PostgreSQL variadic array of arrays of composite type

I want to write a function that accepts array of arrays of my composite type. because of they have different lengths, it can NOT be defined as one array. So, I want to use variadic to accept arrays separately from each other. What can I do to achieve this ?

my type:

create type column_data as
(
    "name"         text,
    "type"         text
);

A function that accepts multiple arrays of my type:

create or replace function delete_from_table(table_name regclass, data jsonb, variadic columns column_data[][]) returns jsonb
    language plpgsql
as
$BODY$
.
.
.
.
.
$BODY$;

call:

select delete_from_table('cm.clients_rate_limits'::regclass,
                         '[]'::jsonb,
                         array [('pk', 'bigint')]::column_data[],
                         array [('id', 'uuid')]::column_data[],
                         array [('client_id', 'bigint'), ('rate_limit_id', 'bigint')]::column_data[]
       );

error:

[42704] ERROR: could not find array type for data type column_data[]