왜 수학이 머신러닝에 필수적인가

Source: Dev.to

Introduction — The Black Box Myth

Machine Learning is often presented as an essentially algorithmic discipline: you load data, choose a model, train it, and “it works.”
This view is partly true, but fundamentally incomplete.

Behind every Machine Learning algorithm lie precise mathematical structures:

• notions of distance
• properties of continuity
• assumptions of convexity
• convergence guarantees and theoretical limits that no model can circumvent

👉 Modern Machine Learning is not an alternative to mathematics; it is a direct application of it.

This article sets the general framework for the series: understanding why mathematical analysis is indispensable for understanding, designing, and mastering Machine Learning algorithms.

1. Machine Learning Is Primarily an Optimization Problem

At a fundamental level, almost all ML algorithms solve the same problem:

Minimize a loss function.

Formally, we search for parameters θ such that

[ \theta^{*} = \arg\min_{\theta} L(\theta) ]

where L(θ) measures the model’s error on the data.

Essential mathematical questions arise immediately:

• What does it mean to minimize?
• Does a minimum exist?
• Is it unique?
• Can it be reached numerically?
• At what speed?

These questions are mathematical, not merely algorithmic.

2. Distance, Norms, and Geometry: Measuring Error Is Not Neutral

Before optimizing anything, we must answer:

How do we measure error?

This leads directly to the notions of distance and norm.

Classic examples:

• MAE (Mean Absolute Error) ↔ L¹ norm
• MSE (Mean Squared Error) ↔ L² norm
• Maximum error ↔ L^∞ norm

These choices are not incidental:

• they change the geometry of the problem
• they affect robustness to outliers
• they influence numerical stability
• they impact gradient descent behavior

👉 Without understanding the geometry induced by a norm, one does not truly understand what the algorithm is optimizing.

3. Convergence: When Can We Say an Algorithm Works?

A Machine Learning algorithm is often iterative:

[ \theta_{0} \rightarrow \theta_{1} \rightarrow \theta_{2} \rightarrow \dots ]

This raises a crucial question:

Does this sequence converge? And if so, to what?

The answer depends on concepts from analysis:

• sequences and limits
• Cauchy sequences
• completeness
• continuity

Without these notions, it is impossible to answer practical questions such as:

• why training diverges
• why it oscillates
• why it is slow
• why two implementations produce different results

4. Continuity, Lipschitz Conditions, and Stability

A Machine Learning model must be stable: a small change in the data or parameters should not cause predictions to explode. This stability is formalized by:

• uniform continuity
• Lipschitz functions

A function f is Lipschitz if

[ |f(x) - f(y)| \le L,|x - y| ]

This inequality lies at the core of:

• model stability
• learning rate selection
• convergence guarantees for gradient descent

👉 The Lipschitz constant is not a theoretical detail; it directly controls the speed and stability of learning.

5. Convexity: Why Some Problems Are Easy… and Others Are Not

Convexity is arguably the most important mathematical property in optimization.

A convex function has:

• a unique global minimum
• no traps in the form of local minima

This is why methods such as:

• linear regression
• support vector machines
• certain regularization problems

benefit from strong theoretical guarantees.

By contrast, deep neural networks are non‑convex, yet they still work thanks to particular structures and effective heuristics.

👉 Understanding convexity makes it possible to know when guarantees exist — and when they do not.

6. Theory vs. Practice: What Mathematics Guarantees (and What It Does Not)

A crucial point to understand from the outset:

Mathematics guarantees properties, not miraculous performance.

It can tell us:

• whether a solution exists
• whether it is unique
• whether an algorithm converges
• how fast it converges

It cannot guarantee:

• good data
• good generalization
• an unbiased model

Without mathematical insight, we proceed blindly.

Conclusion — Understand Before You Optimize

Modern Machine Learning rests on three fundamental mathematical pillars:

• Geometry (norms, distances)
• Analysis (continuity, convergence, Lipschitz conditions)
• Optimization (convexity, gradient descent)

Ignoring these foundations amounts to:

• applying recipes without understanding their limits
• misdiagnosing failures
• overcomplicating simple problems

👉 Understanding the mathematical analysis of Machine Learning is not theory for theory’s sake; it is about gaining control, robustness, and intuition.