arXiv subjects

Andrej Risteski

Publications and source records attributed to Andrej Risteski.

At least 37 records · Page 2Linked to original sources

arXiv2023

Outliers with Opposing Signals Have an Outsized Effect on Neural Network Optimization

We identify a new phenomenon in neural network optimization which arises from the interaction of depth and a particular heavy-tailed structure in natural data. Our result offers intuitive explanations for several previously reported observations about network training dynamics. In particular, it implies a conceptually new cause for progressive sharpening and the edge of stability; we also highlight connections to other concepts in optimization and generalization including grokking, simplicity bias, and Sharpness-Aware Minimization. Experimentally, we demonstrate the significant influence of paired groups of outliers in the training data with strong opposing signals: consistent, large magnitude features which dominate the network output throughout training and provide gradients which point in opposite directions. Due to these outliers, early optimization enters a narrow valley which carefully balances the opposing groups; subsequent sharpening causes their loss to rise rapidly, oscillating between high on one group and then the other, until the overall loss spikes. We describe how to identify these groups, explore what sets them apart, and carefully study their effect on the network's optimization and behavior. We complement these experiments with a mechanistic explanation on a toy example of opposing signals and a theoretical analysis of a two-layer linear network on a simple model. Our finding enables new qualitative predictions of training behavior which we confirm experimentally. It also provides a new lens through which to study and improve modern training practices for stochastic optimization, which we highlight via a case study of Adam versus SGD.

cs.LG↗

arXiv2023

Provable benefits of annealing for estimating normalizing constants: Importance Sampling, Noise-Contrastive Estimation, and beyond

Recent research has developed several Monte Carlo methods for estimating the normalization constant (partition function) based on the idea of annealing. This means sampling successively from a path of distributions that interpolate between a tractable "proposal" distribution and the unnormalized "target" distribution. Prominent estimators in this family include annealed importance sampling and annealed noise-contrastive estimation (NCE). Such methods hinge on a number of design choices: which estimator to use, which path of distributions to use and whether to use a path at all; so far, there is no definitive theory on which choices are efficient. Here, we evaluate each design choice by the asymptotic estimation error it produces. First, we show that using NCE is more efficient than the importance sampling estimator, but in the limit of infinitesimal path steps, the difference vanishes. Second, we find that using the geometric path brings down the estimation error from an exponential to a polynomial function of the parameter distance between the target and proposal distributions. Third, we find that the arithmetic path, while rarely used, can offer optimality properties over the universally-used geometric path. In fact, in a particular limit, the optimal path is arithmetic. Based on this theory, we finally propose a two-step estimator to approximate the optimal path in an efficient way.

stat.ML↗

arXiv2023

How Do Transformers Learn Topic Structure: Towards a Mechanistic Understanding

While the successes of transformers across many domains are indisputable, accurate understanding of the learning mechanics is still largely lacking. Their capabilities have been probed on benchmarks which include a variety of structured and reasoning tasks -- but mathematical understanding is lagging substantially behind. Recent lines of work have begun studying representational aspects of this question: that is, the size/depth/complexity of attention-based networks to perform certain tasks. However, there is no guarantee the learning dynamics will converge to the constructions proposed. In our paper, we provide fine-grained mechanistic understanding of how transformers learn "semantic structure", understood as capturing co-occurrence structure of words. Precisely, we show, through a combination of mathematical analysis and experiments on Wikipedia data and synthetic data modeled by Latent Dirichlet Allocation (LDA), that the embedding layer and the self-attention layer encode the topical structure. In the former case, this manifests as higher average inner product of embeddings between same-topic words. In the latter, it manifests as higher average pairwise attention between same-topic words. The mathematical results involve several assumptions to make the analysis tractable, which we verify on data, and might be of independent interest as well.

cs.LG↗

arXiv2023

Provable benefits of score matching

Score matching is an alternative to maximum likelihood (ML) for estimating a probability distribution parametrized up to a constant of proportionality. By fitting the ''score'' of the distribution, it sidesteps the need to compute this constant of proportionality (which is often intractable). While score matching and variants thereof are popular in practice, precise theoretical understanding of the benefits and tradeoffs with maximum likelihood -- both computational and statistical -- are not well understood. In this work, we give the first example of a natural exponential family of distributions such that the score matching loss is computationally efficient to optimize, and has a comparable statistical efficiency to ML, while the ML loss is intractable to optimize using a gradient-based method. The family consists of exponentials of polynomials of fixed degree, and our result can be viewed as a continuous analogue of recent developments in the discrete setting. Precisely, we show: (1) Designing a zeroth-order or first-order oracle for optimizing the maximum likelihood loss is NP-hard. (2) Maximum likelihood has a statistical efficiency polynomial in the ambient dimension and the radius of the parameters of the family. (3) Minimizing the score matching loss is both computationally and statistically efficient, with complexity polynomial in the ambient dimension.

cs.LG↗

arXiv2023

Neural Network Approximations of PDEs Beyond Linearity: A Representational Perspective

A burgeoning line of research leverages deep neural networks to approximate the solutions to high dimensional PDEs, opening lines of theoretical inquiry focused on explaining how it is that these models appear to evade the curse of dimensionality. However, most prior theoretical analyses have been limited to linear PDEs. In this work, we take a step towards studying the representational power of neural networks for approximating solutions to nonlinear PDEs. We focus on a class of PDEs known as \emph{nonlinear elliptic variational PDEs}, whose solutions minimize an \emph{Euler-Lagrange} energy functional $\mathcal{E}(u) = \int_ΩL(x, u(x), \nabla u(x)) - f(x) u(x)dx$. We show that if composing a function with Barron norm $b$ with partial derivatives of $L$ produces a function of Barron norm at most $B_L b^p$, the solution to the PDE can be $ε$-approximated in the $L^2$ sense by a function with Barron norm $O\left(\left(dB_L\right)^{\max\{p \log(1/ ε), p^{\log(1/ε)}\}}\right)$. By a classical result due to Barron [1993], this correspondingly bounds the size of a 2-layer neural network needed to approximate the solution. Treating $p, ε, B_L$ as constants, this quantity is polynomial in dimension, thus showing neural networks can evade the curse of dimensionality. Our proof technique involves neurally simulating (preconditioned) gradient in an appropriate Hilbert space, which converges exponentially fast to the solution of the PDE, and such that we can bound the increase of the Barron norm at each iterate. Our results subsume and substantially generalize analogous prior results for linear elliptic PDEs over a unit hypercube.

cs.LG↗

arXiv2023

Pitfalls of Gaussians as a noise distribution in NCE

Noise Contrastive Estimation (NCE) is a popular approach for learning probability density functions parameterized up to a constant of proportionality. The main idea is to design a classification problem for distinguishing training data from samples from an easy-to-sample noise distribution $q$, in a manner that avoids having to calculate a partition function. It is well-known that the choice of $q$ can severely impact the computational and statistical efficiency of NCE. In practice, a common choice for $q$ is a Gaussian which matches the mean and covariance of the data. In this paper, we show that such a choice can result in an exponentially bad (in the ambient dimension) conditioning of the Hessian of the loss, even for very simple data distributions. As a consequence, both the statistical and algorithmic complexity for such a choice of $q$ will be problematic in practice, suggesting that more complex noise distributions are essential to the success of NCE.

cs.LG↗

arXiv2022

Statistical Efficiency of Score Matching: The View from Isoperimetry

Deep generative models parametrized up to a normalizing constant (e.g. energy-based models) are difficult to train by maximizing the likelihood of the data because the likelihood and/or gradients thereof cannot be explicitly or efficiently written down. Score matching is a training method, whereby instead of fitting the likelihood $\log p(x)$ for the training data, we instead fit the score function $\nabla_x \log p(x)$ -- obviating the need to evaluate the partition function. Though this estimator is known to be consistent, its unclear whether (and when) its statistical efficiency is comparable to that of maximum likelihood -- which is known to be (asymptotically) optimal. We initiate this line of inquiry in this paper, and show a tight connection between statistical efficiency of score matching and the isoperimetric properties of the distribution being estimated -- i.e. the Poincaré, log-Sobolev and isoperimetric constant -- quantities which govern the mixing time of Markov processes like Langevin dynamics. Roughly, we show that the score matching estimator is statistically comparable to the maximum likelihood when the distribution has a small isoperimetric constant. Conversely, if the distribution has a large isoperimetric constant -- even for simple families of distributions like exponential families with rich enough sufficient statistics -- score matching will be substantially less efficient than maximum likelihood. We suitably formalize these results both in the finite sample regime, and in the asymptotic regime. Finally, we identify a direct parallel in the discrete setting, where we connect the statistical properties of pseudolikelihood estimation with approximate tensorization of entropy and the Glauber dynamics.

cs.LG↗

arXiv2022

Domain-Adjusted Regression or: ERM May Already Learn Features Sufficient for Out-of-Distribution Generalization

A common explanation for the failure of deep networks to generalize out-of-distribution is that they fail to recover the "correct" features. We challenge this notion with a simple experiment which suggests that ERM already learns sufficient features and that the current bottleneck is not feature learning, but robust regression. Our findings also imply that given a small amount of data from the target distribution, retraining only the last linear layer will give excellent performance. We therefore argue that devising simpler methods for learning predictors on existing features is a promising direction for future research. Towards this end, we introduce Domain-Adjusted Regression (DARE), a convex objective for learning a linear predictor that is provably robust under a new model of distribution shift. Rather than learning one function, DARE performs a domain-specific adjustment to unify the domains in a canonical latent space and learns to predict in this space. Under a natural model, we prove that the DARE solution is the minimax-optimal predictor for a constrained set of test distributions. Further, we provide the first finite-environment convergence guarantee to the minimax risk, improving over existing analyses which only yield minimax predictors after an environment threshold. Evaluated on finetuned features, we find that DARE compares favorably to prior methods, consistently achieving equal or better performance.

cs.LG↗

arXiv2022

Variational autoencoders in the presence of low-dimensional data: landscape and implicit bias

Variational Autoencoders are one of the most commonly used generative models, particularly for image data. A prominent difficulty in training VAEs is data that is supported on a lower-dimensional manifold. Recent work by Dai and Wipf (2020) proposes a two-stage training algorithm for VAEs, based on a conjecture that in standard VAE training the generator will converge to a solution with 0 variance which is correctly supported on the ground truth manifold. They gave partial support for that conjecture by showing that some optima of the VAE loss do satisfy this property, but did not analyze the training dynamics. In this paper, we show that for linear encoders/decoders, the conjecture is true-that is the VAE training does recover a generator with support equal to the ground truth manifold-and does so due to an implicit bias of gradient descent rather than merely the VAE loss itself. In the nonlinear case, we show that VAE training frequently learns a higher-dimensional manifold which is a superset of the ground truth manifold.

cs.LG↗

arXiv2022

Contrasting the landscape of contrastive and non-contrastive learning

A lot of recent advances in unsupervised feature learning are based on designing features which are invariant under semantic data augmentations. A common way to do this is contrastive learning, which uses positive and negative samples. Some recent works however have shown promising results for non-contrastive learning, which does not require negative samples. However, the non-contrastive losses have obvious "collapsed" minima, in which the encoders output a constant feature embedding, independent of the input. A folk conjecture is that so long as these collapsed solutions are avoided, the produced feature representations should be good. In our paper, we cast doubt on this story: we show through theoretical results and controlled experiments that even on simple data models, non-contrastive losses have a preponderance of non-collapsed bad minima. Moreover, we show that the training process does not avoid these minima.

cs.LG↗

arXiv2022

Continual learning: a feature extraction formalization, an efficient algorithm, and fundamental obstructions

Continual learning is an emerging paradigm in machine learning, wherein a model is exposed in an online fashion to data from multiple different distributions (i.e. environments), and is expected to adapt to the distribution change. Precisely, the goal is to perform well in the new environment, while simultaneously retaining the performance on the previous environments (i.e. avoid "catastrophic forgetting") -- without increasing the size of the model. While this setup has enjoyed a lot of attention in the applied community, there hasn't be theoretical work that even formalizes the desired guarantees. In this paper, we propose a framework for continual learning through the framework of feature extraction -- namely, one in which features, as well as a classifier, are being trained with each environment. When the features are linear, we design an efficient gradient-based algorithm $\mathsf{DPGD}$, that is guaranteed to perform well on the current environment, as well as avoid catastrophic forgetting. In the general case, when the features are non-linear, we show such an algorithm cannot exist, whether efficient or not.

cs.LG↗

arXiv2022

Masked prediction tasks: a parameter identifiability view

The vast majority of work in self-supervised learning, both theoretical and empirical (though mostly the latter), have largely focused on recovering good features for downstream tasks, with the definition of "good" often being intricately tied to the downstream task itself. This lens is undoubtedly very interesting, but suffers from the problem that there isn't a "canonical" set of downstream tasks to focus on -- in practice, this problem is usually resolved by competing on the benchmark dataset du jour. In this paper, we present an alternative lens: one of parameter identifiability. More precisely, we consider data coming from a parametric probabilistic model, and train a self-supervised learning predictor with a suitably chosen parametric form. Then, we ask whether we can read off the ground truth parameters of the probabilistic model from the optimal predictor. We focus on the widely used self-supervised learning method of predicting masked tokens, which is popular for both natural languages and visual data. While incarnations of this approach have already been successfully used for simpler probabilistic models (e.g. learning fully-observed undirected graphical models), we focus instead on latent-variable models capturing sequential structures -- namely Hidden Markov Models with both discrete and conditionally Gaussian observations. We show that there is a rich landscape of possibilities, out of which some prediction tasks yield identifiability, while others do not. Our results, borne of a theoretical grounding of self-supervised learning, could thus potentially beneficially inform practice. Moreover, we uncover close connections with uniqueness of tensor rank decompositions -- a widely used tool in studying identifiability through the lens of the method of moments.

cs.LG↗

arXiv2022

Sampling Approximately Low-Rank Ising Models: MCMC meets Variational Methods

We consider Ising models on the hypercube with a general interaction matrix $J$, and give a polynomial time sampling algorithm when all but $O(1)$ eigenvalues of $J$ lie in an interval of length one, a situation which occurs in many models of interest. This was previously known for the Glauber dynamics when *all* eigenvalues fit in an interval of length one; however, a single outlier can force the Glauber dynamics to mix torpidly. Our general result implies the first polynomial time sampling algorithms for low-rank Ising models such as Hopfield networks with a fixed number of patterns and Bayesian clustering models with low-dimensional contexts, and greatly improves the polynomial time sampling regime for the antiferromagnetic/ferromagnetic Ising model with inconsistent field on expander graphs. It also improves on previous approximation algorithm results based on the naive mean-field approximation in variational methods and statistical physics. Our approach is based on a new fusion of ideas from the MCMC and variational inference worlds. As part of our algorithm, we define a new nonconvex variational problem which allows us to sample from an exponential reweighting of a distribution by a negative definite quadratic form, and show how to make this procedure provably efficient using stochastic gradient descent. On top of this, we construct a new simulated tempering chain (on an extended state space arising from the Hubbard-Stratonovich transform) which overcomes the obstacle posed by large positive eigenvalues, and combine it with the SGD-based sampler to solve the full problem.

cs.DS↗

arXiv2021

Iterative Feature Matching: Toward Provable Domain Generalization with Logarithmic Environments

Domain generalization aims at performing well on unseen test environments with data from a limited number of training environments. Despite a proliferation of proposal algorithms for this task, assessing their performance both theoretically and empirically is still very challenging. Distributional matching algorithms such as (Conditional) Domain Adversarial Networks [Ganin et al., 2016, Long et al., 2018] are popular and enjoy empirical success, but they lack formal guarantees. Other approaches such as Invariant Risk Minimization (IRM) require a prohibitively large number of training environments -- linear in the dimension of the spurious feature space $d_s$ -- even on simple data models like the one proposed by [Rosenfeld et al., 2021]. Under a variant of this model, we show that both ERM and IRM cannot generalize with $o(d_s)$ environments. We then present an iterative feature matching algorithm that is guaranteed with high probability to yield a predictor that generalizes after seeing only $O(\log d_s)$ environments. Our results provide the first theoretical justification for a family of distribution-matching algorithms widely used in practice under a concrete nontrivial data model.

cs.LG↗

arXiv2021

An Online Learning Approach to Interpolation and Extrapolation in Domain Generalization

A popular assumption for out-of-distribution generalization is that the training data comprises sub-datasets, each drawn from a distinct distribution; the goal is then to "interpolate" these distributions and "extrapolate" beyond them -- this objective is broadly known as domain generalization. A common belief is that ERM can interpolate but not extrapolate and that the latter is considerably more difficult, but these claims are vague and lack formal justification. In this work, we recast generalization over sub-groups as an online game between a player minimizing risk and an adversary presenting new test distributions. Under an existing notion of inter- and extrapolation based on reweighting of sub-group likelihoods, we rigorously demonstrate that extrapolation is computationally much harder than interpolation, though their statistical complexity is not significantly different. Furthermore, we show that ERM -- or a noisy variant -- is provably minimax-optimal for both tasks. Our framework presents a new avenue for the formal analysis of domain generalization algorithms which may be of independent interest.

cs.LG↗

arXiv2021

Analyzing and Improving the Optimization Landscape of Noise-Contrastive Estimation

Noise-contrastive estimation (NCE) is a statistically consistent method for learning unnormalized probabilistic models. It has been empirically observed that the choice of the noise distribution is crucial for NCE's performance. However, such observations have never been made formal or quantitative. In fact, it is not even clear whether the difficulties arising from a poorly chosen noise distribution are statistical or algorithmic in nature. In this work, we formally pinpoint reasons for NCE's poor performance when an inappropriate noise distribution is used. Namely, we prove these challenges arise due to an ill-behaved (more precisely, flat) loss landscape. To address this, we introduce a variant of NCE called "eNCE" which uses an exponential loss and for which normalized gradient descent addresses the landscape issues provably when the target and noise distributions are in a given exponential family.

cs.LG↗

arXiv2021

The Effects of Invertibility on the Representational Complexity of Encoders in Variational Autoencoders

Training and using modern neural-network based latent-variable generative models (like Variational Autoencoders) often require simultaneously training a generative direction along with an inferential(encoding) direction, which approximates the posterior distribution over the latent variables. Thus, the question arises: how complex does the inferential model need to be, in order to be able to accurately model the posterior distribution of a given generative model? In this paper, we identify an important property of the generative map impacting the required size of the encoder. We show that if the generative map is "strongly invertible" (in a sense we suitably formalize), the inferential model need not be much more complex. Conversely, we prove that there exist non-invertible generative maps, for which the encoding direction needs to be exponentially larger (under standard assumptions in computational complexity). Importantly, we do not require the generative model to be layerwise invertible, which a lot of the related literature assumes and isn't satisfied by many architectures used in practice (e.g. convolution and pooling based networks). Thus, we provide theoretical support for the empirical wisdom that learning deep generative models is harder when data lies on a low-dimensional manifold.

cs.LG↗

arXiv2021

Universal Approximation for Log-concave Distributions using Well-conditioned Normalizing Flows

Normalizing flows are a widely used class of latent-variable generative models with a tractable likelihood. Affine-coupling (Dinh et al, 2014-16) models are a particularly common type of normalizing flows, for which the Jacobian of the latent-to-observable-variable transformation is triangular, allowing the likelihood to be computed in linear time. Despite the widespread usage of affine couplings, the special structure of the architecture makes understanding their representational power challenging. The question of universal approximation was only recently resolved by three parallel papers (Huang et al.,2020;Zhang et al.,2020;Koehler et al.,2020) -- who showed reasonably regular distributions can be approximated arbitrarily well using affine couplings -- albeit with networks with a nearly-singular Jacobian. As ill-conditioned Jacobians are an obstacle for likelihood-based training, the fundamental question remains: which distributions can be approximated using well-conditioned affine coupling flows? In this paper, we show that any log-concave distribution can be approximated using well-conditioned affine-coupling flows. In terms of proof techniques, we uncover and leverage deep connections between affine coupling architectures, underdamped Langevin dynamics (a stochastic differential equation often used to sample from Gibbs measures) and Hénon maps (a structured dynamical system that appears in the study of symplectic diffeomorphisms). Our results also inform the practice of training affine couplings: we approximate a padded version of the input distribution with iid Gaussians -- a strategy which Koehler et al.(2020) empirically observed to result in better-conditioned flows, but had hitherto no theoretical grounding. Our proof can thus be seen as providing theoretical evidence for the benefits of Gaussian padding when training normalizing flows.

cs.LG↗