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Theory/Learning Theory

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 for one of the most fundamental distribution families, Gaussian mixture models.We prove that GD 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 GD with random initialization learns mixtures of two spherical Gaussians in $d$ dimensions with $1/\text{poly}(d)$-separated centers.2. We show GD with a warm start learns mixtures of $K$ spherical Gaussians with $\Omega(\sqrt{\log(\min(K,d))})$-separated centers.A key ingredient in our proofs is a new connection between score-based methods and two other approaches to distribution learning, EM and spectral methods.

Theory/Learning Theory

Shay Moran, Hilla Schefler, Jonathan Shafer

We show that many definitions of stability found in the learning theory literature are equivalent to one another. We distinguish between two families of definitions of stability: distribution-dependent and distribution-independent Bayesian stability. Within each family, we establish equivalences between various definitions, encompassing approximate differential privacy, pure differential privacy, replicability, global stability, perfect generalization, TV stability, mutual information stability, KL-divergence stability, and Rényi-divergence stability. Along the way, we prove boosting results that enable the amplification of the stability of a learning rule. This work is a step towards a more systematic taxonomy of stability notions in learning theory, which can promote clarity and an improved understanding of an array of stability concepts that have emerged in recent years.

Theory/Learning Theory

Alexandru Tifrea, Gizem Yüce, Amartya Sanyal, Fanny Yang

Prior theoretical and empirical works have established that semi-supervised learning algorithms can leverage the unlabeled data to improve over the labeled sample complexity of supervised learning (SL) algorithms. However, existing theoretical work focuses on regimes where the unlabeled data is sufficient to learn a good decision boundary using unsupervised learning (UL) alone. This begs the question: Can SSL algorithms simultaneously improve upon both UL and SL? To this end, we derive a tight lower bound for 2-Gaussian mixture models that explicitly depends on the labeled and the unlabeled dataset size as well as the signal-to-noise ratio of the mixture distribution. Surprisingly, our result implies that no SSL algorithm improves upon the minimax-optimal statistical error rates of SL or UL algorithms for these distributions. Nevertheless, in our real-world experiments, SSL algorithms can often outperform UL and SL algorithms. In summary, our work suggests that while it is possible to prove the performance gains of SSL algorithms, this would require careful tracking of constants in the theoretical analysis.

Theory/Learning Theory

Lingjiong Zhu, Mert Gurbuzbalaban, Anant Raj, Umut Simsekli

Algorithmic stability is an important notion that has proven powerful for deriving generalization bounds for practical algorithms. The last decade has witnessed an increasing number of stability bounds for different algorithms applied on different classes of loss functions. While these bounds have illuminated various properties of optimization algorithms, the analysis of each case typically required a different proof technique with significantly different mathematical tools. In this study, we make a novel connection between learning theory and applied probability and introduce a unified guideline for proving Wasserstein stability bounds for stochastic optimization algorithms. We illustrate our approach on stochastic gradient descent (SGD) and we obtain time-uniform stability bounds (i.e., the bound does not increase with the number of iterations) for strongly convex losses and non-convex losses with additive noise, where we recover similar results to the prior art or extend them to more general cases by using a single proof technique. Our approach is flexible and can be generalizable to other popular optimizers, as it mainly requires developing Lyapunov functions, which are often readily available in the literature. It also illustrates that ergodicity is an important component for obtaining time-uniform bounds -- which might not be achieved for convex or non-convex losses unless additional noise is injected to the iterates. Finally, we slightly stretch our analysis technique and prove time-uniform bounds for SGD under convex and non-convex losses (without additional additive noise), which, to our knowledge, is novel.

Theory/Learning Theory

Austin Xu, Andrew McRae, Jingyan Wang, Mark Davenport, Ashwin Pananjady

We introduce a new type of query mechanism for collecting human feedback, called the perceptual adjustment query (PAQ). Being both informative and cognitively lightweight, the PAQ adopts an inverted measurement scheme, and combines advantages from both cardinal and ordinal queries. We showcase the PAQ in the metric learning problem, where we collect PAQ measurements to learn an unknown Mahalanobis distance. This gives rise to a high-dimensional, low-rank matrix estimation problem to which standard matrix estimators cannot be applied. Consequently, we develop a two-stage estimator for metric learning from PAQs, and provide sample complexity guarantees for this estimator. We present numerical simulations demonstrating the performance of the estimator and its notable properties.

Theory/Learning Theory

Aravind Gollakota, Adam Klivans, Konstantinos Stavropoulos, Arsen Vasilyan

We give the first tester-learner for halfspaces that succeeds universally over a wide class of structured distributions. Our universal tester-learner runs in fully polynomial time and has the following guarantee: the learner achieves error $O(\mathrm{opt}) + \epsilon$ on any labeled distribution that the tester accepts, and moreover, the tester accepts whenever the marginal is any distribution that satisfies a Poincare inequality. In contrast to prior work on testable learning, our tester is not tailored to any single target distribution but rather succeeds for an entire target class of distributions. The class of Poincare distributions includes all strongly log-concave distributions, and, assuming the Kannan--Lovasz--Simonovits (KLS) conjecture, includes all log-concave distributions. In the special case where the label noise is known to be Massart, our tester-learner achieves error $\mathrm{opt} + \epsilon$ while accepting all log-concave distributions unconditionally (without assuming KLS).Our tests rely on checking hypercontractivity of the unknown distribution using a sum-of-squares (SOS) program, and crucially make use of the fact that Poincare distributions are certifiably hypercontractive in the SOS framework.

Theory/Learning Theory

Michael Feldman, David Donoho

Many applications seek to recover low-rank approximations of noisy tensor data. We consider several practical and effective matricization strategies which construct specific matrices from such tensors and then apply spectral methods; the strategies include tensor unfolding, partial tracing, power iteration, and recursive unfolding. We settle the behaviors of unfolding and partial tracing, identifying sharp thresholds in signal-to-noise ratio above which the signal is partially recovered. In particular, we extend previous results to a much larger class of tensor shapes where axis lengths may be different. For power iteration and recursive unfolding, we prove that under conditions where previous algorithms partially recovery the signal, these methods achieve (asymptotically) exact recovery. Our analysis deploys random matrix theory to obtain sharp thresholds which elude perturbation and concentration bounds. Specifically, we rely upon recent disproportionate random matrix results, which describe sequences of matrices with diverging aspect ratio.

Theory/Learning Theory

Yifeng Chu, Maxim Raginsky

This paper presents a general methodology for deriving information-theoretic generalization bounds for learning algorithms. The main technical tool is a probabilistic decorrelation lemma based on a change of measure and a relaxation of Young's inequality in $L_{\psi_p}$ Orlicz spaces. Using the decorrelation lemma in combination with other techniques, such as symmetrization, couplings, and chaining in the space of probability measures, we obtain new upper bounds on the generalization error, both in expectation and in high probability, and recover as special cases many of the existing generalization bounds, including the ones based on mutual information, conditional mutual information, stochastic chaining, and PAC-Bayes inequalities. In addition, the Fernique--Talagrand upper bound on the expected supremum of a subgaussian process emerges as a special case.