SlowGuess/NeurIPS_2023_OCR huggingface dataset:
We make extensive use of big- $O$  notation. In this paragraph,  $c$  refers to a positive constant which does not depend on  $n$ , and all statements hold for sufficiently large  $n$ . If  $f(n) = O(g(n))$ , then  $f(n) \leq cg(n)$  for some  $c$ . If  $f(n) = \tilde{O}(g(n))$ , then  $f(n) \leq cg(n)$  poly log  $(n)$  for some  $c$ . If  $f(n) = o(g(n))$ , then for all  $c > 0$  we have  $f(n) \leq cg(n)$ . We write  $f(n) = \Omega(g(n))$  if  $f(n) \geq cg(n)$  for some  $c$ . Finally, we write  $f(n) = \Theta(g(n))$  if there exists positive constants  $c_1$  and  $c_2$  such that  $c_1g(n) \leq f(n) \leq c_2g(n)$ .
Table 2: Notation
Symbol	Definition	Dimension	Source
k	Number of classes	Scalar	Sec. 2
n	Number of training points	Scalar	Sec. 2
d	Dimension of each point — the total number of features	Scalar	Sec. 2
s	The number of favored features	Scalar	Def. 1
a	The constant controlling the favored weights	Scalar	Def. 1
p	Parameter controlling overparameterization (d = np)	Scalar	Def. 1
r	Parameter controlling the number of favored features (s = nr)	Scalar	Def. 1
q	Parameter controlling the favored weights (a = n-q)	Scalar	Def. 1
t	Parameter controlling the number of classes (k = cknt)	Scalar	Def. 1
ck	The number of classes when t = 0 (k = cknt)	Scalar	Def. 1
λj	Squared weight of the jth feature	Scalar	Def. 1
xi	ith training point (unweighted)	Length-d vector	Sec. 2
l i	Class label of ith training point	Scalar	Eqn. 1
wi	ith training point (weighted)	Length-d vector	Eqn. 2
Xw	Weighted feature matrix	(n × d)-matrix	Eqn. 3
zj	The collected jth features of all training points	Length-n vector	Eqn. 3
yohm	One-hot encoding of all the training points for label m	Length-n vector	Eqn. 4
Yoh	One-hot label matrix	(n × k)-matrix	Eqn. 4
ym	Zero-mean encoding of the training points for label m	Length-n vector	Eqn. 5
f̃m	Learned coefficients for label m using min-norm interpolation	Length-d vector	Eqn. 8
xtest	A single test point	Length-d vector	Sec. 2
xwtest	A single weighted test point	Length-d vector	Sec. 2
A	Gram matrix A = Xw(Xw)T	(n × n)-matrix	Sec. 4
μi(A)	The ith eigenvalue of matrix A, sorted in descending order	Scalar	App. A
λF	Squared favored feature weights: λF = ad/s	Scalar	Def. 1
λU	Squared unfavored feature weights: λF = (1-a)d/d-s	Scalar	Def. 1
hα,β	Relative survival hα,β[j] = λj-1/2(ˆfα[j] -ˆfβ[j])	Length-d vector	Eqn. 16
CNα,β	Normalizing factor CNα,β = √(∑j∉{α,β}λj2(ˆhβ,α[j])2)	Scalar	Eqn. 19
||·||ψ2	The subgaussian norm of a scalar random variable	Scalar	Eqn. 24
||·||ψ1	The subexponential norm of a scalar random variable	Scalar	Eqn. 51
μ	Factor controlling whether regression works, μ ≅ nq+r-1	Scalar	App. A.1
ZT	Unweighted subset of favored features, where T ⊆ [s]	(n × |T|)-matrix	App. B.2
WT	Weighted subset of favored features, WT = √λFT	(n × |T|)-matrix	App. B.2
A-T	Leave-T-out Gram matrix, where T ⊆ [s], A-T = A - WTWT	(n × n)-matrix	Eqn. 74
Hk	Hat matrix, Hk = WkA-kWk	(k × k)-matrix	Eqn. 53
Let us now describe the organization of the appendix. In Appendix A.1, we give a more detailed proof sketch and introduce the main propositions that complete the proof of Theorem 3.2. In Appendix B, we introduce the main tools that allow us to prove that the critical bilinear forms  $z_{j}^{\top}A^{-1}\Delta y$  concentrate: our new variant of the Hanson-Wright inequality, the Woodbury inversion formula, and Wishart concentration to bound the spectra of the relevant random matrices that appear. In Appendix C we apply these tools to bound some useful quantities that repeatedly appear in the rest of the proofs. After that, we proceed to bound the survival, contamination, and correlation structure in Appendices D to F. In Appendix J, we present the analysis for the averaging scheme described in Section 5. Finally, we prove our new variant of Hanson-Wright (Theorem 4.1) in Appendix G.
In this section, we fill in some of the details of the proof sketch of Theorem 3.2. After recalling the beginning of the proof, we will split up the proof into two subtheorems: one for the positive result where MNI generalizes (Theorem A.2), and another for the negative result where MNI misclassifies (Theorem A.4).
$$
\begin{array}{l} \widehat {\boldsymbol {h}} _ {\alpha , \beta} [ j ] \triangleq \lambda_ {j} ^ {- 1 / 2} \left(\widehat {\boldsymbol {f}} _ {\alpha} [ j ] - \widehat {\boldsymbol {f}} _ {\beta} [ j ]\right) (27) \\ = \boldsymbol {z} _ {j} ^ {\top} \boldsymbol {A} ^ {- 1} \boldsymbol {\Delta} y, (28) \\ \end{array}
$$
where to obtain the last line we have used the explicit formula for the MNI classifiers (8). By converting the misclassification condition into the unweighted feature space we see that we will have errors when
$$
\lambda_ {\alpha} \widehat {\boldsymbol {h}} _ {\alpha , \beta} [ \alpha ] \boldsymbol {x} _ {\text {t e s t}} [ \alpha ] - \lambda_ {\beta} \widehat {\boldsymbol {h}} _ {\beta , \alpha} [ \beta ] \boldsymbol {x} _ {\text {t e s t}} [ \beta ] <   \sum_ {j \notin \{\alpha , \beta \}} \lambda_ {j} \widehat {\boldsymbol {h}} _ {\beta , \alpha} [ j ] \boldsymbol {x} _ {\text {t e s t}} [ j ]. \tag {29}
$$
$$
\mathrm {C N} _ {\alpha , \beta} \triangleq \sqrt {\sum_ {j \notin \{\alpha , \beta \}} \lambda_ {j} ^ {2} (\hat {\boldsymbol {h}} _ {\beta , \alpha} [ j ]) ^ {2}}. \tag {30}
$$
Note that  $\mathsf{CN}_{\alpha,\beta}$  normalizes the RHS of (29) into a standard Gaussian. Indeed, define
$$
Z ^ {(\beta)} \triangleq \frac {1}{\mathbb {C N} _ {\alpha , \beta}} \sum_ {j \notin \{\alpha , \beta \}} \lambda_ {j} \widehat {\boldsymbol {h}} _ {\beta , \alpha} [ j ] \boldsymbol {x} _ {\text {t e s t}} [ j ] \sim N (0, 1). \tag {31}
$$
Since  $\alpha, \beta \in [k]$  are favored, we have  $\lambda_{\alpha} = \lambda_{\beta} = \lambda_{F}$ . Hence an equivalent condition for misclassification is that there exists some  $\beta \neq \alpha$ ,  $\beta \in [k]$  such that
$$
\frac {\lambda_ {F}}{\mathbb {C N} _ {\alpha , \beta}} \left(\widehat {\boldsymbol {h}} _ {\alpha , \beta} [ \alpha ] \boldsymbol {x} _ {\text {t e s t}} [ \alpha ] - \widehat {\boldsymbol {h}} _ {\beta , \alpha} [ \beta ] \boldsymbol {x} _ {\text {t e s t}} [ \beta ]\right) <   Z ^ {(\beta)}. \tag {32}
$$
We will translate the above criterion into sufficient conditions for correct classification and misclassification and analyze these two cases separately.
First, let us present our tight characterization of the survival and contamination terms, which will be useful for both sides of the theorem. Recall our definition of  $\mu \triangleq n^{q + r - 1}$ ; whether this quantity polynomially shrinks or decays directly determines if regression works or fails.
We defer the proof of Proposition 4.2 to Appendix D and the proof of Proposition 4.3 to Appendix E. Combining Propositions 4.2 and 4.3 yields the following sufficient conditions for when the SU/CN ratio grows or shrinks polynomially.
Proposition A.1 (Regimes for survival-to-contamination). Under the bi-level ensemble model (Definition 1), when the true data generating process is 1-sparse (Assumption 1), as  $n \to \infty$ , with probability at least  $1 - O(1/n)$ , the survival-to-contamination ratio satisfies
$$
\frac {\min  _ {\beta} \lambda_ {F} \widehat {\boldsymbol {h}} _ {\alpha , \beta} [ \alpha ]}{\max  _ {\beta} \mathbb {C N} _ {\alpha , \beta}} \geq n ^ {v} f o r s o m e v > 0 i f t <   \min  \{1 - r, p + 1 - 2 \max  \{1, q + r \} \} \tag {33}
$$
$$
\frac {\operatorname* {m a x} _ {\beta} \lambda_ {F} \left| \widehat {\boldsymbol {h}} _ {\alpha , \beta} [ \alpha ] \right|}{\min  _ {\beta} \mathbb {C N} _ {\alpha , \beta}} \leq n ^ {- w} f o r s o m e w > 0 i f t > \min  \{1 - r, p + 1 - 2 \max  \{1, q + r \} \} \tag {34}
$$
Here, the max and min are being taken over  $\beta \neq \alpha ,\beta \in [k]$
Proof. We do casework on whether we want to prove an upper bound or lower bound on  $\mathsf{SU} / \mathsf{CN}$ . First, suppose we want to prove the lower bound, so assume  $t < \min\{1 - r, p + 1 - 2 \max\{1, q + r\}\}$ . Since  $t < r$  by the definition of the bi-level ensemble (Definition 1), we have that  $t < \frac{1}{2}$ . So by union bounding over  $\beta$ , Proposition 4.2 implies that with probability  $1 - O(1/n)$
$$
\min  _ {\beta} \lambda_ {F} \widehat {\boldsymbol {h}} _ {\alpha , \beta} [ \alpha ] \geq \min  \left\{\mu^ {- 1}, 1 \right\} \Omega (n ^ {- t}) \sqrt {\log k}. \tag {35}
$$
Then from Proposition 4.3, by union bounding over  $\beta$  we see that with probability  $1 - O(1/n)$ ,
$$
\max_{\beta}\mathsf{CN}_{\alpha ,\beta}\leq \underbrace{\min\left\{\mu^{-1},1\right\} \tilde{O}(n^{\frac{r - t - 1}{2}})}_{\text{favored features}} + \underbrace{\tilde{O}(n^{\frac{1 - t - p}{2}})}_{\text{unfavored features}}.
$$
Let us combine these two bounds. If we compare the survival to the contamination coming from favored features, we obtain
$$
\begin{array}{l} \frac {\min  \left\{\mu^ {- 1} , 1 \right\} n ^ {- t} \sqrt {\log k}}{\min  \left\{\mu^ {- 1} , 1 \right\} \tilde {O} \left(n ^ {\frac {r - t - 1}{2}}\right)} \geq \frac {n ^ {- t - \frac {r - t - 1}{2}}}{\log (n s k)} (36) \\ \geq \frac {n ^ {\frac {1 - r - t}{2}}}{\log (n s k)}, (37) \\ \end{array}
$$
where we have included the explicit poly log factors for precision. Hence, if  $t < 1 - r$ , the numerator grows polynomially and dominates the denominator. Now let's compare the survival to the contamination coming from unfavored features. This yields
$$
\begin{array}{l} \frac {\operatorname* {m i n} \left\{\mu^ {- 1} , 1 \right\} n ^ {- t} \sqrt {\log k}}{\tilde {O} \left(n ^ {\frac {1 - t - p}{2}}\right)} \geq \frac {\operatorname* {m i n} \left\{\mu^ {- 1} , 1 \right\} n ^ {- t - \frac {1 - t - p}{2}}}{\log (n s k)} (38) \\ \geq \frac {n ^ {- \max  \{q + r - 1 , 0 \}} \cdot n ^ {\frac {p - t - 1}{2}}}{\log (n s k)} (39) \\ \geq \frac {n ^ {\frac {p + 1 - 2 \max  \{1 , q + r \} - t}{2}}}{\log (n s k)}. (40) \\ \end{array}
$$
Hence, by union bounding, we see that with probability  $1 - O(1/n)$ ,
$$
\min  _ {\beta} \frac {\lambda_ {F} \widehat {\boldsymbol {h}} _ {\alpha , \beta} [ \alpha ]}{\mathbb {C N} _ {\alpha , \beta}} \geq n ^ {v}, \tag {41}
$$
where  $v \triangleq \frac{1}{4} (\min \{1 - r, p + 1 - 2 \max \{1, q + r\} \} - t) > 0$  by assumption.
For the upper bound, suppose  $t > \min \{1 - r, p + 1 - 2\max \{1, q + r\}\}$ . Hence  $t > 0$ , and by union bounding we conclude that with probability at least  $1 - O(1/n)$ ,
$$
\left. \max  _ {\beta} \lambda_ {F} \left| \widehat {\boldsymbol {h}} _ {\alpha , \beta} [ \alpha ] \right| \leq \min  \left\{\mu^ {- 1}, 1 \right\} O \left(n ^ {- \frac {1}{2}}\right) \sqrt {\log k} \right. \tag {42}
$$
$$
\min  _ {\beta} \mathsf {C N} _ {\alpha , \beta} \geq \min  \left\{\mu^ {- 1}, 1 \right\} \Omega \left(n ^ {\frac {r - t - 1}{2}}\right) + \Omega \left(n ^ {\frac {1 - t - p}{2}}\right). \tag {43}
$$
Combining these and union bounding yields that with probability  $1 - O(1/n)$ ,
$$
\min  _ {\beta} \frac {\lambda_ {F} \widehat {\boldsymbol {h}} _ {\alpha , \beta} [ \alpha ]}{\mathsf {C N} _ {\alpha , \beta}} \leq n ^ {- w}, \tag {44}
$$
where  $w \triangleq \frac{1}{4} (t - \min \{1 - r, p + 1 - 2\max \{1, q + r\}\}) > 0$  by assumption.
We now sketch out a proof of both the positive and negative sides of Theorem 3.2. We point out that the regimes for generalization and misclassification exactly match the regimes above for where the SU/CN ratio grows or shrinks polynomially.
Theorem A.2 (Positive side of Theorem 3.2). Under the bi-level ensemble model (Definition 1), when the true data generating process is 1-sparse (Assumption 1), as  $n \to \infty$ , the probability of misclassification for MNI satisfies  $\operatorname{Pr}[\mathcal{E}_{\mathrm{err}}] \to 0$  if
$$
t <   \min  \left\{1 - r, p + 1 - 2 \max  \left\{1, q + r \right\} \right\}.
$$
Proof sketch. For correct classification, it suffices for the minimum value of the LHS of Eq. (32) to outcompete the maximum value of the RHS, where the max is taken over  $\beta \in [k],\beta \neq \alpha$  . Some algebra, as in Subramanian et al. (2022), shows that we correctly classify if
$$
\begin{array}{l} \underbrace {\frac {\min  _ {\beta} \lambda_ {F} \widehat {\boldsymbol {h}} _ {\alpha , \beta} [ \alpha ]}{\max  _ {\beta} \mathsf {C N} _ {\alpha , \beta}}} _ {\text {S U / C N r a t i o}} \left(\underbrace {\min  _ {\beta} (\boldsymbol {x} _ {\text {t e s t}} [ \alpha ] - \boldsymbol {x} _ {\text {t e s t}} [ \beta ])} _ {\text {c l o s e s t f e a t u r e m a r g i n}} - \underbrace {\max  _ {\beta} | \boldsymbol {x} _ {\text {t e s t}} [ \beta ] |} _ {\text {l a r g e s t c o m p e t e n g f e a t u r e}} \cdot \underbrace {\max  _ {\beta} \left| \frac {\widehat {\boldsymbol {h}} _ {\alpha , \beta} [ \alpha ] - \widehat {\boldsymbol {h}} _ {\beta , \alpha} [ \beta ]}{\widehat {\boldsymbol {h}} _ {\alpha , \beta} [ \alpha ]} \right|} _ {\text {s u r v i v a l v a r i a t i o n}}\right) \\ > \underbrace {\max  _ {\beta} Z ^ {(\beta)}} _ {\text {n o r m a l i z e d c o n t a m i n a t i o n}}. \tag {45} \\ \end{array}
$$
By our lower bound on the survival to contamination ratio (Proposition A.1), assuming  $t < \min \{1 - r, p + 1 - 2(q + r)\}$ , then with probability at least  $1 - O(1/n)$  we have that  $\frac{\lambda_F \widehat{h}_{\alpha,\beta}[\alpha]}{\mathsf{CN}_{\alpha,\beta}} \geq n^u$  for some constant  $u > 0$ . By Lemmas B.2 and B.3 in Subramanian et al. (2022) for every  $\epsilon > 0$  with probability at least  $1 - \epsilon$ , we have  $\min_{\beta} x_{\text{test}}[\alpha] - x_{\text{test}}[\beta] \geq \Omega(\frac{1}{\sqrt{\log k}})$ .
Next, by standard subgaussian maxima tail bounds we have that  $|\pmb{x}_{\mathrm{test}}[\beta] \leq 2\sqrt{\log(nk)}$  and  $Z^{(\beta)} \leq 2\sqrt{\log(nk)}$  with probability at least  $1 - O(1 / nk)$ . Finally, applying our upper bound on the relative survival variance (Proposition A.3, which we prove below), the survival variation is at most a polynomially decaying  $n^{-w}$  with probability at least  $1 - O(1 / nk)$ .
By union bounding, we see that with probability at least  $1 - O(1/n) - \epsilon$ , the LHS outcompetes the RHS, implying that the model correctly classifies.
In fact, given Proposition 4.2, it is straightforward to bound the survival variation.
Proposition A.3 (Upper bound on the survival variation). Suppose that  $t < 1 - r$ . With probability at least  $1 - 2/n$ , we have
$$
\left| \frac {\widehat {\boldsymbol {h}} _ {\alpha , \beta} [ \alpha ] - \widehat {\boldsymbol {h}} _ {\beta , \alpha} [ \beta ]}{\widehat {\boldsymbol {h}} _ {\alpha , \beta} [ \alpha ]} \right| \leq c _ {1} n ^ {- w}, \tag {46}
$$
where  $c_{1}$  and  $w$  are both positive constants.
Proof. Since we have  $\widehat{\pmb{h}}_{\alpha,\beta}[\alpha] = \pmb{z}_{\alpha}^{\top}\pmb{A}^{-1}\pmb{\Delta}y$ , the survival variation is
$$
\frac {\widehat {\pmb {h}} _ {\alpha , \beta} [ \alpha ] - \widehat {\pmb {h}} _ {\beta , \alpha} [ \beta ]}{\widehat {\pmb {h}} _ {\alpha , \beta} [ \alpha ]} = \frac {\pmb {z} _ {\alpha} ^ {\top} \pmb {A} ^ {- 1} \pmb {\Delta y} + \pmb {z} _ {\beta} ^ {\top} \pmb {A} ^ {- 1} \pmb {\Delta y}}{\pmb {z} _ {\alpha} ^ {\top} \pmb {A} ^ {- 1} \pmb {\Delta y}}
$$
Since  $t < 1 - r$  and  $t < r$  by definition, we know that  $t < \frac{1}{2}$ , and we can apply Proposition 4.2 to see that with probability at least  $1 - 2/n$  we have
$$
\pmb {z} _ {\alpha} ^ {\top} \pmb {A} ^ {- 1} \pmb {\Delta} y = \max \left\{\mu^ {- 1}, 1 \right\} n ^ {- t} (1 \pm O (n ^ {- \kappa_ {5}})) \sqrt {\log k} = - \pmb {z} _ {\beta} ^ {\top} \pmb {A} ^ {- 1} \pmb {\Delta} y
$$
$$
\left| \frac {\widehat {\boldsymbol {h}} _ {\alpha , \beta} [ \alpha ] - \widehat {\boldsymbol {h}} _ {\beta , \alpha} [ \beta ]}{\widehat {\boldsymbol {h}} _ {\alpha , \beta} [ \alpha ]} \right| \leq c _ {1} n ^ {- \kappa_ {5}} \tag {47}
$$
where  $c_{1}$  is an appropriately defined positive constant.
Theorem A.4 (Negative side of Theorem 3.2). Under the bi-level ensemble model (Definition 1), when the true data generating process is 1-sparse (Assumption 1), as  $n \to \infty$ , the probability of misclassification for MNI satisfies  $\operatorname*{Pr}[\mathcal{E}_{\mathrm{err}}] \to 1$  if
$$
t > \min  \left\{1 - r, p + 1 - 2 \max  \left\{1, q + r \right\} \right\}.
$$
Proof sketch. On the other hand, for misclassification it suffices for the maximum absolute value of the LHS of Eq. (32) to be outcompeted by the maximum value of the RHS. Some manipulations yield the following sufficient condition for misclassification:
$$
\underbrace {\frac {\operatorname* {m a x} _ {\beta} \lambda_ {F} \left(\left| \widehat {\boldsymbol {h}} _ {\alpha , \beta} [ \alpha ] \right| + \left| \widehat {\boldsymbol {h}} _ {\beta , \alpha} [ \beta ] \right|\right)}{\operatorname* {m i n} _ {\beta} \mathsf {C N} _ {\alpha , \beta}}} _ {\text {S U / C N r a t i o}} \cdot \underbrace {\underset {\gamma \in [ k ]} {\operatorname* {m a x}} | \boldsymbol {x} _ {\text {t e s t}} [ \gamma ] |} _ {\text {l a r g e s t l a b e l - d e f i n i n g f e a t u r e}} <   \underbrace {\underset {\beta} {\operatorname* {m a x}} Z ^ {(\beta)}} _ {\text {n o r m a l i z e d c o n t a m i n a t i o n}}. \tag {48}
$$
Within the misclassification regimes in Conjecture 3.1, Proposition A.1 implies that the survival-to-contamination ratio  $\mathrm{SU} / \mathrm{CN}$  shrinks at a polynomial rate  $n^{-w}$  for some  $w > 0$ . By standard subgaussian maximal inequalities, the largest label-defining feature is  $O(\sqrt{\log(nk)})$  with high probability. Gaussian anticoncentration implies that for some  $\beta \neq \alpha, \beta \in [k]$ ,  $Z^{(\beta)}$  outcompetes the LHS, which is bounded above by  $n^{-w}$ , with probability at least  $\frac{1}{2} - o(1)$ . Hence, we conclude that the model will misclassify with rate at least  $\frac{1}{2}$  asymptotically.
Let us now describe how to boost the misclassification rate to  $1 - o(1)$ . Notice that the above argument only considered the competition between the LHS of Eq. (48) and one of the  $Z^{(\beta)}$ 's on the RHS instead of the maximum  $Z^{(\beta)}$ . It's not hard to see from the definition of  $Z^{(\beta)}$  in Eq. (31) that the  $Z^{(\beta)}$  are jointly Gaussian. For intuition's sake, assuming the  $Z^{(\beta)}$  were independent, then  $\max_{\beta} Z^{(\beta)}$  would outcompete with probability  $(\frac{1}{2} - o(1))^{k-1}$ .
In reality, the  $Z^{(\beta)}$  are correlated, but we are able to show that the maximum correlation between the  $Z^{(\beta)}$  is  $\frac{1}{2} + o(1)$  with high probability. An application of Slepian's lemma (Slepian (1962)) and some explicit bounds on orthant probabilities (Pinasco et al. (2021)) implies that  $\max_{\beta} Z^{(\beta)} > 0$  with probability at least  $1 - \frac{1}{k^{1 + o(1)}}$ . An application of anticoncentration for Gaussian maxima (Chernozhukov et al., 2015) implies that  $\max_{\beta} Z^{(\beta)} > n^{-w}$  with probability  $1 - o(1)$ , which finishes off the proof.
To fill in the details of the above proof sketch, we will prove the following proposition in Appendix F.
Proposition A.5 (Correlation bound). Assume we are in the bi-level ensemble model (Definition 1), the true data generating process is 1-sparse (Assumption 1), and the number of classes scales with  $n$  (i.e.  $t > 0$ ). Then for every  $\epsilon > 0$ , we have
$$
\Pr \left[ \max  _ {\beta \in [ k ], \beta \neq \alpha} Z ^ {(\beta)} > n ^ {- u} \right] \geq 1 - \Theta \left(\frac {1}{k ^ {1 + o (1)}}\right) - \epsilon \tag {49}
$$
for sufficiently large  $n$  and any  $u > 0$ .
B Main tools
In this section we introduce our suite of technical tools that allow us to prove the desired rates of growth for survival, contamination, and correlation.
B.1 Hanson-Wright Inequality
As established in Section 4, we need to use the Hanson-Wright inequality to prove our tight characterization of generalization. For the sake of precision, we explicitly state our definitions of subgaussian and subexponential which we use throughout the rest of the paper.
The subgaussian norm  $\| \xi \|_{\psi_2}$  of a random variable  $\xi$  is defined as in Rudelson and Vershynin (2013),
$$
\| \xi \| _ {\psi_ {2}} = \inf  _ {K > 0} \left\{K: \mathbb {E} \exp \left(\xi^ {2} / K ^ {2}\right) \leq 2 \right\}. \tag {50}
$$
The sub-exponential norm  $\| \xi \|_{\psi_1}$  is defined as in Vershynin (2018, Definition 2.7.5):
$$
\left\| \xi \right\| _ {\psi_ {1}} = \inf  _ {K > 0} \{K: \mathbb {E} \exp (| \xi | / K) \leq 2 \}. \tag {51}
$$
We will occasionally need to use the following variant of Hanson-Wright for nonsparse bilinear forms, first proved in Park et al. (2021).
Theorem B.1 (Hanson-Wright for bilinear forms without sparsity). Let  $\pmb{x} = (X_1, \dots, X_n) \in \mathbb{R}^n$  and  $\pmb{y} \in (Y_1, \dots, Y_n)$  be random vectors such that the pairs  $(X_i, Y_i)$  are all independent of each other (however  $X_i$  and  $Y_i$  can be correlated). Assume also that  $\mathbb{E}[X_i] = \mathbb{E}[Y_i] = 0$  and  $\max \left\{\|X_i\|_{\psi_2}, \|Y_i\|_{\psi_2}\right\} \leq K$ . Then there exists an absolute constant  $c > 0$  such that for all  $M \in \mathbb{R}^{n \times n}$  and  $\epsilon \geq 0$  we have
$$
\Pr \left[ | \boldsymbol {x} ^ {\top} \boldsymbol {M} \boldsymbol {y} - \mathbb {E} [ \boldsymbol {x} ^ {\top} \boldsymbol {M} \boldsymbol {y} ] | > \epsilon \right] \leq 2 \exp \left(- c \min  \left\{\frac {\epsilon^ {2}}{K ^ {4} \| \boldsymbol {M} \| _ {F} ^ {2}}, \frac {\epsilon}{K ^ {2} \| \boldsymbol {M} \| _ {2}} \right\}\right). \tag {52}
$$
Finally, we restate our new version of Hanson-Wright for bilinear forms with soft sparsity, which we prove in Appendix G.
B.2 Gram matrices and the Woodbury formula
In order to apply Hanson-Wright to the bilinear form  $\boldsymbol{x}^{\top}\boldsymbol{M}\boldsymbol{y}$ , we need to have a deterministic matrix  $M$  such that the hypotheses are satisfied. However, in our setting we study bilinear forms such as  $z_{j}^{\top}A^{-1}\Delta y$ . Here, the inverse Gram matrix  $A^{-1}$  is not independent of  $z_{j}$  or  $\Delta y$ , so we cannot simply condition on  $A^{-1}$ . The way around this is to cleverly decompose  $A^{-1}$  using the so-called Woodbury inversion formula (stated formally below), which generalizes the leave-one-out trick and Sherman-Morrison used to study binary classification in Muthukumar et al. (2021). To that end, we will explicitly decompose the Gram matrix  $A\triangleq \sum_{j\in [d]}\lambda_jz_jz_j^\top$  based on whether the features  $z_{j}$  are favored or not.
We now introduce some notation to keep track of which matrices contain or leave out which indices. In general, we use subscripts to denote which sets of features we preserve or leave out; we use a minus sign to signify leaving out. The  $k$  label-defining features are represented with a subscript  $k$ , whereas the  $s - k$  favored but not label defining features are represented with a subscript  $F$ . The rest of the  $d - s$  unfavored features are represented with a subscript  $U$ .
For notational convenience, we introduce some new notation for the weighted features, as the superscript  $w$  to denote weighted features is rather cumbersome. We denote the weighted label-defining feature matrix by  $\mathbf{W}_k \triangleq [w_1 \dots w_k] \in \mathbb{R}^{n \times k}$ , where the vectors  $\mathbf{w}_i \triangleq \sqrt{\lambda_i} \mathbf{z}_i \in \mathbb{R}^n$  denote the weighted observations for feature  $i$ . Define the unweighted label-defining feature matrix  $Z_k \triangleq [z_1 \dots z_k] \in \mathbb{R}^{n \times k}$ . Similarly, define  $W_F \triangleq [w_{k+1} \dots w_s] \in \mathbb{R}^{n \times (s-k)}$ , which contains the rest of the weighted favored features and the corresponding unweighted version  $Z_F$ .