Time-Space Tradeoffs for the Memory Game

A single-player game of Memory is played with $n$ distinct pairs of cards, with the cards in each pair bearing identical pictures. The cards are laid face-down. A move consists of revealing two cards, chosen adaptively. If these cards match, i.e., they bear the same picture, they are removed from play; otherwise, they are turned back to face down. The object of the game is to clear all cards while minimizing the number of moves. Past works have thoroughly studied the expected number of moves required, assuming optimal play by a player has that has perfect memory. In this work, we study the Memory game in a space-bounded setting. We prove two time-space tradeoff lower bounds on algorithms (strategies for the player) that clear all cards in $T$ moves while using at most $S$ bits of memory. First, in a simple model where the pictures on the cards may only be compared for equality, we prove that $ST = \Omega(n^2 \log n)$. This is tight: it is easy to achieve $ST = O(n^2 \log n)$ essentially everywhere on this tradeoff curve. Second, in a more general model that allows arbitrary computations, we prove that $ST^2 = \Omega(n^3)$. We prove this latter tradeoff by modeling strategies as branching programs and extending a classic counting argument of Borodin and Cook with a novel probabilistic argument. We conjecture that the stronger tradeoff $ST = \widetilde{\Omega}(n^2)$ in fact holds even in this general model

Noise-adaptive Margin-based Active Learning and Lower Bounds under Tsybakov Noise Condition

We present a simple noise-robust margin-based active learning algorithm to find homogeneous (passing the origin) linear separators and analyze its error convergence when labels are corrupted by noise. We show that when the imposed noise satisfies the Tsybakov low noise condition (Mammen, Tsybakov, and others 1999; Tsybakov 2004) the algorithm is able to adapt to unknown level of noise and achieves optimal statistical rate up to poly-logarithmic factors. We also derive lower bounds for margin based active learning algorithms under Tsybakov noise conditions (TNC) for the membership query synthesis scenario (Angluin 1988). Our result implies lower bounds for the stream based selective sampling scenario (Cohn 1990) under TNC for some fairly simple data distributions. Quite surprisingly, we show that the sample complexity cannot be improved even if the underlying data distribution is as simple as the uniform distribution on the unit ball. Our proof involves the construction of a well separated hypothesis set on the d-dimensional unit ball along with carefully designed label distributions for the Tsybakov noise condition. Our analysis might provide insights for other forms of lower bounds as well.Comment: 16 pages, 2 figures. An abridged version to appear in Thirtieth AAAI Conference on Artificial Intelligence (AAAI), which is held in Phoenix, AZ USA in 201