Statistical Algorithms and a Lower Bound for Detecting Planted Clique (1201.1214v6)

Abstract: We introduce a framework for proving lower bounds on computational problems over distributions against algorithms that can be implemented using access to a statistical query oracle. For such algorithms, access to the input distribution is limited to obtaining an estimate of the expectation of any given function on a sample drawn randomly from the input distribution, rather than directly accessing samples. Most natural algorithms of interest in theory and in practice, e.g., moments-based methods, local search, standard iterative methods for convex optimization, MCMC and simulated annealing can be implemented in this framework. Our framework is based on, and generalizes, the statistical query model in learning theory (Kearns, 1998). Our main application is a nearly optimal lower bound on the complexity of any statistical query algorithm for detecting planted bipartite clique distributions (or planted dense subgraph distributions) when the planted clique has size $O(n^{1/2-\delta})$ for any constant $\delta > 0$. The assumed hardness of variants of these problems has been used to prove hardness of several other problems and as a guarantee for security in cryptographic applications. Our lower bounds provide concrete evidence of hardness, thus supporting these assumptions.

• The paper's main contribution is introducing an SQ framework that generalizes learning theory to establish lower bounds for detecting planted cliques.
• It demonstrates that no SQ algorithm can efficiently detect cliques of size O(n^(1/2-δ)), revealing significant computational limits.
• The work extends its approach to cryptography and related distribution problems, paving the way for new algorithm designs and complexity analyses.

An Analysis of Statistical Algorithms and Lower Bound for Detecting Planted Cliques

The paper presents a sophisticated framework to derive lower bounds for computational problems by utilizing statistical algorithms in detecting planted cliques. Specifically, it focuses on problems where the input is derived from an unknown distribution, and the primary tool is a Statistical Query (SQ) oracle. The fundamental premise is that instead of accessing data samples directly, the algorithm can only obtain estimates of expected values over a randomly sampled input distribution. The SQ framework was initially framed to simplify the process of designing noise-tolerant learning algorithms.

Overview and Contributions

Framework Introduction: A notable contribution is the introduction of a framework that generalizes the SQ model from learning theory to broader computational problems over distributions. This framework elucidates some enduring challenges associated with directly proving computational complexity lower bounds beyond sample complexity in machine learning, cryptography, and related fields. The reliance on SQ algorithms allows for deriving unconditional complexity bounds because most real-world algorithmic approaches can be re-expressed within the confines of this framework.

Lower Bound Results: The main result is achieving nearly optimal lower bounds for the complexity of any statistical query algorithm in detecting planted bipartite cliques. For cases when the planted clique consists of at most $O(n^{1/2-\delta})$ vertices (for some constant $\delta > 0$), the framework indicates that no statistical query algorithm operating with feasible complexity can efficiently detect such cliques. Given the significance of clique detection in proving the hardness of several problems and its applications in cryptography, the framework offers robust evidence supporting the presumed difficulty of these problems.

Statistical Dimension and Complexity Measurement: The research introduces concepts such as statistical dimension and average correlations, enhancing the understanding of the problem’s complexity. It effectively expands the SQ dimension, which characterizes complexity in the traditional SQ learning models, by applying it to arbitrary distribution sets. This presents more comprehensive audit methods, offering potential pathways for complexity evaluations in a variety of problems beyond the conventional structure.

Additional Applications: Beyond the statistical dimension in learning, the paper explores related problems, including planted densest subgraphs and a variant of the MAX-XOR-SAT problem, showcasing the generality and extensibility of their approach. These findings reinforce the intractability of specific distributional problems within statistical frameworks, suggesting that solutions must emerge from outside the current constraints of statistical algorithms.

Potential Implications and Future Directions

1. Algorithm Development: Researchers designing algorithms for problems involving distributions over structures might need to explore beyond statistical sampling techniques due to inherent limitations, as elucidated by the proposed framework. This helps in redirecting focus towards algorithms that leverage alternative heuristic or combinatorial strategies.
2. Cryptographic Applications: The paper underscores the fortitude of certain cryptographic assumptions, lending credence to their security implications. Future cryptographic constructs could leverage these hardness evidences to solidify or even revise security enforcements.
3. Further Exploration of Distribution Packs: Future work can expand these foundational ideas on statistical dimensions to investigate new boundaries for other complex problems linked to distributions — particularly those that interact synergistically across machine learning and cryptographic interfaces.

This paper crafts a critical narrative regarding the boundaries of statistical algorithm capabilities, notably illustrating why detecting planted cliques under specific constraints remains an arduous task. It ventures into cross-sections of learning theory, optimization, and cryptography, offering a pathway for enriched discourse and new research trajectories in computational complexity.