DS Dimension: PAC Learning & Fractal Spectrum

• DS Dimension is a combinatorial invariant that quantifies the richness of multiclass hypothesis classes and the range of fractal dimensions in Cantor-type sets.
• In multiclass PAC learning, DS dimension generalizes the VC dimension via pseudo-cube shattering, leading to refined sample complexity bounds compared to the Natarajan dimension.
• Algorithmic advances using improper compression schemes and multiplicative-weights reduction leverage the DS dimension to optimize learning and elucidate fractal spectra properties.

The term DS dimension refers to several invariants across mathematical and computational contexts, primarily denoting structured notions of “dimension” that capture combinatorial, geometric, or learning-theoretic complexity. The most prominent instantiations of DS dimension arise in statistical learning theory—where it is known as the Daniely–Shalev–Shwartz (DS) dimension for multiclass hypothesis classes—and in the paper of continued fractions, where DS denotes the “dimension spectrum” of a family of Cantor-type sets. This entry rigorously presents the DS dimension in both the PAC learning and fractal geometry settings, elucidating definitions, key properties, theoretical consequences, algorithmic constructions, and its broader significance.

1. DS Dimension in Multiclass PAC Learning

The Daniely–Shalev–Shwartz (DS) dimension is a combinatorial parameter for multiclass hypothesis classes $H \subset Y^X$, generalizing the Vapnik–Chervonenkis (VC) dimension from binary to multiclass settings. The formal definition is as follows:

A finite set $B \subset Y^d$ is a $d$-dimensional pseudo-cube if, for every $b \in B$ and each coordinate $i \in \{1, ..., d\}$, there exists $b' \in B$ with $b'(i) \ne b(i)$ and $b'(j) = b(j)$ for $j \ne i$. That is, from any “vertex,” the set contains all one-coordinate “flips.” For $Y = \{0,1\}$, every pseudo-cube is the full Boolean cube; in the multiclass context, these structures are significantly richer.

A sequence $x = (x_1, ..., x_d) \in X^d$ is DS-shattered by $H$ if the projection $H|_x = \{(h(x_1), ..., h(x_d)) : h \in H\}$ contains a $d$-dimensional pseudo-cube. The DS dimension $d_{DS}(H)$ is the maximal $d$ for which such a shattering exists.

This parameter measures the ability of a class to realize rich collections of label patterns—generalizing the shattering concept familiar from VC-dimension to multiclass labelings via pseudo-cubes—thus quantifying the inherent complexity of multiclass concept classes (Cohen et al., 16 Nov 2025).

2. DS Dimension versus Natarajan Dimension

In multiclass learning, the Natarajan dimension ($\mathrm{Nat} = d_N(H)$) also measures combinatorial richness but is strictly weaker: it requires $H|_x$ to contain copies of the binary cube. Thus, for any $H$, $d_N(H) \le d_{DS}(H)$, but $d_{DS}(H)$ can be arbitrarily larger. For example, there exist hypothesis classes with $d_N(H) = 1$ but $d_{DS}(H) = n$ for any $n$, demonstrating exponential separation (Cohen et al., 16 Nov 2025).

The interplay of these two dimensions is crucial: Daniely & Shalev–Shwartz established that finite DS dimension is necessary for realizable PAC learning, while Brukhim et al. proved it is also sufficient. In agnostic learning, both dimensions jointly control sample complexity: $$m(\epsilon) = \widetilde O\left( \frac{DS^{1.5}}{\epsilon} + \frac{Nat}{\epsilon^2} \right)$$ The first term dominates for moderate $\epsilon$ (the “DS-controlled regime”), while the second term drives behavior for small $\epsilon$ (“Nat-controlled regime”). Thus, multiclass agnostic PAC sample complexity does not reduce to a single combinatorial invariant, but inherently involves two terms—contrasting sharply with the binary theory, where the VC dimension alone suffices (Cohen et al., 16 Nov 2025).

3. Algorithmic and Structural Implications

The agnostic multiclass PAC-learning bounds involving DS involve nontrivial algorithmic constructions. Core methods include:

• Improper compression schemes: A realization-case compression of size $k = O(d_{\mathrm{RE}} \log n)$ is used, with $d_{\mathrm{RE}}$ the smallest $m$ such that $H$ is learnable to constant error in $m$ examples.
• Multiplicative-weights label-space reduction: A novel on-line multiplicative-weights meta-algorithm is used to construct, over multiple rounds, a small collection of functions (“label list”) corresponding to the dominant patterns in the finite cover. This stage reduces the effective label-space to a small, data-driven set, permitting high-probability control over the best-in-class error.
• Compression-based generalization: Once the effective hypothesis list is small, sample complexity is improved further via sample-compression bounds, with the final rates reflecting the underlying Nat and DS dimensions (Cohen et al., 16 Nov 2025).

These algorithmic steps yield, up to logarithmic factors, the sample complexity stated above, and rigorously demonstrate the criticality of DS dimension for learnability.

4. DS (Dimension Spectrum) in Continued Fraction Fractals

In fractal geometry and dynamical systems, DS denotes the “dimension spectrum” of certain continued-fraction Cantor sets associated with infinite IFSs. For an infinite alphabet $E \subset \mathbb{N}$, one considers the IFS $\{\phi_e\}_{e \in E}$ with $\phi_e(x) = 1/(e + x)$, generating the limit set

$$J_E = \{ x \in (0,1) : \text{continued-fraction digits of } x \text{ lie in } E \}$$

For every subset $F \subset E$, $J_F$ is the associated limit set, and the dimension spectrum is

$$DS(\mathcal{CF}_E) = \{ \dim_H J_F : F \subset E \}$$

This set quantifies the full range of Hausdorff dimensions obtainable by restricting continued-fraction expansions to subalphabets (Chousionis et al., 2018).

Main Theorems

• For $E$ an arithmetic progression, the set of primes, or squares, $DS(\mathcal{CF}_E)$ is the full interval $[0, \dim_H J_E]$.
• For $E$ consisting of powers $\{\lambda^n : n \in \mathbb{N}\}$, $DS(\mathcal{CF}_E)$ contains a nontrivial interval $[0, s(\lambda)]$.
• There exist $E$ and intervals $I_1, I_2$ such that $DS(\mathcal{CF}_E) \cap I_1 = I_1$ and $DS(\mathcal{CF}_E) \cap I_2$ is a Cantor set (Chousionis et al., 2018).

Proofs leverage thermodynamic formalism, pressure functions, and computer-assisted rigour (Falk–Nussbaum method) for bounding Hausdorff dimensions.

5. Relations to Other “Dimension” Notions

The term “DS dimension” is unrelated to the Dushnik–Miller (order) dimension, surface fractal dimension, or the “diagonal dimension” of C*-algebra pairs, each of which has its own rigorous meaning:

• Dushnik–Miller dimension arises in order theory and the geometry of complexes (Gonçalves et al., 2018).
• Surface fractal dimension $D_s$ characterizes roughness in porous media (Ghanbarian, 2019).
• Diagonal dimension generalizes nuclear dimension for sub-C*-algebras, connecting to dynamical systems and coarse geometry (Li et al., 2023).

Nevertheless, the DS dimension in learning theory and as the dimension spectrum in fractals both provide quantitative invariants that control structural and quantitative questions—sample complexity and fractal dimensions, respectively—in their respective domains.

6. Impact and Ongoing Research Directions

The DS dimension is now recognized as the critical combinatorial invariant for multiclass PAC learning, with the Natarajan dimension re-emerging only in the high-precision ($\epsilon \to 0$) limit. This fundamentally alters learning theory in multiclass settings, revealing that no single-parameter dimension theory (in the classical VC or Littlestone sense) suffices (Cohen et al., 16 Nov 2025).

The dimension spectrum concept has similarly deepened understanding of the range of fractal dimensions possible in parametrized families of fractals, motivating further research into arithmetic and measure-theoretic properties of these spectra (Chousionis et al., 2018).

Open directions include tighter characterization for agnostic sample complexity, the search for optimal algorithms matching the lower bounds, and further structural paper of spectra for other types of fractals and symbolic dynamics.