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Researcher profile

Cristóbal Guzmán

Cristóbal Guzmán contributes to research discovery and scholarly infrastructure.

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Machine Learningmath.OC

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Published work

6 published item(s)

preprint2026arXiv

Calibeating for general proper losses: A Bregman divergence approach

This work introduces a general framework for calibeating based on regret minimization. As compared to Foster and Hart's seminal calibeating work which had specialized treatments of Brier score (squared loss) and log loss, we consider a large family of proper losses that includes $α$-Tsallis losses (for $α\in [1, 2]$) and Lipschitz losses. Our results for Tsallis losses also hold for an unscaled version of Tsallis loss that recovers log loss. Our analysis is oriented around the Bregman divergence view of a proper loss. Technically, our results for the family of Tsallis losses that we consider are U-calibration results, simultaneously obtaining logarithmic regret for all losses in this family while having a weaker dependence on the dimension compared to previous results. Of potential independent interest, we also show a new regret equality for the regret of Be The Regularized Leader. This regret equality holds for general proper losses and itself is based on two results related to online updating formulas for the generalized variance, the latter being a previously introduced generalization of variance based on Bregman divergences.

0 citations0 reviewsNo code linkedOpen access
preprint2023arXiv

Stochastic Halpern Iteration with Variance Reduction for Stochastic Monotone Inclusions

We study stochastic monotone inclusion problems, which widely appear in machine learning applications, including robust regression and adversarial learning. We propose novel variants of stochastic Halpern iteration with recursive variance reduction. In the cocoercive -- and more generally Lipschitz-monotone -- setup, our algorithm attains $ε$ norm of the operator with $\mathcal{O}(\frac{1}{ε^3})$ stochastic operator evaluations, which significantly improves over state of the art $\mathcal{O}(\frac{1}{ε^4})$ stochastic operator evaluations required for existing monotone inclusion solvers applied to the same problem classes. We further show how to couple one of the proposed variants of stochastic Halpern iteration with a scheduled restart scheme to solve stochastic monotone inclusion problems with ${\mathcal{O}}(\frac{\log(1/ε)}{ε^2})$ stochastic operator evaluations under additional sharpness or strong monotonicity assumptions.

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preprint2022arXiv

Between Stochastic and Adversarial Online Convex Optimization: Improved Regret Bounds via Smoothness

Stochastic and adversarial data are two widely studied settings in online learning. But many optimization tasks are neither i.i.d. nor fully adversarial, which makes it of fundamental interest to get a better theoretical understanding of the world between these extremes. In this work we establish novel regret bounds for online convex optimization in a setting that interpolates between stochastic i.i.d. and fully adversarial losses. By exploiting smoothness of the expected losses, these bounds replace a dependence on the maximum gradient length by the variance of the gradients, which was previously known only for linear losses. In addition, they weaken the i.i.d. assumption by allowing, for example, adversarially poisoned rounds, which were previously considered in the expert and bandit setting. Our results extend this to the online convex optimization framework. In the fully i.i.d. case, our bounds match the rates one would expect from results in stochastic acceleration, and in the fully adversarial case they gracefully deteriorate to match the minimax regret. We further provide lower bounds showing that our regret upper bounds are tight for all intermediate regimes in terms of the stochastic variance and the adversarial variation of the loss gradients.

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preprint2022arXiv

Non-Euclidean Differentially Private Stochastic Convex Optimization: Optimal Rates in Linear Time

Differentially private (DP) stochastic convex optimization (SCO) is a fundamental problem, where the goal is to approximately minimize the population risk with respect to a convex loss function, given a dataset of $n$ i.i.d. samples from a distribution, while satisfying differential privacy with respect to the dataset. Most of the existing works in the literature of private convex optimization focus on the Euclidean (i.e., $\ell_2$) setting, where the loss is assumed to be Lipschitz (and possibly smooth) w.r.t. the $\ell_2$ norm over a constraint set with bounded $\ell_2$ diameter. Algorithms based on noisy stochastic gradient descent (SGD) are known to attain the optimal excess risk in this setting. In this work, we conduct a systematic study of DP-SCO for $\ell_p$-setups under a standard smoothness assumption on the loss. For $1< p\leq 2$, under a standard smoothness assumption, we give a new, linear-time DP-SCO algorithm with optimal excess risk. Previously known constructions with optimal excess risk for $1< p <2$ run in super-linear time in $n$. For $p=1$, we give an algorithm with nearly optimal excess risk. Our result for the $\ell_1$-setup also extends to general polyhedral norms and feasible sets. Moreover, we show that the excess risk bounds resulting from our algorithms for $1\leq p \leq 2$ are attained with high probability. For $2 < p \leq \infty$, we show that existing linear-time constructions for the Euclidean setup attain a nearly optimal excess risk in the low-dimensional regime. As a consequence, we show that such constructions attain a nearly optimal excess risk for $p=\infty$. Our work draws upon concepts from the geometry of normed spaces, such as the notions of regularity, uniform convexity, and uniform smoothness.

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preprint2022arXiv

Optimal Algorithms for Differentially Private Stochastic Monotone Variational Inequalities and Saddle-Point Problems

In this work, we conduct the first systematic study of stochastic variational inequality (SVI) and stochastic saddle point (SSP) problems under the constraint of differential privacy (DP). We propose two algorithms: Noisy Stochastic Extragradient (NSEG) and Noisy Inexact Stochastic Proximal Point (NISPP). We show that a stochastic approximation variant of these algorithms attains risk bounds vanishing as a function of the dataset size, with respect to the strong gap function; and a sampling with replacement variant achieves optimal risk bounds with respect to a weak gap function. We also show lower bounds of the same order on weak gap function. Hence, our algorithms are optimal. Key to our analysis is the investigation of algorithmic stability bounds, both of which are new even in the nonprivate case. The dependence of the running time of the sampling with replacement algorithms, with respect to the dataset size $n$, is $n^2$ for NSEG and $\tilde{O}(n^{3/2})$ for NISPP.

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preprint2020arXiv

Stability of Stochastic Gradient Descent on Nonsmooth Convex Losses

Uniform stability is a notion of algorithmic stability that bounds the worst case change in the model output by the algorithm when a single data point in the dataset is replaced. An influential work of Hardt et al. (2016) provides strong upper bounds on the uniform stability of the stochastic gradient descent (SGD) algorithm on sufficiently smooth convex losses. These results led to important progress in understanding of the generalization properties of SGD and several applications to differentially private convex optimization for smooth losses. Our work is the first to address uniform stability of SGD on {\em nonsmooth} convex losses. Specifically, we provide sharp upper and lower bounds for several forms of SGD and full-batch GD on arbitrary Lipschitz nonsmooth convex losses. Our lower bounds show that, in the nonsmooth case, (S)GD can be inherently less stable than in the smooth case. On the other hand, our upper bounds show that (S)GD is sufficiently stable for deriving new and useful bounds on generalization error. Most notably, we obtain the first dimension-independent generalization bounds for multi-pass SGD in the nonsmooth case. In addition, our bounds allow us to derive a new algorithm for differentially private nonsmooth stochastic convex optimization with optimal excess population risk. Our algorithm is simpler and more efficient than the best known algorithm for the nonsmooth case Feldman et al. (2020).

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