Computational Learning Theory

Computational Learning Theory

Complexity Measures

1. Sample Complexity

PAC learning bound:

\[ m \geq \frac{1}{\epsilon}\left(\ln|\mathcal{H}| + \ln\frac{1}{\delta}\right) \]

VC dimension bound:

\[ m = O\left(\frac{d}{\epsilon^2}\ln\frac{1}{\epsilon} + \frac{1}{\epsilon^2}\ln\frac{1}{\delta}\right) \]

2. Time Complexity

Learning algorithm runtime:

\[ T(m,n,\epsilon,\delta) = \text{poly}(m,n,\frac{1}{\epsilon},\frac{1}{\delta}) \]

Where: - \(m\) is sample size - \(n\) is input dimension - \(\epsilon\) is accuracy - \(\delta\) is confidence

3. Space Complexity

Memory requirements:

\[ S(m,n) = O(mn) \]

Streaming bound:

\[ S = O(\log m + \log n) \]

Learnability Analysis

1. Efficient Learnability

Definition: - Polynomial sample complexity - Polynomial time complexity - Polynomial space complexity

Requirements:

\[ \begin{aligned} m &= \text{poly}(n,\frac{1}{\epsilon},\frac{1}{\delta}) \\ T &= \text{poly}(n,\frac{1}{\epsilon},\frac{1}{\delta}) \\ S &= \text{poly}(n,\frac{1}{\epsilon},\frac{1}{\delta}) \end{aligned} \]

2. Hardness Results

Cryptographic assumptions:

\[ P \neq NP \implies \text{Not efficiently learnable} \]

Reduction techniques:

\[ \text{Problem A} \leq_p \text{Problem B} \]

3. Average-Case Analysis

Expected runtime:

\[ \mathbb{E}[T(X)] = \sum_{x} T(x)P(x) \]

Smoothed analysis:

\[ \max_{\text{input } I} \mathbb{E}_{\text{noise } \xi}[T(I + \xi)] \]

Learning Models

1. Query Learning

Membership queries:

\[ \text{MQ}(x) = c(x) \]

Equivalence queries:

\[ \text{EQ}(h) = \begin{cases} \text{Yes} & \text{if } h \equiv c \\ x & \text{where } h(x) \neq c(x) \end{cases} \]

2. Online Learning

Mistake bound:

\[ M \leq \text{poly}(n,\text{size}(c)) \]

Halving algorithm:

\[ |\mathcal{H}_t| \leq |\mathcal{H}_0|/2^t \]

3. Active Learning

Label complexity:

\[ \Lambda(\epsilon,\delta) = O(\theta d\log(1/\epsilon)\log(1/\delta)) \]

Where: - \(\theta\) is disagreement coefficient - \(d\) is VC dimension

Algorithmic Efficiency

1. Boosting Analysis

AdaBoost iterations:

\[ T = O\left(\frac{\log(1/\epsilon)}{\gamma^2}\right) \]

Where: - \(\gamma\) is edge over random

2. Kernel Methods

Kernel evaluation:

\[ T = O(m^2n) \]

Nyström approximation:

\[ \|K - \tilde{K}\|_2 \leq \epsilon \]

3. Neural Networks

Backpropagation complexity:

\[ T = O(mnd) \]

Where: - \(d\) is network depth

Computational Trade-offs

1. Time-Space Trade-offs

Memory-runtime relationship:

\[ T \cdot S = \Omega(n^2) \]

Streaming algorithms:

\[ S \cdot \text{passes} = \Omega(n) \]

2. Sample-Computation Trade-offs

Active learning:

\[ \text{queries} \cdot \text{computation} = O(m\log m) \]

Parallel speedup:

\[ T_p = \frac{T_1}{p} + O(\log p) \]

3. Accuracy-Complexity Trade-offs

Approximation guarantee:

\[ f(x) \leq (1+\epsilon)\text{OPT} \]

Runtime dependency:

\[ T = O(\text{poly}(n,1/\epsilon)) \]

Advanced Topics

1. Communication Complexity

Two-party protocol:

\[ CC(f) = \min_P \max_{x,y} \text{bits}(P(x,y)) \]

Lower bound:

\[ CC(f) \geq \log_2 \text{rank}(M_f) \]

2. Circuit Complexity

Boolean circuit size:

\[ \text{size}(f) = \min_{C: C \text{ computes } f} \text{gates}(C) \]

Depth bound:

\[ \text{depth}(f) \geq \log_2 \text{sensitivity}(f) \]

3. Information Complexity

Information cost:

\[ IC(P) = I(X;M|Y) + I(Y;M|X) \]

Protocol compression:

\[ |P'| \leq O(IC(P)\log|P|) \]

Practical Implications

1. Algorithm Design

1. Resource Constraints:
   • Time efficiency
   • Memory usage
   • Communication cost
2. Scalability:
   • Input size
   • Dimensionality
   • Sample complexity
3. Parallelization:
   • Task decomposition
   • Communication overhead
   • Load balancing

2. System Implementation

1. Architecture:
   • Processing units
   • Memory hierarchy
   • Network topology
2. Optimization:
   • Caching strategies
   • Data structures
   • Algorithm selection
3. Trade-offs:
   • Accuracy vs speed
   • Memory vs computation
   • Communication vs local processing

Theoretical Frameworks

1. Learning with Errors

LWE assumption:

\[ (A,As+e) \approx_c (A,u) \]

Where: - \(A\) is random matrix - \(s\) is secret - \(e\) is error - \(u\) is uniform

2. Statistical Query Model

Query complexity:

\[ \text{SQ-DIM}(\mathcal{C}) = \min_{D} \max_{f \in \mathcal{C}} |\langle f,D \rangle| \]

3. Property Testing

Query complexity:

\[ Q(\epsilon) = O(\frac{1}{\epsilon}\log\frac{1}{\epsilon}) \]

Distance measure:

\[ \text{dist}(f,g) = \text{Pr}_{x}[f(x) \neq g(x)] \]

Best Practices

1. Algorithm Analysis

1. Complexity Measures:
   • Asymptotic bounds
   • Average case
   • Worst case
2. Resource Usage:
   • Memory footprint
   • CPU utilization
   • I/O operations
3. Scalability:
   • Data size
   • Dimensionality
   • Parallelization

2. Implementation

1. Optimization:
   • Algorithm choice
   • Data structures
   • Memory management
2. Trade-offs:
   • Time vs space
   • Accuracy vs speed
   • Communication vs computation
3. Evaluation:
   • Benchmarking
   • Profiling
   • Performance analysis

References

1. Theory:
   • “Foundations of Machine Learning” by Mohri et al.
   • “Understanding Machine Learning” by Shalev-Shwartz and Ben-David
   • “Computational Learning Theory” by Kearns and Vazirani
2. Complexity:
   • “Computational Complexity” by Arora and Barak
   • “Communication Complexity” by Kushilevitz and Nisan
   • “The Nature of Computation” by Moore and Mertens
3. Applications:
   • “Algorithmic Learning Theory” by Anthony and Biggs
   • “Learning with Kernels” by Schölkopf and Smola
   • “Theoretical Computer Science” by Hopcroft and Ullman