arXiv subjects

Junwen Qiu

Publications and source records attributed to Junwen Qiu.

13 recordsLinked to original sources

arXiv2026

Stochastic Saddle Avoidance Beyond Unit Excitation and Smoothness: A Pathwise Lyapunov-Perron Framework

Unit excitation (UE) is a common assumption in stochastic saddle avoidance: the stochastic error must have a uniformly positive component along every direction, in expectation. This condition gives a direct way to rule out convergence to strict saddles, but it also oversimplifies the actual noise structure, and does not match many stochastic optimization regimes. In overparameterized or interpolation models, the noise may vanish near stationarity. In finite-sum problems, the stochastic gradient noise may lie in a low-dimensional, data-dependent subspace. In these (common) scenarios, UE is naturally not satisfied. In this paper, we prove an abstract almost sure avoidance theorem for stochastic recursions without UE. The theorem replaces UE-type requirements by verifiable pathwise conditions. In applications, these conditions follow, e.g., from local smoothness and finite-moment assumptions under standard i.i.d. sampling, or from the finite-sum structure under without-replacement sampling. Since the stochastically sampled maps generally do not share a fixed point, the celebrated center-stable manifold argument used in deterministic analyses is not directly applicable. Instead, we use a path-dependent change of variables together with a pathwise Lyapunov--Perron-based proof strategy. As applications, we obtain strict saddle avoidance for stochastic mirror descent (including SGD) and for random reshuffling. For nonsmooth composite objectives, we prove avoidance results for a proximal-type stochastic gradient method. Combining these insights with suitable iterate convergence guarantees, this allows establishing convergence to local minimizers of the original objective function.

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arXiv2026

FlexiSLM: A Spoken Language Model with Dynamic and Controllable Frame Rates

Spoken language models (SLMs) extend LLMs to speech input and output. Existing SLMs represent speech at fixed frame rates (e.g., 25 or 12.5 Hz), ignoring the time-varying information density of speech and offering limited flexibility to trade off quality for speed at inference time. Recent audio tokenizer research has proposed dynamic-frame-rate speech coding, which exploits this non-uniformity and enables two new capabilities: very low average frame rates and frame-rate controllability. However, this technique has not yet been applied to SLMs. We introduce FlexiSLM, the first SLM with dynamic and controllable frame rates. FlexiSLM uses the pretrained FlexiCodec to obtain dynamic speech output tokens. The main contributions of this work are threefold: (1) integrating and validating this dynamic-rate representation within a multi-task, speech-to-speech SLM architecture; (2) extending it to frame compression on the input side; and (3) introducing direct frame-rate conditioning to enable accurate and controllable SLM inference. FlexiSLM outperforms fixed-frame-rate 7B models including Qwen2.5-Omni and Kimi-Audio at its 12.5 Hz and 6.25 Hz operating points. We further verify that FlexiSLM can be accurately steered down to 4.0 Hz; at 6.25 Hz, it roughly halves inference time relative to 12.5 Hz while retaining strong speech-to-speech quality. Audio samples are available at: https://flexislm.github.io. Code and data are available at: https://github.com/AmphionTeam/FlexiSLM.

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arXiv2026

Zero-VC: Zero-Lookahead Streaming Voice Conversion via Speaker Anonymization

Streaming zero-shot voice conversion struggles to disentangle timbre from linguistic content without degrading utility or inflating latency. Current methods rely on information bottleneck (IB) or speaker perturbation. While IB filters out timbre, it discards prosody, forcing models to explicitly inject features like fundamental frequency. This often requires buffering future frames, creating algorithmic lookahead latency. On the other hand, existing perturbation methods largely overlook the crucial trade-off between timbre leakage and utility preservation. Recognizing this neglected trade-off, we find that the inherent objective of Speaker Anonymization (SA) aligns well with balancing these factors. Thus, we introduce SA as a novel perturbation mechanism to explicitly mitigate timbre leakage while retaining prosodic utility. Crucially, SA's robust representations significantly alleviate the generator's reliance on future context, enabling our strictly causal, zero-lookahead network. Audio samples are available at https://amphionteam.github.io/Zero-VC-demo/.

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arXiv2026

Shuffling the Stochastic Mirror Descent via Dual Lipschitz Continuity and Kernel Conditioning

The global Lipschitz smoothness condition underlies most convergence and complexity analyses via two key consequences: the descent lemma and the gradient Lipschitz continuity. How to study the performance of optimization algorithms in the absence of Lipschitz smoothness remains an active area. The relative smoothness framework from Bauschke-Bolte-Teboulle (2017) and Lu-Freund-Nesterov (2018) provides an extended descent lemma, ensuring convergence of Bregman-based proximal gradient methods and their vanilla stochastic counterparts. However, many widely used techniques (e.g., momentum schemes, random reshuffling, and variance reduction) additionally require the Lipschitz-type bound for gradient deviations, leaving their analysis under relative smoothness an open area. To resolve this issue, we introduce the dual kernel conditioning (DKC) regularity condition to regulate the local relative curvature of the kernel functions. Combined with the relative smoothness, DKC provides a dual Lipschitz continuity for gradients: even though the gradient mapping is not Lipschitz in the primal space, it preserves Lipschitz continuity in the dual space induced by a mirror map. We verify that DKC is widely satisfied by popular kernels and is closed under affine composition and conic combination. With these novel tools, we establish the first complexity bounds as well as the iterate convergence of random reshuffling mirror descent for constrained nonconvex relative smooth problems.

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arXiv2026

A New Kernel Regularity Condition for Distributed Mirror Descent: Broader Coverage and Simpler Analysis

Existing convergence analyses of distributed optimization methods in non-Euclidean geometries typically rely on kernel assumptions: (i) global Lipschitz smoothness and (ii) bi-convexity of the associated Bregman divergence function. Unfortunately, these conditions are violated by nearly all kernels used in practice, leaving a huge theory-practice gap. This work closes this gap by developing a unified analytical tool that guarantees convergence under mild conditions. Specifically, we introduce Hessian relative uniform continuity (HRUC), a regularity satisfied by nearly all standard kernels. Importantly, HRUC is closed under concatenation, positive scaling, composition, and various kernel combinations. Leveraging the geometric structure induced by HRUC, we derive convergence guarantees for mirror-descent-based gradient tracking without imposing any restrictive assumptions. More broadly, our analysis techniques extend seamlessly to other decentralized optimization methods in genuinely non-Euclidean and non-Lipschitz settings.

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arXiv2024

Convergence of SGD with momentum in the nonconvex case: A time window-based analysis

The stochastic gradient descent method with momentum (SGDM) is a common approach for solving large-scale and stochastic optimization problems. Despite its popularity, the convergence behavior of SGDM remains less understood in nonconvex scenarios. This is primarily due to the absence of a sufficient descent property and challenges in simultaneously controlling the momentum and stochastic errors in an almost sure sense. To address these challenges, we investigate the behavior of SGDM over specific time windows, rather than examining the descent of consecutive iterates as in traditional studies. This time window-based approach simplifies the convergence analysis and enables us to establish the iterate convergence result for SGDM under the Łojasiewicz property. We further provide local convergence rates which depend on the underlying Łojasiewicz exponent and the utilized step size schemes.

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arXiv2024

A Generalized Version of Chung's Lemma and its Applications

Chung's Lemma is a classical tool for establishing asymptotic convergence rates of (stochastic) optimization methods under strong convexity-type assumptions and appropriate polynomial diminishing step sizes. In this work, we develop a generalized version of Chung's Lemma, which provides a simple non-asymptotic convergence framework for a more general family of step size rules. We demonstrate broad applicability of the proposed generalized lemma by deriving tight non-asymptotic convergence rates for a large variety of stochastic methods. In particular, we obtain partially new non-asymptotic complexity results for stochastic optimization methods, such as Stochastic Gradient Descent (SGD) and Random Reshuffling (RR), under a general $(\theta,\mu)$-Polyak-Lojasiewicz (PL) condition and for various step sizes strategies, including polynomial, constant, exponential, and cosine step sizes rules. Notably, as a by-product of our analysis, we observe that exponential step sizes exhibit superior adaptivity to both landscape geometry and gradient noise; specifically, they achieve optimal convergence rates without requiring exact knowledge of the underlying landscape or separate parameter selection strategies for noisy and noise-free regimes. Our results demonstrate that the developed variant of Chung's Lemma offers a versatile, systematic, and streamlined approach to establish non-asymptotic convergence rates under general step size rules.

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arXiv2024

A KL-based Analysis Framework with Applications to Non-Descent Optimization Methods

We propose a novel analysis framework for non-descent-type optimization methodologies in nonconvex scenarios based on the Kurdyka-Lojasiewicz property. Our framework allows covering a broad class of algorithms, including those commonly employed in stochastic and distributed optimization. Specifically, it enables the analysis of first-order methods that lack a sufficient descent property and do not require access to full (deterministic) gradient information. We leverage this framework to establish, for the first time, iterate convergence and the corresponding rates for the decentralized gradient method and federated averaging under mild assumptions. Furthermore, based on the new analysis techniques, we show the convergence of the random reshuffling and stochastic gradient descent method without necessitating typical a priori bounded iterates assumptions.

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arXiv2024

Random Reshuffling with Momentum: Complexity Bounds and Last-iterate Convergence

Random reshuffling with momentum (RRM) corresponds to the SGD optimizer with the 'momentum' option enabled, as found in many machine learning libraries such as PyTorch and TensorFlow. Despite its widespread use, the convergence properties of RRM do not seem to be well understood. This work establishes new complexity bounds and asymptotic convergence guarantees for popular versions of RRM using stochastic heavy-ball momentum, Nesterov acceleration, and mini-batches in a general nonconvex setting. In particular, we prove that the base variant of RRM achieves the complexity $O(n^{-1/3}((1-\beta^n)T)^{-2/3})$, where $n$ denotes the number of samples, $\beta \in [0,1)$ is a momentum parameter, and $T$ is the total number of epochs. On the asymptotic side, we show that every accumulation point of the iterates $\{x^k\}_k$ generated by RRM is a stationary point of the problem. For definable objectives -- a broad and common class of functions including, e.g., semialgebraic, globally subanalytic, and log-exp functions -- we strengthen this subsequential result to last-iterate convergence to a single stationary point. Moreover, improved asymptotic complexity bounds are presented that are based on the additional geometric properties of definable functions.

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arXiv2023

A New Random Reshuffling Method for Nonsmooth Nonconvex Finite-sum Optimization

Random reshuffling techniques are prevalent in large-scale applications, such as training neural networks. While the convergence and acceleration effects of random reshuffling-type methods are fairly well understood in the smooth setting, much less studies seem available in the nonsmooth case. In this work, we design a new normal map-based proximal random reshuffling (norm-PRR) method for nonsmooth nonconvex finite-sum problems. We show that norm-PRR achieves the iteration complexity ${\cal O}(n^{-1/3}T^{-2/3})$ where $n$ denotes the number of component functions $f(\cdot,i)$ and $T$ counts the total number of iterations. This improves the currently known complexity bounds for this class of problems by a factor of $n^{-1/3}$ in terms of the number of gradient evaluations. Additionally, we prove that norm-PRR converges linearly under the (global) Polyak-{\L}ojasiewicz condition and in the interpolation setting. We further complement these non-asymptotic results and provide an in-depth analysis of the asymptotic properties of norm-PRR. Specifically, under the (local) Kurdyka-{\L}ojasiewicz inequality, the whole sequence of iterates generated by norm-PRR is shown to converge to a single stationary point. Moreover, we derive last-iterate convergence rates that can match those in the smooth, strongly convex setting. Finally, numerical experiments are performed on nonconvex classification tasks to illustrate the efficiency of the proposed approach.

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arXiv2023

A Normal Map-Based Proximal Stochastic Gradient Method: Convergence and Identification Properties

The proximal stochastic gradient method (PSGD) is one of the state-of-the-art approaches for stochastic composite-type problems. In contrast to its deterministic counterpart, PSGD has been found to have difficulties with the correct identification of underlying substructures (such as supports, low rank patterns, or active constraints) and it does not possess a finite-time manifold identification property. Existing solutions rely on convexity assumptions or on the additional usage of variance reduction techniques. In this paper, we address these limitations and present a simple variant of PSGD based on Robinson's normal map. The proposed normal map-based proximal stochastic gradient method (NSGD) is shown to converge globally, i.e., accumulation points of the generated iterates correspond to stationary points almost surely. In addition, we establish complexity bounds for NSGD that match the known results for PSGD and we prove that NSGD can almost surely identify active manifolds in finite-time in a general nonconvex setting. Our derivations are built on almost sure iterate convergence guarantees and utilize analysis techniques based on the Kurdyka-Lojasiewicz inequality.

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arXiv2023

Distributed Random Reshuffling over Networks

In this paper, we consider distributed optimization problems where $n$ agents, each possessing a local cost function, collaboratively minimize the average of the local cost functions over a connected network. To solve the problem, we propose a distributed random reshuffling (D-RR) algorithm that invokes the random reshuffling (RR) update in each agent. We show that D-RR inherits favorable characteristics of RR for both smooth strongly convex and smooth nonconvex objective functions. In particular, for smooth strongly convex objective functions, D-RR achieves $\mathcal{O}(1/T^2)$ rate of convergence (where $T$ counts epoch number) in terms of the squared distance between the iterate and the global minimizer. When the objective function is assumed to be smooth nonconvex, we show that D-RR drives the squared norm of gradient to $0$ at a rate of $\mathcal{O}(1/T^{2/3})$. These convergence results match those of centralized RR (up to constant factors) and outperform the distributed stochastic gradient descent (DSGD) algorithm if we run a relatively large number of epochs. Finally, we conduct a set of numerical experiments to illustrate the efficiency of the proposed D-RR method on both strongly convex and nonconvex distributed optimization problems.

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arXiv2023

Convergence of Random Reshuffling Under The Kurdyka-Łojasiewicz Inequality

We study the random reshuffling (RR) method for smooth nonconvex optimization problems with a finite-sum structure. Though this method is widely utilized in practice such as the training of neural networks, its convergence behavior is only understood in several limited settings. In this paper, under the well-known Kurdyka-Lojasiewicz (KL) inequality, we establish strong limit-point convergence results for RR with appropriate diminishing step sizes, namely, the whole sequence of iterates generated by RR is convergent and converges to a single stationary point in an almost sure sense. In addition, we derive the corresponding rate of convergence, depending on the KL exponent and the suitably selected diminishing step sizes. When the KL exponent lies in $[0,\frac12]$, the convergence is at a rate of $\mathcal{O}(t^{-1})$ with $t$ counting the iteration number. When the KL exponent belongs to $(\frac12,1)$, our derived convergence rate is of the form $\mathcal{O}(t^{-q})$ with $q\in (0,1)$ depending on the KL exponent. The standard KL inequality-based convergence analysis framework only applies to algorithms with a certain descent property. We conduct a novel convergence analysis for the non-descent RR method with diminishing step sizes based on the KL inequality, which generalizes the standard KL framework. We summarize our main steps and core ideas in an informal analysis framework, which is of independent interest. As a direct application of this framework, we also establish similar strong limit-point convergence results for the reshuffled proximal point method.

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