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Tarun Kathuria

Tarun Kathuria appears in the imported research catalog. Authorship, coauthor and topic links are available while profile ownership is still unclaimed.

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Machine LearningData Structures and Algorithmsmath.PRmath.OCmath.FA

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Published work

8 published item(s)

preprint2026arXiv

Leveraging Pretrained Language Models as Energy Functions for Glauber Dynamics Text Diffusion

We present a discrete diffusion-based language model using Glauber dynamics from statistical physics. Our main insight is that instead of trying to train a discrete state space diffusion model using Glauber dynamics with a uniform transition kernel as the forward process, one can set up an ``energy function'' based on pretrained causal/masked language models. When viewed as the stationary distribution, this energy function allows us to significantly improve the quality of the generated text. Incorporating UL2 as the pretrained model into our diffusion pipeline, we outperform prior diffusion based LMs and perform competitively with autoregressive models of comparable model sizes. Furthermore, our models are competitive with or outperform prior diffusion models and GPT-2 style auto-regressive models on zero-shot common sense reasoning tasks as well as planning and search tasks like Sudoku and Zebra puzzles.

0 citations0 reviewsNo code linkedOpen access
preprint2020arXiv

A Faster Interior Point Method for Semidefinite Programming

Semidefinite programs (SDPs) are a fundamental class of optimization problems with important recent applications in approximation algorithms, quantum complexity, robust learning, algorithmic rounding, and adversarial deep learning. This paper presents a faster interior point method to solve generic SDPs with variable size $n \times n$ and $m$ constraints in time \begin{align*} \widetilde{O}(\sqrt{n}( mn^2 + m^ω+ n^ω) \log(1 / ε) ), \end{align*} where $ω$ is the exponent of matrix multiplication and $ε$ is the relative accuracy. In the predominant case of $m \geq n$, our runtime outperforms that of the previous fastest SDP solver, which is based on the cutting plane method of Jiang, Lee, Song, and Wong [JLSW20]. Our algorithm's runtime can be naturally interpreted as follows: $\widetilde{O}(\sqrt{n} \log (1/ε))$ is the number of iterations needed for our interior point method, $mn^2$ is the input size, and $m^ω+ n^ω$ is the time to invert the Hessian and slack matrix in each iteration. These constitute natural barriers to further improving the runtime of interior point methods for solving generic SDPs.

0 citations0 reviewsNo code linkedOpen access
preprint2020arXiv

A Potential Reduction Inspired Algorithm for Exact Max Flow in Almost $\widetilde{O}(m^{4/3})$ Time

We present an algorithm for computing $s$-$t$ maximum flows in directed graphs in $\widetilde{O}(m^{4/3+o(1)}U^{1/3})$ time. Our algorithm is inspired by potential reduction interior point methods for linear programming. Instead of using scaled gradient/Newton steps of a potential function, we take the step which maximizes the decrease in the potential value subject to advancing a certain amount on the central path, which can be efficiently computed. This allows us to trace the central path with our progress depending only $\ell_\infty$ norm bounds on the congestion vector (as opposed to the $\ell_4$ norm required by previous works) and runs in $O(\sqrt{m})$ iterations. To improve the number of iterations by establishing tighter bounds on the $\ell_\infty$ norm, we then consider the weighted central path framework of Madry \cite{M13,M16,CMSV17} and Liu-Sidford \cite{LS20}. Instead of changing weights to maximize energy, we consider finding weights which maximize the maximum decrease in potential value. Finally, similar to finding weights which maximize energy as done in \cite{LS20} this problem can be solved by the iterative refinement framework for smoothed $\ell_2$-$\ell_p$ norm flow problems \cite{KPSW19} completing our algorithm. We believe our potential reduction based viewpoint provides a versatile framework which may lead to faster algorithms for max flow.

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preprint2020arXiv

On Concentration Inequalities for Random Matrix Products

Consider $n$ complex random matrices $X_1,\ldots,X_n$ of size $d\times d$ sampled i.i.d. from a distribution with mean $E[X]=μ$. While the concentration of averages of these matrices is well-studied, the concentration of other functions of such matrices is less clear. One function which arises in the context of stochastic iterative algorithms, like Oja's algorithm for Principal Component Analysis, is the normalized matrix product defined as $\prod\limits_{i=1}^{n}\left(I + \frac{X_i}{n}\right).$ Concentration properties of this normalized matrix product were recently studied by \cite{HW19}. However, their result is suboptimal in terms of the dependence on the dimension of the matrices as well as the number of samples. In this paper, we present a stronger concentration result for such matrix products which is optimal in $n$ and $d$ up to constant factors. Our proof is based on considering a matrix Doob martingale, controlling the quadratic variation of that martingale, and applying the Matrix Freedman inequality of Tropp \cite{TroppIntro15}.

0 citations0 reviewsNo code linkedOpen access
preprint2020arXiv

Scalar Poincaré Implies Matrix Poincaré

We prove that every reversible Markov semigroup which satisfies a Poincaré inequality satisfies a matrix-valued Poincaré inequality for Hermitian $d\times d$ matrix valued functions, with the same Poincaré constant. This generalizes recent results [Aoun et al. 2019, Kathuria 2019] establishing such inequalities for specific semigroups and consequently yields new matrix concentration inequalities. The short proof follows from the spectral theory of Markov semigroup generators.

0 citations0 reviewsNo code linkedOpen access
preprint2016arXiv

Batched Gaussian Process Bandit Optimization via Determinantal Point Processes

Gaussian Process bandit optimization has emerged as a powerful tool for optimizing noisy black box functions. One example in machine learning is hyper-parameter optimization where each evaluation of the target function requires training a model which may involve days or even weeks of computation. Most methods for this so-called "Bayesian optimization" only allow sequential exploration of the parameter space. However, it is often desirable to propose batches or sets of parameter values to explore simultaneously, especially when there are large parallel processing facilities at our disposal. Batch methods require modeling the interaction between the different evaluations in the batch, which can be expensive in complex scenarios. In this paper, we propose a new approach for parallelizing Bayesian optimization by modeling the diversity of a batch via Determinantal point processes (DPPs) whose kernels are learned automatically. This allows us to generalize a previous result as well as prove better regret bounds based on DPP sampling. Our experiments on a variety of synthetic and real-world robotics and hyper-parameter optimization tasks indicate that our DPP-based methods, especially those based on DPP sampling, outperform state-of-the-art methods.

0 citations0 reviewsNo code linkedOpen access
preprint2016arXiv

How to be Fair and Diverse?

Due to the recent cases of algorithmic bias in data-driven decision-making, machine learning methods are being put under the microscope in order to understand the root cause of these biases and how to correct them. Here, we consider a basic algorithmic task that is central in machine learning: subsampling from a large data set. Subsamples are used both as an end-goal in data summarization (where fairness could either be a legal, political or moral requirement) and to train algorithms (where biases in the samples are often a source of bias in the resulting model). Consequently, there is a growing effort to modify either the subsampling methods or the algorithms themselves in order to ensure fairness. However, in doing so, a question that seems to be overlooked is whether it is possible to produce fair subsamples that are also adequately representative of the feature space of the data set - an important and classic requirement in machine learning. Can diversity and fairness be simultaneously ensured? We start by noting that, in some applications, guaranteeing one does not necessarily guarantee the other, and a new approach is required. Subsequently, we present an algorithmic framework which allows us to produce both fair and diverse samples. Our experimental results on an image summarization task show marked improvements in fairness without compromising feature diversity by much, giving us the best of both the worlds.

0 citations0 reviewsNo code linkedOpen access
preprint2016arXiv

On Sampling and Greedy MAP Inference of Constrained Determinantal Point Processes

Subset selection problems ask for a small, diverse yet representative subset of the given data. When pairwise similarities are captured by a kernel, the determinants of submatrices provide a measure of diversity or independence of items within a subset. Matroid theory gives another notion of independence, thus giving rise to optimization and sampling questions about Determinantal Point Processes (DPPs) under matroid constraints. Partition constraints, as a special case, arise naturally when incorporating additional labeling or clustering information, besides the kernel, in DPPs. Finding the maximum determinant submatrix under matroid constraints on its row/column indices has been previously studied. However, the corresponding question of sampling from DPPs under matroid constraints has been unresolved, beyond the simple cardinality constrained k-DPPs. We give the first polynomial time algorithm to sample exactly from DPPs under partition constraints, for any constant number of partitions. We complement this by a complexity theoretic barrier that rules out such a result under general matroid constraints. Our experiments indicate that partition-constrained DPPs offer more flexibility and more diversity than k-DPPs and their naive extensions, while being reasonably efficient in running time. We also show that a simple greedy initialization followed by local search gives improved approximation guarantees for the problem of MAP inference from k- DPPs on well-conditioned kernels. Our experiments show that this improvement is significant for larger values of k, supporting our theoretical result.

0 citations0 reviewsNo code linkedOpen access