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Dan Roy

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9 papers

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NeurIPS Conference 2025 Conference Paper

On Traceability in $\ell_p$ Stochastic Convex Optimization

Sasha Voitovych
Mahdi Haghifam
Idan Attias
Gintare Karolina Dziugaite
Roi Livni
Dan Roy

In this paper, we investigate the necessity of traceability for accurate learning in stochastic convex optimization (SCO) under $\ell_p$ geometries. Informally, we say a learning algorithm is \emph{$m$-traceable} if, by analyzing its output, it is possible to identify at least $m$ of its training samples. Our main results uncover a fundamental tradeoff between traceability and excess risk in SCO. For every $p\in [1, \infty)$, we establish the existence of an excess risk threshold below which every sample-efficient learner is traceable with the number of samples which is a \emph{constant fraction} of its training sample. For $p\in [1, 2]$, this threshold coincides with the best excess risk of differentially private (DP) algorithms, i. e. , above this threshold, there exist algorithms that are not traceable, which corresponds to a sharp phase transition. For $p \in (2, \infty)$, this threshold instead gives novel lower bounds for DP learning, partially closing an open problem in this setup. En route to establishing these results, we prove a sparse variant of the fingerprinting lemma, which is of independent interest to the community.

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NeurIPS Conference 2023 Conference Paper

The Shaped Transformer: Attention Models in the Infinite Depth-and-Width Limit

Lorenzo Noci
Chuning Li
Mufan Li
Bobby He
Thomas Hofmann
Chris J. Maddison
Dan Roy

In deep learning theory, the covariance matrix of the representations serves as aproxy to examine the network’s trainability. Motivated by the success of Transform-ers, we study the covariance matrix of a modified Softmax-based attention modelwith skip connections in the proportional limit of infinite-depth-and-width. Weshow that at initialization the limiting distribution can be described by a stochasticdifferential equation (SDE) indexed by the depth-to-width ratio. To achieve awell-defined stochastic limit, the Transformer’s attention mechanism is modifiedby centering the Softmax output at identity, and scaling the Softmax logits by awidth-dependent temperature parameter. We examine the stability of the networkthrough the corresponding SDE, showing how the scale of both the drift and diffu-sion can be elegantly controlled with the aid of residual connections. The existenceof a stable SDE implies that the covariance structure is well-behaved, even for verylarge depth and width, thus preventing the notorious issues of rank degeneracyin deep attention models. Finally, we show, through simulations, that the SDEprovides a surprisingly good description of the corresponding finite-size model. We coin the name shaped Transformer for these architectural modifications.

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NeurIPS Conference 2022 Conference Paper

Adaptively Exploiting d-Separators with Causal Bandits

Blair Bilodeau
Linbo Wang
Dan Roy

Multi-armed bandit problems provide a framework to identify the optimal intervention over a sequence of repeated experiments. Without additional assumptions, minimax optimal performance (measured by cumulative regret) is well-understood. With access to additional observed variables that d-separate the intervention from the outcome (i. e. , they are a d-separator), recent "causal bandit" algorithms provably incur less regret. However, in practice it is desirable to be agnostic to whether observed variables are a d-separator. Ideally, an algorithm should be adaptive; that is, perform nearly as well as an algorithm with oracle knowledge of the presence or absence of a d-separator. In this work, we formalize and study this notion of adaptivity, and provide a novel algorithm that simultaneously achieves (a) optimal regret when a d-separator is observed, improving on classical minimax algorithms, and (b) significantly smaller regret than recent causal bandit algorithms when the observed variables are not a d-separator. Crucially, our algorithm does not require any oracle knowledge of whether a d-separator is observed. We also generalize this adaptivity to other conditions, such as the front-door criterion.

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NeurIPS Conference 2022 Conference Paper

Pruning’s Effect on Generalization Through the Lens of Training and Regularization

Tian Jin
Michael Carbin
Dan Roy
Jonathan Frankle
Gintare Karolina Dziugaite

Practitioners frequently observe that pruning improves model generalization. A long-standing hypothesis based on bias-variance trade-off attributes this generalization improvement to model size reduction. However, recent studies on over-parameterization characterize a new model size regime, in which larger models achieve better generalization. Pruning models in this over-parameterized regime leads to a contradiction -- while theory predicts that reducing model size harms generalization, pruning to a range of sparsities nonetheless improves it. Motivated by this contradiction, we re-examine pruning’s effect on generalization empirically. We show that size reduction cannot fully account for the generalization-improving effect of standard pruning algorithms. Instead, we find that pruning leads to better training at specific sparsities, improving the training loss over the dense model. We find that pruning also leads to additional regularization at other sparsities, reducing the accuracy degradation due to noisy examples over the dense model. Pruning extends model training time and reduces model size. These two factors improve training and add regularization respectively. We empirically demonstrate that both factors are essential to fully explaining pruning's impact on generalization.

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NeurIPS Conference 2022 Conference Paper

The Neural Covariance SDE: Shaped Infinite Depth-and-Width Networks at Initialization

Mufan Li
Mihai Nica
Dan Roy

The logit outputs of a feedforward neural network at initialization are conditionally Gaussian, given a random covariance matrix defined by the penultimate layer. In this work, we study the distribution of this random matrix. Recent work has shown that shaping the activation function as network depth grows large is necessary for this covariance matrix to be non-degenerate. However, the current infinite-width-style understanding of this shaping method is unsatisfactory for large depth: infinite-width analyses ignore the microscopic fluctuations from layer to layer, but these fluctuations accumulate over many layers. To overcome this shortcoming, we study the random covariance matrix in the shaped infinite-depth-and-width limit. We identify the precise scaling of the activation function necessary to arrive at a non-trivial limit, and show that the random covariance matrix is governed by a stochastic differential equation (SDE) that we call the Neural Covariance SDE. Using simulations, we show that the SDE closely matches the distribution of the random covariance matrix of finite networks. Additionally, we recover an if-and-only-if condition for exploding and vanishing norms of large shaped networks based on the activation function.

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NeurIPS Conference 2021 Conference Paper

Minimax Optimal Quantile and Semi-Adversarial Regret via Root-Logarithmic Regularizers

Jeffrey Negrea
Blair Bilodeau
Nicolò Campolongo
Francesco Orabona
Dan Roy

Quantile (and, more generally, KL) regret bounds, such as those achieved by NormalHedge (Chaudhuri, Freund, and Hsu 2009) and its variants, relax the goal of competing against the best individual expert to only competing against a majority of experts on adversarial data. More recently, the semi-adversarial paradigm (Bilodeau, Negrea, and Roy 2020) provides an alternative relaxation of adversarial online learning by considering data that may be neither fully adversarial nor stochastic (I. I. D. ). We achieve the minimax optimal regret in both paradigms using FTRL with separate, novel, root-logarithmic regularizers, both of which can be interpreted as yielding variants of NormalHedge. We extend existing KL regret upper bounds, which hold uniformly over target distributions, to possibly uncountable expert classes with arbitrary priors; provide the first full-information lower bounds for quantile regret on finite expert classes (which are tight); and provide an adaptively minimax optimal algorithm for the semi-adversarial paradigm that adapts to the true, unknown constraint faster, leading to uniformly improved regret bounds over existing methods.

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NeurIPS Conference 2021 Conference Paper

The future is log-Gaussian: ResNets and their infinite-depth-and-width limit at initialization

Mufan Li
Mihai Nica
Dan Roy

Theoretical results show that neural networks can be approximated by Gaussian processes in the infinite-width limit. However, for fully connected networks, it has been previously shown that for any fixed network width, $n$, the Gaussian approximation gets worse as the network depth, $d$, increases. Given that modern networks are deep, this raises the question of how well modern architectures, like ResNets, are captured by the infinite-width limit. To provide a better approximation, we study ReLU ResNets in the infinite-depth-and-width limit, where \emph{both} depth and width tend to infinity as their ratio, $d/n$, remains constant. In contrast to the Gaussian infinite-width limit, we show theoretically that the network exhibits log-Gaussian behaviour at initialization in the infinite-depth-and-width limit, with parameters depending on the ratio $d/n$. Using Monte Carlo simulations, we demonstrate that even basic properties of standard ResNet architectures are poorly captured by the Gaussian limit, but remarkably well captured by our log-Gaussian limit. Moreover, our analysis reveals that ReLU ResNets at initialization are hypoactivated: fewer than half of the ReLUs are activated. Additionally, we calculate the interlayer correlations, which have the effect of exponentially increasing the variance of the network output. Based on our analysis, we introduce \emph{Balanced ResNets}, a simple architecture modification, which eliminates hypoactivation and interlayer correlations and is more amenable to theoretical analysis.

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NeurIPS Conference 2021 Conference Paper

Towards a Unified Information-Theoretic Framework for Generalization

Mahdi Haghifam
Gintare Karolina Dziugaite
Shay Moran
Dan Roy

In this work, we investigate the expressiveness of the "conditional mutual information" (CMI) framework of Steinke and Zakynthinou (2020) and the prospect of using it to provide a unified framework for proving generalization bounds in the realizable setting. We first demonstrate that one can use this framework to express non-trivial (but sub-optimal) bounds for any learning algorithm that outputs hypotheses from a class of bounded VC dimension. We then explore two directions of strengthening this bound: (i) Can the CMI framework express optimal bounds for VC classes? (ii) Can the CMI framework be used to analyze algorithms whose output hypothesis space is unrestricted (i. e. has an unbounded VC dimension)? With respect to Item (i) we prove that the CMI framework yields the optimal bound on the expected risk of Support Vector Machines (SVMs) for learning halfspaces. This result is an application of our general result showing that stable compression schemes Bousquet al. (2020) of size $k$ have uniformly bounded CMI of order $O(k)$. We further show that an inherent limitation of proper learning of VC classes contradicts the existence of a proper learner with constant CMI, and it implies a negative resolution to an open problem of Steinke and Zakynthinou (2020). We further study the CMI of empirical risk minimizers (ERMs) of class $H$ and show that it is possible to output all consistent classifiers (version space) with bounded CMI if and only if $H$ has a bounded star number (Hanneke and Yang (2015)). With respect to Item (ii) we prove a general reduction showing that "leave-one-out" analysis is expressible via the CMI framework. As a corollary we investigate the CMI of the one-inclusion-graph algorithm proposed by Haussler et al. (1994). More generally, we show that the CMI framework is universal in the sense that for every consistent algorithm and data distribution, the expected risk vanishes as the number of samples diverges if and only if its evaluated CMI has sublinear growth with the number of samples.

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NeurIPS Conference 2011 Conference Paper

Complexity of Inference in Latent Dirichlet Allocation

David Sontag
Dan Roy

We consider the computational complexity of probabilistic inference in Latent Dirichlet Allocation (LDA). First, we study the problem of finding the maximum a posteriori (MAP) assignment of topics to words, where the document's topic distribution is integrated out. We show that, when the effective number of topics per document is small, exact inference takes polynomial time. In contrast, we show that, when a document has a large number of topics, finding the MAP assignment of topics to words in LDA is NP-hard. Next, we consider the problem of finding the MAP topic distribution for a document, where the topic-word assignments are integrated out. We show that this problem is also NP-hard. Finally, we briefly discuss the problem of sampling from the posterior, showing that this is NP-hard in one restricted setting, but leaving open the general question.

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