arXiv subjects

Jelena Diakonikolas

Publications and source records attributed to Jelena Diakonikolas.

At least 19 recordsLinked to original sources

arXiv2026

Oracle Complexity of Stochastic Fixed-Point Equations with Nonexpansive Maps

We study the oracle complexity of computing a point with small fixed-point residual $\|T(x)-x\| \leq ε$, for a general norm $\|\cdot\|$ and a self-map $T$ of a compact convex set. We study this problem in the setting where $T$ is nonexpansive with respect to the same norm $\|\cdot\|$ and accessed via an unbiased stochastic oracle with bounded variance $σ^2$. We provide an algorithm that solves such instances for any norm with a weak Rademacher type $q > 1$, with high probability. The algorithm is based on a recursive anchoring technique. For type-$2$ spaces, such as $\ell_p$-spaces for $p \in [2, \infty]$, our algorithm attains stochastic oracle complexity $\tilde O(σ^2 ε^{-3} + ε^{-1})$. We further prove a near-matching lower bound (i.e., matching up to poly-log factors) for such $\ell_{\infty}$-norm instances in high dimensions. Our lower bound holds against any randomized algorithm that succeeds with constant probability. It further extends to settings with ``sparse'' noise, where variance measured with respect to any $\ell_p$ norm is of the same order, ruling out the possibility of improving oracle complexity as a function of $\varepsilon$ by measuring variance in a non-matching $\ell_p$ norm.

math.OC↗

arXiv2026

Efficient Optimization on the Euclidean Sphere: Riemannian Gradient Alignment as a Unifying Principle

Euclidean sphere-constrained optimization problems represent perhaps the most basic class of nonconvex-constrained problems for which there exist efficient optimization algorithms for specific problem instances, such as the eigenvalue problems and learning problems arising in the recent literature. However, when and why we can efficiently optimize on the sphere using simple first-order methods is poorly understood. We introduce Riemannian Gradient Alignment (RGA) as a unifying structural condition enabling fast convergence. RGA is a parameterized directional error bound requiring a tangent vector to sufficiently align with a target solution vector. We prove that Riemannian Gradient Descent with diminishing step sizes converges linearly when RGA holds with order $r=1$, and at rate $O(k^{-1/(2r-2)})$ for $r>1$. The result applies even to tangent vector fields that are not gradients of any objective. Our main technical contribution is a family of derivative-based sufficient conditions for RGA. We prove these conditions for a range of learning problems, including multi-instance learning/max pooling, Gaussian halfspaces, single-index and generalized linear models, phase retrieval, complete orthogonal dictionary learning, and (sparse) PCA. The provided examples span distributional families that include standard Gaussian, conditionally Gaussian, and more general structured distributions. The provided guarantees cover objectives that are generally not (geodesically) convex, may be nonsmooth, and may arise from discrete distributions. These results establish tractability of problem classes where no prior error bound conditions were known, as well as unify, strengthen, and generalize several prior results.

math.OC↗

arXiv2026

Linear Regression under Missing or Corrupted Coordinates

We study multivariate linear regression under Gaussian covariates in two settings, where data may be erased or corrupted by an adversary under a coordinate-wise budget. In the incomplete data setting, an adversary may inspect the dataset and delete entries in up to an $η$-fraction of samples per coordinate; a strong form of the Missing Not At Random model. In the corrupted data setting, the adversary instead replaces values arbitrarily, and the corruption locations are unknown to the learner. Despite substantial work on missing data, linear regression under such adversarial missingness remains poorly understood, even information-theoretically. Unlike the clean setting, where estimation error vanishes with more samples, here the optimal error remains a positive function of the problem parameters. Our main contribution is to characterize this error up to constant factors across essentially the entire parameter range. Specifically, we establish novel information-theoretic lower bounds on the achievable error that match the error of (computationally efficient) algorithms. A key implication is that, perhaps surprisingly, the optimal error in the missing data setting matches that in the corruption setting-so knowing the corruption locations offers no general advantage.

cs.DS↗

arXiv2026

Pushing the Complexity Boundaries of Fixed-Point Equations: Adaptation to Contraction and Controlled Expansion

Fixed-point equations with Lipschitz operators are central to areas such as optimization, game theory, economics, and dynamical systems. When the operator is contractive or nonexpansive (i.e., when its Lipschitz constant is $γ\leq 1$), decades of work have established efficient algorithms with tight oracle complexity guarantees. In sharp contrast, even mildly expansive operators ($γ> 1$) render fixed-point computation intractable in the worst case, with exponential oracle lower bounds. This dichotomy leaves open a fundamental question: are there intermediate regimes where efficient approximation is still possible? Our main contribution is in establishing such regimes. First, we show that a seemingly misguided idea -- running Halpern iteration with a fixed step size -- provably finds $ε$-approximate fixed points at near-optimal oracle complexity for contractive and nonexpansive operators, and it succeeds for mildly expansive operators up to the hardness frontier. Building on this insight, we design the Gradual Halpern Algorithm (GHAL) and its parameter-free variant AdaGHAL, which adapt automatically to the operator's Lipschitz constant and recover optimal complexity across the contractive and nonexpansive regimes, while extending guarantees into the mildly expansive case. Finally, we introduce the class of gradually expansive operators, permitting constant expansion (up to $\approx 1.4$), and prove that AdaGHAL finds $ε$-approximate fixed points in $O(1/ε)$ iterations -- demonstrating for the first time that efficient fixed-point computation is possible under controlled expansion significantly beyond the nonexpansive regime. Our results apply in general normed vector spaces -- including infinite-dimensional Banach spaces, and generalize to non-positively curved geodesic metric spaces.

math.OC↗

arXiv2026

Solving Stochastic Fixed-Point Equations with High Probability

We study stochastic fixed-point equations $\mathbf{T}(\mathbf{x}) = \mathbf{x}$ over normed spaces $(\mathcal{E}, \|\cdot\|)$, where the operator $\mathbf{T}$ is nonexpansive or contractive and is accessed only through unbiased stochastic evaluations with bounded second central moment. Given $ε> 0, δ\in (0, 1)$, the goal is to output $\mathbf{x} \in \mathcal{E}$ such that $\|\mathbf{T}(\mathbf{x}) - \mathbf{x}\| \leq ε$ with probability at least $1-δ$. We introduce VR-GHAL, a variance-reduced gradual Halpern method for quadratically smoothable Banach spaces. The key algorithmic ingredient is a recursive stochastic estimator based on clipped differences of oracle evaluations: instead of clipping $τ(\mathbf{x}; ξ)$ itself, we clip stochastic differences at the Lipschitz scale $γ\|\mathbf{x} - \mathbf{y}\|$. This makes the estimator pathwise Lipschitz along the algorithmic trajectory while permitting martingale concentration under finite second moments in the native norm. Our main theorem gives an anytime high-probability residual bound: on a single event of probability at least $1 - δ$, the residual decreases nearly geometrically across epochs, up to lower-order logarithmic factors. Under only bounded variance, displaying only the dependence on the target error $ε$ and Lipschitz constant $γ\in (0, 1]$ of $\mathbf{T}$, the resulting oracle complexity is $\min\{ε^{-5}, (1-γ)^{-3}ε^{-2}\}$. Under a Lipschitz-in-expectation oracle, the dependence improves to the corresponding $ε^{-3}$ nonexpansive rate (i.e., for $γ= 1$), and under samplewise nonexpansiveness to $ε^{-2}$.

math.OC↗

arXiv2026

Efficient Methods for Min-Max Optimization with Dual-Linear Coupling

We study a class of convex-concave min-max problems in which the coupled component of the objective is linear in at least one of the two decision vectors. We identify such problem structure as interpolating between the bilinearly and nonbilinearly coupled problems, motivated by key applications in areas such as distributionally robust optimization and convex optimization with functional constraints. Leveraging the considered nonlinear-linear coupling of the primal and the dual decision vectors, we develop a general algorithmic framework leading to fine-grained complexity bounds exploiting separability properties of the problem, whenever present. The obtained complexity bounds offer potential improvements over state-of-the-art scaling with $\sqrt{n}$ or $n$ in some of the considered problem settings, which even include bilinearly coupled problems, where $n$ is the dimension of the dual decision vector. On the algorithmic front, our work provides novel strategies for combining randomization with extrapolation and multi-point anchoring in the mirror descent-style updates in the primal and the dual, which we hope will find further applications in addressing related optimization problems.

math.OC↗

arXiv2026

Efficiently Learning Drifting Halfspaces with Massart Noise

We study the problem of learning a drifting concept in the presence of Massart noise. In this framework, an online learner has access to a history of independent samples whose labels are noisy versions of a target concept that may change from round to round. The goal is to output, in each round, a hypothesis with small prediction error. We study the complexity of this learning problem for the fundamental class of margin-separable linear classifiers (halfspaces). On the positive side, we give a computationally efficient learner achieving error $η+ \tilde O(Δ^{1/3}/γ)$, where $η$ upper bounds the Massart noise rate, $Δ$ is the drift rate, and $γ$ is the margin. Interestingly, in the realizable setting, an adaptation of our techniques yields an efficient learner with an improved error rate over prior work. On the lower-bound side, we provide formal evidence of an information-computation tradeoff, strongly suggesting that our algorithm's performance is essentially optimal. Specifically, while the information-theoretically optimal error scales with $Δ^{1/2}$, we prove that $Δ^{1/3}$-scaling is unavoidable for low-degree polynomial tests, even in the special case of random classification noise.

cs.LG↗

arXiv2026

Robust Learning of a Group DRO Neuron

We study the problem of learning a single neuron under standard squared loss in the presence of arbitrary label noise and group-level distributional shifts, for a broad family of covariate distributions. Our goal is to identify a ''best-fit'' neuron parameterized by $\mathbf{w}_*$ that performs well under the most challenging reweighting of the groups. Specifically, we address a Group Distributionally Robust Optimization problem: given sample access to $K$ distinct distributions $\mathcal p_{[1]},\dots,\mathcal p_{[K]}$, we seek to approximate $\mathbf{w}_*$ that minimizes the worst-case objective over convex combinations of group distributions $\boldsymbolλ \in Δ_K$, where the objective is $\sum_{i \in [K]}λ_{[i]}\,\mathbb E_{(\mathbf x,y)\sim\mathcal p_{[i]}}(σ(\mathbf w\cdot\mathbf x)-y)^2 - νd_f(\boldsymbolλ,\frac{1}{K}\mathbf1)$ and $d_f$ is an $f$-divergence that imposes (optional) penalty on deviations from uniform group weights, scaled by a parameter $ν\geq 0$. We develop a computationally efficient primal-dual algorithm that outputs a vector $\widehat{\mathbf w}$ that is constant-factor competitive with $\mathbf{w}_*$ under the worst-case group weighting. Our analytical framework directly confronts the inherent nonconvexity of the loss function, providing robust learning guarantees in the face of arbitrary label corruptions and group-specific distributional shifts. The implementation of the dual extrapolation update motivated by our algorithmic framework shows promise on LLM pre-training benchmarks.

cs.LG↗

arXiv2026

Adaptive Delayed-Update Cyclic Algorithm for Variational Inequalities

Cyclic block coordinate methods are a fundamental class of first-order algorithms, widely used in practice for their simplicity and strong empirical performance. Yet, their theoretical behavior remains challenging to explain, and setting their step sizes -- beyond classical coordinate descent for minimization -- typically requires careful tuning or line-search machinery. In this work, we develop $\texttt{ADUCA}$ (Adaptive Delayed-Update Cyclic Algorithm), a cyclic algorithm addressing a broad class of Minty variational inequalities with monotone Lipschitz operators. $\texttt{ADUCA}$ is parameter-free: it requires no global or block-wise Lipschitz constants and uses no per-epoch line search, except at initialization. A key feature of the algorithm is using operator information delayed by a full cycle, which makes the algorithm compatible with parallel and distributed implementations, and attractive due to weakened synchronization requirements across blocks. We prove that $\texttt{ADUCA}$ attains (near) optimal global oracle complexity as a function of target error $ε>0,$ scaling with $1/ε$ for monotone operators, or with $\log^2(1/ε)$ for operators that are strongly monotone.

math.OC↗

arXiv2025

Distributionally Robust Optimization with Adversarial Data Contamination

Distributionally Robust Optimization (DRO) provides a framework for decision-making under distributional uncertainty, yet its effectiveness can be compromised by outliers in the training data. This paper introduces a principled approach to simultaneously address both challenges. We focus on optimizing Wasserstein-1 DRO objectives for generalized linear models with convex Lipschitz loss functions, where an $ε$-fraction of the training data is adversarially corrupted. Our primary contribution lies in a novel modeling framework that integrates robustness against training data contamination with robustness against distributional shifts, alongside an efficient algorithm inspired by robust statistics to solve the resulting optimization problem. We prove that our method achieves an estimation error of $O(\sqrtε)$ for the true DRO objective value using only the contaminated data under the bounded covariance assumption. This work establishes the first rigorous guarantees, supported by efficient computation, for learning under the dual challenges of data contamination and distributional shifts.

cs.LG↗

arXiv2025

Robustly Learning Monotone Single-Index Models

We consider the basic problem of learning Single-Index Models with respect to the square loss under the Gaussian distribution in the presence of adversarial label noise. Our main contribution is the first computationally efficient algorithm for this learning task, achieving a constant factor approximation, that succeeds for the class of {\em all} monotone activations with bounded moment of order $2 + ζ,$ for $ζ> 0.$ This class in particular includes all monotone Lipschitz functions and even discontinuous functions like (possibly biased) halfspaces. Prior work for the case of unknown activation either does not attain constant factor approximation or succeeds for a substantially smaller family of activations. The main conceptual novelty of our approach lies in developing an optimization framework that steps outside the boundaries of usual gradient methods and instead identifies a useful vector field to guide the algorithm updates by directly leveraging the problem structure, properties of Gaussian spaces, and regularity of monotone functions.

cs.LG↗

arXiv2025

Robustly Learning Monotone Generalized Linear Models via Data Augmentation

We study the task of learning Generalized Linear models (GLMs) in the agnostic model under the Gaussian distribution. We give the first polynomial-time algorithm that achieves a constant-factor approximation for \textit{any} monotone Lipschitz activation. Prior constant-factor GLM learners succeed for a substantially smaller class of activations. Our work resolves a well-known open problem, by developing a robust counterpart to the classical GLMtron algorithm (Kakade et al., 2011). Our robust learner applies more generally, encompassing all monotone activations with bounded $(2+ζ)$-moments, for any fixed $ζ>0$ -- a condition that is essentially necessary. To obtain our results, we leverage a novel data augmentation technique with decreasing Gaussian noise injection and prove a number of structural results that may be useful in other settings.

cs.LG↗

arXiv2025

Near-Linear Runtime for a Classical Matrix Preconditioning Algorithm

In 1960, Osborne proposed a simple iterative algorithm for matrix balancing with outstanding numerical performance. Today, it is the default preconditioning procedure before eigenvalue computation and other linear algebra subroutines in mainstream software packages such as Python, Julia, MATLAB, EISPACK, LAPACK, and more. Despite its widespread usage, Osborne's algorithm has long resisted theoretical guarantees for its runtime: the first polynomial-time guarantees were obtained only in the past decade, and recent near-linear runtimes remain confined to variants of Osborne's algorithm with important differences that make them simpler to analyze but empirically slower. In this paper, we address this longstanding gap between theory and practice by proving that Osborne's original algorithm -- the de facto preconditioner in practice -- in fact has a near-linear runtime. This runtime guarantee (1) is optimal in the input size up to at most a single logarithm, (2) is the first runtime for Osborne's algorithm that does not dominate the runtime of downstream tasks like eigenvalue computation, and (3) improves upon the theoretical runtimes for all other variants of Osborne's algorithm.

math.OC↗

arXiv2025

Drago: Primal-Dual Coupled Variance Reduction for Faster Distributionally Robust Optimization

We consider the penalized distributionally robust optimization (DRO) problem with a closed, convex uncertainty set, a setting that encompasses learning using $f$-DRO and spectral/$L$-risk minimization. We present Drago, a stochastic primal-dual algorithm that combines cyclic and randomized components with a carefully regularized primal update to achieve dual variance reduction. Owing to its design, Drago enjoys a state-of-the-art linear convergence rate on strongly convex-strongly concave DRO problems with a fine-grained dependency on primal and dual condition numbers. Theoretical results are supported by numerical benchmarks on regression and classification tasks.

stat.ML↗

arXiv2025

A Block Coordinate and Variance-Reduced Method for Generalized Variational Inequalities of Minty Type

Block coordinate methods have been extensively studied for minimization problems, where they come with significant complexity improvements whenever the considered problems are compatible with block decomposition and, moreover, block Lipschitz parameters are highly nonuniform. For the more general class of variational inequalities with monotone operators, essentially none of the existing methods transparently shows potential complexity benefits of using block coordinate updates in such settings. Motivated by this gap, we develop a new randomized block coordinate method and study its oracle complexity and runtime. We prove that in the setting where block Lipschitz parameters are highly nonuniform -- the main setting in which block coordinate methods lead to high complexity improvements in any of the previously studied settings -- our method can lead to complexity improvements by a factor order-$m$, where $m$ is the number of coordinate blocks. The same method further applies to the more general problem with a finite-sum operator with $m$ components, where it can be interpreted as performing variance reduction. Compared to the state of the art, the method leads to complexity improvements up to a factor $\sqrt{m},$ obtained when the component Lipschitz parameters are highly nonuniform.

math.OC↗

arXiv2024

Learning a Single Neuron Robustly to Distributional Shifts and Adversarial Label Noise

We study the problem of learning a single neuron with respect to the $L_2^2$-loss in the presence of adversarial distribution shifts, where the labels can be arbitrary, and the goal is to find a ``best-fit'' function. More precisely, given training samples from a reference distribution $\mathcal{p}_0$, the goal is to approximate the vector $\mathbf{w}^*$ which minimizes the squared loss with respect to the worst-case distribution that is close in $χ^2$-divergence to $\mathcal{p}_{0}$. We design a computationally efficient algorithm that recovers a vector $ \hat{\mathbf{w}}$ satisfying $\mathbb{E}_{\mathcal{p}^*} (σ(\hat{\mathbf{w}} \cdot \mathbf{x}) - y)^2 \leq C \, \mathbb{E}_{\mathcal{p}^*} (σ(\mathbf{w}^* \cdot \mathbf{x}) - y)^2 + ε$, where $C>1$ is a dimension-independent constant and $(\mathbf{w}^*, \mathcal{p}^*)$ is the witness attaining the min-max risk $\min_{\mathbf{w}~:~\|\mathbf{w}\| \leq W} \max_{\mathcal{p}} \mathbb{E}_{(\mathbf{x}, y) \sim \mathcal{p}} (σ(\mathbf{w} \cdot \mathbf{x}) - y)^2 - νχ^2(\mathcal{p}, \mathcal{p}_0)$. Our algorithm follows a primal-dual framework and is designed by directly bounding the risk with respect to the original, nonconvex $L_2^2$ loss. From an optimization standpoint, our work opens new avenues for the design of primal-dual algorithms under structured nonconvexity.

cs.LG↗

arXiv2024

Sample and Computationally Efficient Robust Learning of Gaussian Single-Index Models

A single-index model (SIM) is a function of the form $σ(\mathbf{w}^{\ast} \cdot \mathbf{x})$, where $σ: \mathbb{R} \to \mathbb{R}$ is a known link function and $\mathbf{w}^{\ast}$ is a hidden unit vector. We study the task of learning SIMs in the agnostic (a.k.a. adversarial label noise) model with respect to the $L^2_2$-loss under the Gaussian distribution. Our main result is a sample and computationally efficient agnostic proper learner that attains $L^2_2$-error of $O(\mathrm{OPT})+ε$, where $\mathrm{OPT}$ is the optimal loss. The sample complexity of our algorithm is $\tilde{O}(d^{\lceil k^{\ast}/2\rceil}+d/ε)$, where $k^{\ast}$ is the information-exponent of $σ$ corresponding to the degree of its first non-zero Hermite coefficient. This sample bound nearly matches known CSQ lower bounds, even in the realizable setting. Prior algorithmic work in this setting had focused on learning in the realizable case or in the presence of semi-random noise. Prior computationally efficient robust learners required significantly stronger assumptions on the link function.

cs.LG↗

arXiv2024

Optimization on a Finer Scale: Bounded Local Subgradient Variation Perspective

We initiate the study of nonsmooth optimization problems under bounded local subgradient variation, which postulates bounded difference between (sub)gradients in small local regions around points, in either average or maximum sense. The resulting class of objective functions encapsulates the classes of objective functions traditionally studied in optimization, which are defined based on either Lipschitz continuity of the objective or Hölder/Lipschitz continuity of its gradient. Further, the defined class contains functions that are neither Lipschitz continuous nor have a Hölder continuous gradient. When restricted to the traditional classes of optimization problems, the parameters defining the studied classes lead to more fine-grained complexity bounds, recovering traditional oracle complexity bounds in the worst case but generally leading to lower oracle complexity for functions that are not ``worst case.'' Some highlights of our results are that: (i) it is possible to obtain complexity results for both convex and nonconvex problems with the (local or global) Lipschitz constant being replaced by a constant of local subgradient variation and (ii) mean width of the subdifferential set around the optima plays a role in the complexity of nonsmooth optimization, particularly in parallel settings. A consequence of (ii) is that for any error parameter $ε> 0$, parallel oracle complexity of nonsmooth Lipschitz convex optimization is lower than its sequential oracle complexity by a factor $\tildeΩ\big(\frac{1}ε\big)$ whenever the objective function is piecewise linear with polynomially many pieces in the input size. This is particularly surprising as existing parallel complexity lower bounds are based on such classes of functions. The seeming contradiction is resolved by considering the region in which the algorithm is allowed to query the objective.

math.OC↗