Multistage Conditional Compositional Optimization

We study stochastic composite nonconvex optimization over a compact convex set when gradient samples arrive along a single trajectory of a fixed ergodic Markov chain. Existing single-trajectory variance-reduction theory covers smooth unconstrained objectives; we address the projection-free composite setting using the generalized Frank-Wolfe gap. We propose MC-ALFCG, which combines a momentum conditional-gradient method with coupled capped multilevel Monte Carlo estimation and per-iteration clipping. The deepest nested average uses consecutive states from the same trajectory, yielding conditional bias $O(τ_{\mathrm{mix}}/T)$ uniformly over the starting state, while coupling controls the gradient-difference second moment through the iterate displacement. Clipping enforces the pathwise bounds needed by the adaptive analysis. We reduce the Markovian recursion to its independent-sampling counterpart under $σ^2\mapsto 2ΛG_σ^2$ and $L^2\mapsto 2ΛL^2$, where $Λ=O(τ_{\mathrm{mix}}\log T)$. For positive centered noise, the tuned method achieves expected sample complexity $\widetilde{O}((τ_{\mathrm{mix}}^2G_σ+τ_{\mathrm{mix}}^{5/2}G_σ^2)\varepsilon^{-3}+τ_{\mathrm{mix}}^5\varepsilon^{-2})$. The exactly noiseless specialization achieves $\widetilde{O}(\varepsilon^{-2})$ with mixing-time-free constants, while a mixing-time-oblivious variant achieves $\widetilde{O}(τ_{\mathrm{mix}}^6\varepsilon^{-3}+τ_{\mathrm{mix}}^3\varepsilon^{-2})$. All guarantees are in expectation under a fixed transition kernel. Controlled numerical studies examine dependence sensitivity, a nonconvex composite instance, and clipping behavior.

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.

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.