arXiv subjects

Alp Yurtsever

Publications and source records attributed to Alp Yurtsever.

At least 19 recordsLinked to original sources

arXiv2026

Rethinking MoE Load Balancing as Feedback Control

Mixture-of-Experts (MoE) training is often not stationary: the data mixture changes at curriculum boundaries, when a model is continually pretrained into a new domain, or when the corpus is refreshed. Routing is learned against the current mixture, so each such change disrupts the per-expert load the router had converged to, and existing balancing methods have only been evaluated on stationary data. We rethink load balancing as a feedback control problem, in which the deviation of each expert's load from its fair share is the control error and the per-expert routing bias is the control input, and define a family of controllers, the PID-Balancer, from the proportional, integral, and derivative terms of classical PID control. Two widely used methods belong to this family: the auxiliary balancing loss behaves like a proportional controller, and the loss-free bias update is an integral controller acting on the sign of the error. Reading the router as a static map with a one-step delay, we show that the integral term is the only one that both removes persistent imbalance and averages out the step-to-step fluctuation of the measured load, whereas the proportional and derivative terms pass or amplify that fluctuation into the routing bias. We therefore adopt PID (I), the pure-integral member of the family with a single gain. Two settings are evaluated: (i) training an OLMoE backbone from scratch with 16 to 64 experts per layer, through a mid-run mixture switch, and (ii) continual pretraining of the released OLMoE-1B-7B on a new corpus. PID (I) holds balance before the shift and restores it more tightly afterwards than either the auxiliary loss or loss-free balancing, at comparable perplexity.

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arXiv2026

The Multi-Block DC Function Class: Theory, Algorithms, and Applications

We present the Multi-Block DC (BDC) class, a rich class of structured nonconvex functions that admit a DC ("difference-of-convex") decomposition across parameter blocks. This multi-block class not only subsumes the usual DC programming, but also turns out to be provably more powerful. Specifically, we demonstrate how standard models (e.g., polynomials and tensor factorization) must have DC decompositions of exponential size, while their BDC formulation is polynomial. This separation in complexity also underscores another key aspect: unlike DC formulations, obtaining BDC formulations for problems is vastly easier and constructive. We illustrate this aspect by presenting explicit BDC formulations for modern tasks such as deep ReLU networks, a result with no known equivalent in the DC class. Moreover, we complement the theory by developing algorithms with non-asymptotic convergence theory, including both batch and stochastic settings, and demonstrate the broad applicability of our method through several applications.

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arXiv2026

Universal Adaptive Proximal Gradient Methods via Gradient Mapping Accumulation

We propose an adaptive proximal gradient method for minimizing the sum of two functions, where one is a simple convex function, and the other belongs to one of the three classes: nonconvex smooth, convex nonsmooth, or convex smooth. The key feature of the method is an adaptive step size that accumulates historical gradient mapping norms in the denominator. Without any modification or knowledge of problem parameters, the method converges across all three problem classes under mild bounded-iterates and bounded-variance assumptions, with rates matching those of the proximal gradient method up to logarithmic factors, in both deterministic and stochastic settings. For the convex setting, we further propose an accelerated variant. It retains a similar near-optimal convergence rate for the nonsmooth case and achieves an improved rate of order $\widetilde{O}\big(1/t^2 + σ/\sqrt{t}\big)$ for the smooth case, which is optimal up to logarithmic factors. Notably, we develop new techniques for controlling the effect of stochastic noise, which are applicable across all three problem classes in the stochastic setting and enable a simplified analysis.

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arXiv2026

Adaptive Federated Optimization Across Smooth and Nonsmooth Regimes

AdaGrad-style methods can simultaneously adapt to both the order and constants of smoothness in centralized optimization. However, existing extensions to federated optimization with heterogeneous client objectives and multiple local steps require the smoothness order to be specified in advance. We address this limitation with AdaFed, a federated method that adapts across several regimes under a single choice of parameters. In the deterministic setting, AdaFed achieves an $O(1/T)$ convergence rate for both smooth nonconvex and smooth convex objectives, and an $O(1/\sqrt{T})$ rate for Lipschitz nonsmooth convex objectives up to logarithmic factors. Under a light-tailed stochastic-gradient noise assumption, these guarantees extend to the stochastic setting with additional logarithmic factors and noise-dependent terms. Experiments across the three problem classes further support our theoretical findings.

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arXiv2026

Primal-Dual Adaptive Subgradient Methods

We propose adaptive primal-dual subgradient methods for constrained convex optimization. Our first method applies an AdaGrad-style step-sizes to the dual problem and recovers primal solutions by averaging the associated Fenchel-type oracle outputs. It achieves $\widetilde{\mathcal{O}}(T^{-1/2})$ convergence for the dual objective residual, primal objective residual, and feasibility gap in the nonsmooth regime, while automatically improving to $\mathcal{O}(T^{-1})$ when the dual objective is smooth. We further develop an accelerated variant, which retains the $\widetilde{\mathcal{O}}(T^{-1/2})$ nonsmooth rate while achieving $\mathcal{O}(T^{-2})$ convergence in the smooth regime under mild assumptions. These guarantees require no prior knowledge of the underlying regularity. We demonstrate the empirical performance of the proposed methods on optimal transport and semidefinite relaxation of the Max-Cut problem.

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arXiv2026

Exact Convex Reformulations of Linear Neural Networks via Completely Positive Lifting

We show that the training problem of a deep linear neural network under the squared loss admits an exact convex reformulation in a lifted space over a generalized completely positive cone. The reformulation has the same optimal value as the original nonconvex problem and is linear in the lifted variables, with all nonconvexity encoded in the cone constraint. Its ambient lifted dimension depends only on the input and output dimensions, independent of the network depth and the number of data points, and the bottleneck width enters only through scalar constraints. The construction proceeds by reducing the multilayer parameterization to a bilinear factorization, lifting it to a rank-constrained semidefinite program, expressing the rank constraint via a complementarity condition, and applying a completely positive lifting. The resulting formulation gives a conic representation of the nonconvexity induced by linear factorization and connects linear neural network training with copositive programming.

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arXiv2026

Generalized Stochastic Gradient Descent with Momentum Methods for Smooth Optimization

Stochastic gradient descent with momentum (SGDM) methods have become fundamental optimization tools in machine learning, combining the computational efficiency of stochastic gradients with the acceleration benefits of momentum. Despite their widespread use in practice, the theoretical understanding of SGDM remains incomplete, with most existing analyses focusing on specific momentum schemes or requiring restrictive assumptions. In this paper, we introduce a generalized SGDM framework that unifies a broad class of momentum-based methods, including SGD with Polyak's momentum, SGD with Nesterov's momentum, and many others. We provide comprehensive convergence analyses for both convex and nonconvex optimization problems under mild smoothness and bounded variance assumptions. For convex problems, we establish general ergodic convergence results with constant parameters and derive improved iterate convergence rates with time-varying parameters. For nonconvex problems, we prove sublinear convergence to stationary points and establish linear convergence to a neighborhood of the optimum under the Polyak--Łojasiewicz condition. Notably, our analysis allows flexible parameter choices and thus provides convergence guarantees for many existing momentum methods as special cases.

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arXiv2025

Randomized Block Coordinate DC Programming

We introduce an extension of the Difference of Convex Algorithm (DCA) in the form of a randomized block coordinate approach for problems with separable structure. For $n$ coordinate-blocks and $k$ iterations, our main result proves a non-asymptotic convergence rate of $O(n/k)$ in expectation, with respect to a stationarity measure based on a Forward-Backward envelope. Furthermore, leveraging the connection between DCA and Expectation Maximization (EM), we propose a randomized block coordinate EM algorithm.

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arXiv2025

Revisiting Frank-Wolfe for Structured Nonconvex Optimization

We introduce a new projection-free (Frank-Wolfe) method for optimizing structured nonconvex functions that are expressed as a difference of two convex functions. This problem class subsumes smooth nonconvex minimization, positioning our method as a promising alternative to the classical Frank-Wolfe algorithm. DC decompositions are not unique; by carefully selecting a decomposition, we can better exploit the problem structure, improve computational efficiency, and adapt to the underlying problem geometry to find better local solutions. We prove that the proposed method achieves a first-order stationary point in $O(1/ε^2)$ iterations, matching the complexity of the standard Frank-Wolfe algorithm for smooth nonconvex minimization in general. Specific decompositions can, for instance, yield a gradient-efficient variant that requires only $O(1/ε)$ calls to the gradient oracle. Finally, we present numerical experiments demonstrating the effectiveness of the proposed method compared to other projection-free algorithms.

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arXiv2025

Implicit Bias in Matrix Factorization and its Explicit Realization in a New Architecture

Gradient descent for matrix factorization exhibits an implicit bias toward approximately low-rank solutions. While existing theories often assume the boundedness of iterates, empirically the bias persists even with unbounded sequences. This reflects a dynamic where factors develop low-rank structure while their magnitudes increase, tending to align with certain directions. To capture this behavior in a stable way, we introduce a new factorization model: $X\approx UDV^\top$, where $U$ and $V$ are constrained within norm balls, while $D$ is a diagonal factor allowing the model to span the entire search space. Experiments show that this model consistently exhibits a strong implicit bias, yielding truly (rather than approximately) low-rank solutions. Extending the idea to neural networks, we introduce a new model featuring constrained layers and diagonal components that achieves competitive performance on various regression and classification tasks while producing lightweight, low-rank representations.

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arXiv2025

Improved Rates for Stochastic Variance-Reduced Difference-of-Convex Algorithms

In this work, we propose and analyze DCA-PAGE, a novel algorithm that integrates the difference-of-convex algorithm (DCA) with the ProbAbilistic Gradient Estimator (PAGE) to solve structured nonsmooth difference-of-convex programs. In the finite-sum setting, our method achieves a gradient computation complexity of $O(N + N^{1/2}\varepsilon^{-2})$ with sample size $N$, surpassing the previous best-known complexity of $O(N + N^{2/3}\varepsilon^{-2})$ for stochastic variance-reduced (SVR) DCA methods. Furthermore, DCA-PAGE readily extends to online settings with a similar optimal gradient computation complexity $O(b + b^{1/2}\varepsilon^{-2})$ with batch size $b$, a significant advantage over existing SVR DCA approaches that only work for the finite-sum setting. We further refine our analysis with a gap function, which enables us to obtain comparable convergence guarantees under milder assumptions.

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arXiv2025

Provable Reduction in Communication Rounds for Non-Smooth Convex Federated Learning

Multiple local steps are key to communication-efficient federated learning. However, theoretical guarantees for such algorithms, without data heterogeneity-bounding assumptions, have been lacking in general non-smooth convex problems. Leveraging projection-efficient optimization methods, we propose FedMLS, a federated learning algorithm with provable improvements from multiple local steps. FedMLS attains an $ε$-suboptimal solution in $\mathcal{O}(1/ε)$ communication rounds, requiring a total of $\mathcal{O}(1/ε^2)$ stochastic subgradient oracle calls.

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arXiv2025

A Unified Model for High-Resolution ODEs: New Insights on Accelerated Methods

Recent work on high-resolution ordinary differential equations (HR-ODEs) captures fine nuances among different momentum-based optimization methods, leading to accurate theoretical insights. However, these HR-ODEs often appear disconnected, each targeting a specific algorithm and derived with different assumptions and techniques. We present a unifying framework by showing that these diverse HR-ODEs emerge as special cases of a general HR-ODE derived using the Forced Euler-Lagrange equation. Discretizing this model recovers a wide range of optimization algorithms through different parameter choices. Using integral quadratic constraints, we also introduce a general Lyapunov function to analyze the convergence of the proposed HR-ODE and its discretizations, achieving significant improvements across various cases, including new guarantees for the triple momentum method$'$s HR-ODE and the quasi-hyperbolic momentum method, as well as faster gradient norm minimization rates for Nesterov$'$s accelerated gradient algorithm, among other advances.

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arXiv2025

Convex Formulations for Training Two-Layer ReLU Neural Networks

Solving non-convex, NP-hard optimization problems is crucial for training machine learning models, including neural networks. However, non-convexity often leads to black-box machine learning models with unclear inner workings. While convex formulations have been used for verifying neural network robustness, their application to training neural networks remains less explored. In response to this challenge, we reformulate the problem of training infinite-width two-layer ReLU networks as a convex completely positive program in a finite-dimensional (lifted) space. Despite the convexity, solving this problem remains NP-hard due to the complete positivity constraint. To overcome this challenge, we introduce a semidefinite relaxation that can be solved in polynomial time. We then experimentally evaluate the tightness of this relaxation, demonstrating its competitive performance in test accuracy across a range of classification tasks.

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arXiv2024

Federated Frank-Wolfe Algorithm

Federated learning (FL) has gained a lot of attention in recent years for building privacy-preserving collaborative learning systems. However, FL algorithms for constrained machine learning problems are still limited, particularly when the projection step is costly. To this end, we propose a Federated Frank-Wolfe Algorithm (FedFW). FedFW features data privacy, low per-iteration cost, and communication of sparse signals. In the deterministic setting, FedFW achieves an $\varepsilon$-suboptimal solution within $O(\varepsilon^{-2})$ iterations for smooth and convex objectives, and $O(\varepsilon^{-3})$ iterations for smooth but non-convex objectives. Furthermore, we present a stochastic variant of FedFW and show that it finds a solution within $O(\varepsilon^{-3})$ iterations in the convex setting. We demonstrate the empirical performance of FedFW on several machine learning tasks.

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arXiv2024

Personalized Multi-tier Federated Learning

The key challenge of personalized federated learning (PerFL) is to capture the statistical heterogeneity properties of data with inexpensive communications and gain customized performance for participating devices. To address these, we introduced personalized federated learning in multi-tier architecture (PerMFL) to obtain optimized and personalized local models when there are known team structures across devices. We provide theoretical guarantees of PerMFL, which offers linear convergence rates for smooth strongly convex problems and sub-linear convergence rates for smooth non-convex problems. We conduct numerical experiments demonstrating the robust empirical performance of PerMFL, outperforming the state-of-the-art in multiple personalized federated learning tasks.

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arXiv2023

A Variational Perspective on High-Resolution ODEs

We consider unconstrained minimization of smooth convex functions. We propose a novel variational perspective using forced Euler-Lagrange equation that allows for studying high-resolution ODEs. Through this, we obtain a faster convergence rate for gradient norm minimization using Nesterov's accelerated gradient method. Additionally, we show that Nesterov's method can be interpreted as a rate-matching discretization of an appropriately chosen high-resolution ODE. Finally, using the results from the new variational perspective, we propose a stochastic method for noisy gradients. Several numerical experiments compare and illustrate our stochastic algorithm with state of the art methods.

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arXiv2022

CCCP is Frank-Wolfe in disguise

This paper uncovers a simple but rather surprising connection: it shows that the well-known convex-concave procedure (CCCP) and its generalization to constrained problems are both special cases of the Frank-Wolfe (FW) method. This connection not only provides insight of deep (in our opinion) pedagogical value, but also transfers the recently discovered convergence theory of nonconvex Frank-Wolfe methods immediately to CCCP, closing a long-standing gap in its non-asymptotic convergence theory. We hope the viewpoint uncovered by this paper spurs the transfer of other advances made for FW to both CCCP and its generalizations.

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