Dmitriy Kunisky

Afonso S. Bandeira, Dmitriy Kunisky, Petar Nizić-Nikolac et al.

The hypergraph Moore bound conjectured by Feige (2008) controls the size of the smallest even cover in a $k$-uniform hypergraph in terms of the average density of hyperedges. An even cover is a set of hyperedges covering each vertex an even number of times, generalizing the notion of a cycle in a graph, so the size of the smallest non-trivial even cover provides a notion of hypergraph girth. Recent work starting from the breakthrough result of Guruswami, Kothari, and Manohar (2022) proved the conjecture up to polylogarithmic factors, whose exponents were later gradually improved. We give a simple proof of Feige's original hypergraph Moore bound conjecture for all even $k\ge 4$, with no superfluous polylogarithmic factors. Our proof roughly follows the proof of the graph Moore bound, but works with colored walks in a Kikuchi graph built from a hypergraph and controls their growth using a polynomial interpolation method.

Yuxin Ma, Nan Chen, Mateo Díaz et al.

Modern large-scale neural networks are often trained and released in multiple sizes to accommodate diverse inference budgets. To improve efficiency, recent work has explored model upscaling: initializing larger models from trained smaller ones in order to transfer knowledge and accelerate convergence. However, this method can be sensitive to hyperparameters that need to be tuned at the target upscaled model size, which is prohibitively costly to do directly. It remains unclear whether the most common workaround -- tuning on smaller models and extrapolating via hyperparameter scaling laws -- is still sound when using upscaling. We address this with principled approaches to upscaling with respect to model widths and efficiently tuning hyperparameters in this setting. First, motivated by $μ$P and any-dimensional architectures, we introduce a general upscaling method applicable to a broad range of architectures and optimizers, backed by theory guaranteeing that models are equivalent to their widened versions and allowing for rigorous analysis of infinite-width limits. Second, we extend the theory of $μ$Transfer to a hyperparameter transfer technique for models upscaled using our method and empirically demonstrate that this method is effective on realistic datasets and architectures.

Debsurya De, Dmitriy Kunisky

Recent work has generalized several results concerning the well-understood spiked Wigner matrix model of a low-rank signal matrix corrupted by additive i.i.d. Gaussian noise to the inhomogeneous case, where the noise has a variance profile. In particular, for the special case where the variance profile has a block structure, a series of results identified an effective spectral algorithm for detecting and estimating the signal, identified the threshold signal strength required for that algorithm to succeed, and proved information-theoretic lower bounds that, for some special signal distributions, match the above threshold. We complement these results by studying the computational optimality of this spectral algorithm. Namely, we show that, for a much broader range of signal distributions, whenever the spectral algorithm cannot detect a low-rank signal, then neither can any low-degree polynomial algorithm. This gives the first evidence for a computational hardness conjecture of Guionnet, Ko, Krzakala, and Zdeborová (2023). With similar techniques, we also prove sharp information-theoretic lower bounds for a class of signal distributions not treated by prior work. Unlike all of the above results on inhomogeneous models, our results do not assume that the variance profile has a block structure, and suggest that the same spectral algorithm might remain optimal for quite general profiles. We include a numerical study of this claim for an example of a smoothly-varying rather than piecewise-constant profile. Our proofs involve analyzing the graph sums of a matrix, which also appear in free and traffic probability, but we require new bounds on these quantities that are tighter than existing ones for non-negative matrices, which may be of independent interest.

Yunzi Ding, Dmitriy Kunisky, Alexander S. Wein et al.

In compressed sensing, the restricted isometry property (RIP) on $M \times N$ sensing matrices (where $M < N$) guarantees efficient reconstruction of sparse vectors. A matrix has the $(s,δ)$-$\mathsf{RIP}$ property if behaves as a $δ$-approximate isometry on $s$-sparse vectors. It is well known that an $M\times N$ matrix with i.i.d. $\mathcal{N}(0,1/M)$ entries is $(s,δ)$-$\mathsf{RIP}$ with high probability as long as $s\lesssim δ^2 M/\log N$. On the other hand, most prior works aiming to deterministically construct $(s,δ)$-$\mathsf{RIP}$ matrices have failed when $s \gg \sqrt{M}$. An alternative way to find an RIP matrix could be to draw a random gaussian matrix and certify that it is indeed RIP. However, there is evidence that this certification task is computationally hard when $s \gg \sqrt{M}$, both in the worst case and the average case. In this paper, we investigate the exact average-case time complexity of certifying the RIP property for $M\times N$ matrices with i.i.d. $\mathcal{N}(0,1/M)$ entries, in the "possible but hard" regime $\sqrt{M} \ll s\lesssim M/\log N$. Based on analysis of the low-degree likelihood ratio, we give rigorous evidence that subexponential runtime $N^{\tildeΩ(s^2/M)}$ is required, demonstrating a smooth tradeoff between the maximum tolerated sparsity and the required computational power. This lower bound is essentially tight, matching the runtime of an existing algorithm due to Koiran and Zouzias. Our hardness result allows $δ$ to take any constant value in $(0,1)$, which captures the relevant regime for compressed sensing. This improves upon the existing average-case hardness result of Wang, Berthet, and Plan, which is limited to $δ= o(1)$.

Dmitriy Kunisky, Alexander S. Wein, Afonso S. Bandeira

These notes survey and explore an emerging method, which we call the low-degree method, for predicting and understanding statistical-versus-computational tradeoffs in high-dimensional inference problems. In short, the method posits that a certain quantity -- the second moment of the low-degree likelihood ratio -- gives insight into how much computational time is required to solve a given hypothesis testing problem, which can in turn be used to predict the computational hardness of a variety of statistical inference tasks. While this method originated in the study of the sum-of-squares (SoS) hierarchy of convex programs, we present a self-contained introduction that does not require knowledge of SoS. In addition to showing how to carry out predictions using the method, we include a discussion investigating both rigorous and conjectural consequences of these predictions. These notes include some new results, simplified proofs, and refined conjectures. For instance, we point out a formal connection between spectral methods and the low-degree likelihood ratio, and we give a sharp low-degree lower bound against subexponential-time algorithms for tensor PCA.

Yunzi Ding, Dmitriy Kunisky, Alexander S. Wein et al.

We study the computational cost of recovering a unit-norm sparse principal component $x \in \mathbb{R}^n$ planted in a random matrix, in either the Wigner or Wishart spiked model (observing either $W + λxx^\top$ with $W$ drawn from the Gaussian orthogonal ensemble, or $N$ independent samples from $\mathcal{N}(0, I_n + βxx^\top)$, respectively). Prior work has shown that when the signal-to-noise ratio ($λ$ or $β\sqrt{N/n}$, respectively) is a small constant and the fraction of nonzero entries in the planted vector is $\|x\|_0 / n = ρ$, it is possible to recover $x$ in polynomial time if $ρ\lesssim 1/\sqrt{n}$. While it is possible to recover $x$ in exponential time under the weaker condition $ρ\ll 1$, it is believed that polynomial-time recovery is impossible unless $ρ\lesssim 1/\sqrt{n}$. We investigate the precise amount of time required for recovery in the "possible but hard" regime $1/\sqrt{n} \ll ρ\ll 1$ by exploring the power of subexponential-time algorithms, i.e., algorithms running in time $\exp(n^δ)$ for some constant $δ\in (0,1)$. For any $1/\sqrt{n} \ll ρ\ll 1$, we give a recovery algorithm with runtime roughly $\exp(ρ^2 n)$, demonstrating a smooth tradeoff between sparsity and runtime. Our family of algorithms interpolates smoothly between two existing algorithms: the polynomial-time diagonal thresholding algorithm and the $\exp(ρn)$-time exhaustive search algorithm. Furthermore, by analyzing the low-degree likelihood ratio, we give rigorous evidence suggesting that the tradeoff achieved by our algorithms is optimal.