Finite Sample Analysis of Approximate Message Passing Algorithms

I Introduction

Approximate Message Passing (AMP) is a class of low-complexity, scalable algorithms to solve the above problem, under suitable assumptions on $A$ and $\beta_{0}$. AMP algorithms are derived as Gaussian or quadratic approximations of loopy belief propagation algorithms (e.g., min-sum, sum-product) on the dense factor graph corresponding to (1.1).

For a Gaussian measurement matrix $A$ with entries that are i.i.d. $\sim\mathcal{N}(0,1/n)$, it was rigorously proven that the performance of AMP can be characterized in the large system limit via a simple scalar iteration called state evolution. This result was extended to the class of matrices with i.i.d. sub-Gaussian entries in . In particular, these results imply that performance measures such as the $L^{2}$-error $\frac{1}{N}\lVert\beta_{0}-\beta^{t}\rVert^{2}$ and the $L^{1}$-error $\frac{1}{N}\lVert\beta_{0}-\beta^{t}\rVert_{1}$ converge almost surely to constants that can be computed via the distribution of $\beta_{0}$. (The large system limit is defined as $n,N\to\infty$ such that $\frac{n}{N}=\delta$, a constant.)

AMP has also been applied to a variety of other high-dimensional estimation problems. Some examples are low-rank matrix estimation , decoding of sparse superposition codes , matrix factorization , and estimation in generalized linear and bilinear models .

Main Contributions: In this paper, we obtain a non-asymptotic result for the performance of the AMP iteration in (1.2)–(1.3), when the measurement matrix $A$ has i.i.d. Gaussian entries $\sim\mathcal{N}(0,1/n)$. We derive a concentration inequality (Theorem 1) that implies that the probability of $\epsilon$-deviation between various performance measures (such as $\frac{1}{N}\lVert\beta_{0}-\beta^{t}\rVert^{2}$) and their limiting constant values fall exponentially in $n$. Our result provides theoretical support for empirical findings that have demonstrated excellent agreement of AMP performance with state evolution predictions for moderately large dimensions, e.g., $n$ of the order of several hundreds .

In addition to refining earlier asymptotic results, the concentration inequality in Theorem 1 also clarifies the effect of the iteration number $t$ versus the problem dimension $n$. One implication is that the actual AMP performance is close to the state evolution prediction with high probability as long as $t$ is of order smaller than $\frac{\log n}{\log\log n}$. This is particularly relevant for settings where the number of AMP iterations and the problem dimension are both large, e.g., solving the LASSO via AMP .

We prove the concentration result in Theorem 1 by analyzing the following general recursion:

Since the publication of the conference version of this paper, the analysis described here has been used in a couple of recent papers: an error exponent for sparse regression codes with AMP decoding was obtained in , and a non-asymptotic result for AMP with non-separable denoisers was given in .

Before proceeding, we state the assumptions on the model (1.1) and the functions used to define the AMP. In what follows, $K,\kappa>0$ are generic positive constants whose values are not exactly specified but do not depend on $n$. We use the notation $[n]$ to denote the set $\{1,2,\ldots,n\}$.

Functions defined with scalar inputs are assumed to act component-wise when applied to vectors.

The remainder of the paper is organized as follows. In Section II we review state evolution, the formalism predicting the performance of AMP, and discuss how knowledge of the signal distribution $p_{\beta}$ and the noise distribution $p_{w}$ can help choose good denoising functions $\{\eta_{t}\}$. However, we emphasize that our result holds for the AMP with any choice of $\{\eta_{t}\}$ satisfying the above condition, even those that do not depend on $p_{\beta}$ and $p_{w}$. In Section II-A, we introduce a stopping criterion for termination of the AMP. In Section III, we give our main result (Theorem 1) which proves that the performance of AMP can be characterized accurately via state evolution for large but finite sample size $n$. Section IV gives the proof of Theorem 1. The proof is based on two technical lemmas: Lemmas 3 and 5. The proof of Lemma 5 is long; we therefore give a brief summary of the main ideas in Section IV-F and then the full proof in Section V. In the appendices, we list a number of concentration inequalities that are used in the proof of Lemma 5. Some of these, such as the concentration inequality for the sum of pseudo-Lipschitz functions of i.i.d. sub-Gaussian random variables (Lemma B.4), may be of independent interest.

In this section, we briefly describe state evolution, the formalism that predicts the behavior of AMP in the large system limit. We only review the main points followed by a few examples; a more detailed treatment can be found in .

where $\beta\sim p_{\beta}$ and $Z\sim\mathcal{N}(0,1)$ are independent random variables.

The AMP update (1.3) is underpinned by the following key property of the vector $A^{*}z^{t}+\beta^{t}$: for large $n$, $A^{*}z^{t}+\beta^{t}$ is approximately distributed as $\beta_{0}+\tau_{t}Z$, where $Z$ is an i.i.d. $\mathcal{N}(0,1)$ random vector independent of $\beta_{0}$. In light of this property, a natural way to generate $\beta^{t+1}$ from the “effective observation” $A^{*}z^{t}+\beta^{t}=s$ is via the conditional expectation:

i.e., $\beta^{t+1}$ is the MMSE estimate of $\beta_{0}$ given the noisy observation $\beta_{0}+\tau_{t}Z$. Thus if $p_{\beta}$ is known, the Bayes optimal choice for $\eta_{t}(s)$ is the conditional expectation in (2.2).

In the definition of the “modified residual” $z^{t}$, the third term on the RHS of (1.2) is crucial to ensure that the effective observation $A^{*}z^{t}+\beta^{t}$ has the above distributional property. For intuition about the role of this ‘Onsager term’, the reader is referred to [1, Section I-C].

We review two examples to illustrate how full or partial knowledge of $p_{\beta}$ can guide the choice of the denoising function $\eta_{t}$. In the first example, suppose we know that each element of $\beta_{0}$ is chosen uniformly at random from the set $\{+1,-1\}$. Computing the conditional expectation in (2.2) with this $p_{\beta}$, we obtain $\eta_{t}(s)=\tanh(s/\tau_{t}^{2})$ . The constants $\tau^{2}_{t}$ are determined iteratively from the state evolution equations (2.1).

As a second example, consider the compressed sensing problem, where $\delta<1$, and $p_{\beta}$ is such that $P(\beta_{0}=0)=1-\xi$. The parameter $\xi\in(0,1)$ determines the sparsity of $\beta_{0}$. For this problem, the authors in suggested the choice $\eta_{t}(s)=\eta(s;\theta_{t})$, where the soft-thresholding function $\eta$ is defined as

The threshold $\theta_{t}$ at step $t$ is set to $\theta_{t}=\alpha\tau_{t}$, where $\alpha$ is a tunable constant and $\tau_{t}$ is determined by (2.1), making the threshold value proportional to the standard deviation of the noise in the effective observation. However, computing $\tau_{t}$ using (2.1) requires knowledge of $p_{\beta}$. In the absence of such knowledge, we can estimate $\tau_{t}^{2}$ by $\frac{1}{n}\lVert z^{t}\rVert^{2}$: our concentration result (Lemma 5(e)) shows that this approximation is increasingly accurate as $n$ grows large. To fix $\alpha$, one could run the AMP with several different values of $\alpha$, and choose the one that gives the smallest value of $\frac{1}{n}\lVert z^{t}\rVert^{2}$ for large $t$.

We note that in each of the above examples $\eta_{t}$ is Lipschitz, and its derivative satisfies the assumption stated in Section I-A.

To obtain a concentration result that clearly highlights the dependence on the iteration $t$ and the dimension $n$, we include a stopping criterion for the AMP algorithm. The intuition is that the AMP algorithm can be terminated once the expected squared error of the estimates (as predicted by state evolution equations in (2.1)) is either very small or stops improving appreciably.

For Bayes-optimal AMP where the denoising function $\eta_{t}(\cdot)$ is the conditional expectation given in (2.2), the stopping criterion is as follows. Terminate the algorithm at the first iteration $t>0$ for which either

where $\varepsilon_{0}>0$ and $\varepsilon^{\prime}_{0}\in(0,1)$ are pre-specified constants. Recall from (2.1) that $\sigma^{2}_{t}$ is expected squared error in the estimate. Therefore, for suitably chosen values of $\varepsilon_{0},\varepsilon_{0}^{\prime}$, the AMP will terminate when the expected squared error is either small enough, or has not significantly decreased from the previous iteration.

For the general case where $\eta_{t}(\cdot)$ is not the Bayes-optimal choice, the stopping criterion is: terminate the algorithm at the first iteration $t>0$ for which at least one of the following is true:

where $\varepsilon_{1},\varepsilon_{2},\varepsilon_{3}>0$ are pre-specified constants, and $(\sigma^{\perp}_{t})^{2},(\tau^{\perp}_{t})^{2}$ are defined in (4.19). The precise definitions of the scalars $(\sigma^{\perp}_{t})^{2},(\tau^{\perp}_{t})^{2}$ are postponed to Sec. IV-B as a few other definitions are needed first. For now, it suffices to note that $(\sigma^{\perp}_{t})^{2},(\tau^{\perp}_{t})^{2}$ are measures of how close $\sigma_{t}^{2}$ and $\tau_{t}^{2}$ are to $\sigma_{t-1}^{2}$ and $\tau_{t-1}^{2}$, respectively. Indeed, for the Bayes-optimal case, we show in Sec IV-C that

Let $T^{*}>0$ be the first value of $t>0$ for which at least one of the conditions is met. Then the algorithm is run only for $0\leq t<T^{*}$. It follows that for $0\leq t<T^{*}$,

In the rest of the paper, we will use the stopping criterion to implicitly assume that $\sigma_{t}^{2},\tau_{t}^{2},(\sigma^{\perp}_{t})^{2},(\tau^{\perp}_{t})^{2}$ are bounded below by positive constants.

III Main Result

In the expectation in (3.1), $\beta\sim p_{\beta}$ and $Z\sim\mathcal{N}(0,1)$ are independent, and $\tau_{t}$ is given by (2.1). The constants $K_{t},\kappa_{t}$ are given by $K_{t}=C^{2t}(t!)^{10},\kappa_{t}=\frac{1}{c^{2t}(t!)^{22}}$, where $C,c>0$ are universal constants (not depending on $t$, $n$, or $\epsilon$) that are not explicitly specified.

The probability in (3.1) is with respect to the product measure on the space of the measurement matrix $A$, signal $\beta_{0}$, and the noise $w$.

1. By considering the pseudo-Lipschitz function $\phi(a,b)=(a-b)^{2}$, Theorem 1 proves that state evolution tracks the mean square error of the AMP estimates with exponentially small probability of error in the sample size $n$. Indeed, for all $t\geq 0$,

2. Asymptotic convergence results of the kind given in are implied by Theorem 1. Indeed, from Theorem 1, the sum

is finite for any fixed $t\geq 0$. Therefore the Borel-Cantelli lemma implies that for any fixed $t\geq 0$:

3. Theorem 1 also refines the asymptotic convergence result by specifying how large $t$ can be (compared to the dimension $n$) for the state evolution predictions to be meaningful. Indeed, if we require the bound in (3.1) to go to zero with growing $n$, we need $\kappa_{t}n\epsilon^{2}\to\infty$ as $n\to\infty$. Using the expression for $\kappa_{t}$ from the theorem then yields $t=o\left(\frac{\log n}{\log\log n}\right)$.

Thus, when the AMP is run for a growing number of iterations, the state evolution predictions are guaranteed to be valid until iteration $t$ if the problem dimension grows faster than exponentially in $t$. Though the constants $K_{t},\kappa_{t}$ in the bound have not been optimized, we believe that the dependence of these constants on $t!$ is inevitable in any induction-based proof of the result. An open question is whether this relationship between $t$ and $n$ is fundamental, or a different analysis of the AMP can yield constants which allow $t$ to grow faster with $n$.

4. As mentioned in the introduction, we expect that non-asymptotic results similar to Theorem 1 can be obtained for other estimation problems (with Gaussian matrices) for which rigorous asymptotic results have been proven for AMP. Examples of such problems include low-rank matrix estimation , robust high-dimensional M-estimation , AMP with spatially coupled matrices , and generalized AMP .

As our proof technique depends heavily on $A$ being i.i.d. Gaussian, extending Theorem 1 to AMP with sub-Gaussian matrices and to variants of AMP with structured measurement matrices (e.g., ) is non-trivial, and an interesting direction for future work.

IV Proof of Theorem 1

We first lay down the notation that will be used in the proof, then state two technical lemmas (Lemmas 3 and 5) and use them to prove Theorem 1.

where the scalars $\xi_{t}$ and $\lambda_{t}$ are defined as

Define the state evolution scalars $\{\tau_{t}^{2}\}_{t\geq 0}$ and $\{\sigma_{t}^{2}\}_{t\geq 1}$ for the general recursion as follows.

where $\beta\sim p_{\beta},W\sim p_{w}$, and $Z\sim\mathcal{N}(0,1)$ are independent random variables. We assume that both $\sigma_{0}^{2}$ and $\tau_{0}^{2}$ are strictly positive.

The AMP algorithm is a special case of the general recursion in (4.1) and (4.2). Indeed, the AMP can be recovered by defining the following vectors recursively for $t\geq 0$, starting with $\beta^{0}=0$ and $z^{0}=y$.

It can be verified that these vectors satisfy (4.1) and (4.2) with

Using this choice of $f_{t},g_{t}$ in (4.4) yields the expressions for $\sigma_{t}^{2},\tau_{t}^{2}$ given in (2.1). Using (4.6) in (4.2), we also see that for AMP,

For the analysis, we work with the general recursion given by (4.1) and (4.2). Notice from (4.1) that for all $t$,

Thus we have the matrix equations $X_{t}=A^{*}M_{t}$ and $Y_{t}=AQ_{t},$ where

The notation $[c_{1}\mid c_{2}\mid\ldots\mid c_{k}]$ is used to denote a matrix with columns $c_{1},\ldots,c_{k}$. Note that $M_{0}$ and $Q_{0}$ are the all-zero vector. Additionally define the matrices

Note that $B_{0}$, $H_{0}$, $\Lambda_{0}$, and $\Xi_{0}$ are all-zero vectors. Using the above we see that $Y_{t}=B_{t}+[0|M_{t-1}]\Lambda_{t}$ and $X_{t}=H_{t}+Q_{t}\Xi_{t}.$

We use the notation $m^{t}_{\|}$ and $q^{t}_{\|}$ to denote the projection of $m^{t}$ and $q^{t}$ onto the column space of $M_{t}$ and $Q_{t}$, respectively. Let

be the coefficient vectors of these projections, i.e.,

The projections of $m^{t}$ and $q^{t}$ onto the orthogonal complements of $M_{t}$ and $Q_{t}$, respectively, are denoted by

Lemma 5 shows that for large $n$, the entries of $\alpha^{t}$ and ${\gamma}^{t}$ are concentrated around constants. We now specify these constants and provide some intuition about their values in the special case where the denoising function in the AMP recursion is the Bayes-optimal choice, as in (2.2).

IV-B Concentrating Values

Let $(\sigma^{\perp}_{0})^{2}:=\sigma_{0}^{2}$ and $(\tau^{\perp}_{0})^{2}:=\tau_{0}^{2}$, and for $t>0$ define

Finally, we define the concentrating values for $\lambda_{t}$ and $\xi_{t}$ as

Since $\{f_{t}\}_{t\geq 0}$ and $\{g_{t}\}_{t\geq 0}$ are assumed to be Lipschitz continuous, the derivatives $\{f^{\prime}_{t}\}$ and $\{g^{\prime}_{t}\}$ are bounded for $t\geq 0$. Therefore $\lambda_{t},\xi_{t}$ defined in (4.2) and $\hat{\lambda}_{t},\hat{\xi}_{t}$ defined in (4.20) are also bounded. For the AMP recursion, it follows from (4.6) that

IV-C Bayes-optimal AMP

The concentrating constants in (4.14)–(4.19) have simple representations in the special case where the denoising function $\eta_{t}(\cdot)$ is chosen to be Bayes-optimal, i.e., the conditional expectation of $\beta$ given the noisy observation $\beta+\tau_{t}Z$, as in (2.2). In this case:

From the orthogonality principle, it also follows that for $0\leq r\leq t$,

For the AMP, $m^{t}=-z^{t}$ is the modified residual in iteration $t$, and $q^{t}=\beta^{t}-\beta$ is the error in the estimate $\beta^{t}$. Also recall that $\gamma^{t}$ and $\alpha^{t}$ are the coefficients of the projection of $m^{t}$ and $q^{t}$ onto $\{m^{0},\ldots,m^{t-1}\}$ and $\{q^{0},\ldots,q^{t-1}\}$, respectively. The fact that only the last entry of $\hat{\gamma}^{t}$ is non-zero in the Bayes-optimal case indicates that residual $z^{t}$ can be well approximated as a linear combination of $z^{t-1}$ and a vector that is independent of $\{z^{0},\ldots,z^{t-1}\}$; a similar interpretation holds for the error $q^{t}=\beta^{t}-\beta$.

IV-D Conditional Distribution Lemma

We next characterize the conditional distribution of the vectors $h^{t+1}$ and $b^{t}$ given the matrices in (4.9) as well as $\beta_{0},w$. Lemmas 3 and 4 show that the conditional distributions of $h^{t+1}$ and $b^{t}$ can each be expressed in terms of a standard normal vector and a deviation vector. Lemma 5 shows that the norms of the deviation vectors are small with high probability, and provides concentration inequalities for various inner products and functions involving $\{h^{t+1},q^{t},b^{t},m^{t}\}$.

Define $\mathscr{S}_{t_{1},t_{2}}$ to be the sigma-algebra generated by

A key ingredient in the proof is the distribution of $A$ conditioned on the sigma algebra $\mathscr{S}_{t_{1},t}$ where $t_{1}$ is either $t+1$ or $t$ from which we are able to specify the conditional distributions of $b^{t}$ and $h^{t+1}$ given $\mathscr{S}_{t,t}$ and $\mathscr{S}_{t+1,t}$, respectively. Observing that conditioning on $\mathscr{S}_{t_{1},t}$ is equivalent to conditioning on the linear constraintsWhile conditioning on the linear constraints, we emphasize that only $A$ is treated as random.

the following lemma from specifies the conditional distribution of $A|_{\mathscr{S}_{t_{1},t}}$.

[1, Lemma $10$, Lemma $12$] The conditional distributions of the vectors in (4.8) satisfy the following, provided $n>t$ and $M_{t},Q_{t}$ have full column rank.

For the vectors $h^{t+1}$ and $b^{t}$ defined in (4.1), the following hold for $t\geq 1$, provided $n>t$ and $M_{t},Q_{t}$ have full column rank.

and for $t>0$, defining $\mathbf{Q}_{t}:=Q_{t}^{*}Q_{t}$ and $\mathbf{M}_{t}:=M_{t}^{*}M_{t}$,

We begin by demonstrating (4.24). By (4.1) it follows that

For the case $t\geq 1$, we use Lemma 2 to write

All the quantities in the RHS of (4.30) except $Z^{\prime}_{t}$ are in the conditioning sigma-field. We can rewrite (4.30) with the following pair of values:

The above definition of $\Delta_{t,t}$ equals that given in (4.28) since

This completes the proof of (4.24). Result (4.25) can be shown similarly. ∎

The conditional distribution representation in Lemma 3 implies that for each $t\geq 0$, $h^{t+1}$ is the sum of an i.i.d. $\mathcal{N}(0,\tau_{t}^{2})$ random vector plus a deviation term. Similarly $b^{t}$ is the sum of an i.i.d. $\mathcal{N}(0,\sigma_{t}^{2})$ random vector and a deviation term. This is made precise in the following lemma.

Then for $t\geq 0$, the following statements hold.

where the constants $\{\mathsf{c}^{t}_{i}\}_{0\leq i\leq t}$ and $\{\mathsf{d}^{t}_{i}\}_{0\leq i\leq t}$ are recursively defined as follows, starting with $\mathsf{c}^{0}_{0}=1$ and $\mathsf{d}^{0}_{0}=1$. For $t>0$,

The conditional distributions in Lemma 3 can be expressed as

where the last equality follows from rewriting the double sum as follows using the definitions in Section IV-A:

Next we show the expression for $b_{\rm pure}^{t}$ in (4.33) using induction; the proof for $h_{\rm pure}^{t}$ is similar. The base case of $t=0$ holds by definition because $\sigma_{1}^{\perp}=\sigma_{1}$. Using the induction hypothesis that (4.33) holds for $b_{\rm pure}^{0},\ldots,b_{\rm pure}^{t-1}$, the defintion (4.31) can be written as

where the last inequality follows from the definition of $c^{t}_{i}$ for $0\leq i\leq t$ in (4.35). This proves (4.33).

The expressions for the conditional distribution of $b^{t}$ and $h^{t+1}$ in (4.36) can be similarly obtained from (4.24) and (4.25) using an induction argument. ∎

IV-E Main Concentration Lemma

where $C,c>0$ are universal constants (not depending on $t$, $n$, or $\epsilon$). To keep the notation compact, we use $K,\kappa,\kappa^{\prime}$ to denote generic positive universal constants whose values may change through the lemma statement and the proof.

The following statements hold for $1\leq t<T^{*}$ and $\epsilon\in(0,1)$.

The random variables $\breve{Z}_{0},\ldots,\breve{Z}_{t}$ are jointly Gaussian with zero mean and covariance given by (4.14), and are independent of $W\sim p_{w}$.

As above, $\breve{Z}_{t}\sim\mathcal{N}(0,1)$ and $W\sim p_{w}$ are independent.

Let $\mathbf{Q}_{t+1}:=\frac{1}{n}Q_{t+1}^{*}Q_{t+1}$ and $\mathbf{M}_{t}:=\frac{1}{n}M_{t}^{*}M_{t}$. Then,

When the inverses of $\mathbf{Q}_{t+1},\mathbf{M}_{t}$ exist, for $1\leq i,j\leq t+1$,

where $\hat{\gamma}^{t+1}$ and $\hat{\alpha}^{t}$ are defined in (4.17).

With $\sigma_{t+1}^{\perp},\tau_{t}^{\perp}$ defined in (4.19),

IV-F Remarks on Lemma 5

The proof of Theorem 1 below only requires the concentration result in part $(b)$.(i) of Lemma 5, but the proof of part $(b)$.(i) hinges on the other parts of the lemma. The proof of Lemma 5, given in Section V, uses induction starting at time $t=0$, sequentially proving the concentration results in parts $(a)-(h)$. The proof is long, but is based on a sequence of a few key steps which we summarize here.

The concentration constants $\kappa_{t},K_{t}$: The concentration results in Lemma 5 and Theorem 1 for AMP iteration $t\geq 1$ are of the form $K_{t}e^{-\kappa_{t}n\epsilon^{2}}$, where $\kappa_{t},K_{t}$ are given in (4.38). Due to the inductive nature of the proof, the concentration results for step $t$ depend on those corresponding to all the previous steps — this determines how $\kappa_{t},K_{t}$ scale with $t$.

The $t!$ terms in $\kappa_{t},K_{t}$ can be understood as follows. Suppose that we want prove a concentration result for a quantity that can be expressed as a sum of $t$ terms with step indices $1,\ldots,t$. (A typical example is $\Delta_{t+1,t}$ in (3).) For such a term, the deviation from the deterministic concentrating value is less than $\epsilon$ if the deviation in each of the terms in the sum is less than $\epsilon/t$. The induction hypothesis (for steps $1,\ldots,t$) is then used to bound the $\epsilon/t$-deviation probability for each term in the sum. This introduces factors of $1/t$ and $t$ multiplying the exponent and pre-factor, respectively, in each step $t$ (see Lemma A.2), which results in the $t!$ terms in $K_{t}$ and $\kappa_{t}$.

The $(C_{2})^{t}$ and $(c_{2})^{t}$ terms in $\kappa_{t},K_{t}$ arise due to quantities that can be expressed as the product of two terms, for each of which we have a concentration result available (due to the induction hypothesis). This can be used to bound the $\epsilon$-deviation probability of the product, but with a smaller exponent and a larger prefactor (see Lemma A.3). Since this occurs in each step of the induction, the constants $K_{t},\kappa_{t}$ have terms of the form $(C_{2})^{t},(c_{2})^{t}$, respectively.

Comparison with earlier work: Lemmas 3 and 5 are similar to the main technical lemma in [1, Lemma $1$], in that they both analyze the behavior of similar functions and inner products arising in the AMP. The key difference is that Lemma 5 replaces the asymptotic convergence statements in with concentration inequalities. Other differences from [1, Lemma 1] include:

Lemma 5 gives explicit values for the deterministic limits in parts $(c)$–$(h)$, which are needed in other parts of our proof.

Lemma 3 characterizes the the conditional distribution of the vectors $h^{t+1}$ and $b^{t}$ as the sum of an ideal distribution and a deviation term. [1, Lemma $1$(a)] is a similar distributional characterization of $h^{t+1}$ and $b^{t}$, however it does not use the ideal distribution. We found that working with the ideal distribution throughout Lemma 5 simplified our proof.

IV-G Proof of Theorem 1

Applying Part $(b)$.(i) of Lemma 5 to a pseudo-Lipschitz function of the form $\phi_{h}(h^{t+1},\beta_{0})$, for $0\leq t\leq T^{*}$ we have

where the random variables $Z\sim{N}(0,1)$ and $\beta\sim p_{\beta}$ are independent. (Though Lemma 5 is stated for $1\leq t\leq T^{*}$, one can see that (4.59) holds for $t=0$ by considering the pseudo-Lipschitz (PL) function $\phi_{h}(h^{1},\beta_{0})$.) Now let $\phi_{h}(h^{t+1}_{i},\beta_{0_{i}}):=\phi(\eta_{t}(\beta_{0_{i}}-h^{t+1}_{i}),\beta_{0_{i}}),$ where $\phi$ is the PL function in the statement of the theorem. The function $\phi_{h}(h^{t+1}_{i},\beta_{0_{i}})$ is PL since $\phi$ is PL and $\eta_{t}$ is Lipschitz. We therefore obtain

The proof is completed by noting from (1.3) and (4.5) that $\beta^{t+1}=\eta_{t}(A^{*}z^{t}+\beta^{t})=\eta_{t}(\beta_{0}-h^{t+1})$. ∎

V Proof of Lemma 5

Some of the results below can be found in [1, Section III.G], but we summarize them here for completeness.

We also use several concentration results listed in Appendices A and B, with proofs provided for the results that are non-standard. Some of these may be of independent interest, e.g., concentration of sums of a pseudo-Lipschitz function of sub-Gaussians (Lemma B.4).

The proof of Lemma 5. proceeds by induction on $t$. We label as $\mathcal{H}_{t+1}$ the results (4.39), (4.41), (4.42), (4.45), (4.47), (4.49), (4.51), (4.53), (4.55), (4.57) and similarly as $\mathcal{B}_{t}$ the results (4.40), (4.43), (4.44), (4.46), (4.48), (4.50), (4.52), (4.54), (4.56), (4.58). The proof consists of showing four steps:

If $\mathcal{B}_{r},\mathcal{H}_{s}$ holds for all $r<t$ and $s\leq t$, then $\mathcal{B}_{t}$ holds.

if $\mathcal{B}_{r},\mathcal{H}_{s}$ holds for all $r\leq t$ and $s\leq t$, then $\mathcal{H}_{t+1}$ holds.

For the proofs of parts $(b)$.(ii) and $(b)$.(iv), for brevity we assume that the functions $\psi_{h}$ and $\psi_{b}$ are differentiable everywhere. The case where they are not differentiable at a finite number of points involves additional technical details; see Appendix D.

We wish to show results (a)-(h) in (4.40), (4.43), (4.44), (4.46), (4.48), (4.50), (4.52), (4.54), (4.56), (4.58).

Step (a) is obtained using the definition of $\Delta_{0,0}$ in (4.26), and then applying Lemma A.3. For step (b), we use (4.3), Lemma A.4, and Lemma B.2.

(b).(iii) For $t=0$, the LHS of (4.43) can be bounded as

Step (a) uses the conditional distribution of $b^{0}$ given in (4.24), and step (b) follows from Lemma A.2. Label the terms on the RHS of (5.1) as $T_{1}$ and $T_{2}$. Term $T_{1}$ can be upper bounded by $Ke^{-\kappa n\epsilon^{2}}$ using Lemma B.4. We now show a similar upper bound for term $T_{2}$.

where inequality (a) holds because $\phi_{b}$ is pseudo-Lipschitz with constant $L>0$. Inequality (b) follows from Cauchy-Schwarz (with $\mathbf{1}$ denoting the all-ones vector). Inequality $(c)$ is obtained by applying Lemma C.3. From (5.2), we have

where to obtain $(a)$, we use assumption (1.6), Lemma B.2, and $\mathcal{B}_{0}(a)$ proved above.

(b).(iv) For $t=0$, the probability in (4.44) can be bounded as

Step $(a)$ uses the conditional distribution of $b^{0}$ given in (4.24), and step (b) follows from Lemma A.2. Label the two terms on the RHS of (5.4) as $T_{1}$ and $T_{2}$, respectively. We now show that each term is bounded by $Ke^{-\kappa n\epsilon^{2}}$. Since $\lvert\psi_{b}\rvert$ is bounded (say it takes values in an interval of length $B$), the term $T_{2}$ can be bounded using Hoeffding’s inequality (Lemma A.1) by $2e^{-n{\epsilon^{2}}/(2B^{2})}$.

Next, consider $T_{1}$. Let $\Pi_{0}$ be the event under consideration, so that $T_{1}=P(\Pi_{0})$, and define an event $\mathcal{F}$ as follows.

where $\epsilon_{0}>0$ will be specified later. With this definition,

The final inequality in (5.6) follows from the concentration of $\lVert q^{0}\rVert$ in (4.3). To bound the last term $P(\Pi_{0}|\mathcal{F}^{c})$, we write it as

where $\mathsf{I}\{\cdot\}$ denotes the indicator function, and $P\left(\Pi_{0}|\mathcal{F}^{c},\mathscr{S}_{0,0}\right)$ equals

To obtain (5.8), we use the fact that $\sigma_{0}Z^{\prime}_{0_{i}}+[\Delta_{0,0}]_{i}=\frac{1}{\sqrt{n}}\lVert q^{0}\rVert Z^{\prime}_{0_{i}}$ which follows from the definition of $\Delta_{0,0}$ in Lemma 3. Recall from Section IV-D that $\mathscr{S}_{0,0}$ is the sigma-algebra generated by $\{w,\beta_{0},q^{0}\}$; so in (5.8), only $Z^{\prime}_{0}$ is random — all other terms are in $\mathscr{S}_{0,0}$. We now derive a bound for the upper tail of the probability in (5.8); the lower tail bound is similarly obtained. From here on, we suppress the conditioning on $\mathcal{F}^{c},\mathscr{S}_{0,0}$ for brevity.

Define the shorthand $\textsf{diff}(Z^{\prime}_{0_{i}}):=\psi_{b}(\frac{1}{\sqrt{n}}\lVert q^{0}\rVert Z^{\prime}_{0_{i}},w_{i})-\psi_{b}(\sigma_{0}Z^{\prime}_{0_{i}},w_{i})$. Since $\psi_{b}$ is bounded, so is $\textsf{diff}(Z^{\prime}_{0_{i}})$. Let $\lvert\psi_{b}\rvert\leq B/2$, so that $\lvert\textsf{diff}(Z^{\prime}_{0_{i}})\rvert\leq B$ for all $i$. Then the upper tail of the probability in (5.8) can be written as

The above is bounded by $\frac{1}{4}\epsilon$ if we choose $\epsilon_{0}\leq\epsilon/8C$. In the chain above, $(a)$ follows by Fact 4 for a suitable constant $C>0$ as $\psi_{b}$ is bounded and assumed to be differentiable. Step $(b)$ follows since $\lvert\frac{1}{\sqrt{n}}\lVert q^{0}\rVert-\sigma_{0}\rvert\leq\epsilon_{0}$ under $\mathcal{F}^{c}$.

The probability in (5.9) can then be bounded using Hoeffding’s inequality (Lemma A.1):

Substituting in (5.8) and using a similar bound for the lower tail, we have shown via (5.7) that $P(\Pi_{0}\mid\mathcal{F}^{c})\leq 2e^{-n\epsilon^{2}/(8B^{2})}$. Using this in (5.6) with $\epsilon_{0}\leq\epsilon/8C$ proves that the first term in (5.4) is bounded by $Ke^{-n\kappa\epsilon^{2}}$.

(c) The function $\phi_{b}(b^{0}_{i},w_{i}):=b^{0}_{i}w_{i}\in PL(2)$ by Lemma C.1. By $\mathcal{B}_{0}(b).\text{(iii)}$,

(d) The function $\phi_{b}(b^{0}_{i},w_{i}):=(b^{0}_{i})^{2}\in PL(2)$ by Lemma C.1. By $\mathcal{B}_{0}(b).\text{(iii)}$,

(e) Since $g_{0}$ is Lipschitz, the function $\phi_{b}(b^{0}_{i},w_{i}):=(g_{0}(b^{0}_{i},w_{i}))^{2}\in PL(2)$ by Lemma C.1. By $\mathcal{B}_{0}(b).\text{(iii)}$,

(f) The concentration of $\xi_{0}$ around $\hat{\xi}_{0}$ follows from $\mathcal{B}_{0}(b).$(iv) applied to the function $\psi_{b}(b^{0}_{i},w_{i}):=g_{0}^{\prime}(b^{0}_{i},w_{i})$. Next, the function $\phi_{b}(b^{0}_{i},w_{i}):=b^{0}_{i}\,g_{0}(b^{0}_{i},w_{i})\in PL(2)$ by Lemma C.1. Then by $\mathcal{B}_{0}(b).\text{(iii)}$,

(h) The result is equivalent to $\mathcal{B}_{0}(e)$ since $\lVert m^{0}_{\perp}\rVert=\lVert m^{0}\rVert$ and $(\tau_{0}^{\perp})^{2}=\tau_{0}^{2}$.

We wish to show results (a)–(h) in (4.39), (4.41), (4.42), (4.45), (4.47), (4.49), (4.51), (4.53), (4.55), (4.57).

(a) From the definition of $\Delta_{1,0}$ in (4.27) of Lemma 3, we have

Step (a) follows from Lemma C.3 applied to $\Delta_{1,0}$ in (5.10) and Lemma A.2. Label the terms on the RHS of (5.11) as $T_{1}-T_{3}$. To complete the proof, we show that each term is bounded by $Ke^{-\kappa n\epsilon}$ for generic positive constants $K,\kappa$ that do not depend on $n,\epsilon$.

Indeed, $T_{1}\leq Ke^{-\kappa n\epsilon}$ using Lemma A.3, Lemma A.4, result $\mathcal{B}_{0}(e)$, and Lemma B.2. Similarly, $T_{2}\leq Ke^{-\kappa n\epsilon}$ using Lemma A.3, Lemma A.4, result $\mathcal{B}_{0}(e)$, and Lemma B.1. Finally,

Step (a) follows from Lemma A.2, and step (b) from Lemma A.3, $\mathcal{B}_{0}(f)$, the concentration of $\lVert q^{0}\rVert$ given in (4.3), and Lemma A.6.

(b)(i) The proof of (4.41) is similar to analogous $\mathcal{B}_{0}(b)$(iii) result (4.43).

Step $(a)$ follows from the conditional distribution of $h^{1}$ stated in (4.25) and step $(b)$ from Lemma A.2. Label the two terms on the RHS as $T_{1}$ and $T_{2}$. Term $T_{2}$ is upper bounded by $Ke^{-\kappa n\epsilon^{2}}$ by Hoeffding’s inequality (Lemma A.1). To complete the proof, we show that $T_{1}$ has the same bound.

Consider the first term in (5.12). From the definition of $\Delta_{1,0}$ in Lemma 3,

. For $\epsilon_{0}>0$ to be specified later, define event $\mathcal{F}$ as

Denoting the event we are considering in $T_{1}$ by $\Pi_{1}$, so that $T_{1}=P(\Pi_{1})$, we write

Note that in (5.16), only $Z_{0}$ is random as the other terms are all in $\mathscr{S}_{1,0}$. Label the two terms on the RHS of (5.16) as $T_{1,a}$ and $T_{1,b}$. To complete the proof we show that both are bounded by $Ke^{-\kappa n\epsilon^{2}}$.

Step $(a)$ holds by Fact 4 for a suitable constant $C>0$. Step $(b)$ follows because we are conditioning on $\mathcal{F}^{c}$ defined in (5.14). Step $(c)$ is obtained by writing out the expression for the vector $\mathsf{P}^{\parallel}_{q^{0}}Z_{0}$:

Considering $T_{1,b}$, the second term of (5.16), and noting that all quantities except $Z_{0}$ are in $\mathscr{S}_{1,0}$, define the shorthand $\textsf{diff}(Z_{0_{i}}):=\psi_{h}(\frac{1}{\sqrt{n}}\lVert m^{0}\rVert Z_{0_{i}}+u_{i},\beta_{0_{i}})-\psi_{h}(\tau_{0}Z_{0_{i}},\beta_{0,i})$. Then the upper tail of $T_{1,b}$ can be written as

(c),(d),(e),(f) These results can be proved by appealing to $\mathcal{H}_{1}(b)$ in a manner similar to $\mathcal{B}_{0}(c)(d)(e)(f)$.

(h) From the definitions in Section IV-A, we have $\lVert q^{1}_{\perp}\rVert^{2}=\lVert q^{1}\rVert^{2}-\lVert q^{1}_{\parallel}\rVert^{2}=\lVert q^{1}\rVert^{2}-(\gamma_{0}^{1})^{2}\lVert q^{0}\rVert^{2}$, and $(\sigma_{1}^{\perp})^{2}=\sigma_{1}^{2}-(\hat{\gamma}^{1}_{0})^{2}\sigma_{0}^{2}$. We therefore have

In the chain above, $(a)$ uses Lemma A.2 and $(b)$ is obtained using $\mathcal{H}_{1}(e)$ for bounding the first term and by applying Lemma A.3 to the second term along with the concentration of $\lVert q^{0}\rVert$ in (4.3), $\mathcal{H}_{1}(g)$, and Lemma A.5 (for concentration of the square).

We prove the statements in $\mathcal{B}_{t}$ assuming that $\mathcal{B}_{0},\ldots,\mathcal{B}_{t-1}$, and $\mathcal{H}_{1},\ldots,\mathcal{H}_{t}$ hold due to the induction hypothesis. The induction hypothesis implies that for $0\leq r\leq(t-1)$, the deviation probabilities $P(\frac{1}{n}\|\Delta_{r,r}\|^{2}\geq\epsilon)$ in (4.40) and $P(\frac{1}{n}\|\Delta_{r+1,r}\|^{2}\geq\epsilon)$ in (4.39) are each bounded by $K_{r}e^{-\kappa_{r}n\epsilon}$. Similarly, the LHS in each of (4.41) – (4.58) is bounded by $K_{r}e^{-\kappa_{r}n\epsilon^{2}}$.

We begin with a lemma that is required to prove $\mathcal{B}_{t}(a)$. The lemma as well as other parts of $\mathcal{B}_{t}$ assume the invertibility of $\mathbf{M}_{1},\ldots,\mathbf{M}_{t}$, but for the sake of brevity, we do not explicitly specify the conditioning.

Let $v:=\frac{1}{n}H_{t}^{*}q^{t}_{\perp}-\frac{1}{n}M_{t}^{*}[\lambda_{t}m^{t-1}-\sum_{i=1}^{t-1}\lambda_{i}\gamma^{t}_{i}m^{i-1}]$ and $\mathbf{M}_{t}:=\frac{1}{n}M_{t}^{*}M_{t}$. If $\mathbf{M}_{1},\ldots,\mathbf{M}_{t}$ are invertible, we have for $j\in[t]$,

Then, if $\mathbf{M}_{t-1}$ is invertible, by the block inversion formula we have

where we have used $\alpha^{t-1}=\frac{1}{n}\mathbf{M}_{t-1}^{-1}M_{t-1}^{*}m^{t-1}$ and $(M_{t-1}^{*}m^{t-1})^{*}\alpha^{t-1}=(m^{t-1})^{*}m^{t-1}_{\parallel}$. Therefore,

Continuing in this fashion, we can express each element of $\mathbf{M}_{t}^{-1}v$ as follows:

We will prove that each entry of $\mathbf{M}_{t}^{-1}v$ concentrates around by showing that each entry of $v$ concentrates around zero, and the entries of $\alpha^{j},\mathsf{a}_{j}$ concentrate around constants for $j\in[t]$.

For $k\in[t]$, bound $\lvert v_{k}\rvert$ as follows. Substituting $q^{t}_{\perp}=q^{t}-\sum_{j=0}^{t-1}\gamma^{t}_{j}q^{j}$ in the definition of $v$ and using the triangle inequality, we have

where $\epsilon^{\prime}=\frac{\epsilon}{t+1}$. The first term in (5.23) can be bounded using Lemma A.3 and induction hypotheses $\mathcal{H}_{t}(f)$ and $\mathcal{B}_{t-1}(e)$ as follows.

For $k\in[t]$, the second term in (5.23) can be bounded as

where the last inequality follows from induction hypotheses $\mathcal{H}_{t}(g)$ and $\mathcal{H}_{t}(c)$. Similarly, for $k\in[t],\,i\in[t-1]$, the third term in (5.23) can be bounded as

Substituting $\epsilon^{\prime}=\frac{\epsilon}{t+1}$ in each of the above bounds and using them in (5.23),

Furthermore, from induction hypotheses $\mathcal{B}_{0}(g)-\mathcal{B}_{t-1}(g)$, for $0\leq i<j\leq(t-1)$:

Also, using induction hypotheses $\mathcal{B}_{0}(h)-\mathcal{B}_{t-1}(h)$ and Lemma A.6, for $0\leq r\leq(t-1)$:

Finally, from (5.21), we have for $k\in[t]$,

where in step $(a)$, $\kappa_{1},\kappa_{2}$ are appropriately chosen positive constants, and step $(b)$ follows from the bounds in (5.24), (5.25), and (5.26). ∎

For $0\leq r\leq t-1$, the first term is bounded as

where step $(a)$ follows from induction hypotheses $\mathcal{H}_{t}(g)$, $\mathcal{B}_{0}(d)-\mathcal{B}_{t-1}(d)$, and Lemma A.4. Next, the third term in (5.27) is bounded as

where step $(b)$ is obtained using induction hypothesis $\mathcal{H}_{t}(h)$, Lemma A.4, and Lemma B.2. Since $\frac{1}{\sqrt{n}}\lVert q^{t}_{\perp}\rVert$ concentrates on $\sigma_{t}^{\perp}$ by $\mathcal{H}_{t}(h)$, the second term in (5.27) can be bounded as

Step $(c)$ is obtained from Lemma A.2, and step $(d)$ from Lemma B.1. This yields the second term in (5.28).

Finally, for $0\leq j\leq(t-1)$, the last term in (5.27) can be bounded by

Lemma 4 (Eq. (4.32)) shows the joint distribution of $({b_{\rm pure}^{0}}_{i},...,{b_{\rm pure}^{t}}_{i})$ is jointly Gaussian for $i\in[N]$. The first term in (5.30) can therefore be bounded as

where the last inequality is obtained from Lemma B.4. Here $\kappa>0$ is a generic absolute constant.

We now bound the second term in (5.30) using the pseudo-Lipschitz property of $\phi_{b}$. Denoting the pseudo-Lipschitz constant by $L$, we have

where the last inequality is obtained by first applying Cauchy-Schwarz, and then using Lemma C.3.

Label the two terms above as $T_{1}$ and $T_{2}$. We bound $T_{2}$ as

for an absolute constant $\kappa>0$, where the last inequality is obtained by applying the concentration result in Lemma B.4 to the pseudo-Lipschitz function $\phi_{b}(c_{j})=\|c_{j}\|^{2}$.

where the inequality is obtained by applying Cauchy-Schwarz.

Comparing (4.32) and (4.33) in Lemma 4, we observe that for $k\geq 0$ and $j\in[n]$,

where the last inequality follows from the stopping criterion in (2.5). Using (5.37) and (5.35) we have

Therefore we can bound the first term $T_{1}$ in (5.33) as follows.

where $K,\kappa>0$ are some absolute constants. The inequality $(a)$ follows from steps $\mathcal{B}_{0}(a)-\mathcal{B}_{t}(a)$.

Finally, substituting (5.38) and (5.34) in (5.33), and then combining with (5.31) and (5.30), we obtain

(b).(iv) For brevity, we write $\mathsf{b}_{t,i}:=\sum_{r=0}^{t-1}\hat{\gamma}^{t}_{r}b^{r}_{i}$. Then using the conditional distribution of $b^{t}$ in (4.24) and Lemma A.2, we write

Label the terms of (5.40) as $T_{1}-T_{3}$. First consider $T_{2}$. Since $\psi_{b}$ is bounded, Hoeffding’s inequality yields $T_{2}\leq 2e^{-\kappa n\epsilon^{2}}$.

In the above, the expectation in each term is over the random variables denoted in upper case. Recall from the proof of Lemma 4 above that $\sum_{r^{\prime}=1}^{t-1}\hat{\gamma}^{t}_{t-1}\sigma_{t-1}\breve{Z}_{t-1}+\sigma^{\perp}_{t}Z^{\prime}_{t_{i}}\stackrel{{\scriptstyle d}}{{=}}\sigma_{t}\breve{Z}_{t}$. Thus $T_{3}$, the third term in (5.40), can be bounded by the probability of the union of the events in (5.41), which is no larger than $tK_{t-1}\exp\{-\kappa_{t-1}n\epsilon^{2}/t^{2}\}$.

Finally, consider $T_{1}$, the first term of (5.40). From the definition of $\Delta_{t,t}$ in Lemma 3, we have $\mathsf{b}_{t,i}+\sigma^{\perp}_{t}Z^{\prime}_{t_{i}}+[\Delta_{t,t}]_{i}=\mathsf{b}_{t,i}+\frac{1}{n}\lVert q^{t}_{\perp}\rVert[(\mathsf{I}-\mathsf{P}^{\parallel}_{M_{t}})Z^{\prime}_{t}]_{i}+u_{i},$ where $u=(u_{1},\ldots,u_{n})$ is defined $u:=\sum_{r=0}^{t-1}(\gamma^{t}_{r}-\hat{\gamma}^{t}_{r})b^{r}+\sum_{j=0}^{t-1}m^{j}[\textbf{M}_{t}^{-1}v]_{j+1}$, with $v$ and $\textbf{M}_{t}$ defined as in Lemma 6. For $\epsilon_{0}>0$ to be specified later, define the event $\mathcal{F}$ as

Denoting the event we are considering in $T_{1}$ by $\Pi_{t}$ and following steps analogous to (5.15)–(5.16) in $\mathcal{H}_{1}(b)$.(ii), we obtain

where the bound on $P(\mathcal{F})$ is obtained by the induction hypotheses $\mathcal{H}_{t}(h)$, $\mathcal{B}_{0}(d)-\mathcal{B}_{t-1}(d)$, Lemma A.4, and steps similar to the proof of $\mathcal{B}_{t}(a)$ for the concentration of $\lVert u\rVert^{2}/n$ (cf. (5.27)).

where we have omitted the conditioning on the RHS to shorten notation. Label the two terms in (5.44) as $T_{1,a}$ and $T_{1,b}$. To complete the proof we show that both terms are bounded by $Ke^{-\kappa n\epsilon^{2}/t}$.

Finally $T_{1,a}$, the first term in (5.44), can be bounded using Hoeffding’s inequality. Noting that all quantities except $Z^{\prime}_{t}$ are in $\mathscr{S}_{t,t}$, define the shorthand $\textsf{diff}(Z^{\prime}_{t_{i}}):=\psi_{b}(\sum_{r=0}^{t-1}\hat{\gamma}^{t}_{r}b^{r}_{i}+\frac{1}{\sqrt{n}}\lVert q^{t}_{\perp}\rVert Z^{\prime}_{t_{i}}+u_{i},w_{i})-\psi_{b}(\sum_{r=0}^{t-1}\hat{\gamma}^{t}_{r}b^{r}_{i}+\sigma^{\perp}_{t}Z^{\prime}_{t_{i}},w_{i})$. Then the upper tail of $T_{1,a}$ can be written as

The proof is completed by collecting the above bounds for each of the terms in (5.40), and observing that the overall bound is dominated by $P(T_{1})$ in (5.43). Hence the final bound is of the form $Kt^{2}K_{t-1}\exp\{-\kappa\kappa_{t-1}n\epsilon^{2}/t^{4}\}$.

(d) The function $\phi_{b}(b^{r}_{i},b^{t}_{i},w_{i}):=b^{r}_{i}b^{t}_{i}\in PL(2)$ by Lemma C.1. The result then follows from $\mathcal{B}_{t}(b).\text{(iii)}$.