M. Pawan Kumar

ICLR Conference 2025 Conference Paper

Do as I do (Safely): Mitigating Task-Specific Fine-tuning Risks in Large Language Models

• Francisco Eiras
• Aleksandar Petrov
• Philip H. S. Torr
• M. Pawan Kumar
• Adel Bibi

Recent research shows that fine-tuning on benign instruction-following data can inadvertently undo the safety alignment process and increase a model's propensity to comply with harmful queries. While instruction-following fine-tuning is important, task-specific fine-tuning-where models are trained on datasets with clear ground truth answers (e.g., multiple choice questions)-can enhance model performance on specialized downstream tasks. Understanding and mitigating safety risks in the task-specific setting remains distinct from the instruction-following context due to structural differences in the data. Our work demonstrates how malicious actors can subtly manipulate the structure of almost *any* task-specific dataset to foster significantly more dangerous model behaviors, while maintaining an appearance of innocuity and reasonable downstream task performance. To address this issue, we propose a novel mitigation strategy that mixes in safety data which *mimics* the task format and prompting style of the user data, showing this is significantly more effective and efficient than existing baselines at re-establishing safety alignment while maintaining similar task performance.

ICML Conference 2024 Conference Paper

Efficient Error Certification for Physics-Informed Neural Networks

• Francisco Eiras
• Adel Bibi
• Rudy Bunel
• Krishnamurthy Dvijotham
• Philip H. S. Torr
• M. Pawan Kumar

Recent work provides promising evidence that Physics-Informed Neural Networks (PINN) can efficiently solve partial differential equations (PDE). However, previous works have failed to provide guarantees on the worst-case residual error of a PINN across the spatio-temporal domain - a measure akin to the tolerance of numerical solvers - focusing instead on point-wise comparisons between their solution and the ones obtained by a solver on a set of inputs. In real-world applications, one cannot consider tests on a finite set of points to be sufficient grounds for deployment, as the performance could be substantially worse on a different set. To alleviate this issue, we establish guaranteed error-based conditions for PINNs over their continuous applicability domain. To verify the extent to which they hold, we introduce $\partial$-CROWN: a general, efficient and scalable post-training framework to bound PINN residual errors. We demonstrate its effectiveness in obtaining tight certificates by applying it to two classically studied PINNs – Burgers’ and Schrödinger’s equations –, and two more challenging ones with real-world applications – the Allan-Cahn and Diffusion-Sorption equations.

ICLR Conference 2024 Conference Paper

Expressive Losses for Verified Robustness via Convex Combinations

• Alessandro De Palma
• Rudy Bunel
• Krishnamurthy Dvijotham
• M. Pawan Kumar
• Robert Stanforth
• Alessio Lomuscio

In order to train networks for verified adversarial robustness, it is common to over-approximate the worst-case loss over perturbation regions, resulting in networks that attain verifiability at the expense of standard performance. As shown in recent work, better trade-offs between accuracy and robustness can be obtained by carefully coupling adversarial training with over-approximations. We hypothesize that the expressivity of a loss function, which we formalize as the ability to span a range of trade-offs between lower and upper bounds to the worst-case loss through a single parameter (the over-approximation coefficient), is key to attaining state-of-the-art performance. To support our hypothesis, we show that trivial expressive losses, obtained via convex combinations between adversarial attacks and IBP bounds, yield state-of-the-art results across a variety of settings in spite of their conceptual simplicity. We provide a detailed analysis of the relationship between the over-approximation coefficient and performance profiles across different expressive losses, showing that, while expressivity is essential, better approximations of the worst-case loss are not necessarily linked to superior robustness-accuracy trade-offs.

JMLR Journal 2024 Journal Article

Scaling the Convex Barrier with Sparse Dual Algorithms

• Alessandro De Palma
• Harkirat Singh Behl
• Rudy Bunel
• Philip H.S. Torr
• M. Pawan Kumar

Tight and efficient neural network bounding is crucial to the scaling of neural network verification systems. Many efficient bounding algorithms have been presented recently, but they are often too loose to verify more challenging properties. This is due to the weakness of the employed relaxation, which is usually a linear program of size linear in the number of neurons. While a tighter linear relaxation for piecewise-linear activations exists, it comes at the cost of exponentially many constraints and currently lacks an efficient customized solver. We alleviate this deficiency by presenting two novel dual algorithms: one operates a subgradient method on a small active set of dual variables, the other exploits the sparsity of Frank-Wolfe type optimizers to incur only a linear memory cost. Both methods recover the strengths of the new relaxation: tightness and a linear separation oracle. At the same time, they share the benefits of previous dual approaches for weaker relaxations: massive parallelism, GPU implementation, low cost per iteration and valid bounds at any time. As a consequence, we can obtain better bounds than off-the-shelf solvers in only a fraction of their running time, attaining significant formal verification speed-ups. [abs] [ pdf ][ bib ] [ code ] &copy JMLR 2024. ( edit, beta )

JMLR Journal 2022 Journal Article

A Stochastic Bundle Method for Interpolation

• Alasdair Paren
• Leonard Berrada
• Rudra P. K. Poudel
• M. Pawan Kumar

We propose a novel method for training deep neural networks that are capable of interpolation, that is, driving the empirical loss to zero. At each iteration, our method constructs a stochastic approximation of the learning objective. The approximation, known as a bundle, is a pointwise maximum of linear functions. Our bundle contains a constant function that lower bounds the empirical loss. This enables us to compute an automatic adaptive learning rate, thereby providing an accurate solution. In addition, our bundle includes linear approximations computed at the current iterate and other linear estimates of the DNN parameters. The use of these additional approximations makes our method significantly more robust to its hyperparameters. Based on its desirable empirical properties, we term our method Bundle Optimisation for Robust and Accurate Training (BORAT). In order to operationalise BORAT, we design a novel algorithm for optimising the bundle approximation efficiently at each iteration. We establish the theoretical convergence of BORAT in both convex and non-convex settings. Using standard publicly available data sets, we provide a thorough comparison of BORAT to other single hyperparameter optimisation algorithms. Our experiments demonstrate BORAT matches the state-of-the-art generalisation performance for these methods and is the most robust. [abs] [ pdf ][ bib ] [ code ] &copy JMLR 2022. ( edit, beta )

TMLR Journal 2022 Journal Article

ANCER: Anisotropic Certification via Sample-wise Volume Maximization

• Francisco Eiras
• Motasem Alfarra
• Philip Torr
• M. Pawan Kumar
• Puneet K. Dokania
• Bernard Ghanem
• Adel Bibi

Randomized smoothing has recently emerged as an effective tool that enables certification of deep neural network classifiers at scale. All prior art on randomized smoothing has focused on isotropic $\ell_p$ certification, which has the advantage of yielding certificates that can be easily compared among isotropic methods via $\ell_p$-norm radius. However, isotropic certification limits the region that can be certified around an input to worst-case adversaries, i.e., it cannot reason about other "close", potentially large, constant prediction safe regions. To alleviate this issue, (i) we theoretically extend the isotropic randomized smoothing $\ell_1$ and $\ell_2$ certificates to their generalized anisotropic counterparts following a simplified analysis. Moreover, (ii) we propose evaluation metrics allowing for the comparison of general certificates - a certificate is superior to another if it certifies a superset region - with the quantification of each certificate through the volume of the certified region. We introduce ANCER, a framework for obtaining anisotropic certificates for a given test set sample via volume maximization. We achieve it by generalizing memory-based certification of data-dependent classifiers. Our empirical results demonstrate that ANCER achieves state-of-the-art $\ell_1$ and $\ell_2$ certified accuracy on CIFAR-10 and ImageNet in the data-dependence setting, while certifying larger regions in terms of volume, highlighting the benefits of moving away from isotropic analysis.

TMLR Journal 2022 Journal Article

Faking Interpolation Until You Make It

• Alasdair Paren
• Rudra P. K. Poudel
• M. Pawan Kumar

Deep over-parameterized neural networks exhibit the interpolation property on many data sets. Specifically, these models can achieve approximately zero loss on all training samples simultaneously. This property has been exploited to develop optimisation algorithms for this setting. These algorithms use the fact that the optimal loss value is known to employ a variation of a Polyak step size calculated on each stochastic batch of data. We introduce a novel extension of this idea to tasks where the interpolation property does not hold. As we no longer have access to the optimal loss values a priori, we instead estimate them for each sample online. To realise this, we introduce a simple but highly effective heuristic for approximating the optimal value based on previous loss evaluations. We provide rigorous experimentation on a range of problems. From our empirical analysis we demonstrate the effectiveness of our approach, which outperforms other single hyperparameter optimisation methods.

TMLR Journal 2022 Journal Article

Lookback for Learning to Branch

• Prateek Gupta
• Elias Boutros Khalil
• Didier Chételat
• Maxime Gasse
• Andrea Lodi
• Yoshua Bengio
• M. Pawan Kumar

The expressive and computationally inexpensive bipartite Graph Neural Networks (GNN) have been shown to be an important component of deep learning based Mixed-Integer Linear Program (MILP) solvers. Recent works have demonstrated the effectiveness of such GNNs in replacing the branching (variable selection) heuristic in branch-and-bound B&B solvers. These GNNs are trained, offline and on a collection of MILPs, to imitate a very good but computationally expensive branching heuristic, strong branching. Given that B&B results in a tree of sub-MILPs, we ask (a) whether there are strong dependencies exhibited by the target heuristic among the neighboring nodes of the B&B tree, and (b) if so, whether we can incorporate them in our training procedure. Specifically, we find that with the strong branching heuristic, a child node's best choice was often the parent's second-best choice. We call this the "lookback" phenomenon. Surprisingly, the typical branching GNN of Gasse et al. (2019) often misses this simple "answer". To imitate the target behavior more closely by incorporating the lookback phenomenon in GNNs, we propose two methods: (a) target smoothing for the standard cross-entropy loss function, and (b) adding a Parent-as-Target (PAT) Lookback regularizer term. Finally, we propose a model selection framework to incorporate harder-to-formulate objectives such as solving time in the final models. Through extensive experimentation on standard benchmark instances, we show that our proposal results in up to 22% decrease in the size of the B&B tree and up to 15% improvement in the solving times.

UAI Conference 2021 Conference Paper

Generating adversarial examples with graph neural networks

• Florian Jaeckle
• M. Pawan Kumar

Recent years have witnessed the deployment of adversarial attacks to evaluate the robustness of Neural Networks. Past work in this field has relied on traditional optimization algorithms that ignore the inherent structure of the problem and data, or generative methods that rely purely on learning and often fail to generate adversarial examples where they are hard to find. To alleviate these deficiencies, we propose a novel attack based on a graph neural network (GNN) that takes advantage of the strengths of both approaches; we call it AdvGNN. Our GNN architecture closely resembles the network we wish to attack. During inference, we perform forward-backward passes through the GNN layers to guide an iterative procedure towards adversarial examples. During training, its parameters are estimated via a loss function that encourages the efficient computation of adversarial examples over a time horizon. We show that our method beats state-of-the-art adversarial attacks, including PGD-attack, MI-FGSM, and Carlini and Wagner attack, reducing the time required to generate adversarial examples with small perturbation norms by over 65%. Moreover, AdvGNN achieves good generalization performance on unseen networks. Finally, we provide a new challenging dataset specifically designed to allow for a more illustrative comparison of adversarial attacks.

NeurIPS Conference 2021 Conference Paper

Make Sure You're Unsure: A Framework for Verifying Probabilistic Specifications

• Leonard Berrada
• Sumanth Dathathri
• Krishnamurthy Dvijotham
• Robert Stanforth
• Rudy R. Bunel
• Jonathan Uesato
• Sven Gowal
• M. Pawan Kumar

Most real world applications require dealing with stochasticity like sensor noise or predictive uncertainty, where formal specifications of desired behavior are inherently probabilistic. Despite the promise of formal verification in ensuring the reliability of neural networks, progress in the direction of probabilistic specifications has been limited. In this direction, we first introduce a general formulation of probabilistic specifications for neural networks, which captures both probabilistic networks (e. g. , Bayesian neural networks, MC-Dropout networks) and uncertain inputs (distributions over inputs arising from sensor noise or other perturbations). We then propose a general technique to verify such specifications by generalizing the notion of Lagrangian duality, replacing standard Lagrangian multipliers with "functional multipliers" that can be arbitrary functions of the activations at a given layer. We show that an optimal choice of functional multipliers leads to exact verification (i. e. , sound and complete verification), and for specific forms of multipliers, we develop tractable practical verification algorithms. We empirically validate our algorithms by applying them to Bayesian Neural Networks (BNNs) and MC Dropout Networks, and certifying properties such as adversarial robustness and robust detection of out-of-distribution (OOD) data. On these tasks we are able to provide significantly stronger guarantees when compared to prior work -- for instance, for a VGG-64 MC-Dropout CNN trained on CIFAR-10 in a verification-agnostic manner, we improve the certified AUC (a verified lower bound on the true AUC) for robust OOD detection (on CIFAR-100) from $0 \% \rightarrow 29\%$. Similarly, for a BNN trained on MNIST, we improve on the $\ell_\infty$ robust accuracy from $60. 2 \% \rightarrow 74. 6\%$. Further, on a novel specification -- distributionally robust OOD detection -- we improve on the certified AUC from $5\% \rightarrow 23\%$.

NeurIPS Conference 2021 Conference Paper

Overcoming the Convex Barrier for Simplex Inputs

• Harkirat Singh Behl
• M. Pawan Kumar
• Philip Torr
• Krishnamurthy Dvijotham

Recent progress in neural network verification has challenged the notion of a convex barrier, that is, an inherent weakness in the convex relaxation of the output of a neural network. Specifically, there now exists a tight relaxation for verifying the robustness of a neural network to $\ell_\infty$ input perturbations, as well as efficient primal and dual solvers for the relaxation. Buoyed by this success, we consider the problem of developing similar techniques for verifying robustness to input perturbations within the probability simplex. We prove a somewhat surprising result that, in this case, not only can one design a tight relaxation that overcomes the convex barrier, but the size of the relaxation remains linear in the number of neurons, thereby leading to simpler and more efficient algorithms. We establish the scalability of our overall approach via the specification of $\ell_1$ robustness for CIFAR-10 and MNIST classification, where our approach improves the state of the art verified accuracy by up to $14. 4\%$. Furthermore, we establish its accuracy on a novel and highly challenging task of verifying the robustness of a multi-modal (text and image) classifier to arbitrary changes in its textual input.

ICLR Conference 2021 Conference Paper

Scaling the Convex Barrier with Active Sets

• Alessandro De Palma
• Harkirat Singh Behl
• Rudy Bunel
• Philip H. S. Torr
• M. Pawan Kumar

Tight and efficient neural network bounding is of critical importance for the scaling of neural network verification systems. A number of efficient specialised dual solvers for neural network bounds have been presented recently, but they are often too loose to verify more challenging properties. This lack of tightness is linked to the weakness of the employed relaxation, which is usually a linear program of size linear in the number of neurons. While a tighter linear relaxation for piecewise linear activations exists, it comes at the cost of exponentially many constraints and thus currently lacks an efficient customised solver. We alleviate this deficiency via a novel dual algorithm that realises the full potential of the new relaxation by operating on a small active set of dual variables. Our method recovers the strengths of the new relaxation in the dual space: tightness and a linear separation oracle. At the same time, it shares the benefits of previous dual approaches for weaker relaxations: massive parallelism, GPU implementation, low cost per iteration and valid bounds at any time. As a consequence, we obtain better bounds than off-the-shelf solvers in only a fraction of their running time and recover the speed-accuracy trade-offs of looser dual solvers if the computational budget is small. We demonstrate that this results in significant formal verification speed-ups.

JMLR Journal 2020 Journal Article

Branch and Bound for Piecewise Linear Neural Network Verification

• Rudy Bunel
• Jingyue Lu
• Ilker Turkaslan
• Philip H.S. Torr
• Pushmeet Kohli
• M. Pawan Kumar

The success of Deep Learning and its potential use in many safety-critical applicationshas motivated research on formal verification of Neural Network (NN) models. In thiscontext, verification involves proving or disproving that an NN model satisfies certaininput-output properties. Despite the reputation of learned NN models as black boxes,and the theoretical hardness of proving useful properties about them, researchers havebeen successful in verifying some classes of models by exploiting their piecewise linearstructure and taking insights from formal methods such as Satisifiability Modulo Theory.However, these methods are still far from scaling to realistic neural networks. To facilitateprogress on this crucial area, we exploit the Mixed Integer Linear Programming (MIP) formulation of verification to propose a family of algorithms based on Branch-and-Bound (BaB). We show that our family contains previous verification methods as special cases.With the help of the BaB framework, we make three key contributions. Firstly, we identifynew methods that combine the strengths of multiple existing approaches, accomplishingsignificant performance improvements over previous state of the art. Secondly, we introducean effective branching strategy on ReLU non-linearities. This branching strategy allows usto efficiently and successfully deal with high input dimensional problems with convolutionalnetwork architecture, on which previous methods fail frequently. Finally, we proposecomprehensive test data sets and benchmarks which includes a collection of previouslyreleased testcases. We use the data sets to conduct a thorough experimental comparison ofexisting and new algorithms and to provide an inclusive analysis of the factors impactingthe hardness of verification problems. [abs] [ pdf ][ bib ] &copy JMLR 2020. ( edit, beta )

UAI Conference 2020 Conference Paper

Lagrangian Decomposition for Neural Network Verification

• Rudy Bunel
• Alessandro De Palma
• Alban Desmaison
• Krishnamurthy Dvijotham
• Pushmeet Kohli
• Philip H. S. Torr
• M. Pawan Kumar

A fundamental component of neural network verification is the computation of bounds on the values their outputs can take. Previous methods have either used off-the-shelf solvers, discarding the problem structure, or relaxed the problem even further, making the bounds unnecessarily loose. We propose a novel approach based on Lagrangian Decomposition. Our formulation admits an efficient supergradient ascent algorithm, as well as an improved proximal algorithm. Both the algorithms offer three advantages: (i) they yield bounds that are provably at least as tight as previous dual algorithms relying on Lagrangian relaxations; (ii) they are based on operations analogous to forward/backward pass of neural networks layers and are therefore easily parallelizable, amenable to GPU implementation and able to take advantage of the convolutional structure of problems; and (iii) they allow for anytime stopping while still providing valid bounds. Empirically, we show that we obtain bounds comparable with off-the-shelf solvers in a fraction of their running time, and obtain tighter bounds in the same time as previous dual algorithms. This results in an overall speed-up when employing the bounds for formal verification. Code for our algorithms is available at https: //github. com/oval-group/decomposition-plnn-bounds.

ICLR Conference 2020 Conference Paper

Neural Network Branching for Neural Network Verification

• Jingyue Lu
• M. Pawan Kumar

Formal verification of neural networks is essential for their deployment in safety-critical areas. Many available formal verification methods have been shown to be instances of a unified Branch and Bound (BaB) formulation. We propose a novel framework for designing an effective branching strategy for BaB. Specifically, we learn a graph neural network (GNN) to imitate the strong branching heuristic behaviour. Our framework differs from previous methods for learning to branch in two main aspects. Firstly, our framework directly treats the neural network we want to verify as a graph input for the GNN. Secondly, we develop an intuitive forward and backward embedding update schedule. Empirically, our framework achieves roughly $50\%$ reduction in both the number of branches and the time required for verification on various convolutional networks when compared to the best available hand-designed branching strategy. In addition, we show that our GNN model enjoys both horizontal and vertical transferability. Horizontally, the model trained on easy properties performs well on properties of increased difficulty levels. Vertically, the model trained on small neural networks achieves similar performance on large neural networks.

ICML Conference 2020 Conference Paper

Training Neural Networks for and by Interpolation

• Leonard Berrada
• Andrew Zisserman
• M. Pawan Kumar

In modern supervised learning, many deep neural networks are able to interpolate the data: the empirical loss can be driven to near zero on all samples simultaneously. In this work, we explicitly exploit this interpolation property for the design of a new optimization algorithm for deep learning, which we term Adaptive Learning-rates for Interpolation with Gradients (ALI-G). ALI-G retains the two main advantages of Stochastic Gradient Descent (SGD), which are (i) a low computational cost per iteration and (ii) good generalization performance in practice. At each iteration, ALI-G exploits the interpolation property to compute an adaptive learning-rate in closed form. In addition, ALI-G clips the learning-rate to a maximal value, which we prove to be helpful for non-convex problems. Crucially, in contrast to the learning-rate of SGD, the maximal learning-rate of ALI-G does not require a decay schedule. This makes ALI-G considerably easier to tune than SGD. We prove the convergence of ALI-G in various stochastic settings. Notably, we tackle the realistic case where the interpolation property is satisfied up to some tolerance. We also provide experiments on a variety of deep learning architectures and tasks: (i) learning a differentiable neural computer; (ii) training a wide residual network on the SVHN data set; (iii) training a Bi-LSTM on the SNLI data set; and (iv) training wide residual networks and densely connected networks on the CIFAR data sets. ALI-G produces state-of-the-art results among adaptive methods, and even yields comparable performance with SGD, which requires manually tuned learning-rate schedules. Furthermore, ALI-G is simple to implement in any standard deep learning framework and can be used as a drop-in replacement in existing code.

JMLR Journal 2016 Journal Article

Rounding-based Moves for Semi-Metric Labeling

• M. Pawan Kumar
• Puneet K. Dokania

Semi-metric labeling is a special case of energy minimization for pairwise Markov random fields. The energy function consists of arbitrary unary potentials, and pairwise potentials that are proportional to a given semi-metric distance function over the label set. Popular methods for solving semi-metric labeling include (i) move-making algorithms, which iteratively solve a minimum $st$-cut problem; and (ii) the linear programming ( LP) relaxation based approach. In order to convert the fractional solution of the LP relaxation to an integer solution, several randomized rounding procedures have been developed in the literature. We consider a large class of parallel rounding procedures, and design move-making algorithms that closely mimic them. We prove that the multiplicative bound of a move-making algorithm exactly matches the approximation factor of the corresponding rounding procedure for any arbitrary distance function. Our analysis includes all known results for move- making algorithms as special cases. [abs] [ pdf ][ bib ] &copy JMLR 2016. ( edit, beta )

NeurIPS Conference 2014 Conference Paper

Efficient Optimization for Average Precision SVM

• Pritish Mohapatra
• C. V. Jawahar
• M. Pawan Kumar

The accuracy of information retrieval systems is often measured using average precision (AP). Given a set of positive (relevant) and negative (non-relevant) samples, the parameters of a retrieval system can be estimated using the AP-SVM framework, which minimizes a regularized convex upper bound on the empirical AP loss. However, the high computational complexity of loss-augmented inference, which is required for learning an AP-SVM, prohibits its use with large training datasets. To alleviate this deficiency, we propose three complementary approaches. The first approach guarantees an asymptotic decrease in the computational complexity of loss-augmented inference by exploiting the problem structure. The second approach takes advantage of the fact that we do not require a full ranking during loss-augmented inference. This helps us to avoid the expensive step of sorting the negative samples according to their individual scores. The third approach approximates the AP loss over all samples by the AP loss over difficult samples (for example, those that are incorrectly classified by a binary SVM), while ensuring the correct classification of the remaining samples. Using the PASCAL VOC action classification and object detection datasets, we show that our approaches provide significant speed-ups during training without degrading the test accuracy of AP-SVM.

NeurIPS Conference 2014 Conference Paper

Rounding-based Moves for Metric Labeling

• M. Pawan Kumar

Metric labeling is a special case of energy minimization for pairwise Markov random fields. The energy function consists of arbitrary unary potentials, and pairwise potentials that are proportional to a given metric distance function over the label set. Popular methods for solving metric labeling include (i) move-making algorithms, which iteratively solve a minimum st-cut problem; and (ii) the linear programming (LP) relaxation based approach. In order to convert the fractional solution of the LP relaxation to an integer solution, several randomized rounding procedures have been developed in the literature. We consider a large class of parallel rounding procedures, and design move-making algorithms that closely mimic them. We prove that the multiplicative bound of a move-making algorithm exactly matches the approximation factor of the corresponding rounding procedure for any arbitrary distance function. Our analysis includes all known results for move-making algorithms as special cases.

ICML Conference 2012 Conference Paper

Modeling Latent Variable Uncertainty for Loss-based Learning

• M. Pawan Kumar
• Benjamin Packer
• Daphne Koller

JMLR Journal 2011 Journal Article

Improved Moves for Truncated Convex Models

• M. Pawan Kumar
• Olga Veksler
• Philip H.S. Torr

We consider the problem of obtaining an approximate maximum a posteriori estimate of a discrete random field characterized by pairwise potentials that form a truncated convex model. For this problem, we propose two st -MINCUT based move making algorithms that we call Range Swap and Range Expansion. Our algorithms can be thought of as extensions of αβ -Swap and α -Expansion respectively that fully exploit the form of the pairwise potentials. Specifically, instead of dealing with one or two labels at each iteration, our methods explore a large search space by considering a range of labels (that is, an interval of consecutive labels). Furthermore, we show that Range Expansion provides the same multiplicative bounds as the standard linear programming (LP) relaxation in polynomial time. Compared to previous approaches based on the LP relaxation, for example interior-point algorithms or tree-reweighted message passing (TRW), our methods are faster as they use only the efficient st -MINCUT algorithm in their design. We demonstrate the usefulness of the proposed approaches on both synthetic and standard real data problems. [abs] [ pdf ][ bib ] &copy JMLR 2011. ( edit, beta )

JMLR Journal 2009 Journal Article

An Analysis of Convex Relaxations for MAP Estimation of Discrete MRFs

• M. Pawan Kumar
• Vladimir Kolmogorov
• Philip H.S. Torr

The problem of obtaining the maximum a posteriori estimate of a general discrete Markov random field (i.e., a Markov random field defined using a discrete set of labels) is known to be NP-hard. However, due to its central importance in many applications, several approximation algorithms have been proposed in the literature. In this paper, we present an analysis of three such algorithms based on convex relaxations: (i) LP-S: the linear programming (LP) relaxation proposed by Schlesinger (1976) for a special case and independently in Chekuri et al. (2001), Koster et al. (1998), and Wainwright et al. (2005) for the general case; (ii) QP-RL: the quadratic programming (QP) relaxation of Ravikumar and Lafferty (2006); and (iii) SOCP-MS: the second order cone programming (SOCP) relaxation first proposed by Muramatsu and Suzuki (2003) for two label problems and later extended by Kumar et al. (2006) for a general label set. We show that the SOCP-MS and the QP-RL relaxations are equivalent. Furthermore, we prove that despite the flexibility in the form of the constraints/objective function offered by QP and SOCP, the LP-S relaxation strictly dominates (i.e., provides a better approximation than) QP-RL and SOCP-MS. We generalize these results by defining a large class of SOCP (and equivalent QP) relaxations which is dominated by the LP-S relaxation. Based on these results we propose some novel SOCP relaxations which define constraints using random variables that form cycles or cliques in the graphical model representation of the random field. Using some examples we show that the new SOCP relaxations strictly dominate the previous approaches. [abs] [ pdf ][ bib ] &copy JMLR 2009. ( edit, beta )

UAI Conference 2009 Conference Paper

MAP Estimation of Semi-Metric MRFs via Hierarchical Graph Cuts

• M. Pawan Kumar
• Daphne Koller

We consider the task of obtaining the maximum a posteriori estimate of discrete pairwise random fields with arbitrary unary potentials and semimetric pairwise potentials. For this problem, we propose an accurate hierarchical move making strategy where each move is computed efficiently by solving an st-MINCUT problem. Unlike previous move making approaches, e. g. the widely used α-expansion algorithm, our method obtains the guarantees of the standard linear programming ( LP) relaxation for the important special case of metric labeling. Unlike the existing LP relaxation solvers, e. g. interior-point algorithms or tree-reweighted message passing, our method is significantly faster as it uses only the efficient st-MINCUT algorithm in its design. Using both synthetic and real data experiments, we show that our technique outperforms several commonly used algorithms.

ICML Conference 2008 Conference Paper

Efficiently solving convex relaxations for MAP estimation

• M. Pawan Kumar
• Philip H. S. Torr