Lucas Slot

Adam Kurpisz, Lucas Slot, Mikhail Zaytsev

We analyze the sum-of-squares rank of unweighted instances of the Minimum Knapsack (MK) problem, i.e., minimization of $\sum_{i=1}^n x_i$ for 0/1 variables under the constraint $\sum_{i=1}^n x_i \geq q$, with $q \in \mathbb{R}$. Such instances have long served as a testbed for understanding the limitations of lift-and-project methods in Boolean optimization. For example, both the Lovász-Schrijver and Sherali-Adams hierarchies require (maximal) rank $n$ to solve them, already when $q=1/2$ is constant. The SOS hierarchy requires only \emph{sublinear} rank $O(\sqrt{n})$ to solve unweighted MK when $q=1/2$. On the other hand, when $q$ is allowed to vary with~$n$, the SOS rank of the problem may become linear. Interestingly, this is known to happen both when $q$ is large, and when $q$ is very small ($0<q \leq 2^{-n}$). This raises the question of whether we should think of hard instances of unweighted MK as being typical for the SOS hierarchy, or as a consequence of very specific choices of the threshold parameter $q$. In this paper, we address this question by showing new upper and lower bounds on the SOS rank of unweighted MK in the whole regime of the parameter $q$. For $n-q \leq O(1)$, we show that the SOS rank is constant. In contrast, when $q \leq O(1)$, a linear rank is needed if $q$ is exponentially close to an integer. As our main positive result, we show that linear rank is very rare for $q \leq O(1)$. This can be expressed in the language of smoothed analysis: after perturbing $q$ by a Gaussian with mean $0$ and variance $σ^2$, the expected SOS rank of MK is $O(\sqrt{n} \log (n/σ))$.

Lucas Pesenti, Lucas Slot, Manuel Wiedmer

The complexity of learning a concept class under Gaussian marginals in the difficult agnostic model is closely related to its $L_1$-approximability by low-degree polynomials. For any concept class with Gaussian surface area at most $Γ$, Klivans et al. (2008) show that degree $d = O(Γ^2 / \varepsilon^4)$ suffices to achieve an $\varepsilon$-approximation. This leads to the best-known bounds on the complexity of learning a variety of concept classes. In this note, we improve their analysis by showing that degree $d = \tilde O (Γ^2 / \varepsilon^2)$ is enough. In light of lower bounds due to Diakonikolas et al. (2021), this yields (near) optimal bounds on the complexity of agnostically learning polynomial threshold functions in the statistical query model. Our proof relies on a direct analogue of a construction of Feldman et al. (2020), who considered $L_1$-approximation on the Boolean hypercube.

Lucas Slot, Stefan Tiegel, Manuel Wiedmer

Rubinfeld & Vasilyan recently introduced the framework of testable learning as an extension of the classical agnostic model. It relaxes distributional assumptions which are difficult to verify by conditions that can be checked efficiently by a tester. The tester has to accept whenever the data truly satisfies the original assumptions, and the learner has to succeed whenever the tester accepts. We focus on the setting where the tester has to accept standard Gaussian data. There, it is known that basic concept classes such as halfspaces can be learned testably with the same time complexity as in the (distribution-specific) agnostic model. In this work, we ask whether there is a price to pay for testably learning more complex concept classes. In particular, we consider polynomial threshold functions (PTFs), which naturally generalize halfspaces. We show that PTFs of arbitrary constant degree can be testably learned up to excess error $\varepsilon > 0$ in time $n^{\mathrm{poly}(1/\varepsilon)}$. This qualitatively matches the best known guarantees in the agnostic model. Our results build on a connection between testable learning and fooling. In particular, we show that distributions that approximately match at least $\mathrm{poly}(1/\varepsilon)$ moments of the standard Gaussian fool constant-degree PTFs (up to error $\varepsilon$). As a secondary result, we prove that a direct approach to show testable learning (without fooling), which was successfully used for halfspaces, cannot work for PTFs.