Efficient Quantum Agnostic Improper Learning of Decision Trees

TL;DR

This paper gives a quantum algorithm for learning size decision trees with uniform marginal over instances, in the agnostic setting, without membership queries, and has a polynomial improvement in the dependence of the bias of the weak learner over all adaptive quantum boosting algorithms while retaining the standard speedup in the VC dimension over classical boosting algorithms.

Abstract

The agnostic setting is the hardest generalization of the PAC model since it is akin to learning with adversarial noise. In this paper, we give a poly$(n,t,{\frac{1}{\varepsilon}})$ quantum algorithm for learning size $t$ decision trees with uniform marginal over instances, in the agnostic setting, without membership queries. Our algorithm is the first algorithm (classical or quantum) for learning decision trees in polynomial time without membership queries. We show how to construct a quantum agnostic weak learner by designing a quantum version of the classical Goldreich-Levin algorithm that works with strongly biased function oracles. We show how to quantize the agnostic boosting algorithm by Kalai and Kanade (NIPS 2009) to obtain the first efficient quantum agnostic boosting algorithm. Our quantum boosting algorithm has a polynomial improvement in the dependence of the bias of the weak learner over all adaptive quantum boosting algorithms while retaining the standard speedup in the VC dimension over classical boosting algorithms. We then use our quantum boosting algorithm to boost the weak quantum learner we obtained in the previous step to obtain a quantum agnostic learner for decision trees. Using the above framework, we also give quantum decision tree learning algorithms for both the realizable setting and random classification noise model, again without membership queries.

Figures (2)

• Figure 1: Agnostically learning polynomial-sized decision trees without MQ.
• Figure 2: A partial QGL tree (up to $3$ levels) indicating the level ordered traversal of "good" prefixes, that have Pw above threshold $\tau$. Bad prefixes are indicated using red arrows. The sub-trees of bad prefixes are not explored further. The set of good prefixes for level $i+1$ is decided in superposition over the set of good prefixes of level $i$. Shaded boxes indicate nodes evaluated in superposition.

Theorems & Definitions (43)

• Theorem 1
• Definition 1: Correlation kalai-kanade
• Definition 2: $(m,\kappa,\eta)$-weak Agnostic Learner kalai-kanade
• Definition 3: $\beta$-optimal $(\varepsilon,\delta)$-agnostic PAC learner gavinsky2002
• Definition 4: $(m,\kappa,\eta)$-Weak Quantum Agnostic Learner
• Lemma 2: Amplitude Amplification bhmt
• Lemma 3: Relative Error Estimation bhmt
• Lemma 4: Multidistribution Amplitude Estimation. Theorem 4 of bera2022few
• Theorem 5: Quantum Agnostic Boosting
• Theorem 6: Weak Agnostic Learner for size-$t$ Decision Trees