# Towards a Combinatorial Characterization of Bounded-Memory Learning

NIPS 2020, 2020.

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Abstract:

Combinatorial dimensions play an important role in the theory of machine learning. For example, VC dimension characterizes PAC learning, SQ dimension characterizes weak learning with statistical queries, and Littlestone dimension characterizes online learning. In this paper we aim to develop combinatorial dimensions that characterize boun...More

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Introduction

- Characterization of different learning tasks using a combinatorial condition has been investigated in depth in machine learning.
- 2. If the class C is PAC-learnable under P with accuracy 0.99 using b bits and m sample√s, for every distribution Q ∈ PΘ(1)(P ) its SQ dimension is bounded by SQQ(C) ≤ max(poly(m), 2O( b)).
- A class C under a distribution P is learnable with bounded memory with accuracy 1 − if there is a learning algorithm that uses only m = (|C|/ )o(1) samples and b = o(log |C|(log |X | + log(1/ ))) bits4.

Highlights

- Characterization of different learning tasks using a combinatorial condition has been investigated in depth in machine learning
- Learning a class in an unconstrained fashion is characterized by a finite VC dimension [8,38], and weakly learning in the statistical query (SQ) framework is characterized by a small SQ dimension [6]
- Is there a simple combinatorial condition that characterizes learnability with bounded memory? In this paper we propose a candidate condition, prove upper and lower bounds that match in some of the regime of parameters, and conjecture that they match in a much wider regime of parameters
- If the class C is PAC-learnable under P with accuracy 0.99 using b bits and m sample√s, for every distribution Q ∈ PΘ(1)(P ) its SQ dimension is bounded by SQQ(C) ≤ max(poly(m), 2O( b))
- We can transform any improper learner into a proper learner without significantly increasing the neither the sample nor the space complexity
- We prove similar conditions for SQ learning, implying equivalence between bounded memory learning and SQ learning for small enough

Results

- The authors state the main results for a combinatorial characterization of bounded memory PAC learning in terms of the SQ dimension of distributions close to the underlying distribution.
- There exists an algorithm that learns the class C with accuracy 1 − under the distribution P using b = O(log(d/ ) · log |C|) bits and m = poly(d/ ) · log(|C|) · log log(|C|) samples.
- Recall that the class is bounded memory learnable if there is a learning algorithm with sample complexity m = N o(1) and space complexity b = o(log2 N ).
- For any , the class C is bounded memory learnable under distribution P with accuracy 1 − ⇐⇒ ∀Q ∈ Ppoly(1/ )(P ), SQQ(C) ≤ poly(1/ ).
- There exists an SQ-learner that learns the class C with accuracy 1 − under the distribution P using q = poly(d/ ) statistical queries with tolerance τ ≥ poly( /d).
- In Section 3 the authors construct learning algorithms based on the assumption that close distributions have bounded SQ dimensions, and prove Theorem 5 and Theorem 8.
- To prove Theorem 6, the authors would like to use a recent result by [17] that establishes an upper bound on SQQ(C) given memory-efficient learner.

Conclusion

- The few claims establish the fact that if a class C is learnable with bounded memory under distribution Q, the statistical dimension SQQ(C) is low.
- Assume that the concept class C can be learned with accuracy 1 − 0.1 , m samples, and b bits under distribution P .
- Lemma 17 states that any probability distribution Q that is (1/ )-close to P can be learned with accuracy 0.9, O(m/ 2) samples, and b bits.

Summary

- Characterization of different learning tasks using a combinatorial condition has been investigated in depth in machine learning.
- 2. If the class C is PAC-learnable under P with accuracy 0.99 using b bits and m sample√s, for every distribution Q ∈ PΘ(1)(P ) its SQ dimension is bounded by SQQ(C) ≤ max(poly(m), 2O( b)).
- A class C under a distribution P is learnable with bounded memory with accuracy 1 − if there is a learning algorithm that uses only m = (|C|/ )o(1) samples and b = o(log |C|(log |X | + log(1/ ))) bits4.
- The authors state the main results for a combinatorial characterization of bounded memory PAC learning in terms of the SQ dimension of distributions close to the underlying distribution.
- There exists an algorithm that learns the class C with accuracy 1 − under the distribution P using b = O(log(d/ ) · log |C|) bits and m = poly(d/ ) · log(|C|) · log log(|C|) samples.
- Recall that the class is bounded memory learnable if there is a learning algorithm with sample complexity m = N o(1) and space complexity b = o(log2 N ).
- For any , the class C is bounded memory learnable under distribution P with accuracy 1 − ⇐⇒ ∀Q ∈ Ppoly(1/ )(P ), SQQ(C) ≤ poly(1/ ).
- There exists an SQ-learner that learns the class C with accuracy 1 − under the distribution P using q = poly(d/ ) statistical queries with tolerance τ ≥ poly( /d).
- In Section 3 the authors construct learning algorithms based on the assumption that close distributions have bounded SQ dimensions, and prove Theorem 5 and Theorem 8.
- To prove Theorem 6, the authors would like to use a recent result by [17] that establishes an upper bound on SQQ(C) given memory-efficient learner.
- The few claims establish the fact that if a class C is learnable with bounded memory under distribution Q, the statistical dimension SQQ(C) is low.
- Assume that the concept class C can be learned with accuracy 1 − 0.1 , m samples, and b bits under distribution P .
- Lemma 17 states that any probability distribution Q that is (1/ )-close to P can be learned with accuracy 0.9, O(m/ 2) samples, and b bits.

Related work

- Characterization of bounded memory learning. Many works have proved lower bounds under memory constraints [3, 10, 11, 16, 17, 22, 24, 25, 27, 28, 32, 33]. Some of these works even provide a necessary condition for learnability with bounded memory. As for upper bounds, not many works have tried to give a general property that implies learnability under memory constraints. One work suggested such property [26] but this did not lead to a full characterization of bounded memory learning.

Statistical query learning. After Kearns’s introduction of statistical query [20], Blum et al [6] characterized weak learnability using SQ dimension. Specifically, if SQP (C) = d, then poly(d) queries are both needed and sufficient to learn with accuracy 1/2 + poly(1/d). Note that the advantage is very small, only poly(1/d). Subsequently several works [2, 12, 34, 36] suggested a few characterizations of strong SQ learnability.

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- 2. Go over all hypothesis in C and return one that agrees with h on 1 − 2 of the examples by testing consistency on O(log |C|/ 2) random examples.

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