Averaged Annealed Markov Chains Achieve the Algorithmic Threshold for Tensor PCA, Part II: Langevin Dynamics

We study simultaneous alternating power iteration for fixed-order asymmetric rank-one spiked tensor models. Our main contribution is a finite-iteration local theory that is independent of any particular initialization. Once the iterates enter a sufficiently small neighborhood of the planted rank-one direction, their error decomposes into a geometrically decaying transient and an intrinsic noise floor caused by fixed orthogonal noise contractions at the planted point. The deterministic finite-sample conditions are stated explicitly, but under a coarse fixed-order multilinear noise event they reduce to a conservative high-signal regime for fixed or slowly expanding local radii. We then separate the warm-start mechanism from any specific spectral construction. A generic one-sweep principle shows that, if a sign-compatible initializer has correlation $γ_N$, first-sweep noise level $a_N$, and $a_N/(γ_N^{d-1}ω_{N,d})\to0$, then one can choose an expanding radius $r_N=o(ω_{N,d})$ for which the first sweep enters the local basin. After entry, the local affine contraction yields convergence to the unique informative local fixed point in that basin. For centered-Gram initialization, we verify the required correlation and same-sample first-sweep noise bound under i.i.d. finite-fourth-moment noise by a signal-preserving noise-only leave-one comparison and an averaged leave-one slice-contraction estimate, which we call a pressed-back estimate. The leave-one comparison keeps the spike fixed and averages over the deleted coordinate, so planted coordinates enter through $\ell_2$-weighted sums rather than worst-case incoherence bounds.

Significant recent work has studied the ability of gradient descent to recover a hidden planted direction $θ^\star \in S^{d-1}$ in different high-dimensional settings, including tensor PCA and single-index models. The key quantity that governs the ability of gradient descent to traverse these landscapes is the information exponent $k^\star$ (Ben Arous et al., (2021)), which corresponds to the order of the saddle at initialization in the population landscape. Ben Arous et al., (2021) showed that $n \gtrsim d^{\max(1, k^\star-1)}$ samples were necessary and sufficient for online SGD to recover $θ^\star$, and Ben Arous et al., (2020) proved a similar lower bound for Langevin dynamics. More recently, Damian et al., (2023) showed it was possible to circumvent these lower bounds by running gradient descent on a smoothed landscape, and that this algorithm succeeds with $n \gtrsim d^{\max(1, k^\star/2)}$ samples, which is optimal in the worst case. This raises the question of whether it is possible to achieve the same rate without explicit smoothing. In this paper, we show that Langevin dynamics can succeed with $n \gtrsim d^{ k^\star/2 }$ samples if one considers the average iterate, rather than the last iterate. The key idea is that the combination of noise-injection and iterate averaging is able to emulate the effect of landscape smoothing. We apply this result to both the tensor PCA and single-index model settings. Finally, we conjecture that minibatch SGD can also achieve the same rate without adding any additional noise.

Asymmetric Tensor PCA (ATPCA) is a prototypical model for studying the trade-offs between sample complexity, computation, and memory. Existing algorithms for this problem typically require at least $d^{\left\lceil\overline{k}/2\right\rceil}$ state memory cost to recover the signal, where $d$ is the vector dimension and $\overline{k}$ is the tensor order. We focus on the setting where $\overline{k} \geq 4$ is even and consider (stochastic) gradient descent-based algorithms under a limited memory budget, which permits only mild over-parameterization of the model. We propose a matrix-parameterized method (in $d^{2}$ state memory cost) using a novel three-phase alternating-update algorithm to address the problem and demonstrate how mild over-parameterization facilitates learning in two key aspects: (i) it improves sample efficiency, allowing our method to achieve near-optimal $d^{\overline{k}-2}$ sample complexity in our limited memory setting; and (ii) it enhances adaptivity to problem structure, a previously unrecognized phenomenon, where the required sample size naturally decreases as consecutive vectors become more aligned, and in the symmetric limit attains $d^{\overline{k}/2}$, matching the best known polynomial-time complexity. To our knowledge, this is the first tractable algorithm for ATPCA with $d^{\overline{k}}$-independent memory costs.