodular functions and their generalizations; example discrete constraints; a discussion of basic algorithms for maximization, minimization, and other operations; a brief overview of continuous submodular extensions; and some historical applications. We then turn to how submodularity is useful in machine learning and artificial intelligence. This includes summarization, and we offer a complete account of the differences between and commonalities amongst sketching, coresets, extractive and abstractive summarization in NLP, data distillation and condensation, and data subset selection and feature selection. We discuss a variety of ways to produce a submodular function useful for machine learning, including heuristic hand-crafting, learning or approximately learning a submodular function or aspects thereof, and some advantages of the use of a submodular function as a coreset producer. We discuss submodular combinatorial information functions, and how submodularity is useful for clustering, data partitioning, parallel machine learning, active and semi-supervised learning, probabilistic modeling, and structured norms and loss functions.

Sunil Thulasidasan, Sushil Thapa, Sayera Dhaubhadel et al.

Refraining from confidently predicting when faced with categories of inputs different from those seen during training is an important requirement for the safe deployment of deep learning systems. While simple to state, this has been a particularly challenging problem in deep learning, where models often end up making overconfident predictions in such situations. In this work we present a simple, but highly effective approach to deal with out-of-distribution detection that uses the principle of abstention: when encountering a sample from an unseen class, the desired behavior is to abstain from predicting. Our approach uses a network with an extra abstention class and is trained on a dataset that is augmented with an uncurated set that consists of a large number of out-of-distribution (OoD) samples that are assigned the label of the abstention class; the model is then trained to learn an effective discriminator between in and out-of-distribution samples. We compare this relatively simple approach against a wide variety of more complex methods that have been proposed both for out-of-distribution detection as well as uncertainty modeling in deep learning, and empirically demonstrate its effectiveness on a wide variety of of benchmarks and deep architectures for image recognition and text classification, often outperforming existing approaches by significant margins. Given the simplicity and effectiveness of this method, we propose that this approach be used as a new additional baseline for future work in this domain.

Suraj Kothawade, Vishal Kaushal, Ganesh Ramakrishnan et al.

With the rapid growth of data, it is becoming increasingly difficult to train or improve deep learning models with the right subset of data. We show that this problem can be effectively solved at an additional labeling cost by targeted data subset selection(TSS) where a subset of unlabeled data points similar to an auxiliary set are added to the training data. We do so by using a rich class of Submodular Mutual Information (SMI) functions and demonstrate its effectiveness for image classification on CIFAR-10 and MNIST datasets. Lastly, we compare the performance of SMI functions for TSS with other state-of-the-art methods for closely related problems like active learning. Using SMI functions, we observe ~20-30% gain over the model's performance before re-training with added targeted subset; ~12% more than other methods.

Suraj Kothawade, Vishal Kaushal, Ganesh Ramakrishnan et al.

With ever-increasing dataset sizes, subset selection techniques are becoming increasingly important for a plethora of tasks. It is often necessary to guide the subset selection to achieve certain desiderata, which includes focusing or targeting certain data points, while avoiding others. Examples of such problems include: i)targeted learning, where the goal is to find subsets with rare classes or rare attributes on which the model is underperforming, and ii)guided summarization, where data (e.g., image collection, text, document or video) is summarized for quicker human consumption with specific additional user intent. Motivated by such applications, we present PRISM, a rich class of PaRameterIzed Submodular information Measures. Through novel functions and their parameterizations, PRISM offers a variety of modeling capabilities that enable a trade-off between desired qualities of a subset like diversity or representation and similarity/dissimilarity with a set of data points. We demonstrate how PRISM can be applied to the two real-world problems mentioned above, which require guided subset selection. In doing so, we show that PRISM interestingly generalizes some past work, therein reinforcing its broad utility. Through extensive experiments on diverse datasets, we demonstrate the superiority of PRISM over the state-of-the-art in targeted learning and in guided image-collection summarization

Vishal Kaushal, Suraj Kothawade, Ganesh Ramakrishnan et al.

We study submodular information measures as a rich framework for generic, query-focused, privacy sensitive, and update summarization tasks. While past work generally treats these problems differently ({\em e.g.}, different models are often used for generic and query-focused summarization), the submodular information measures allow us to study each of these problems via a unified approach. We first show that several previous query-focused and update summarization techniques have, unknowingly, used various instantiations of the aforesaid submodular information measures, providing evidence for the benefit and naturalness of these models. We then carefully study and demonstrate the modelling capabilities of the proposed functions in different settings and empirically verify our findings on both a synthetic dataset and an existing real-world image collection dataset (that has been extended by adding concept annotations to each image making it suitable for this task) and will be publicly released. We employ a max-margin framework to learn a mixture model built using the proposed instantiations of submodular information measures and demonstrate the effectiveness of our approach. While our experiments are in the context of image summarization, our framework is generic and can be easily extended to other summarization settings (e.g., videos or documents).

Rishabh Iyer, Ninad Khargonkar, Jeff Bilmes et al.

Information-theoretic quantities like entropy and mutual information have found numerous uses in machine learning. It is well known that there is a strong connection between these entropic quantities and submodularity since entropy over a set of random variables is submodular. In this paper, we study combinatorial information measures that generalize independence, (conditional) entropy, (conditional) mutual information, and total correlation defined over sets of (not necessarily random) variables. These measures strictly generalize the corresponding entropic measures since they are all parameterized via submodular functions that themselves strictly generalize entropy. Critically, we show that, unlike entropic mutual information in general, the submodular mutual information is actually submodular in one argument, holding the other fixed, for a large class of submodular functions whose third-order partial derivatives satisfy a non-negativity property. This turns out to include a number of practically useful cases such as the facility location and set-cover functions. We study specific instantiations of the submodular information measures on these, as well as the probabilistic coverage, graph-cut, and saturated coverage functions, and see that they all have mathematically intuitive and practically useful expressions. Regarding applications, we connect the maximization of submodular (conditional) mutual information to problems such as mutual-information-based, query-based, and privacy-preserving summarization -- and we connect optimizing the multi-set submodular mutual information to clustering and robust partitioning.

Rishabh Iyer, Jeff Bilmes

Submodular Functions are a special class of set functions, which generalize several information-theoretic quantities such as entropy and mutual information [1]. Submodular functions have subgradients and subdifferentials [2] and admit polynomial-time algorithms for minimization, both of which are fundamental characteristics of convex functions. Submodular functions also show signs similar to concavity. Submodular function maximization, though NP-hard, admits constant-factor approximation guarantees, and concave functions composed with modular functions are submodular. In this paper, we try to provide a more complete picture of the relationship between submodularity with concavity. We characterize the super-differentials and polyhedra associated with upper bounds and provide optimality conditions for submodular maximization using the-super differentials. This paper is a concise and shorter version of our longer preprint [3].

Baharan Mirzasoleiman, Jeff Bilmes, Jure Leskovec

Incremental gradient (IG) methods, such as stochastic gradient descent and its variants are commonly used for large scale optimization in machine learning. Despite the sustained effort to make IG methods more data-efficient, it remains an open question how to select a training data subset that can theoretically and practically perform on par with the full dataset. Here we develop CRAIG, a method to select a weighted subset (or coreset) of training data that closely estimates the full gradient by maximizing a submodular function. We prove that applying IG to this subset is guaranteed to converge to the (near)optimal solution with the same convergence rate as that of IG for convex optimization. As a result, CRAIG achieves a speedup that is inversely proportional to the size of the subset. To our knowledge, this is the first rigorous method for data-efficient training of general machine learning models. Our extensive set of experiments show that CRAIG, while achieving practically the same solution, speeds up various IG methods by up to 6x for logistic regression and 3x for training deep neural networks.

Sunil Thulasidasan, Gopinath Chennupati, Jeff Bilmes et al.

Mixup~\cite{zhang2017mixup} is a recently proposed method for training deep neural networks where additional samples are generated during training by convexly combining random pairs of images and their associated labels. While simple to implement, it has been shown to be a surprisingly effective method of data augmentation for image classification: DNNs trained with mixup show noticeable gains in classification performance on a number of image classification benchmarks. In this work, we discuss a hitherto untouched aspect of mixup training -- the calibration and predictive uncertainty of models trained with mixup. We find that DNNs trained with mixup are significantly better calibrated -- i.e., the predicted softmax scores are much better indicators of the actual likelihood of a correct prediction -- than DNNs trained in the regular fashion. We conduct experiments on a number of image classification architectures and datasets -- including large-scale datasets like ImageNet -- and find this to be the case. Additionally, we find that merely mixing features does not result in the same calibration benefit and that the label smoothing in mixup training plays a significant role in improving calibration. Finally, we also observe that mixup-trained DNNs are less prone to over-confident predictions on out-of-distribution and random-noise data. We conclude that the typical overconfidence seen in neural networks, even on in-distribution data is likely a consequence of training with hard labels, suggesting that mixup be employed for classification tasks where predictive uncertainty is a significant concern.

Tianyi Zhou, Hua Ouyang, Yi Chang et al.

We propose a new random pruning method (called "submodular sparsification (SS)") to reduce the cost of submodular maximization. The pruning is applied via a "submodularity graph" over the $n$ ground elements, where each directed edge is associated with a pairwise dependency defined by the submodular function. In each step, SS prunes a $1-1/\sqrt{c}$ (for $c>1$) fraction of the nodes using weights on edges computed based on only a small number ($O(\log n)$) of randomly sampled nodes. The algorithm requires $\log_{\sqrt{c}}n$ steps with a small and highly parallelizable per-step computation. An accuracy-speed tradeoff parameter $c$, set as $c = 8$, leads to a fast shrink rate $\sqrt{2}/4$ and small iteration complexity $\log_{2\sqrt{2}}n$. Analysis shows that w.h.p., the greedy algorithm on the pruned set of size $O(\log^2 n)$ can achieve a guarantee similar to that of processing the original dataset. In news and video summarization tasks, SS is able to substantially reduce both computational costs and memory usage, while maintaining (or even slightly exceeding) the quality of the original (and much more costly) greedy algorithm.

Tianyi Zhou, Jeff Bilmes

We propose a streaming submodular maximization algorithm "stream clipper" that performs as well as the offline greedy algorithm on document/video summarization in practice. It adds elements from a stream either to a solution set $S$ or to an extra buffer $B$ based on two adaptive thresholds, and improves $S$ by a final greedy step that starts from $S$ adding elements from $B$. During this process, swapping elements out of $S$ can occur if doing so yields improvements. The thresholds adapt based on if current memory utilization exceeds a budget, e.g., it increases the lower threshold, and removes from the buffer $B$ elements below the new lower threshold. We show that, while our approximation factor in the worst case is $1/2$ (like in previous work, and corresponding to the tight bound), we show that there are data-dependent conditions where our bound falls within the range $[1/2, 1-1/e]$. In news and video summarization experiments, the algorithm consistently outperforms other streaming methods, and, while using significantly less computation and memory, performs similarly to the offline greedy algorithm.

Weiran Wang, Raman Arora, Karen Livescu et al.

We consider learning representations (features) in the setting in which we have access to multiple unlabeled views of the data for learning while only one view is available for downstream tasks. Previous work on this problem has proposed several techniques based on deep neural networks, typically involving either autoencoder-like networks with a reconstruction objective or paired feedforward networks with a batch-style correlation-based objective. We analyze several techniques based on prior work, as well as new variants, and compare them empirically on image, speech, and text tasks. We find an advantage for correlation-based representation learning, while the best results on most tasks are obtained with our new variant, deep canonically correlated autoencoders (DCCAE). We also explore a stochastic optimization procedure for minibatch correlation-based objectives and discuss the time/performance trade-offs for kernel-based and neural network-based implementations.

Jennifer Gillenwater, Rishabh Iyer, Bethany Lusch et al.

We show that there is a largely unexplored class of functions (positive polymatroids) that can define proper discrete metrics over pairs of binary vectors and that are fairly tractable to optimize over. By exploiting submodularity, we are able to give hardness results and approximation algorithms for optimizing over such metrics. Additionally, we demonstrate empirically the effectiveness of these metrics and associated algorithms on both a metric minimization task (a form of clustering) and also a metric maximization task (generating diverse k-best lists).

Kai Wei, Rishabh Iyer, Shengjie Wang et al.

We study two mixed robust/average-case submodular partitioning problems that we collectively call Submodular Partitioning. These problems generalize both purely robust instances of the problem (namely max-min submodular fair allocation (SFA) and min-max submodular load balancing (SLB) and also generalize average-case instances (that is the submodular welfare problem (SWP) and submodular multiway partition (SMP). While the robust versions have been studied in the theory community, existing work has focused on tight approximation guarantees, and the resultant algorithms are not, in general, scalable to very large real-world applications. This is in contrast to the average case, where most of the algorithms are scalable. In the present paper, we bridge this gap, by proposing several new algorithms (including those based on greedy, majorization-minimization, minorization-maximization, and relaxation algorithms) that not only scale to large sizes but that also achieve theoretical approximation guarantees close to the state-of-the-art, and in some cases achieve new tight bounds. We also provide new scalable algorithms that apply to additive combinations of the robust and average-case extreme objectives. We show that these problems have many applications in machine learning (ML). This includes: 1) data partitioning and load balancing for distributed machine algorithms on parallel machines; 2) data clustering; and 3) multi-label image segmentation with (only) Boolean submodular functions via pixel partitioning. We empirically demonstrate the efficacy of our algorithms on real-world problems involving data partitioning for distributed optimization of standard machine learning objectives (including both convex and deep neural network objectives), and also on purely unsupervised (i.e., no supervised or semi-supervised learning, and no interactive segmentation) image segmentation.

Shengjie Wang, John T. Halloran, Jeff A. Bilmes et al.

Liquid chromatography coupled with tandem mass spectrometry, also known as shotgun proteomics, is a widely-used high-throughput technology for identifying proteins in complex biological samples. Analysis of the tens of thousands of fragmentation spectra produced by a typical shotgun proteomics experiment begins by assigning to each observed spectrum the peptide hypothesized to be responsible for generating the spectrum, typically done by searching each spectrum against a database of peptides. We have recently described a machine learning method---Dynamic Bayesian Network for Rapid Identification of Peptides (DRIP)---that not only achieves state-of-the-art spectrum identification performance on a variety of datasets but also provides a trainable model capable of returning valuable auxiliary information regarding specific peptide-spectrum matches. In this work, we present two significant improvements to DRIP. First, we describe how to use word lattices, which are widely used in natural language processing, to significantly speed up DRIP's computations. To our knowledge, all existing shotgun proteomics search engines compute independent scores between a given observed spectrum and each possible candidate peptide from the database. The key idea of the word lattice is to represent the set of candidate peptides in a single data structure, thereby allowing sharing of redundant computations among the different candidates. We demonstrate that using lattices in conjunction with DRIP leads to speedups on the order of tens across yeast and worm data sets. Second, we introduce a variant of DRIP that uses a discriminative training framework, performing maximum mutual entropy estimation rather than maximum likelihood estimation. This modification improves DRIP's statistical power, enabling us to increase the number of identified spectrum at a 1% false discovery rate on yeast and worm data sets.

Rishabh Iyer, Jeff A. Bilmes

We extend the recently introduced theory of Lovasz-Bregman (LB) divergences (Iyer & Bilmes 2012) in several ways. We show that they represent a distortion between a "score" and an "ordering", thus providing a new view of rank aggregation and order based clustering with interesting connections to web ranking. We show how the LB divergences have a number of properties akin to many permutation based metrics, and in fact have as special cases forms very similar to the Kendall-tau metric. We also show how the LB divergences subsume a number of commonly used ranking measures in information retrieval, like NDCG and AUC. Unlike the traditional permutation based metrics, however, the LB divergence naturally captures a notion of "confidence" in the orderings, thus providing a new representation to applications involving aggregating scores as opposed to just orderings. We show how a number of recently used web ranking models are forms of Lovasz-Bregman rank aggregation and also observe that a natural form of Mallow's model using the LB divergence has been used as conditional ranking models for the "Learning to Rank" problem.

Rishabh Iyer, Jeff A. Bilmes

We extend the work of Narasimhan and Bilmes [30] for minimizing set functions representable as a dierence between submodular functions. Similar to [30], our new algorithms are guaranteed to monotonically reduce the objective function at every step. We empirically and theoretically show that the per-iteration cost of our algorithms is much less than [30], and our algorithms can be used to efficiently minimize a dierence between submodular functions under various combinatorial constraints, a problem not previously addressed. We provide computational bounds and a hardness result on the multiplicative inapproximability of minimizing the dierence between submodular functions. We show, however, that it is possible to give worst-case additive bounds by providing a polynomial time computable lower-bound on the minima. Finally we show how a number of machine learning problems can be modeled as minimizing the dierence between submodular functions. We experimentally show the validity of our algorithms by testing them on the problem of feature selection with submodular cost features.

Tianyi Zhou, Jeff Bilmes, Carlos Guestrin

We reduce a broad class of machine learning problems, usually addressed by EM or sampling, to the problem of finding the $k$ extremal rays spanning the conical hull of a data point set. These $k$ "anchors" lead to a global solution and a more interpretable model that can even outperform EM and sampling on generalization error. To find the $k$ anchors, we propose a novel divide-and-conquer learning scheme "DCA" that distributes the problem to $\mathcal O(k\log k)$ same-type sub-problems on different low-D random hyperplanes, each can be solved by any solver. For the 2D sub-problem, we present a non-iterative solver that only needs to compute an array of cosine values and its max/min entries. DCA also provides a faster subroutine for other methods to check whether a point is covered in a conical hull, which improves algorithm design in multiple dimensions and brings significant speedup to learning. We apply our method to GMM, HMM, LDA, NMF and subspace clustering, then show its competitive performance and scalability over other methods on rich datasets.

Stefanie Jegelka, Jeff Bilmes

We study an extension of the classical graph cut problem, wherein we replace the modular (sum of edge weights) cost function by a submodular set function defined over graph edges. Special cases of this problem have appeared in different applications in signal processing, machine learning, and computer vision. In this paper, we connect these applications via the generic formulation of "cooperative graph cuts", for which we study complexity, algorithms, and connections to polymatroidal network flows. Finally, we compare the proposed algorithms empirically.

Rishabh Iyer, Stefanie Jegelka, Jeff Bilmes

We investigate three related and important problems connected to machine learning: approximating a submodular function everywhere, learning a submodular function (in a PAC-like setting [53]), and constrained minimization of submodular functions. We show that the complexity of all three problems depends on the 'curvature' of the submodular function, and provide lower and upper bounds that refine and improve previous results [3, 16, 18, 52]. Our proof techniques are fairly generic. We either use a black-box transformation of the function (for approximation and learning), or a transformation of algorithms to use an appropriate surrogate function (for minimization). Curiously, curvature has been known to influence approximations for submodular maximization [7, 55], but its effect on minimization, approximation and learning has hitherto been open. We complete this picture, and also support our theoretical claims by empirical results.

Rishabh Iyer, Jeff Bilmes

We investigate two new optimization problems -- minimizing a submodular function subject to a submodular lower bound constraint (submodular cover) and maximizing a submodular function subject to a submodular upper bound constraint (submodular knapsack). We are motivated by a number of real-world applications in machine learning including sensor placement and data subset selection, which require maximizing a certain submodular function (like coverage or diversity) while simultaneously minimizing another (like cooperative cost). These problems are often posed as minimizing the difference between submodular functions [14, 35] which is in the worst case inapproximable. We show, however, that by phrasing these problems as constrained optimization, which is more natural for many applications, we achieve a number of bounded approximation guarantees. We also show that both these problems are closely related and an approximation algorithm solving one can be used to obtain an approximation guarantee for the other. We provide hardness results for both problems thus showing that our approximation factors are tight up to log-factors. Finally, we empirically demonstrate the performance and good scalability properties of our algorithms.

Rishabh Iyer, Jeff Bilmes

We extend the recently introduced theory of Lovasz-Bregman (LB) divergences (Iyer & Bilmes, 2012) in several ways. We show that they represent a distortion between a 'score' and an 'ordering', thus providing a new view of rank aggregation and order based clustering with interesting connections to web ranking. We show how the LB divergences have a number of properties akin to many permutation based metrics, and in fact have as special cases forms very similar to the Kendall-$τ$ metric. We also show how the LB divergences subsume a number of commonly used ranking measures in information retrieval, like the NDCG and AUC. Unlike the traditional permutation based metrics, however, the LB divergence naturally captures a notion of "confidence" in the orderings, thus providing a new representation to applications involving aggregating scores as opposed to just orderings. We show how a number of recently used web ranking models are forms of Lovasz-Bregman rank aggregation and also observe that a natural form of Mallow's model using the LB divergence has been used as conditional ranking models for the 'Learning to Rank' problem.

Rishabh Iyer, Stefanie Jegelka, Jeff Bilmes

We present a practical and powerful new framework for both unconstrained and constrained submodular function optimization based on discrete semidifferentials (sub- and super-differentials). The resulting algorithms, which repeatedly compute and then efficiently optimize submodular semigradients, offer new and generalize many old methods for submodular optimization. Our approach, moreover, takes steps towards providing a unifying paradigm applicable to both submodular min- imization and maximization, problems that historically have been treated quite distinctly. The practicality of our algorithms is important since interest in submodularity, owing to its natural and wide applicability, has recently been in ascendance within machine learning. We analyze theoretical properties of our algorithms for minimization and maximization, and show that many state-of-the-art maximization algorithms are special cases. Lastly, we complement our theoretical analyses with supporting empirical experiments.

Jeff A. Bilmes

In this work, dynamic Bayesian multinets are introduced where a Markov chain state at time t determines conditional independence patterns between random variables lying within a local time window surrounding t. It is shown how information-theoretic criterion functions can be used to induce sparse, discriminative, and class-conditional network structures that yield an optimal approximation to the class posterior probability, and therefore are useful for the classification task. Using a new structure learning heuristic, the resulting models are tested on a medium-vocabulary isolated-word speech recognition task. It is demonstrated that these discriminatively structured dynamic Bayesian multinets, when trained in a maximum likelihood setting using EM, can outperform both HMMs and other dynamic Bayesian networks with a similar number of parameters.

Jeff A. Bilmes, Chris Bartels

This paper introduces new methodology to triangulate dynamic Bayesian networks (DBNs) and dynamic graphical models (DGMs). While most methods to triangulate such networks use some form of constrained elimination scheme based on properties of the underlying directed graph, we find it useful to view triangulation and elimination using properties only of the resulting undirected graph, obtained after the moralization step. We first briefly introduce the Graphical model toolkit (GMTK) and its notion of dynamic graphical models, one that slightly extends the standard notion of a DBN. We next introduce the 'boundary algorithm', a method to find the best boundary between partitions in a dynamic model. We find that using this algorithm, the notions of forward- and backward-interface become moot - namely, the size and fill-in of the best forward- and backward- interface are identical. Moreover, we observe that finding a good partition boundary allows for constrained elimination orders (and therefore graph triangulations) that are not possible using standard slice-by-slice constrained eliminations. More interestingly, with certain boundaries it is possible to obtain constrained elimination schemes that lie outside the space of possible triangulations using only unconstrained elimination. Lastly, we report triangulation results on invented graphs, standard DBNs from the literature, novel DBNs used in speech recognition research systems, and also random graphs. Using a number of different triangulation quality measures (max clique size, state-space, etc.), we find that with our boundary algorithm the triangulation quality can dramatically improve.

Hui Lin, Jeff A. Bilmes

We introduce a method to learn a mixture of submodular "shells" in a large-margin setting. A submodular shell is an abstract submodular function that can be instantiated with a ground set and a set of parameters to produce a submodular function. A mixture of such shells can then also be so instantiated to produce a more complex submodular function. What our algorithm learns are the mixture weights over such shells. We provide a risk bound guarantee when learning in a large-margin structured-prediction setting using a projected subgradient method when only approximate submodular optimization is possible (such as with submodular function maximization). We apply this method to the problem of multi-document summarization and produce the best results reported so far on the widely used NIST DUC-05 through DUC-07 document summarization corpora.

Mukund Narasimhan, Jeff A. Bilmes

We show that the class of strongly connected graphical models with treewidth at most k can be properly efficiently PAC-learnt with respect to the Kullback-Leibler Divergence. Previous approaches to this problem, such as those of Chow ([1]), and Ho gen ([7]) have shown that this class is PAC-learnable by reducing it to a combinatorial optimization problem. However, for k > 1, this problem is NP-complete ([15]), and so unless P=NP, these approaches will take exponential amounts of time. Our approach differs significantly from these, in that it first attempts to find approximate conditional independencies by solving (polynomially many) submodular optimization problems, and then using a dynamic programming formulation to combine the approximate conditional independence information to derive a graphical model with underlying graph of the tree-width specified. This gives us an efficient (polynomial time in the number of random variables) PAC-learning algorithm which requires only polynomial number of samples of the true distribution, and only polynomial running time.

Mukund Narasimhan, Jeff A. Bilmes

In this paper, we present an algorithm for minimizing the difference between two submodular functions using a variational framework which is based on (an extension of) the concave-convex procedure [17]. Because several commonly used metrics in machine learning, like mutual information and conditional mutual information, are submodular, the problem of minimizing the difference of two submodular problems arises naturally in many machine learning applications. Two such applications are learning discriminatively structured graphical models and feature selection under computational complexity constraints. A commonly used metric for measuring discriminative capacity is the EAR measure which is the difference between two conditional mutual information terms. Feature selection taking complexity considerations into account also fall into this framework because both the information that a set of features provide and the cost of computing and using the features can be modeled as submodular functions. This problem is NP-hard, and we give a polynomial time heuristic for it. We also present results on synthetic data to show that classifiers based on discriminative graphical models using this algorithm can significantly outperform classifiers based on generative graphical models.

Rishabh Iyer, Jeff Bilmes

We extend the work of Narasimhan and Bilmes [30] for minimizing set functions representable as a difference between submodular functions. Similar to [30], our new algorithms are guaranteed to monotonically reduce the objective function at every step. We empirically and theoretically show that the per-iteration cost of our algorithms is much less than [30], and our algorithms can be used to efficiently minimize a difference between submodular functions under various combinatorial constraints, a problem not previously addressed. We provide computational bounds and a hardness result on the mul- tiplicative inapproximability of minimizing the difference between submodular functions. We show, however, that it is possible to give worst-case additive bounds by providing a polynomial time computable lower-bound on the minima. Finally we show how a number of machine learning problems can be modeled as minimizing the difference between submodular functions. We experimentally show the validity of our algorithms by testing them on the problem of feature selection with submodular cost features.

Amarnag Subramanya, Alvin Raj, Jeff A. Bilmes et al.

We introduce a new dynamic model with the capability of recognizing both activities that an individual is performing as well as where that ndividual is located. Our model is novel in that it utilizes a dynamic graphical model to jointly estimate both activity and spatial context over time based on the simultaneous use of asynchronous observations consisting of GPS measurements, and measurements from a small mountable sensor board. Joint inference is quite desirable as it has the ability to improve accuracy of the model. A key goal, however, in designing our overall system is to be able to perform accurate inference decisions while minimizing the amount of hardware an individual must wear. This minimization leads to greater comfort and flexibility, decreased power requirements and therefore increased battery life, and reduced cost. We show results indicating that our joint measurement model outperforms measurements from either the sensor board or GPS alone, using two types of probabilistic inference procedures, namely particle filtering and pruned exact inference.

Chris Bartels, Jeff A. Bilmes

We observe that certain large-clique graph triangulations can be useful to reduce both computational and space requirements when making queries on mixed stochastic/deterministic graphical models. We demonstrate that many of these large-clique triangulations are non-minimal and are thus unattainable via the variable elimination algorithm. We introduce ancestral pairs as the basis for novel triangulation heuristics and prove that no more than the addition of edges between ancestral pairs need be considered when searching for state space optimal triangulations in such graphs. Empirical results on random and real world graphs show that the resulting triangulations that yield significant speedups are almost always non-minimal. We also give an algorithm and correctness proof for determining if a triangulation can be obtained via elimination, and we show that the decision problem associated with finding optimal state space triangulations in this mixed stochastic/deterministic setting is NP-complete.

Marina Meila, Kapil Phadnis, Arthur Patterson et al.

We analyze the generalized Mallows model, a popular exponential model over rankings. Estimating the central (or consensus) ranking from data is NP-hard. We obtain the following new results: (1) We show that search methods can estimate both the central ranking pi0 and the model parameters theta exactly. The search is n! in the worst case, but is tractable when the true distribution is concentrated around its mode; (2) We show that the generalized Mallows model is jointly exponential in (pi0; theta), and introduce the conjugate prior for this model class; (3) The sufficient statistics are the pairwise marginal probabilities that item i is preferred to item j. Preliminary experiments confirm the theoretical predictions and compare the new algorithm and existing heuristics.

Jeff Bilmes, Andrew Ng

This is the Proceedings of the Twenty-Fifth Conference on Uncertainty in Artificial Intelligence, which was held in Montreal, QC, Canada, June 18 - 21 2009.

Andrew Guillory, Jeff A. Bilmes

We consider active, semi-supervised learning in an offline transductive setting. We show that a previously proposed error bound for active learning on undirected weighted graphs can be generalized by replacing graph cut with an arbitrary symmetric submodular function. Arbitrary non-symmetric submodular functions can be used via symmetrization. Different choices of submodular functions give different versions of the error bound that are appropriate for different kinds of problems. Moreover, the bound is deterministic and holds for adversarially chosen labels. We show exactly minimizing this error bound is NP-complete. However, we also introduce for any submodular function an associated active semi-supervised learning method that approximately minimizes the corresponding error bound. We show that the error bound is tight in the sense that there is no other bound of the same form which is better. Our theoretical results are supported by experiments on real data.