arXiv subjects

Adrien Taylor

Publications and source records attributed to Adrien Taylor.

At least 19 recordsLinked to original sources

arXiv2026

Mirror Polyak and a Primal-Dual Lifting

First-order methods typically require a specific step-size that depends on the regularity conditions of the objective function, such as the smoothness, Lipschitz continuity, or strong convexity constants. The Polyak step-size is a classical alternative for subgradient descent on convex functions that only uses knowledge of the optimal value of the objective function and automatically adapts to the above-mentioned regimes. However, many optimization problems are better described by non-Euclidean geometries and are more amenable to mirror descent. Extending this adaptivity to mirror descent is subtle. Some existing generalizations of the Polyak step-size rely on norms instead of purely on relative geometry, excluding many of the use cases of mirror descent. In this work, we revisit a variant of the Polyak step-size based on Bregman projections due to Kiwiel (1997), which we call mirror Polyak. This method is known to converge asymptotically, but its convergence rate is not known. We show that mirror Polyak enjoys guarantees similar to its Euclidean counterpart, automatically adapting to relative notions of smoothness, Lipschitz continuity, or strong convexity. We then leverage mirror Polyak to avoid having to know the optimal value in some structured optimization problems such as regularized linear and logistic regression. We propose a lifted formulation based on convex duality with optimal value exactly zero and a natural mirror map given by the problem's structure. Mirror Polyak applied to the lifted problem enjoys the same worst-case guarantees as the Polyak step-size in the original problem if we knew the optimal value.

math.OC↗

arXiv2026

Finding Simple Proofs for First-Order Optimization

Progress in mathematics often requires more than a certificate of truth: it requires proof structures that are transparent, checkable, and reusable. Automated systems can increasingly certify that a result is true; what they typically return, however, is a dense certificate rather than an interpretable, reusable proof structure. Recent work on performance estimation problems has shown that performance bounds and complexity analyses of first-order optimization methods can be discovered by searching over a structured space of Lagrangian dual certificates. We cast the search for simpler proof structures as a second-stage optimization problem over these certificates. Starting from dual certificates, we develop post-processing procedures using tools from sparse optimization and statistical learning. We measure complexity through features such as active hypotheses and residual structure, and introduce methods based on exhaustive sparsification, weighted $\ell_1$-type heuristics, and semidefinite programming (SDP) formulations for discovering simple proofs and intermediate lemmas. Examples on gradient descent, proximal methods, and fast-gradient methods show that these procedures can autonomously prune redundant inequalities, reveal structured proof patterns, and, in the proximal setting, recover Lyapunov functions as intermediate lemmas that lead to simple, streamlined proofs. By distilling dense machine-generated certificates into compact proof structures, this workflow acts as a pre-processing step for the final proof, reducing the complexity that must be managed during human interpretation, reuse, and formalization.

math.OC↗

arXiv2026

A Tight Theory of Error Feedback Algorithms in Distributed Optimization

Communication costs are a major bottleneck in distributed learning and first-order optimization. A common approach to alleviate this issue is to compress the gradient information exchanged between agents. However, such compression typically degrades the convergence guarantees of gradient-based methods. Error feedback mechanisms provide a simple and computationally cheap remedy for this issue, but numerous variants have been proposed, and their relative performance remains poorly understood. This paper provides tight convergence analyses for two of the main error-feedback algorithms from the literature, the classic Error Feedback method (EF) and Error Feedback 21 (EF21), by identifying optimal step-size choices and constructing optimal Lyapunov functions tailored to each method. The results hold independently of the number of agents and recover the known best guarantees possible in the single-agent regime.

cs.LG↗

arXiv2026

Step-Size Stability in Stochastic Optimization: A Theoretical Perspective

We present a theoretical analysis of stochastic optimization methods in terms of their sensitivity with respect to the step size. We identify a key quantity that, for each method, describes how the performance degrades as the step size becomes too large. For convex problems, we show that this quantity directly impacts the suboptimality bound of the method. Most importantly, our analysis provides direct theoretical evidence that adaptive step-size methods, such as SPS or NGN, are more robust than SGD. This allows us to quantify the advantage of these adaptive methods beyond empirical evaluation. Finally, we show through experiments that our theoretical bound qualitatively mirrors the actual performance as a function of the step size, even for non-convex problems.

math.OC↗

arXiv2025

On the convergence rate of the boosted Difference-of-Convex Algorithm (DCA)

The difference-of-convex algorithm (DCA) is a well-established nonlinear programming technique that solves successive convex optimization problems. These sub-problems are obtained from the difference-of-convex~(DC) decompositions of the objective and constraint functions. We investigate the worst-case performance of the unconstrained DCA, with and without boosting, where boosting simply performs an additional step in the direction generated by the usual DCA method. We show that, for certain classes of DC decompositions, the boosted DCA is provably better in the worst-case than the usual DCA. While several numerical studies have reported that boosted DCA outperforms classical DCA, a theoretical explanation for this behavior has, to the best of our knowledge, not been given until now. Our proof technique relies on semidefinite programming (SDP) performance estimation.

math.OC↗

arXiv2025

Augmented Lagrangian methods for infeasible convex optimization problems and diverging proximal-point algorithms

This work investigates the convergence behavior of augmented Lagrangian methods (ALMs) when applied to convex optimization problems that may be infeasible. ALMs are a popular class of algorithms for solving constrained optimization problems. We demonstrate that, under mild assumptions, the sequences of iterates generated by ALMs converge to solutions of the ``closest feasible problem''. We establish progressively stronger convergence results, ranging from basic sequence convergence to more precise convergence rates, under a hierarchy of assumptions. This study leverages the classical relationship between ALMs and the proximal-point algorithm applied to the dual problem. A key technical contribution is a set of concise results on the behavior of the proximal-point algorithm when applied to functions that may lack minimizers. These results pertain to its convergence in terms of its subgradients and of the values of the convex conjugate.

math.OC↗

arXiv2025

Optimized projection-free algorithms for online learning: construction and worst-case analysis

This work studies and develop projection-free algorithms for online learning with linear optimization oracles (a.k.a. Frank-Wolfe) for handling the constraint set. More precisely, this work (i) provides an improved (optimized) variant of an online Frank-Wolfe algorithm along with its conceptually simple potential-based proof, and (ii) shows how to leverage semidefinite programming to jointly design and analyze online Frank-Wolfe-type algorithms numerically in a variety of settings-that include the design of the variant (i). Based on the semidefinite technique, we conclude with strong numerical evidence suggesting that no pure online Frank-Wolfe algorithm within our model class can have a regret guarantee better than O(T^3/4) (T is the time horizon) without additional assumptions, that the current algorithms do not have optimal constants, that the algorithm benefits from similar anytime properties O(t^3/4) not requiring to know T in advance, and that multiple linear optimization rounds do not generally help to obtain better regret bounds.

math.OC↗

arXiv2025

Tight analyses of first-order methods with error feedback

Communication between agents often constitutes a major computational bottleneck in distributed learning. One of the most common mitigation strategies is to compress the information exchanged, thereby reducing communication overhead. To counteract the degradation in convergence associated with compressed communication, error feedback schemes -- most notably $\mathrm{EF}$ and $\mathrm{EF}^{21}$ -- were introduced. In this work, we provide a tight analysis of both of these methods. Specifically, we find the Lyapunov function that yields the best possible convergence rate for each method -- with matching lower bounds. This principled approach yields sharp performance guarantees and enables a rigorous, apples-to-apples comparison between $\mathrm{EF}$, $\mathrm{EF}^{21}$, and compressed gradient descent. Our analysis is carried out in the simplified single-agent setting, which allows for clean theoretical insights and fair comparison of the underlying mechanisms.

cs.LG↗

arXiv2025

A constructive approach to strengthen algebraic descriptions of function and operator classes

It is well known that functions (resp. operators) satisfying a property~$p$ on a subset $Q\subset \mathbb{R}^d$ cannot necessarily be extended to a function (resp. operator) satisfying~$p$ on the whole of~$\mathbb{R}^d$. Given $Q \subseteq \mathbb{R}^d$, this work considers the problem of obtaining necessary and ideally sufficient conditions to be satisfied by a function (resp. operator) on $Q$, ensuring the existence of an extension of this function (resp. operator) satisfying $p$ on $\mathbb{R}^d$. More precisely, given some property $p$, we present a refinement procedure to obtain stronger necessary conditions to be imposed on $Q$. This procedure can be applied iteratively until the stronger conditions are also sufficient. We illustrate the procedure on a few examples, including the strengthening of existing descriptions for the classes of smooth functions satisfying a \L{}ojasiewicz condition, convex blockwise smooth functions, Lipschitz monotone operators, strongly monotone cocoercive operators, and uniformly convex functions. In most cases, these strengthened descriptions can be represented, or relaxed, to semi-definite constraints, which can be used to formulate tractable optimization problems on functions (resp. operators) within those classes.

math.OC↗

arXiv2025

Solving generic parametric linear matrix inequalities

We consider linear matrix inequalities (LMIs) $A = A_0 + x_1 A_1 + ... + x_n A_n \succeq 0$ with the $A_i$'s being $m \times m$ symmetric matrices, with entries in a ring $\mathcal{R}$. When $\mathcal{R} = \mathbb{R}$, the feasibility problem consists in deciding whether the $x_i$'s can be instantiated to obtain a positive semidefinite matrix. When $\mathcal{R} = \mathbb{Q}[y_1, ... , y_t]$, the problem asks for a formula on the parameters $y_1, ..., y_t$, which describes the values of the parameters for which the specialized LMI is feasible. This problem can be solved using general quantifier elimination algorithms, with a complexity that is exponential in n. In this work, we leverage the LMI structure of the problem to design an algorithm that computes a formula $\Phi$ describing a dense subset of the feasible region of parameters, under genericity assumptions. The complexity of this algorithm is exponential in n, m and t but becomes polynomial in $n$ when $m$ and $t$ are fixed. We apply the algorithm to a parametric sum-of-squares problem and to the convergence analyses of certain first-order optimization methods, which are both known to be equivalent to the feasibility of certain parametric LMIs, hence demonstrating its practical interest.

cs.SC↗

arXiv2025

Open Problem: Two Riddles in Heavy-Ball Dynamics

This short paper presents two open problems on the widely used Polyak's Heavy-Ball algorithm. The first problem is the method's ability to exactly \textit{accelerate} in dimension one exactly. The second question regards the behavior of the method for parameters for which it seems that neither a Lyapunov nor a cycle exists. For both problems, we provide a detailed description of the problem and elements of an answer.

math.OC↗

arXiv2025

The Surprising Agreement Between Convex Optimization Theory and Learning-Rate Scheduling for Large Model Training

We show that learning-rate schedules for large model training behave surprisingly similar to a performance bound from non-smooth convex optimization theory. We provide a bound for the constant schedule with linear cooldown; in particular, the practical benefit of cooldown is reflected in the bound due to the absence of logarithmic terms. Further, we show that this surprisingly close match between optimization theory and practice can be exploited for learning-rate tuning: we achieve noticeable improvements for training 124M and 210M Llama-type models by (i) extending the schedule for continued training with optimal learning-rate, and (ii) transferring the optimal learning-rate across schedules.

cs.LG↗

arXiv2024

Constructive approaches to concentration inequalities with independent random variables

Concentration inequalities, a major tool in probability theory, quantify how much a random variable deviates from a certain quantity. This paper proposes a systematic convex optimization approach to studying and generating concentration inequalities with independent random variables. Specifically, we extend the generalized problem of moments to independent random variables. We first introduce a variational approach that extends classical moment-generating functions, focusing particularly on first-order moment conditions. Second, we develop a polynomial approach, based on a hierarchy of sum-of-square approximations, to extend these techniques to higher-moment conditions. Building on these advancements, we refine Hoeffding's, Bennett's and Bernstein's inequalities, providing improved worst-case guarantees compared to existing results.

math.PR↗

arXiv2024

Geometry-dependent matching pursuit: a transition phase for convergence on linear regression and LASSO

Greedy first-order methods, such as coordinate descent with Gauss-Southwell rule or matching pursuit, have become popular in optimization due to their natural tendency to propose sparse solutions and their refined convergence guarantees. In this work, we propose a principled approach to generating (regularized) matching pursuit algorithms adapted to the geometry of the problem at hand, as well as their convergence guarantees. Building on these results, we derive approximate convergence guarantees and describe a transition phenomenon in the convergence of (regularized) matching pursuit from underparametrized to overparametrized models.

math.OC↗

arXiv2023

On Fundamental Proof Structures in First-Order Optimization

First-order optimization methods have attracted a lot of attention due to their practical success in many applications, including in machine learning. Obtaining convergence guarantees and worst-case performance certificates for first-order methods have become crucial for understanding ingredients underlying efficient methods and for developing new ones. However, obtaining, verifying, and proving such guarantees is often a tedious task. Therefore, a few approaches were proposed for rendering this task more systematic, and even partially automated. In addition to helping researchers finding convergence proofs, these tools provide insights on the general structures of such proofs. We aim at presenting those structures, showing how to build convergence guarantees for first-order optimization methods.

math.OC↗

arXiv2023

Provable non-accelerations of the heavy-ball method

In this work, we show that the heavy-ball ($\HB$) method provably does not reach an accelerated convergence rate on smooth strongly convex problems. More specifically, we show that for any condition number and any choice of algorithmic parameters, either the worst-case convergence rate of $\HB$ on the class of $L$-smooth and $\mu$-strongly convex \textit{quadratic} functions is not accelerated (that is, slower than $1 - \mathcal{O}(\kappa)$), or there exists an $L$-smooth $\mu$-strongly convex function and an initialization such that the method does not converge. To the best of our knowledge, this result closes a simple yet open question on one of the most used and iconic first-order optimization technique. Our approach builds on finding functions for which $\HB$ fails to converge and instead cycles over finitely many iterates. We analytically describe all parametrizations of $\HB$ that exhibit this cycling behavior on a particular cycle shape, whose choice is supported by a systematic and constructive approach to the study of cycling behaviors of first-order methods. We show the robustness of our results to perturbations of the cycle, and extend them to class of functions that also satisfy higher-order regularity conditions.

math.OC↗

arXiv2023

Counter-examples in first-order optimization: a constructive approach

While many approaches were developed for obtaining worst-case complexity bounds for first-order optimization methods in the last years, there remain theoretical gaps in cases where no such bound can be found. In such cases, it is often unclear whether no such bound exists (e.g., because the algorithm might fail to systematically converge) or simply if the current techniques do not allow finding them. In this work, we propose an approach to automate the search for cyclic trajectories generated by first-order methods. This provides a constructive approach to show that no appropriate complexity bound exists, thereby complementing the approaches providing sufficient conditions for convergence. Using this tool, we provide ranges of parameters for which some of the famous heavy-ball, Nesterov accelerated gradient, inexact gradient descent, and three-operator splitting algorithms fail to systematically converge, and show that it nicely complements existing tools searching for Lyapunov functions.

math.OC↗

arXiv2023

Nonlinear conjugate gradient methods: worst-case convergence rates via computer-assisted analyses

We propose a computer-assisted approach to the analysis of the worst-case convergence of nonlinear conjugate gradient methods (NCGMs). Those methods are known for their generally good empirical performances for large-scale optimization, while having relatively incomplete analyses. Using our computer-assisted approach, we establish novel complexity bounds for the Polak-Ribi\`ere-Polyak (PRP) and the Fletcher-Reeves (FR) NCGMs for smooth strongly convex minimization. In particular, we construct mathematical proofs that establish the first non-asymptotic convergence bound for FR (which is historically the first developed NCGM), and a much improved non-asymptotic convergence bound for PRP. Additionally, we provide simple adversarial examples on which these methods do not perform better than gradient descent with exact line search, leaving very little room for improvements on the same class of problems.

math.OC↗