# Lasso and its dual: rates of regularisations

Let us consider the following lasso estimator: $$\hat{\beta}_{L} = \arg\min \, \frac{1}{n}\sum_{i}^{n}||y_{i} - \textbf{x}_{i}\beta||_{2}^{2} + \frac{\lambda_{n}}{n}\sum_{j=1}^{p}|\beta_{j}|$$ For any $$\lambda_{n}$$ the problem above is equivalent to $$\hat{\beta}_{L} = \arg\min \, \frac{1}{n}\sum_{i}^{n}||y_{i} - \textbf{x}_{i}\beta||_{2}^{2},$$ subject to $$\sum_{j=1}^{p}|\beta_{j}| \leq t_{n}$$ for some $$t_{n}$$.

Next, assume that the sequence $$\{\lambda_{n}\}$$ is $$o(n)$$. What is about $$\{t_{n}\}$$?

Edit: For the start, we can assume for simplicity, that $$\textbf{X} = \textbf{I}$$.

• you mean, "What is the assumption about {t_n}? Mar 2, 2020 at 14:42
• I mean what would be the sequence $\{t_{n}\}$ (the rate of it) corresponding to $\{\lambda_{n}\}$
– ABK
Mar 2, 2020 at 15:04
• IIRC, the equivalence depends on the data, so you may need some assumptions about $\mathbf{x}, y$? Jan 7, 2021 at 17:06
• Dear @BenReiniger, I have updated the question. Thank you.
– ABK
Jan 8, 2021 at 10:13

The SSE of the LASSO solution $$\tilde\beta$$ in comparison to the SSE of the OLS solution $$\hat{\beta}$$ can be expressed as

$$(X (\tilde\beta-\hat{\beta})) \cdot (X (\tilde\beta-\hat{\beta})) = (\tilde\beta-\hat{\beta}) X^TX (\tilde\beta-\hat{\beta})$$

You can see this graphically as some ellipsoid surface (as the image below which I copied from this question).

• Limit behaviour of $$\frac{1}{n}\sum_{i}^{n}||y_{i} - \textbf{x}_{i}\tilde\beta||_{2}^{2}$$

This ellipsoid will depend on the particular sample (based on which $$\hat\beta$$ and $$X^TX$$ will vary) but for $$n \to \infty$$ you will get that the variation in this surface becomes smaller.

• Limit behaviour of $$\frac{\lambda_{n}}{n}\sum_{j=1}^{p}|\tilde\beta_{j}|$$

If $$\{\lambda_{n}\}$$ is $$o(n)$$ then $$\lbrace\frac{\lambda_{n}}{n}\rbrace$$ is $$o(1)$$ and approaches zero.

So the first term in the cost function will approach some quadratic function of $$\tilde\beta$$ and the second term will approach zero. The lasso solution that minimises the sum of these terms will approach the true $$\beta$$ (The reasoning here is very intuitive but I am sure there is some reference for that).

The consequence is that $$\{t_n\} - \sum_{j=1}^{p}|\beta_{j}|$$ will approach zero (where $$\beta_j$$ refers to the true coefficients).

• Dear @Sextus Empiricus, I didn't understand the conclusion about $t_{n}$. Could you, please, clarify it?
– ABK
Jan 12, 2021 at 13:31
• @ABK I had a misunderstanding about your meaning of $\{\lambda_n\} \in o(n)$ and thought of $\lambda_n/n$ approaching a constant instead of zero. I have edited the answer now. Jan 12, 2021 at 14:27
• ok, again it is a bit confusing. If I understood correctly, the conclusion is that both $\{ \frac{\lambda_{n}}{n}\}$ and $\{t_{n} \}$ have the same rate of convergence
– ABK
Jan 12, 2021 at 15:49
• @ABK I am not sure about the rate of convergence of $t_n$, but if $n \to \infty$ and $\frac{\lambda_n}{n} \to 0$ then $t_n \to \sum_{j=1}^{p}|\beta_{j}|$. Jan 12, 2021 at 16:21