arXiv subjects

Lijun Zhang

Publications and source records attributed to Lijun Zhang.

At least 19 recordsLinked to original sources

arXiv2026

Adaptive Variance Reduction across Smoothness and Noise Conditions

In this paper, we study adaptive stochastic non-convex optimization under unknown smoothness and noise conditions. Our algorithms require no smoothness constant, noise bound, initial function gap, or oracle-regime label. We first propose a fixed-budget method that combines normalized momentum, recursive variance reduction, and a fresh gradient stage, followed by fresh validation of the candidates. We prove that the same output has expected gradient norm $O(T^{-1/4})$ under objective smoothness and bounded noise variance, $O(T^{-1/3})$ under average smoothness, and $O(T^{-1/2})$ for noiseless gradients, where $T$ counts individual gradient evaluations. We then develop an anytime algorithm whose three optimization paths preserve their endpoints across reporting and budget-extension boundaries. The same three rates hold at every deterministic query budget, although the algorithm receives no information about the budget $T$. The analysis combines a scale potential, a uniform second-moment bound, and accounting for candidates retained through unfinished blocks. An optional sample-mean selector further reduces cumulative validation work from $O(T)$ to $O(T^{2/3})$ while preserving these rates.

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arXiv2026

Optimal Competitive Ratios for Chasing Convex Functions with Quadratic Growth

Chasing convex function has attracted much attention in recent years, which can be formulated as a game between a learner and an adversary. In each round, the adversary reveals a convex function, and the learner selects a decision and incurs a hitting cost and a movement cost for changing its decision. The performance of the learner is measured by the \emph{competitive ratio}, which compares its cumulative cost with that of an optimal offline decision sequence. For Euclidean movement, existing upper and lower bounds for $κ$-well-conditioned functions exhibit a polynomial gap. For squared Euclidean movement, dimension-free competitive guarantees have been established under quadratic growth, but the optimal competitive ratio remains unclear. In this paper, we establish optimal dimension-free competitive ratios for convex function chasing under two types of movement costs. For Euclidean movement, we propose Clipped Proximal Descent (CPD), which combines a balanced proximal step with a clipped correction toward the current minimizer. CPD achieves an $O(κ^{1/3})$ competitive ratio for smooth convex functions satisfying quadratic growth, improving the previous $O(\sqrtκ)$ upper bound and matching the existing $Ω(κ^{1/3})$ lower bound. For squared Euclidean movement, we establish the exact optimal dimension-free competitive ratio for convex functions satisfying quadratic growth. Our result is based on a new tight competitive analysis of Regularized Online Balanced Descent (R-OBD) and a matching lower bound against randomized algorithms.

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arXiv2026

Group Distributionally Robust Optimization with Flexible Sample Queries

Group distributionally robust optimization (GDRO) aims to learn models that perform well across $m$ distributions simultaneously. Existing methods either (i) query $m$ samples per round, which imposes strict implementation requirements, or (ii) query only a single sample per round, which requires many rounds to converge and therefore leads to long runtime. To avoid these limitations, we introduce a novel setting called \emph{flexible sample queries} (FSQ), which allows the number of samples that can be queried to vary across rounds. Under the FSQ setting, we cast GDRO as a two-player game: one performs non-oblivious online convex optimization, while the other tackles a non-oblivious prediction with limited advice (PLA) problem. In this game, the first player updates its decision using follow-the-regularized-leader with stochastic gradients. For the second player, we develop a new PLA method equipped with an adaptive exploration strategy that can leverage an arbitrary per-round sample size. We then establish the \emph{first} high-probability regret bound for non-oblivious PLA. By integrating these components, we develop an anytime GDRO algorithm that supports FSQ, and derive a high-probability optimization error bound of $O\left(\frac{1}{t}\sqrt{\sum_{j=1}^t \frac{m}{r_j}\log m}\right)$, where $r_j$ is the sample size at round $j$. This result demonstrates that the optimization error decreases as the per-round sample sizes increase, and implies the same near-optimal sample complexity of $O(m\log (m)/ε^2)$ for any fixed sample size $r\in[m]$.

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arXiv2026

Strongly Adaptive Online Learning Achieves Uniform Discounted Regret

Discounted regret measures the performance of an online learner by assigning exponentially decaying weights to past losses, reflecting the greater importance of recent losses in non-stationary online learning. While previous work obtains uniform discounted regret bounds over a continuous range of discount factors using a specialized meta-algorithm, its results are limited to convex functions. In this paper, we demonstrate that strongly adaptive regret bounds can directly imply uniform discounted regret guarantees for convex, strongly convex, and exp-concave loss functions. Our key observation is that discounted regret against a fixed comparator admits an exact decomposition into a convex combination of interval regrets. For convex functions, this reduction yields an $O(\sqrt{\log T/(1-λ)})$ discounted regret bound for all discount factors $λ\in(0,1)$, matching the previous rate over a broader range of discount factors. For strongly convex and exp-concave functions, we provide regret bounds of $O\left(\log T\left(1+\log\frac{1}{1-λ}\right)\right)$.

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arXiv2026

Optimal Regret for Communication-Efficient Decentralized Online Learning

In this paper, we investigate decentralized online convex optimization with compressed communication, where $n$ learners over a network jointly minimize a sequence of global loss functions based only on their local losses and compressed messages from their neighbors. The best known regret bounds for this problem are $O(ω^{-1/2}nρ^{-1}\sqrt{T\ln n})$ and $O(ω^{-1}nρ^{-2}\ln n\ln T)$ for convex and strongly convex loss functions, respectively, where $ρ\in(0,1)$ is the spectral gap of the communication matrix and $ω\in(0,1]$ is the compression quality factor. Compared with the corresponding lower bounds of $Ω(ω^{-1/2}nρ^{-1/4}\sqrt{T})$ and $Ω(ω^{-1}nρ^{-1/2}\ln T)$, these upper bounds exhibit polynomial gaps in their dependence on $ρ^{-1}$. To close these gaps, we first develop Ascent-gossip, which combines an online compression strategy with a Chebyshev recursion and attains a faster consensus rate than existing compressed gossip methods. Furthermore, by adopting Ascent-gossip, we propose an online algorithm that achieves optimal regret bounds of $\widetilde O(ω^{-1/2}nρ^{-1/4}\sqrt{T})$ and $\widetilde O(ω^{-1}nρ^{-1/2}\ln T)$ for convex and strongly convex loss functions, respectively. Up to logarithmic factors in $n$ and $ρ^{-1}$, these results match the lower bounds in terms of $T$, $n$, $ρ$, and $ω$.

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arXiv2026

Group Distributionally Robust Optimization over Decentralized Networks

We study group distributionally robust optimization (GDRO) over decentralized networks, where workers collaborate to learn robust models by minimizing the maximum local risk. Existing methods optimize only dual-regularized surrogates and establish guarantees for a globally averaged model that is not available to workers. In this paper, we directly optimize the original GDRO objective and establish guarantees for local models in semi-decentralized (SemiDec) and fully decentralized (Dec) settings. A key difficulty is that applying the primal-dual decomposition from centralized GDRO yields worker-dependent losses whose evaluation requires unavailable cross-worker information. To address this challenge, we develop a novel error decomposition that recasts decentralized GDRO as (i) a decentralized online convex optimization (D-OCO) problem and a prediction-with-expert-advice problem in SemiDec, and as (ii) two coupled D-OCO problems in Dec, with discrepancy terms capturing inconsistencies among local decisions. By isolating consensus errors in analysis, we develop modular frameworks that support different consensus routines. We then establish the \emph{first} per-worker convergence guarantees for optimizing the original GDRO objective over decentralized networks. Instantiations with accelerated gossip and Choco-gossip achieve an $O(\sqrt{(\log m)/T})$ convergence rate over $T$ update rounds, while experiments demonstrate improved worst-worker performance over baselines.

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arXiv2026

Revisiting Smoothed Online Learning: Competitive Ratio, Dynamic Regret with Switching Cost, and Best of Both Worlds

We revisit smoothed online learning and develop algorithms with improved competitive ratios, optimal dynamic regret with switching cost, and best-of-both-worlds guarantees. Surprisingly, minimizing the hitting cost alone is $\max(1,\frac{2}α)$-competitive for $α$-polyhedral functions under the $\ell_2$-norm switching cost and $1+\frac{4}λ$-competitive for $λ$-quadratic growth functions under the squared $\ell_2$-norm switching cost, significantly improving existing guarantees. The polyhedral bound is tight for this naive algorithm, and we establish a matching lower bound for $α\geq1$. For convex $λ$-quadratic growth hitting costs, minimizing a suitably weighted sum of hitting and squared $\ell_2$-norm switching costs yields a competitive ratio of $1+\frac{2}{\sqrtλ}$. For dynamic regret with the $\ell_2$-norm switching cost, we develop Smoothed Ader (SAder), which achieves the optimal $O(\sqrt{T(1+P_T)})$ rate in the standard online convex optimization setting, where $P_T$ is the comparator's path-length. Its lookahead version removes the bounded-gradient assumption, and a matching lower bound establishes optimality in this setting. Finally, for convex hitting costs on a bounded convex domain, we develop UDNP to aggregate any two experts in the lookahead setting under both switching costs. Its additional cumulative cost is at most $O(1)$ relative to the first expert and $O(\sqrt{T\log T})$ relative to the second, yielding best-of-both-worlds guarantees without bounded gradients.

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arXiv2026

Memory-Efficient Subspace Optimization with Residual Correction: From Momentum SGD to Muon

Memory-efficient subspace optimization has emerged as a practical approach to reducing the memory cost of LLM training while preserving optimization performance. Previous algorithms typically rely on the leading singular subspace of stochastic gradients, yet the gradient noise may dominate the subspace in later training stages and can prevent convergence. To overcome this limitation, we adopt a residual correction framework, which maintains the optimizer states in the subspace and adds the orthogonal residual gradient to the parameter update. We first integrate this scheme into momentum SGD to develop SMRC, which attains an $O(T^{-1/4})$ convergence rate for a fixed orthonormal basis, matching that of full-space momentum SGD. Our analysis introduces a subspace virtual iterate to resolve the coupling between the momentum and the residual correction. Furthermore, we extend SMRC to a multi-stage variant which periodically recomputes bases to track the evolving dominant subspace and retains the same rate. Since each basis refresh breaks the consistency of the virtual iterates, we devise a boundary correction approach that aligns them across stages. Beyond momentum SGD, we incorporate our framework into Muon and propose PMRC, which applies the matrix sign to the projected momentum. Through a weighted momentum tracking analysis that accounts for the residual gradient, we establish an $O(T^{-1/4})$ convergence rate for PMRC, which matches the rate of full-space Muon. We also design a multi-stage PMRC with periodically refreshed bases, which maintains a convergence rate of the same order. Extensive experimental results confirm the effectiveness and efficiency of our algorithms.

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arXiv2026

Efficient Preconditioned Optimization Methods with Frequent Directions

Adaptive optimization methods have received considerable attention in recent years due to their strong empirical performance. However, they generally require maintaining a preconditioning matrix, which incurs prohibitive computational overhead in large-scale applications. To address these limitations, we investigate the acceleration of preconditioned optimization methods through frequent directions (FD), a widely used matrix sketching technique. We first propose FTSL by integrating FD with adaptive subgradient methods under the primal-dual framework and adding back the cumulative information discarded by FD. FTSL obtains a tighter regret bound than existing state-of-the-art results. We then apply a doubling trick that further reduces the time complexity of FTSL, while maintaining the same regret guarantee. We also consider optimization over matrix-valued decision variables and exploit their structural properties to construct sketched preconditioners. We develop FTSL-Shampoo by adapting FD to Shampoo and give a novel analysis under the primal-dual framework. To further mitigate its computational complexity, we design one-sided variants of FTSL-Shampoo under both mirror descent and primal-dual frameworks. Our algorithms not only reduce the computational cost of FTSL-Shampoo, but also achieve tighter regret bounds. We extend our methods to offline stochastic convex optimization and exploit smoothness to achieve faster convergence rates. Finally, we introduce practical variants of our algorithms and empirically demonstrate their efficiency and effectiveness across a broad range of tasks.

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arXiv2026

Better Convergence Guarantees for Sign-Based Momentum Methods

This paper presents an improved analysis for sign-based methods with momentum updates. Traditional sign-based methods obtain a convergence rate of $\mathcal{O}(T^{-1/4})$ under the separable smoothness assumption, but they typically require large batch sizes or assume unimodal symmetric stochastic noise. To address these limitations, we demonstrate that signSGD with momentum can achieve the same convergence rate using constant batch sizes without additional assumptions. We also establish a convergence rate under the $l_2$-smoothness condition, improving upon the result of prior work by a factor of $\mathcal{O}(d^{1/2})$, where $d$ is the problem dimension. Furthermore, we explore sign-based methods in distributed settings and show that the proposed methods yield convergence rates of $\mathcal{O}\left( d^{1/2}T^{-1/2} + dn^{-1/2} \right)$ and $\mathcal{O}\left(d^{1/4}T^{-1/4}\right)$, which outperform the previous results of $\mathcal{O}\left( dT^{-1/4} + dn^{-1/2} \right)$ and $\mathcal{O}\left( d^{3/8}T^{-1/8} \right)$, respectively. Numerical experiments also validate the effectiveness of the proposed methods.

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arXiv2026

From Switching to Dynamic Regret: A Simple Reduction via Unbiased Random Sequences

In non-stationary online learning, dynamic regret has attracted increasing attention as a measure of how well an online learner performs against a time-varying comparator sequence. Despite considerable advances, attaining optimal bounds for strongly convex and exp-concave losses often involves intricate analysis. In this paper, we present a \textit{simple} framework that reduces dynamic regret minimization to switching regret minimization. As a result, we can derive dynamic regret bounds by using off-the-shelf algorithms with switching regret guarantees. The key idea of our reduction is to construct, for \textit{any} comparator sequence, an auxiliary random sequence that is unbiased at each round, with the controlled variance and a manageable number of switches. Combining this construction with suitable surrogate losses, we can decompose dynamic regret into the expected switching regret against the random sequence and its controlled variance. Theoretically, for strongly convex and exp-concave losses, we establish the $\widetilde{O}(T^{1/3}P_T^{2/3})$ dynamic regret bounds, where $T$ denotes the time horizon and $P_T$ denotes the path-length of the comparator sequence. Moreover, for general convex losses, the same reduction also recovers the $O(\sqrt{T(1+P_T)})$ dynamic regret bound. Notably, all our findings match the minimax optimal results for these three types of losses, highlighting the versatility of our proposed framework.

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arXiv2026

Optimal Momentum Methods for Stochastic Multilevel Compositional Optimization

This paper investigates stochastic multi-level optimization where the objective is a nested composition of several smooth non-convex functions. We assume that only stochastic estimates of the gradient and function values for each level are accessible. Consequently, obtaining an accurate estimate of the overall gradient is challenging due to the nested structure. To address this, we employ a momentum-based estimator with mini-batches to track the function values of each level, which are subsequently used to construct momentum gradient estimators. We establish an optimal sample complexity of $\mathcal{O}(ε^{-4})$ for finding an $ε$-stationary point, avoiding the stronger average smoothness assumption commonly relied upon in prior literature. Furthermore, by employing a normalization technique, we attain the same rate without requiring problem-dependent constants to set hyperparameters. To achieve the optimal rate without mini-batches, we further develop a batch-free method that incorporates a first-order approximation and a clipping technique for function value estimation. Finally, we validate the effectiveness of our proposed methods through experiments on risk-averse portfolio optimization and hierarchical tilted empirical risk minimization.

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arXiv2026

Convergence Analysis of STORM Under Different Geometries

Stochastic recursive momentum (STORM) achieves fast convergence for nonconvex optimization via the variance reduction effect, but existing analyses rely on the strong average smoothness assumption. In this paper, we study the convergence of STORM for different objectives without average smoothness. We first revisit the results under average smoothness, obtaining the $O(T^{-1/3})$ bound for nonconvex objectives and the $O(σ^2/(μT))$ bound for last-iterate output under the $μ$-Polyak--Łojasiewicz~(PL) condition. Without average smoothness, we design an auxiliary sequence and compare the STORM update with it in the analysis. With the help of this sequence, we prove that STORM still attains an $O(T^{-1/4})$ rate for nonconvex objectives, which is optimal under standard smoothness. For convex and $λ$-strongly convex objectives, we further prove averaged and last-iterate bounds with optimal rates of $O(σR/\sqrt T)$ and $O(σ^2/(λT))$, respectively. All the obtained results use the same STORM recursion with different hyperparameter choices.

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arXiv2026

Communication-Efficient Agnostic Federated Learning via Faster Convergence and Compression

Agnostic federated learning (AFL) seeks a model that performs reliably across $m$ heterogeneous workers, but communication remains a bottleneck. We improve communication efficiency by reducing the number of synchronization rounds via faster convergence and the communication cost per round via compression. We first propose AFL-BR, which updates the dual weights over workers using online mirror ascent with KL divergence and blockwise restarts. It achieves an $O((\log m)^{1/4}T^{-1/8})$ stationarity rate after $T$ update rounds, reducing the $m$-dependence of the synchronization rounds required for convergence from polynomial to logarithmic order. Building on AFL-BR, we develop AFL-Com by applying bidirectional compression with error feedback (EF). Instead of compressing local gradients, workers apply EF to their dual-weighted gradients, enabling direct control of the aggregated compression error under time-varying weights. We then establish an $O((δ^{-1}+(\log m)^{1/4})T^{-1/8})$ stationarity rate for AFL-Com under general $δ$-approximate compressors and improve the $δ$-dependence from $δ^{-1}$ to $δ^{-1/2}$ for additive-and-idempotent compressors with shared randomness (SR). With suitable compression levels, AFL-Com retains the same convergence rate as AFL-BR at a lower per-round communication cost, yielding reductions in total communication complexity by factors of $(\log m)^{1/4}$ with Top-$k$ and $(\log m)^{1/2}$ with Rand-$k$ and SR. Experiments validate the improved synchronization and communication efficiency of our methods.

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arXiv2026

Distribution-Conditioned Task Routing for Class-Incremental Learning

Parameter-efficient adaptation enables continual learners to acquire task-specific knowledge through compact model updates while maintaining strong within-task performance. However, class-incremental inference requires each input to be classified among all classes seen so far without access to its task identity. For learners equipped with task-specific parameter-efficient modules, this introduces a critical task-routing challenge beyond catastrophic forgetting. We study post-hoc task routing without retraining the learner or introducing a separately trained router. Such training-free inference-time calibration remains comparatively underexplored in parameter-efficient class-incremental learning. We identify three sources of routing error (feature-level, task-level, and class-level misalignment) and propose Feature Distribution Calibration (FDC). Its three components address these misalignments: Task Subspace Filtering (TSF) suppresses feature components outside each task's principal subspace, Residual Likelihood Calibration (RLC) evaluates the typicality of its subspace residual, and Prototype Affinity Calibration (PAC) measures compatibility with the task's class prototypes. Experiments demonstrate plug-and-play applicability to eight parameter-efficient class-incremental methods using a shared encoder. With one component configuration selected per method across all five benchmarks, FDC improves final accuracy in all 40 method-dataset pairs by 4.39 percentage points on average. Enabling all components improves 35 of the 40 pairs, with an average gain of 4.45 points. When applied to a simple baseline, FDC achieves strong overall performance.

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arXiv2026

Revisiting Distributed Sign-Based Variance Reduction

Sign-based methods reduce communication costs in distributed environments, but aggregating local signs can introduce bias when data are heterogeneous. As a result, existing sign-based variance reduction methods fail to obtain the optimal convergence rates. In this paper, we solve this problem and obtain optimal rates for both nonconvex stochastic and finite-sum optimization. We first give a counterexample showing that majority voting can fail to approach stationary points even with exact local gradients. Motivated by this limitation, we propose tracking the global gradient at the server through unbiased compression of recursive gradient increments. As a result, we can obtain the convergence rates of $O(\sqrt{d/K}+\sqrt d (a/(nK))^{1/3})$ for the $\ell_1$-norm and $O(\sqrt{a/K}+\sqrt a/(nK)^{1/3})$ for the $\ell_2$-norm. Here, $K$ is the iteration number, $n$ is the number of workers, $d$ is the dimension, and $a=1+ω$, with $ω$ denoting the compressor's relative variance. For finite-sum problems with $M$ components, we combine periodic exact gradient refreshes with compressed component-gradient differences. The resulting total sample complexities are $O(M+d\sqrt{aM}ε^{-2})$ and $O(M+a\sqrt M\ epsilon^{-2})$ for $\ell_1$ and $\ell_2$ gradient norms at most $ε$, matching the corresponding bounds in centralized settings.

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arXiv2026

Projection-Free Multi-level Algorithms for Stochastic Constrained Compositional Optimization

This paper studies projection-free algorithms for stochastic constrained multi-level compositional optimization. In this context, the objective function is a nested composition of several smooth functions, and the decision set is closed and convex. Since projection onto the constraint set can be computationally expensive, we develop projection-free methods that rely on linear minimization oracles. For non-convex objectives, we propose variance-reduced projection-free algorithms and establish complexity guarantees under both the Frank-Wolfe gap and the gradient mapping criteria. We also develop momentum-based methods that achieve convergence guarantees under weaker smoothness assumptions. Additionally, by using a stage-wise design, we derive a parameter-free variant that preserves the same complexities for the Frank-Wolfe gap. Such a design can be further used to develop algorithms for convex and strongly convex functions whose rates match those of single-level projection-free counterparts. Finally, we consider finite-sum problems and derive complexities for non-convex, convex, and strongly convex objectives. Numerical experiments across multiple tasks demonstrate the effectiveness of the proposed methods.

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arXiv2026

Solving Finite-sum Coupled Compositional Optimization via Multi-block-Single-probe Estimator

Traditional variance reduction methods (e.g., SPIDER, SARAH, STORM) have been extensively investigated for improving the convergence rates of stochastic optimization. These techniques typically maintain a sequence of estimators for a single function (or gradient) across iterations. However, what if we need to track multiple functions, but can only access stochastic samples of $\mathcal{O}(1)$ functions at each iteration? This scenario arises in an important emerging family of finite-sum coupled compositional optimization (FCCO) problems of the form $\frac{1}{m}\sum_{i=1}^m f_i(g_i(\mathbf{w}))$, where each $g_i$ is accessible only through a stochastic oracle. The key challenge is to track $\mathbf g(\mathbf{w})=(g_1(\mathbf{w}), \ldots, g_m(\mathbf{w}))$ over time, where $\mathbf g(\mathbf{w})$ has $m$ blocks but only $\mathcal{O}(1)$ blocks can be probed for their stochastic values at each step. To address this challenge, we propose a novel Multi-block-Single-probe Variance Reduction (MSVR) estimator to efficiently trace $\mathbf g(\mathbf{w})$ under partial block sampling. Building on the MSVR estimator, we develop several algorithms for FCCO problems, achieving improved sample complexities for non-convex, convex, strongly convex, and Polyak-Łojasiewicz (PL) objectives. We further obtain an improved dependence on $m$ when the outer function gradients $\nabla f_i$ are linear. Empirical studies on multi-task deep AUC maximization further demonstrate the superior performance of the proposed estimators.

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