Tyler Farghly

selected papers

1. 

   Diffusion Models and the Manifold Hypothesis: Log-Domain Smoothing is Geometry Adaptive

   T Farghly, P Potaptchik, S Howard, G Deligiannidis, and J Pidstrigach

   Draft available on request 2025

   Abs

   Diffusion models have achieved state-of-the-art performance, demonstrating remarkable generalisation capabilities across diverse domains. However, the mechanisms underpinning these strong capabilities remain only partially understood. A leading conjecture, based on the manifold hypothesis, attributes this success to their ability to adapt to low-dimensional geometric structure within the data. This work provides evidence for this conjecture, focusing on how such phenomena could result from the formulation of the learning problem through score matching. We inspect the role of implicit regularisation by investigating the effect of smoothing minimisers of the empirical score matching objective. Our theoretical and empirical results confirm that smoothing the score function—or equivalently, smoothing in the log-density domain—produces smoothing tangential to the data manifold. In addition, we show that the manifold along which the diffusion model generalises can be controlled by choosing an appropriate smoothing.

2. 

   Implicit Regularisation in Diffusion Models: An Algorithm-Dependent Generalisation Analysis

   T Farghly, P Rebeschini, G Deligiannidis, and A Doucet

   arXiv preprint 2025

   Abs URL Bib

   The success of denoising diffusion models raises important questions regarding their generalisation behaviour, particularly in high-dimensional settings. Notably, it has been shown that when training and sampling are performed perfectly, these models memorise training data – implying that some form of regularisation is essential for generalisation. Existing theoretical analyses primarily rely on algorithm-independent techniques such as uniform convergence, heavily utilising model structure to obtain generalisation bounds. In this work, we instead leverage the algorithmic aspects that promote generalisation in diffusion models, developing a general theory of algorithm-dependent generalisation for this setting. Borrowing from the framework of algorithmic stability, we introduce the notion of score stability, which quantifies the sensitivity of score-matching algorithms to dataset perturbations. We derive generalisation bounds in terms of score stability, and apply our framework to several fundamental learning settings, identifying sources of regularisation. In particular, we consider denoising score matching with early stopping (denoising regularisation), sampler-wide coarse discretisation (sampler regularisation) and optimising with SGD (optimisation regularisation). By grounding our analysis in algorithmic properties rather than model structure, we identify multiple sources of implicit regularisation unique to diffusion models that have so far been overlooked in the literature.

   @article{farghly2025implicit,
     title = {Implicit Regularisation in Diffusion Models: An Algorithm-Dependent Generalisation Analysis},
     author = {Farghly, Tyler and Rebeschini, Patrick and Deligiannidis, George and Doucet, Arnaud},
     journal = {arXiv preprint arXiv:2507.03756},
     year = {2025},
   }

3. 

   Towards a Complete Analysis of Langevin Monte Carlo: Beyond Poincaré Inequality

   A Mousavi-Hosseini, T Farghly, Y He, K Balasubramanian, and M Erdogdu

   In COLT 2023

   Abs URL Bib

   Langevin diffusions are rapidly convergent under appropriate functional inequality assumptions. Hence, it is natural to expect that with additional smoothness conditions to handle the discretization errors, their discretizations like the Langevin Monte Carlo (LMC) converge in a similar fashion. This research program was initiated by Vempala and Wibisono (2019), who established results under log-Sobolev inequalities. Chewi et al. (2022a) extended the results to handle the case of Poincaré inequalities. In this paper, we go beyond Poincaré inequalities, and push this research program to its limit. We do so by establishing upper and lower bounds for Langevin diffusions and LMC under weak Poincaré inequalities that are satisfied by a large class of densities including polynomially-decaying heavy-tailed densities (i.e., Cauchy-type). Our results explicitly quantify the effect of the initializer on the performance of the LMC algorithm. In particular, we show that as the tail goes from sub-Gaussian, to sub-exponential, and finally to Cauchy-like, the dependency on the initial error goes from being logarithmic, to polynomial, and then finally to being exponential. This three-step phase transition is in particular unavoidable as demonstrated by our lower bounds, clearly defining the boundaries of LMC.

   @inproceedings{pmlr-v195-mousavi-hosseini23a,
     title = {Towards a Complete Analysis of Langevin Monte Carlo: Beyond Poincaré Inequality},
     author = {Mousavi-Hosseini, Alireza and Farghly, Tyler K. and He, Ye and Balasubramanian, Krishna and Erdogdu, Murat A.},
     booktitle = {Proceedings of Thirty Sixth Conference on Learning Theory},
     pages = {1--35},
     year = {2023},
     editor = {Neu, Gergely and Rosasco, Lorenzo},
     volume = {195},
     series = {Proceedings of Machine Learning Research},
     month = {12--15 Jul},
     publisher = {PMLR},
   }

4. 

   Time-independent Generalization Bounds for SGLD in Non-convex Settings

   T Farghly, and P Rebeschini

   In NeurIPS 2021

   Abs URL Bib

   We establish generalization error bounds for stochastic gradient Langevin dynamics (SGLD) with constant learning rate under the assumptions of dissipativity and smoothness, a setting that has received increased attention in the sampling/optimization literature. Unlike existing bounds for SGLD in non-convex settings, ours are time-independent and decay to zero as the sample size increases. Using the framework of uniform stability, we establish time-independent bounds by exploiting the Wasserstein contraction property of the Langevin diffusion, which also allows us to circumvent the need to bound gradients using Lipschitz-like assumptions. Our analysis also supports variants of SGLD that use different discretization methods, incorporate Euclidean projections, or use non-isotropic noise.

   @inproceedings{NEURIPS2021_a4ee59dd,
     author = {Farghly, Tyler and Rebeschini, Patrick},
     booktitle = {Advances in Neural Information Processing Systems},
     editor = {Ranzato, M. and Beygelzimer, A. and Dauphin, Y. and Liang, P.S. and Vaughan, J. Wortman},
     pages = {19836--19846},
     publisher = {Curran Associates, Inc.},
     title = {Time-independent Generalization Bounds for SGLD in Non-convex Settings},
     volume = {34},
     year = {2021},
   }