Kulin Shah

Kulin Shah, Sitan Chen, Adam Klivans

Recent works have shown that diffusion models can learn essentially any distribution provided one can perform score estimation. Yet it remains poorly understood under what settings score estimation is possible, let alone when practical gradient-based algorithms for this task can provably succeed. In this work, we give the first provably efficient results along these lines for one of the most fundamental distribution families, Gaussian mixture models. We prove that gradient descent on the denoising diffusion probabilistic model (DDPM) objective can efficiently recover the ground truth parameters of the mixture model in the following two settings: 1) We show gradient descent with random initialization learns mixtures of two spherical Gaussians in $d$ dimensions with $1/\text{poly}(d)$-separated centers. 2) We show gradient descent with a warm start learns mixtures of $K$ spherical Gaussians with $Ω(\sqrt{\log(\min(K,d))})$-separated centers. A key ingredient in our proofs is a new connection between score-based methods and two other approaches to distribution learning, the EM algorithm and spectral methods.

Kulin Shah, Nishanth Dikkala, Xin Wang et al.

Causal language modeling using the Transformer architecture has yielded remarkable capabilities in Large Language Models (LLMs) over the last few years. However, the extent to which fundamental search and reasoning capabilities emerged within LLMs remains a topic of ongoing debate. In this work, we study if causal language modeling can learn a complex task such as solving Sudoku puzzles. To solve a Sudoku, the model is first required to search over all empty cells of the puzzle to decide on a cell to fill and then apply an appropriate strategy to fill the decided cell. Sometimes, the application of a strategy only results in thinning down the possible values in a cell rather than concluding the exact value of the cell. In such cases, multiple strategies are applied one after the other to fill a single cell. We observe that Transformer models trained on this synthetic task can indeed learn to solve Sudokus (our model solves $94.21\%$ of the puzzles fully correctly) when trained on a logical sequence of steps taken by a solver. We find that training Transformers with the logical sequence of steps is necessary and without such training, they fail to learn Sudoku. We also extend our analysis to Zebra puzzles (known as Einstein puzzles) and show that the model solves $92.04 \%$ of the puzzles fully correctly. In addition, we study the internal representations of the trained Transformer and find that through linear probing, we can decode information about the set of possible values in any given cell from them, pointing to the presence of a strong reasoning engine implicit in the Transformer weights.

Aayush Karan, Kulin Shah, Sitan Chen et al.

Much of Bayesian inference centers around the design of estimators for inverse problems which are optimal assuming the data comes from a known prior. But what do these optimality guarantees mean if the prior is unknown? In recent years, algorithm unrolling has emerged as deep learning's answer to this age-old question: design a neural network whose layers can in principle simulate iterations of inference algorithms and train on data generated by the unknown prior. Despite its empirical success, however, it has remained unclear whether this method can provably recover the performance of its optimal, prior-aware counterparts. In this work, we prove the first rigorous learning guarantees for neural networks based on unrolling approximate message passing (AMP). For compressed sensing, we prove that when trained on data drawn from a product prior, the layers of the network approximately converge to the same denoisers used in Bayes AMP. We also provide extensive numerical experiments for compressed sensing and rank-one matrix estimation demonstrating the advantages of our unrolled architecture - in addition to being able to obliviously adapt to general priors, it exhibits improvements over Bayes AMP in more general settings of low dimensions, non-Gaussian designs, and non-product priors.

Yuanzhi Li, Raghu Meka, Rina Panigrahy et al.

Supervised and unsupervised learning using deep neural networks typically aims to exploit the underlying structure in the training data; this structure is often explained using a latent generative process that produces the data, and the generative process is often hierarchical, involving latent concepts. Despite the significant work on understanding the learning of the latent structure and underlying concepts using theory and experiments, a framework that mathematically captures the definition of a concept and provides ways to operate on concepts is missing. In this work, we propose to characterize a simple primitive concept by the zero set of a collection of polynomials and use moment statistics of the data to uniquely represent the concepts; we show how this view can be used to obtain a signature of the concept. These signatures can be used to discover a common structure across the set of concepts and could recursively produce the signature of higher-level concepts from the signatures of lower-level concepts. To utilize such desired properties, we propose a method by keeping a dictionary of concepts and show that the proposed method can learn different types of hierarchical structures of the data.

Satwik Bhattamishra, Kulin Shah, Michael Hahn et al.

We study the problem of learning Transformer-based sequence models with black-box access to their outputs. In this setting, a learner may adaptively query the oracle with any sequence of vectors and observe the corresponding real-valued output. We begin with the simplest case, a single-head softmax-attention regressor. We show that for a model with width $d$, there is an elementary algorithm to learn the parameters of single-head attention exactly with $O(d^2)$ queries. Further, we show that if there exists an algorithm to learn ReLU feedforward networks (FFNs), then the single-head algorithm can be easily adapted to learn one-layer Transformers with single-head attention. Next, motivated by the regime where the head dimension $r \ll d$, we provide a randomised algorithm that learns single-head attention-based models with $O(rd)$ queries via compressed sensing arguments. We also study robustness to noisy oracle access, proving that under mild norm and margin conditions, the parameters can be estimated to $\varepsilon$ accuracy with a polynomial number of queries even when outputs are only provided up to additive tolerance. Finally, we show that multi-head attention parameters are not identifiable from value queries in general -- distinct parameterisations can induce the same input-output map. Hence, guarantees analogous to the single-head setting are impossible without additional structural assumptions.

Jaeyeon Kim, Kulin Shah, Vasilis Kontonis et al.

In recent years, masked diffusion models (MDMs) have emerged as a promising alternative approach for generative modeling over discrete domains. Compared to autoregressive models (ARMs), MDMs trade off complexity at training time with flexibility at inference time. At training time, they must learn to solve an exponentially large number of infilling problems, but at inference time, they can decode tokens in essentially arbitrary order. In this work, we closely examine these two competing effects. On the training front, we theoretically and empirically demonstrate that MDMs indeed train on computationally intractable subproblems compared to their autoregressive counterparts. On the inference front, we show that a suitable strategy for adaptively choosing the token decoding order significantly enhances the capabilities of MDMs, allowing them to sidestep hard subproblems. On logic puzzles like Sudoku, we show that adaptive inference can boost solving accuracy in pretrained MDMs from $<7$% to $\approx 90$%, even outperforming ARMs with $7\times$ as many parameters and that were explicitly trained via teacher forcing to learn the right order of decoding.

Sitan Chen, Vasilis Kontonis, Kulin Shah

We study the problem of learning mixtures of $k$ Gaussians in $d$ dimensions. We make no separation assumptions on the underlying mixture components: we only require that the covariance matrices have bounded condition number and that the means and covariances lie in a ball of bounded radius. We give an algorithm that draws $d^{\mathrm{poly}(k/\varepsilon)}$ samples from the target mixture, runs in sample-polynomial time, and constructs a sampler whose output distribution is $\varepsilon$-far from the unknown mixture in total variation. Prior works for this problem either (i) required exponential runtime in the dimension $d$, (ii) placed strong assumptions on the instance (e.g., spherical covariances or clusterability), or (iii) had doubly exponential dependence on the number of components $k$. Our approach departs from commonly used techniques for this problem like the method of moments. Instead, we leverage a recently developed reduction, based on diffusion models, from distribution learning to a supervised learning task called score matching. We give an algorithm for the latter by proving a structural result showing that the score function of a Gaussian mixture can be approximated by a piecewise-polynomial function, and there is an efficient algorithm for finding it. To our knowledge, this is the first example of diffusion models achieving a state-of-the-art theoretical guarantee for an unsupervised learning task.

Kulin Shah, Alkis Kalavasis, Adam R. Klivans et al.

There is strong empirical evidence that the state-of-the-art diffusion modeling paradigm leads to models that memorize the training set, especially when the training set is small. Prior methods to mitigate the memorization problem often lead to a decrease in image quality. Is it possible to obtain strong and creative generative models, i.e., models that achieve high generation quality and low memorization? Despite the current pessimistic landscape of results, we make significant progress in pushing the trade-off between fidelity and memorization. We first provide theoretical evidence that memorization in diffusion models is only necessary for denoising problems at low noise scales (usually used in generating high-frequency details). Using this theoretical insight, we propose a simple, principled method to train the diffusion models using noisy data at large noise scales. We show that our method significantly reduces memorization without decreasing the image quality, for both text-conditional and unconditional models and for a variety of data availability settings.

Aayush Karan, Kulin Shah, Sitan Chen

There has been a flurry of activity around using pretrained diffusion models as informed data priors for solving inverse problems, and more generally around steering these models using reward models. Training-free methods like diffusion posterior sampling (DPS) and its many variants have offered flexible heuristic algorithms for these tasks, but when the reward is not informative enough, e.g., in hard inverse problems with low signal-to-noise ratio, these techniques veer off the data manifold, failing to produce realistic outputs. In this work, we devise a simple wrapper, ReGuidance, for boosting both the sample realism and reward achieved by these methods. Given a candidate solution $\hat{x}$ produced by an algorithm of the user's choice, we propose inverting the solution by running the unconditional probability flow ODE in reverse starting from $\hat{x}$, and then using the resulting latent as an initialization for DPS. We evaluate our wrapper on hard inverse problems like large box in-painting and super-resolution with high upscaling. Whereas state-of-the-art baselines visibly fail, we find that applying our wrapper on top of these baselines significantly boosts sample quality and measurement consistency. We complement these findings with theory proving that on certain multimodal data distributions, ReGuidance simultaneously boosts the reward and brings the candidate solution closer to the data manifold. To our knowledge, this constitutes the first rigorous algorithmic guarantee for DPS.

Kulin Shah, Bhuvesh Kumar, Neil Shah et al.

Generative recommendation (GR) with semantic IDs (SIDs) has emerged as a promising alternative to traditional recommendation approaches due to its performance gains, capitalization on semantic information provided through language model embeddings, and inference and storage efficiency. Existing GR with SIDs works frame the probability of a sequence of SIDs corresponding to a user's interaction history using autoregressive modeling. While this has led to impressive next item prediction performances in certain settings, these autoregressive GR with SIDs models suffer from expensive inference due to sequential token-wise decoding, potentially inefficient use of training data and bias towards learning short-context relationships among tokens. Inspired by recent breakthroughs in NLP, we propose to instead model and learn the probability of a user's sequence of SIDs using masked diffusion. Masked diffusion employs discrete masking noise to facilitate learning the sequence distribution, and models the probability of masked tokens as conditionally independent given the unmasked tokens, allowing for parallel decoding of the masked tokens. We demonstrate through thorough experiments that our proposed method consistently outperforms autoregressive modeling. This performance gap is especially pronounced in data-constrained settings and in terms of coarse-grained recall, consistent with our intuitions. Moreover, our approach allows the flexibility of predicting multiple SIDs in parallel during inference while maintaining superior performance to autoregressive modeling.

Giannis Daras, Kulin Shah, Yuval Dagan et al.

We present the first diffusion-based framework that can learn an unknown distribution using only highly-corrupted samples. This problem arises in scientific applications where access to uncorrupted samples is impossible or expensive to acquire. Another benefit of our approach is the ability to train generative models that are less likely to memorize individual training samples since they never observe clean training data. Our main idea is to introduce additional measurement distortion during the diffusion process and require the model to predict the original corrupted image from the further corrupted image. We prove that our method leads to models that learn the conditional expectation of the full uncorrupted image given this additional measurement corruption. This holds for any corruption process that satisfies some technical conditions (and in particular includes inpainting and compressed sensing). We train models on standard benchmarks (CelebA, CIFAR-10 and AFHQ) and show that we can learn the distribution even when all the training samples have $90\%$ of their pixels missing. We also show that we can finetune foundation models on small corrupted datasets (e.g. MRI scans with block corruptions) and learn the clean distribution without memorizing the training set.

Bhavya Kalra, Kulin Shah, Naresh Manwani

In this paper, we propose deep architectures for learning instance specific abstain (reject option) binary classifiers. The proposed approach uses double sigmoid loss function as described by Kulin Shah and Naresh Manwani in ("Online Active Learning of Reject Option Classifiers", AAAI, 2020), as a performance measure. We show that the double sigmoid loss is classification calibrated. We also show that the excess risk of 0-d-1 loss is upper bounded by the excess risk of double sigmoid loss. We derive the generalization error bounds for the proposed architecture for reject option classifiers. To show the effectiveness of the proposed approach, we experiment with several real world datasets. We observe that the proposed approach not only performs comparable to the state-of-the-art approaches, it is also robust against label noise. We also provide visualizations to observe the important features learned by the network corresponding to the abstaining decision.

Kulin Shah, Amit Deshpande, Navin Goyal

In supervised learning, it is known that overparameterized neural networks with one hidden layer provably and efficiently learn and generalize, when trained using stochastic gradient descent with a sufficiently small learning rate and suitable initialization. In contrast, the benefit of overparameterization in unsupervised learning is not well understood. Normalizing flows (NFs) constitute an important class of models in unsupervised learning for sampling and density estimation. In this paper, we theoretically and empirically analyze these models when the underlying neural network is a one-hidden-layer overparametrized network. Our main contributions are two-fold: (1) On the one hand, we provide theoretical and empirical evidence that for constrained NFs (this class of NFs underlies many NF constructions) with the one-hidden-layer network, overparametrization hurts training. (2) On the other hand, we prove that unconstrained NFs, a recently introduced model, can efficiently learn any reasonable data distribution under minimal assumptions when the underlying network is overparametrized and has one hidden-layer.

Kulin Shah, Pooja Gupta, Amit Deshpande et al.

Group-fairness in classification aims for equality of a predictive utility across different sensitive sub-populations, e.g., race or gender. Equality or near-equality constraints in group-fairness often worsen not only the aggregate utility but also the utility for the least advantaged sub-population. In this paper, we apply the principles of Pareto-efficiency and least-difference to the utility being accuracy, as an illustrative example, and arrive at the Rawls classifier that minimizes the error rate on the worst-off sensitive sub-population. Our mathematical characterization shows that the Rawls classifier uniformly applies a threshold to an ideal score of features, in the spirit of fair equality of opportunity. In practice, such a score or a feature representation is often computed by a black-box model that has been useful but unfair. Our second contribution is practical Rawlsian fair adaptation of any given black-box deep learning model, without changing the score or feature representation it computes. Given any score function or feature representation and only its second-order statistics on the sensitive sub-populations, we seek a threshold classifier on the given score or a linear threshold classifier on the given feature representation that achieves the Rawls error rate restricted to this hypothesis class. Our technical contribution is to formulate the above problems using ambiguous chance constraints, and to provide efficient algorithms for Rawlsian fair adaptation, along with provable upper bounds on the Rawls error rate. Our empirical results show significant improvement over state-of-the-art group-fair algorithms, even without retraining for fairness.

Kulin Shah, Naresh Manwani

Active learning is an important technique to reduce the number of labeled examples in supervised learning. Active learning for binary classification has been well addressed in machine learning. However, active learning of the reject option classifier remains unaddressed. In this paper, we propose novel algorithms for active learning of reject option classifiers. We develop an active learning algorithm using double ramp loss function. We provide mistake bounds for this algorithm. We also propose a new loss function called double sigmoid loss function for reject option and corresponding active learning algorithm. We offer a convergence guarantee for this algorithm. We provide extensive experimental results to show the effectiveness of the proposed algorithms. The proposed algorithms efficiently reduce the number of label examples required.

Kulin Shah, P. S. Sastry, Naresh Manwani

In this paper, we propose a novel mixture of expert architecture for learning polyhedral classifiers. We learn the parameters of the classifierusing an expectation maximization algorithm. Wederive the generalization bounds of the proposedapproach. Through an extensive simulation study, we show that the proposed method performs comparably to other state-of-the-art approaches.

Kulin Shah, Naresh Manwani

In this paper, we propose an approach for learning sparse reject option classifiers using double ramp loss $L_{dr}$. We use DC programming to find the risk minimizer. The algorithm solves a sequence of linear programs to learn the reject option classifier. We show that the loss $L_{dr}$ is Fisher consistent. We also show that the excess risk of loss $L_d$ is upper bounded by the excess risk of $L_{dr}$. We derive the generalization error bounds for the proposed approach. We show the effectiveness of the proposed approach by experimenting it on several real world datasets. The proposed approach not only performs comparable to the state of the art but it also successfully learns sparse classifiers.